vij'i> V^ T?5> :■■:: Digitized by the Internet Archive in 2008 with funding from. IVIicrosdft Corporation http://www.archive.org/details/elementaryalgebrOOmyerrich ELEMENTARY ALGEBRA BY GEORGE W. MYERS THE UNIVERSITY OF CHICAGO AND GEORGE E. ATWOOD NEWBURGH, NEW YORK if SCOTT, FORESMAN AND COMPANY CHICAGO NEW YORK Ml COPYRIGHT 1916 BY SCOTT, FORESMAN AND COMPANY cLUUCm i ION OEf'L PREFACE The authors make no apology for offering another algebra to the school public. In influential places algebra has been challenged as a suitable subject for high school pupils. Is it not the part of wisdom, before eliminating a subject of so long and undisputed standing as algebra, to try reconstruct- ing and improving its form and even some of its substance? The authors believe that this text has accomplished much in both of these particulars. This book is not written, however, with the thought of defending an unworthy claimant to a place in the curriculum. The true view is that the high educational merit of school algebra may be raised even higher by a treatment whose language and mode of exposition are in accord with the possi- bilities and appreciations of youth, and whose scientific soundness is at the same time not seriously compromised. It is the authors' conviction that rightly taught, algebra is of great educational value, and that to most high school students it is not distasteful. In carrying out their views on this line, the authors have attempted several specific things. Some of these stated briefly are as follows: 1. To present the material in a language and mode that are simple and at the same time mathematically sound, without resort to mathematical technicalities. 2. To motivate the various topics of algebra either through special problematic situations, or through the gradually rising demands of the equation for particular phases of alge- braic technique. As examples see pages 27, 32, 59, 266, etc. 3. Persistently to make the first steps into the treatments of algebraic subjects through the analogous subjects of arith- metic. (See pages 20, 41, 91, 107, 180, 229, etc.) __ iii 54! 29 J iv PREFACE 4. To give the pupil some really valuable help in learning to read, to comprehend, and to interpret algebraic language, and to express mathematic principles and rules in this lan- guage. Chapter XIII on General Numbers, Formulas, and Type-forms may be cited as a good illustration of this treatment. 5. To give an early introduction to simultaneous simple equations and to complete their study by recurrent treatments as the course develops. 6. To make early and frequent use of the graph freed from analytical technicalities, as an aid to the development of alge- bra through clarifying and vivifying meanings of algebraic processes and technique in the beginning stages of teaching and learning them. 7. To seek diligently for such an order of treatment of the special topics as is dictated by the highest economy in the mastery of the elements of the science of algebra. By this means it is hoped to give a stronger and a more highly edu- cative first-year course in the customary time. (See Table of Contents.) 8. Carefully to grade as to difficulty and to balance as to quaUty and quantity the problems and exercises of the book, again with an eye single to the unfolding needs of algebra. (See problem-lists given under the different topics.) 9. To correlate with arithmetic, geometry, general science, and everyday life to as great a degree as the best school interests of first-year algebra require. 10. To heighten the workability of the text by a synoptic table of contents, a summary of definitions (page 322), and a good working index. A little' of the pedagogical background of the organiza- tion of this text may be stated here. Tiie authors hold the view that teachers of present-day secondary algebra should recognize that they are under three significant professional obligations to their pupils, viz. : PREFACE V I. To rationalize the analogous arithmetic of the algebraic topics taught. It is hardly reasonable to expect of beginners in secondary algebra that they really understand their arithmetic, even as arithmetic. Still less may secondary teachers rightfully expect that beginning pupils have grasped their arithmetic in such form that it can be made the basis for algebra. This is a much more difficult matter because, although both arith- metic and algebra are abstract sciences, algebra involves a much higher order of abstractness than arithmetic. In view of the scope and complexity of modern elementary school arithmetic, of the sUght emphasis of school officials, examiners, and surveyors, and even of school programs upon rationalizing processes, it is worse than useless to expect, let the most conscientious teacher strive as he may, that more be done in the elementary school than to rationalize the most elementary notions and processes of arithmetic. In fact, for several years elementary teachers have been urged by some authorities to renounce rationalizing for mere habituating and drill procedures. These things, coupled with the fact that arithmetic of the sort covered in our grammar grades is one of the most difficult of all mathematical branches, and with limitations of program time and immaturity of pupils, hope- lessly preclude any attempts at those far-reaching inductions and generalizations that are essential at the very beginning of rational algebra. Therefore, this fundamental work for the highly specialized needs of the several algebraic topics belongs properly to the algebra teacher. This text supplies the initiatory arithmetical rationalizing for the algebraic topics and subjects at the precise places where it is needed and of the sort that is appropriate. II. To show that many algebraic things can be done geomet- rically, i.e., by the aid of the concrete space material of diagrams, pictures, — of any graphical helps to clear thinking. To see, to calculate, and to comprehend is the true order of vi PREFACE steps in mastering algebraic tasks. The concepts of lines, rectilinear figures, and solids are so much space material, always and everywhere available for concreting, visualizing, and vivifying number laws and relations, at no great cost in money or effort. The high school youth has lived long enough in this world of space to have become familiar with it, and his spatial experiences need only to be drawn upon to enable him to lay firm hold on the highly abstract fundamen- tals of beginning algebra. Really to see that algebra merely generalizes mensuration laws, that algebraic numbers, laws, and problems picture into vivid forms, and to learn the secret of laying before his eyes diagrammatically the conditions of algebraic problems as an aid in formulating these conditions into algebraic language and technique, are of the highest interest and value to the beginner. The professional duty of employing the concreting agencies of pictures, diagrams, geometrical figures, and graphs to vivify and vitalize algebra will be readily accepted by the teacher who strives to realize in practice the educational merits of well-taught algebra. No clumsy laboratory equipment of extensive and expensive apparatus is required to enable the algebra teacher through space-materials to supply genetic backgrounds for algebraic problems, truths, and laws. III. To show the pupil that algebra will enable him to do much more than he can do with either arithmetic or geometry, or both. , The first and second professional duties are really prelimi- nary, through which motivating and clearing the way for effective attack are accomplished. This third duty is pecu- liarly due to algebra. It is in fact due to both pupil and sub- ject that the particular gains to be secured by a mastery of the subject-matter shall appear in the learning acts. For example, the pupil should see such things as, that by arithmetic he cannot subtract if the subtrahend happens to be greater than the minuend; that he cannot solve so simple an equation as x+9 = 3; but that if he include the negative PREFACE vii numbers among his number notions he can do both easily. He should see that he can square and cube numbers geo- metrically, but that he can go no further with involution than this. If, however, he will learn the symbohsm of algebra he may easily express and work with 4th, 5th, 6th, even with nth powers. He should be shown that while he can solve equa- tions in one, two, and perhaps in three unknowns with graphical pictures, i.e., geometrically, the great power he gains by mastering the algebraic way enables him to go right on easily to the solution of simultaneous equations in 4, 5, 6, and even n unknowns. He should be made to feel that while arithmetic would enable him, by a slow process of feeling about, to find one solution of many problems, algebra, if he will learn its language and method, will lead him directly not to one, but to all possible solutions. It will thus enable him to know when he has solved his problem completely. These and similar gains of power over quantitative problems are the real reasons why the educated man of today cannot afford not to know algebra. Let teachers perform this professional duty well and the foes of algebra as a school subject will be confined to those who are ignorant of it. The one who has learned the subject will then regard it as the emancipator of quantitative thinking. It is desired to call particular attention to the introductory pages on Reasons for Studying Algebra, and to Suggestions on Problem-solving on page 113, and to the careful treatment of factoring. The treatment of the function notion, on pages 50-56, will appeal to many teachers. It will be noted also that this elementary course is divided into half-year units. The problem and exercise lists are full, varied, and carefully chosen. Teachers who employ supplementary lists of exer- cises with the regular text should not require pupils to try to solve all the problems and exercises given here. These hsts are made full and varied to afford choice and range of material. Great care has been exercised to cover all the standard diffi- viii PREFACE culties of first-year algebra, for this book makes its primal task to teach good algebra. This text is to be followed presently by a second course on Intermediate Algebra. The two together will cover the standard requirements of secondary algebra. The pleasant task now remains to acknowledge the assist- ance the authors have received from Mr. John DeQ. Briggs of St. Paul Academy, St. Paul, Minn. ; from the Misses Ellen Golden and Estelle Fenno of Central High School, Washing- ton, D. C; and from Professor H. C. Cobb of Lewis Institute, Chicago, all of whom read and criticized the proofs of the book. Their criticisms and suggestions have resulted in numerous improvements. May this book find friends amongst teachers and pupils, and a deserving place amongst the influences now making for the improvement of the educational results of high school ^^Sebra. rp^^^ Authors. Chicago, September, 1916. CONTENTS FIRST HALF-YEAR chapter page Introduction. Reasons for Studying Algebra 1 I. Notation in Algebra. The Equation 7 Notation 7 The Equation 11 Axioms 13 Directions for Making Statements and Solving Problems 16 II. Positive and Negative Numbers. Definitions 20 Positive and Negative Numbers. 20 Definitions 24 III. Addition ■ 27 Addition of Monomials 27 Adding Similar Terms 28 Adding Dissimilar Terms 30 Addition of Polynomials 32 IV. Subtraction. Symbols of Aggregation 35 Subtraction of Monomials 35 Subtracting Similar Terms 36 Subtracting Dissimilar Terms 37 Subtraction of Polynomials 39 Symbols of Aggregation 41 Addition of Terms Partly Similar 48 Subtraction of Terms Partly Similar 49 V. Graphing Functions. Solving Equations in One Unknown Graphically 50 Graphing Functions 50 Solving Equations in One Unknown Graphically 55 Summary 58 VI. Equations. General Review 59 Equations 59 Clearing Equations of Fractions 66 General Review 70 VII. Graphing Data. Solving Simultaneous Equations Graphically 74 Graphing Data 74 Solving Simultaneous Equations Graphically 82 ix X CONTENTS CHAPTER PAGE VIII. Simultaneous Simple Equations. Elimination by Addition or Subtraction 85 Simultaneous Simple Equations 85 Elimination by Addition or Subtraction 87 IX. Multiplication 91 The Sign of the Product 91 The Exponent in the Product 93 Multiplying One Monomial by Another 93 Powers of Monomials 95 Multiplying a Polynomial by a Monomial 96 Multiplying a Polynomial by a Polynomial 97 X. Simple Equations 100 XI. Division 107 Dividing a Monomial by a Monomial 107 Dividing a Polynomial by a Monomial .' 109 Dividing a Polynomial by a Polynomial 110 XII. Applications of Simple Equations. Elimination by Substitution 113 Suggestions on Problem-Solving 113 Elimination by Substitution 120 XIII. General Numbers. Formulas. Type-forms 123 General Numbers 123 Formulas 124 Forms and Type-forms of Algebraic Numbers 130 XIV. Factoring 134 Monomial Factors (Type-form: ax-\-ay-\-az) 134 Common Compound Factor: {ax-{-ay-\-bx-\-by) 135 Square of the Sum of Two Numbers: {a^ -\-2ab i-b^) ... 137 Square of the Difference of Two Numbers (a^ -2ab+b'^) 1 38 Trinomial Squares: {x^=i=2xy-\-y^) I40 Product of the Sum and Difference of Two Numbers : (o+6)(a-6) 142 Difference of Two Squares (a^-b^) 143 Product of Two Binomials with a Common Term : (x4-a)(x+&) 147 Special Quadratic Trinomials: {x^-\-ax-\-b) 148 The General Quadratic Trinomial: (ax^+bx+c) 149 Incomplete Trinomial Squares: (x*+xV+?/^) 151 Difference of the Same Odd Powers: (x^—y^) 153 Sum of the Same Odd Powers: (x^-^y^) 154 Review 156 CONTENTS xi CHAPTER PAGE SECOND HALF-YEAR XV. Equations. Exercises for Review and Practice 158 Solution of Equations by Factoring 158 Exercises for Review and Practice 164 XVI. Highest Common Factor. Lowest Common Multiple 172 Highest Common Factor 172 Highest Common Factor of Monomials . . 172 Highest Common Factor of Polynomials by Factoring 173 Lowest Common Multiple 175 Lowest Common Multiple of Monomials. 175 Lowest Common Multiple of Polynomials by Fac- toring 176 XVIL Fractions 179 Reduction of Improper Fractions 184 Reduction of Mixed Expressions 186 Lowest Common Denominator 187 Addition and Subtraction of Fractions 188 MultipHcation of Fractions 191 Division of Fractions 193 XVIII. Literal and Fractional Equations. Solution of Formulas 198 Literal and Fractional Equations 198 Special Methods 201 General Problems 207 Solution of Formulas 210 XIX. Simultaneous Simple Equations 213 Elimination by Comparison . . . , 213 Problems in Simultaneous Simple Equations 219 Three or More Unknown Numbers 226 XX. Proportion. Variation 229 Ratio 229 Proportion 232 Principles of Proportion 235 Variation 241 XXI. Powers. Roots 244 Involution 244 Power of a Power 245 Power of a Product 246 Power of a Fraction 246 Powers of Binomials 247 Xll CONTENTS CHAPTER PAGE Powers. Roots— Continued Evolution 250 Root of a Power 251 Root of a Product. 251 Root of a Fraction 252 Number of Roots 252 Imaginary Roots 253 Signs of Real Roots 253 To Find the Real Roots of Monomials 254 Square Root of a Polynomial 254 Square Root of Numbers 259 To Find the Square Root of a Decimal 261 To Find the Square Root of a Common Fraction . . . 262 XXII. Exponents. Radicals 263 Exponents 263 Radicals 264 SimpUfication of Radicals 267 To Reduce a Mixed Number to an Entire Surd 270 Addition and Subtraction of Surds 270 To Reduce Surds to the Same Order 271 Multiplication of Surds 272 Division of Surds 274 Rationalizing Surds 275 Square Root of Binomial Surds 277 Approximate Values of Surds 278 Irrational Equations in One Unknown 278 XXIII. Quadratic Equations 282 The Graphical Method of Solution 282 Solving Quadratics by Factoring 284 Square Root Method of Solution 287 To Complete the Square When a is 1 288 To Complete the Square when a is not 1 289 Solution by Formula 291 To Find Approximate Values of Roots of Quadratic Equations 292 Equations in Quadratic Form 293 Graphical Solution of Quadratics 295 Character of the Roots of Quadratic Equations 298 To Form a Quadratic Equation with Given Roots 300 Factoring by Principles of Quadratics 301 Problems in Quadratic Equations 302 XXIV. Simultaneous Systems Solved by Quadratics — , — 305 Summary op Definitions 322 Index 331 INTRODUCTION REASONS FOR STUDYING ALGEBRA The high school pupil should become convinced, as early as possible, that there are strong reasons why he should learn algebra. The kind of work the pupil will do and his consequent sense of its actual value to him, depend so largely on the approval he gives to its study that it seems worth while, even before beginning it, to consider the reasons for studying algebra. ALL TASKS REGARDED AS PROBLEMS TO BE SOLVED Whether a pupil continues in school or leaves early for the work of life, he will soon learn that the best way to deal with the questions and difficulties that arise, is to regard them as problems to be solved, and to attack them as such. How to learn his lessons, to write a composition, to do an experi- ment, to debate a question, to win in a contest, to do any- thing the first few times, are famiUar problems to the high school pupil. How to earn more and waste less, to manage affairs more economically, to get more out of and to put more into life, how to conduct household affairs more economically, to learn to appreciate and to understand more of the really good and true in books and in life, are actual problems to every right-minded man and woman. Right living is little more than solving a continuous chain of problems. The question for every young person should be, ''How far can I advance in the problem-book of the great world before the problems get too hard for me?" 2 ELEMENTARY ALGEBRA IMPORTANT TO ACQUIRE POWER AND SKILL IN PROBLEM WORK Clearly then, it is of great importance to learn what it means to «olve a problem and to acquire whatever skill we may in the art of problem-solving, and this too, not merely because our teacher, or our parents want us to do so, but for our own sakes purely. In an especial sense algebra teaches the tactics and the technique of problem-solving. The tools by which both the science and the art are wrought out are the algebraic number and the algebraic equation. To be without the ability to use the equation skillfully is to be without the ability to do much problem-thinking. Power to use the equation with skill and insight is the main part of the equipment of an accurate thinker, and algebra is essentially the science and the art of the equation. TWO REASONS WHY ALGEBRA SHOULD BE STUDIED BY ALL Every person who has his way to make in the world must succeed or fail in his struggle with life's problems. The world's problems are harder than those of algebra, but the best way to acquire ability to grapple with harder problems is first to get some skill with easier ones. Algebra starts with comparatively simple difficulties that gradually increase in complexity as one's skill grows, to difficulties great enough to tax the powers of even the brightest pupils. For two reasons at least, the problem-solving of algebra is easier than that of everyday life. In the first place, the language of algebra makes reasoning easier than does any other language men have yet devised. Before algebraic language was invented, the ancient mathematicians used ordinary words and sentences in the problems they attempted. The form of their work, which was largely sentence-making, is now called rhetorical algebra. It never amounted to much as a problem-solving instrument. REASONS FOR STUDYING ALGEBRA 3 Mathematicians later made use of abbreviated words, phrases, and even sentences that occurred frequently in problems, of initial letters, suggestive symbols, and thus formed what is now called abbreviational, or syncopated algebra. This was a real advance, and a very fair sort of algebra now developed as the need for it came along, and men grew interested in it. But it was still cumbersome, and men continued trying to improve it in this way and that, until finally after many centuries, they hit upon the modern form of writing algebraic numbers and relations. From this time forth, symbolic algebra, as we now know it, step-by-step, but rapidly, grew up. The advance in mathe- matics and mathematical science that soon followed is almost incredible. Thus- the history of mathematics shows two things, viz.: 1. That advance in mathematical thought depends greatly on the kind of language employed, and 2. That the language of modern symbolic algebra is the most powerful aid to precise thinking that the world has yet found. Every civiUzed race uses this language today. Of all existing languages of the world it is best entitled to be called the universal language of man. In the second place, algebraic problems have definite answers, so that the beginner may always have a complete check on his thinking during the apprenticeship-period while he is necessarily somewhat doubtful about its relia- bility. On the other hand, the problems of life have no answers, or the answers are of the general nature of success or failure in one's enterprises. With the latter there is no chance to go back and correct errors before the errors have resulted fatally. This is a strong reason why algebra is a good early training in problem-study and problem-strategy. We can do hard things by virtue of the power and skill acquired in doing similar, but easier, things. '4 " ELEMENTARY ALGEBRA ALGEBRA NOT CREATED FOR A MERE SCHOOL DISCIPLINE It thus appears that algebra was not created, as pupils are sometimes prone to think, merely as a severe discipline for school boys and girls. Algebra was formed through the united efforts of a long succession of scientific men to devise a tool and technique for solving the problems of science that arose from age to age — problems that no known subject or device could conquer. It was created as a necessity to win even the little scientific knowledge the race acquired from age to age. After algebra had revealed the desired solutions, sometimes another mathematical subject was found capable of yielding a solution also, but algebra was usually the pioneer, and it is only rarely that any science furnishes easier and more reliable ways of solving problems than algebra. To be ignorant of algebra is to be deprived of the most effective problem-solving engine yet invented. Why not seize the opportunity to acquire some mastery over this powerful tool? The beginnings of the subject are easily within the comprehension of the twelve-year old boy or girl. ALGEBRA IS FUNDAMENTAL TO ALL MATHEMATICAL SCIENCES One of the strongest reasons for studying algebra is that it is fundamentally necessary to so many fields of higher scien- tific work. Aside from a little elementary geometry, almost no mathematics beyond the simplest arithmetic is possible without a knowledge of algebra. To attempt to get on in mathematics without algebra is verily ''to try to walk without feet." Perhaps the most widely useful mathematical subject within reach of high school students is trigonometry. Trigonometry is the science of the triangle, and is made up very largely of compact practical rules, or laws, expressed in the language of algebraic formulas and equations. The transformations of these formulas that lead to the most REASONS FOR STUDYING ALGEBRA 5 practical forms for calculating the parts and properties of triangles, are all algebraic transforniations. Much the greater part of trigonometry is algebraic. All mathematical subjects of collegiate rank require both algebra and trigo- nometry. Algebra is indispensable to work in all branches of mathematics beyond elementary geometry, and nearly all of higher arithmetic is algebra. ALGEBRA IS MORE POWERFUL THAN ARITHMETIC Many problems of ordinary life that are commonly solved by arithmetic, wpuld be much more simply handled by alge- bra if the solver only knew algebra. Even before elementary arithmetic of the seventh and eighth grades is completed the modern teacher finds many ways of simplifying the diffi- culties through equational modes of solution. Old fashioned analyses are today replaced by the use of the equation by all well-qualified teachers. The use of the equation is alge- braic. In a very large percentage of the problems of higher arithmetic, long and cumbersome arithmetical methods can often be replaced by simple mental algebra. Why be con- tent with a dull and bungling tool when a sharp and handy one is so easily available? The modern farmer would scoff at the idea of using the sickle rather than the self-binder to harvest his wheat-crop. PRACTICAL MEN REGRET NOT KNOWING ENOUGH HIGH SCHOOL ALGEBRA It is a common occurrence for business and professional men who have been out of school twenty years or more, to express great regret that they did not give more attention to their high school mathematics when they were in school. They say many Of their most important problems are too difficult for them, though admitting that if their high school mathematics had been well done, they could now solve many of these problems. A prominent business man said 6 ELEMENTARY ALGEBRA recently: ''O, that I knew enough algebra to enable me to understand the formulas of Kent's Engineers' Pocket-Book, to be able to make proper substitutions in these formulas, and to know the meaning of the results!" It is the weak- ness of their problem-solving ability that men of practical affairs seem most to regret. These men often contend that much of what they had to study in the high school has been of little or no use to them, but that they could not have been given too much mathematics for the work they have since had to do. They tell us the leaders today are not the great orators and charming talkers of a generation ago, but the mathematized thinkers. It is the latter, they tell us, that are carrying off the prizes of this commercial and indus- trial age. Let boys and girls who have not yet lost the opportunity to profit by school-work in mathematics make this study as profitable as possible to themselves, by taking up the fundamental subject of algebra with energy and determina- tion. Dismiss the idea, if you hold it, that you are studying this subject as a favor to your teacher or parents. Embrace and cherish the true idea that you are studying it for your own benefit, to raise your own efficiency, and that you are only cheating yourself if you do poor work. The chances are many to one that the tasks of after-life will be found to make stronger demands on your problem-solving ability than algebra requires. Do not forget that algebra is in a peculiar sense the subject which can best develop and perfect ability of this type. Therefore take up the work vigorously the first day, never relaxing your efforts to master the subject until the last lesson is learned. ELEMENTARY ALGEBRA First Half- Year CHAPTER I NOTATION IN ALGEBRA. THE EQUATION NOTATION 1. The power of algebra is due mainly to its language and its symbols. You have already made some start with this language, for everything you have correctly learned about the language and the symbols of arithmetic holds good also in algebra. But because algebra is a sort of general arith- metic, it adds something to the language and symbols of arithmetic and employs them more generally than arith- metic does. Perhaps the most important things for the beginner to keep in mind from the outset are that what the algebraic language talks about and what the algebraic symbols stand for are numbers and number relations. Though the book or the teacher may talk about algebraic expressions, or quantities, or monomials, or polynomials, it is important to remember that all these terms, and many others, are only other names for numbers. Algebra, like arithmetic, treats of numbers. It adds, subtracts, multiplies, and divides numbers, raises them to powers, and extracts their roots. 2. Notation is the method of expressing numbers by figures or letters. 7 8 ELEMENTARY ALGEBRA In arithmetic, numbers are represented by ten Arabic characters called digits or figures. Thus, 453 = 400+50+3 The Number. The whole of the number is the sum of the parts represented by the several digits. Representing Numbers. In algebra, numbers are repre- sented by figures, by letters, and by a combination of both. 3. Products. When letters and figures are written to- gether in algebra, their product is indicated. Thus, 4a means 4 times a, and 7ab means 7XaXb If a number is the product of two or more numbers, those numbers are factors* of the product. The numbers represented by the digit and the letters in 7ab are therefore factors of the whole number, 7ab. 4. Using Algebraic Language. The expression, 5x yards, means 5 times the number of yards represented by x. Exercise 1 1. What is meant by the expression, 9a; quarts? 12y cents? Sn miles? 8x square feet? 2. If n represents a certain number, what does 4w repre- sent? 9n? 6n? 3. If X represents the price of a yard of silk, what does 5x represent? 8a;? 12a;? 4. A boy bought 8 oranges at h cents apiece. How many cents did he pay for them? 5. What is meant by the expression, 6a; yards? 4a dollars? Sy bushels? 7x square rods? *The word factor means maker. The factors of a number are its makers by multiplication. NOTATION 9 6. If a man works for n dollars a day, how much does he earn in 8 days? In x days? 7. If a square is x inches on each side, what does 4x represent? What does xx represent? 8. How many square inches are there in a rectangle x inches long and y inches wide? 9. If you are c years old today, how old is your father who is three times as old? 10. If n represents a certain number, what represents 6 times the number? m times the number? 5. Algebraic Signs. The signs of addition, subtraction, multiplication, and division mean the same as in arithmetic. 6. Indicating Multiplication. Multiplication is often indi- cated in algebra by placing a dot between the factors. Thus, 5XaXc = 5rtC = 5*a*c 7. Indicating Division. Division is often indicated by writing the dividend over the divisor in the form of a fraction. Exercise 2 1. Indicate the sum of 8 and 7. Of x and 5. Of a and h. Of X, y, and z. Of 2a, 36, and 12. 2. A man is n years old today. How old was he 7 years ago? Eighteen years ago? 3. If a man has p sheep in one field and q sheep in another, how many has he in both fields? 4. What is meant by the expression, 7x feet? 9y square feet? n+8days? 10 ELEMENTARY ALGEBRA 6. When n represents an odd number, what will represent the next larger odd number? 6. What represents the number of square inches in a rectangle x feet by y inches? 12x 7. If n represents an even number, what will represent the next smaller even number? 8. How many square inches are there in a rectangle m yards long and n inches wide? ^^^ 9. If the sum of two numbers is x and the larger number is y, what is the smaller number? 10. If a rectangular piece of land is x rods long and y rods wide, what does 2x-\-2y represent? 11. Indicate in two ways the product of 5 and x. Of h and y. Of 5, a, and h. Of n, x, and 3. 12. A man bought x cows at $35 apiece and had $85 left. How much money did he have at first? 13. Indicate the difference between a and b when a is greater than h. When b is greater than a. 14. A boy had a cents. He earned b cents and then spent 8 cents for candy. How many cents did he have left? 15. A boy has m quarters and n dimes. What expression represents the number of cents he has? 16. What represents the number of square yards in a ceiling x feet long and y feet wide? 17. What will denote the number of acres in a rectangular ' field L rods long and W rods wide? 18. A man sold a horse for b dollars and gained c dollars. How much did the horse cost him? 19. A man bought x sheep at m dollars a head and y lambs at n dollars a head. What did all cost him? THE EQUATION U 20. A boy bought n apples at x cents apiece and sold them at y cents apiece. If he gained, what was his gain? Since 5 times any number +4 times the same number = 9 times that number, 5a+4o = 9a, also 9x+ix = lSx 4a-j-a = 5a 4n-\-2n-\-n = 7n 21. If a man gets x dollars for corn and 4a; dollars for wheat, how much does he get for both? Since 9 times any number — 3 times the same number = 6 times that number, 96 — 36 = 66, also I0a-Sa = 7a 5x-x = 4x 86-36-6 = 46 22. If the larger of two numbers is 8s and the smaller is 3s, what represents the difference between the two numbers? 23. A has n sheep. B has twice as many as A, and C has twice as many as A and B. How many have all? 24. If a set of harness costs x dollars, a carriage 3a; dollars, and a horse 5x dollars, what do all cost? 25. If a man has 5y dollars and spends x dollars for a suit of clothes, how many dollars has he left? THE EQUATION 8. The sign of equality is = . It indicates that the numbers between which it is placed are equal. The expression 9.t — 5 = 2x+9, means that the difference between 9x and 5 is the same as the sum of 2x and 9. 9. An equation is an expression of equality between two equal numbers. Thus, 8+7 = 5X3 8a: + 6 = 3x+36 4n+27i = 54 10. The first member of an equation is the number on the left of the sign. The second member is the number on the right. 12 ELEMENTARY ALGEBRA 7H8 -11. The equation expresses balance of values just as. the horizontal position of the bar of the balances shows balance of weights. To put = between two number expressions is to say that if the numbers were weights and the ex- pressions in the two members were 7+8 = 5X3 represented by proper weights, one in each pan, the balance- bar would stand horizontal. 12. The value of any letter in a number expression is the number or numbers which it represents. If 3 unknown weights of x lb. each in one pan are balanced by 6 weights of. 3 lb. each in the other pan, we may say 3a: =18. Leaving ^ of the weights on each side on the pans, and removing the rest, the bar will remain horizontal, or we may say, x = 6 lb. That is, the bar can be horizontal when the pans are loaded one with 3x lb. and the other with 18 lb. only if a; = 6 lb. But without troubling with the balance, by merely apply- ing the division principle that equal numbers divided by the same number give equal numbers, to the equation 3a; = 18, we find x = 6. In this way algebra makes reasoning take the* place of the weighing apparatus. In the equation, 3x = 18, since 3x means 3 times x, 3x and 18 are equal only when x represents 6. In the equation, 3n+2n = 35, since Sn-{-2n is 5n, 3n+2n and 35 are equal only when n represents 7; or think thus: 3a; = 18 3n 2n I, whence n = 7. 35 THE EQUATION 13 Solve these exercises mentally: 1. If 7a; = 28, what is the value of x? What is the value of n in the equation, 3n+n = 24? 2. If 2a;+3x = 27, what is the value of x? What is the value of s in the equation, 3s — s = 17? 13. The Unknown Number. Any letter whose value in an equation is to be found, is called an unknown number. 14. Solving an equation is the process of finding the value of the unknown numbers. All changes in solutions of equations are based on simple statements, called axioms. AXIOMS 15. An axiom is a statement the truth of which is so evident that it may be accepted without proof. Addition Axiom. — // the same number or equal numbers are added to equal numbers, the sums are equal. If x = a, x+5=a+5, and if x = a and 7/ = 3, x-{-y = a+3 Subtraction Axiom. — // the same number or equal numbers are subtracted from equal numbers, the remainders are equal. If x = y, x — 7 = y — 7, and if x. = y and a = 5, x—a = y — 5 Multiplication Axiom.—// equ,al numbers are multiplied by the same number, or by equal numbers, the products are equal. If 2n = 4, 4n = 8, and if n = 7 and x = 3, nx = 3X7 Division Axiom. — // equal numbers are divided by the same number or by equal numbers {except zero), the quotients are equal. If 3a; = 15, X = 5, andif a: = 12and?/ = 3, - =^ = 4 Comparison Axiom. — Numbers that are equal to the same number, or to equal numbers, are equal to each other. If a+6 = 4 and x—y = 4, a-\-b=x—y 14 ELEMENTARY ALGEBRA In algebra when the reason for a change in an equation is asked, the pupil is expected to quote or to cite an axiom that justifies the change. 16. Give the reason for the conclusion in each of the following : 1. a; = 7and?/ = 4; thena;+2/ = ll 2. c = 21 and d = Q; then c — d = 15 3. a = xand6 = 3; then a — 6 = x — 3 4. c = 2n; then 10c = 20n 5. d = S2; then- = 4 8 6. x — 7 and y = 7; then x = y 7. Sy = 27; then ?/ = 9 8. If n = 5; then 8n = 40 9. Ifm = 9andn = 4; then mn = 36 TTl 10. If m = 28 andn = 4; then— = 7 n 11. If a;-3 = 5; thena: = 8 12. If — = 5; thena = 80 16 13. Ifmn = 7n; then m = 7 Exercise 3 1. Solve 8x-3x+2x-a: = 30. Hx-3x-\-2x-x = S0 6a; = 30 By the division axiom, x = 5 Checking, 40-15+10-5 = 30 or 30 = 30 Always check or test the value of the unknown number after it is found, by substituting it for the unknown number in the given equation. THE EQUATION 15 Solve and check: 2. Qx-2x+3x = 49 3. 5n-2n+4n-n = 48 4. 5s+6s-3s = 48 ■ 5. 9a-3a-2a+a = 45 6. Sy-^y-2y = 24: 7. 86+76-6-46 = 55 8. 7x-\-2x-3x = 54: 9. 4n-3n+6n-n = 72 17. Solving a problem is the process of finding the values of the unknown numbers involved in the problem. In arithmetic the unknown numbers are found by one or more of the fundamental processes. In algebra the unknown numbers are represented by letters and their values are found by the use of equations. Solving a problem in algebra involves three steps : notation , statement, solving an equation. Exercise 4 — Solving Problems 1. The sum of two numbers is 252, and the larger number is 6 times the smaller. Find the numbers. _-. . j Let s = the smaller number; ' \ then 6s = the larger number. Hence s+6s and 252 are two number expressions, each of which represents the sum of the two numbers. Statement, s+6s = 252 The notation is the representation in algebraic symbols of the unknown numbers in the problem. The statement is the expression of the conditions of the problem in one or more equations. f7s = 252 Solving the equation, I s = 36 6s = 216 16 ELEMENTARY ALGEBRA To cheeky substitute in the statement. Thus, 36+216 = 252, or 252 = 252 Even the statement itself may be wrong. To test whether this is the case, substitute in the conditions of the problem itself. 2. The sum of two numbers is 846, and the larger number is 8 times the smaller. Find the numbers. 3. Seven times a certain number plus 6 times the number minus 8 times the number equals 175. Find the number. 18. Obtaining Statements of Problems. To obtain the statement in a problem is to translate the conditions of the problem into an equation. DIRECTIONS FOR MAKING STATEMENTS AND SOLVING PROBLEMS 1. Let any appropriate letter represent one of the unknown numbers to be found. 2. From the conditions of the problem express, in terms of the same letter, the other unknown numbers. 3. Find two number expressions that represent the same num- ber and place them equal, forming an equation. 4. Solve the equation and determine whether the result satis- fies the conditions of the problem. Exercise 5 1. One number is 5 times another, and the difference between them is 48. Find the numbers. ^ . jLet s = the smaller number; ' \then 5s = the larger number. Hence, 5s — s and 48 are two number expressions, each of which represents the difference between the numbers. Statement, 5s — s = 48 Solving this equation, s = 12 and 5s = 60 Checking, 60-12=48, or 48 = 48 THE EQUATION 17 2. A is six times as old as B, and the difference between their ages is 75 years. Find B's age. 3. In a school of 855 pupils there are twice as many girls as boys. How many girls are there? 4. A earned five times as much as B. If B earned $648 less than A, how much did both together earn? 5. The length" of a rectangle is 3 times its width, and the perimeter is 224 inches. Find the dimensions. 19. Letters Represent Numbers. In solving problems, always let the letter represent some number. It must not represent money, but a number of dollars or cents; not time, but a number of days or hours; not weight but a number of pounds or ounces; not distance, but a number of miles, rods, or other units of measure. Exercise 6 — Problems 1. A horse, carriage, and harness cost $450. The carriage cost 3 times as much as the harness, the horse twice as much as the carriage. Find the cost of each. I n I 3« I 6re i I I ! I I I I I I ! 450 Let n = the number of dollars the harness cost; then 3n = the number of dollars the carriage cost; and 6ri = the number of dollars the horse cost. Hence n+3n+6n and 450 are two number expressions, each of which represents the cost of all. n+3n+6n = 450 2. One number is 9 times another, and the difference between them is 624. Find the numbers. 624 18 ELEMENTARY ALGEBRA 3. A has twice as many sheep as C, and B has 4 times as many as C. If all have 665, how many has B? C's A' 8 665 4. A house and lot cost $7250, the house costing 4 times as much as the lot. Find the cost of each. I lot I co,it of house , 7250 5. If twice a number is added to six times the same number, the sum is 192. Find the number. 192 6n 6. The sum of the ages of father and son is 96 years, and the difference between their ages is twice the son's age. What is the father's age? father's age 2 times sons age 7. A rectangle formed by placing two equal squares side by side has a perim- eter of 270 feet. Find the side of each square and the area of th^ rectangle. 8. If two rectangles of the same width and twice as long as wide are placed end to end, the perimeter of the rectangle formed is 180 inches. Find their dimensions. 2w 2w 9. One number is 4 times another, and 4 times their difference is 576. Find the numbers. 10. A man sold a horse and carriage for $340, receiving 3 times as much for the horse as for the carriage. How much did he get for the carriage? THE EQUATION 19 11. The sum of two numbers is 322, and their difference is 5 times the smaller. Find the larger number. 12. A, B, and C own 840 sheep. A owns 3 times as many as B, and C owns twice as many as A and B together. How many do A and C together own? 13. A's age exceeds B's by 3 times B's age, and the sum of their ages is 75 years. Find A's age. 14. In a mixture of 228 bushels of corn and oats there are twice as many bushels of corn as of oats. How many bushels of oats are there in the mixture? 15. A number increased by 3 times itself, 4 times itself, and 5 times itself is 650. Find the number. 16. A man sold some lambs at $3 a head and three times as many sheep at $5 a head, receiving $324 for all of them. How many of each did he sell? 17. The length of a rectangle is 4 times its width, and the perimeter is 280 yards. Find the dimensions. 18. A, B, and C own 600 acres of land. B owns 3 times as many acres as A, and C owns half as many acres as A and B together. How many acres have B and C? 19. A merchant paid $50 for two pieces of silk of equal value, paying 80ff a yard for one piece and $1.20 a yard for the other. How many yards were in each piece? 20. Two equal rectangles whose length is 3 times the width, if placed end to end, form a rectangle whose perimeter is 196 inches. Find the length of each rectangle. CHAPTER II POSITIVE AND NEGATIVE NUMBERS. DEFINITIONS POSITIVE AND NEGATIVE NUMBERS 20. Numbers of Arithmetic. The only relation of numbers considered in arithmetic is the relation of size. A boy starts from 0, takes 12 steps toward the right, then turns, and takes 7 steps toward the left. How far is he then from the starting-place, 0? In arithmetic we would solve this problem thus : 12-7 = 5 But suppose after taking 12 steps to the right and turning back, he had taken 20 steps toward the left. Where would he then be with regard to the starting-point? I ... I I I ... I I I I I I I I I I I I I I I I .. I I I -15 -10 -5-4-3-2-1 OM+2+3+4+5 H-IO +15 An Algebraic Scale We know that in arithmetic we cannot subtract 20 from 12. Still by using the algebraic scale above, we can easily solve the problem, and learn that the boy will be 8 steps to the left of the starting-point, 0. If we agree that the sign — ", instead of meaning ''subtract" shall mean "go leftward,'' we may write: 12-20= -8. It will be more complete also to agree that the sign, -f, instead of always meaning "add," as it did in arithmetic, may mean also "go rightward'' ; hence we write: + 12-20= -8, which means *'12 steps right ward, followed by 20 steps left- ward, leaves one 8 steps left of the starting-point." 20 POSITIVE AND NEGATIVE NUMBERS 21 This is what we do in algebra, thus making it possible to solve numerous problems that cannot be solved by arith- metic. Hence, to learn algebra is to add greatly to our problem-solving power. 21. Such numbers as +12, —20, and —8 are called directed numbers, or signed numbers, and the + or the — is just as much a part of the number as is the 12, the 20, or the 8. There is now nothing impossible about such a problem as: +6-15 = ? By referring to the algebraic scale, tell what the answer is. Thus, in algebra the signs + and — may mean add and subtract, or the direction, or kind, or quality of the number, i.e., they are verbs or adjectives. 22. Algebraic Nimibers have Opposite Qualities. We shall learn, that many numbers in their relation to each other are opposite in quality. Gains and losses, owns and owes, dates before and after, and distances in opposite directions serve as examples. In algebra, numbers are considered with reference to the two relations of size and opposite quality. ILLUSTRATIONS If a man makes $4500 one year and loses $2500 the next year, his net gain for the two years is $2000. If a merchant's assets are $14,000 and his liabilities are $6000, he is really worth only $8000. Numbers are of opposite quality; therefore in combining them, the smaller number united with an equal part of the larger number gives zero. 23. Positive and Negative Numbers. To describe the opposite quality of numbers, the terms positive and negative 22 ELEMENTARY ALGEBRA are used in algebra, and the quality of a number is denoted by the sign + or — . The sign + before a number denotes that it is positive, and the sign — that it is negative, as +5, —6. 24. The absolute value of a number is the number of units in it, independent of their quality. The absolute value of +9 is 9. The absolute value of —8 is 8. Exercise 7 Let us consider distance north from a certain point as posi- tive and distance south as negative. 1. If a man walks north 12 miles one day and north 13 miles the next day, what is the result? 2. If a man walks south 11 miles one day and south 10 miles the next day, what is the result? 3. If a man walks north 14 miles one day and south 10 miles the next day, what is the result? 4. If a man walks north 10 miles one day and south 15 miles the next day, what is the result? 6. If a man walks south 14 miles one day and north 11 miles the next day, what is the result? 6. If a man walks south 10 miles one day and north 17 miles the next day, what is the result? You have probably answered these six questions as follows: He is 25 miles north of the starting-point; 21 miles south; 4 miles north; 5 miles south; 3 miles south; 7 miles north. Here are the algebraic solutions of the six problems. Tell how each result is obtained and what it represents. + 12 -11 +14 +10 -14 -10 + 13 -10 -10 -15 +11 +17 +25 -21 +4 -5 -3 +7 POSITIVE AND NEGATIVE NUMBERS 23 The results of uniting these positive and negative numbers show the following principles : 25. The sum of two numbers with like signs is the sum of their absolute values with the common sign prefixed. 26. The sum of two numbers with unlike signs is the differ- ence between their absolute values with the sign of the number having the greater absolute value prefixed. Exercise 8 Applying these principles, write the sums in the following examples, giving each its proper sign: +23 +33 -31 -41 +29 +19 + 15 -14 -16 +17 -14 -37 +75 +83 -67 +43 -28 -73 +68 -38 -49 -82 +74 +37 +85 +39 * -49 +93 -65 -34 +78 -75 -68 -45 +29 +73 27. Double Meaning of + and — . Thus it appears that the signs + and — are used in algebra to denote quality of numbers as well as to denote operations. Exercise 9 — Problems with Positive and Negative Numbers Assign quahty to the numbers in these problems, solve them algebraically, and interpret the results : 1. A man's property amounts to $18,750 and his debts to $23,250. Find his net debt or property. 2. A merchant gains $2365 one year and loses $1790 the next year. Find the net gain or loss. 3. If a man travels east 58 miles one day and west 73 miles the next day, what is the net result? 24 ELEMENTARY ALGEBRA 4. A man's annual income is $3675 and his expenses $2395. How much does he save annually? 5. If a ship sails north 53 miles one day and south 39 miles the next day, what is the net result? 6. A real estate dealer gains $1465 on one sale and $2375 on another. Find the result of both sales. 7. Draw a line representing a thermometer scale; mark the zero point, 24°, and — 12°. What is the difference between the +24° and the -12°? 8. If the weight of a stone is regarded as positive, what would represent the weight of a balloon? 9. If a speculator makes $2765 one month and loses $2875 the next month, what is his net gain or loss? 10. If a stone weighs 34 pounds, and a balloon pulls upward with a iorce of 8 pounds, what is the combined weight of both, if they are fastened together? DEFINITIONS 28. A system of notation is a system of symbols by means of which numbers, the relations between them, and the operations to be performed upon them can be more concisely expressed than by the use of words. 29. Algebraic notation is the method of expressing numbers by figures and letters. 30. An algebraic expression is the representation of any number in algebraic notation. 31. A term is a number expression whose parts are not separated by the sign + or — , thus, bx 2aX46, 3a6, xy^ box, and — DEFINITIONS ' 25 32. A monomial is an expression oi ^one term. A poly- nomial is an expression of two or 7nore terms, as, 2a+46-3c-5d The signs + and — between the terms of a polynomial may be regarded as signs of operation or of quality. When monomials and the first term of a polynomial^ are ^itten without any sign before them, they are positive. 33. A binomial is a polynomial of two terms. A trinomial is a polynomial of three terms. 34. A coefficient of a term is any factor of the term which shows how many times the other factor is taken as an addend. Thus, ' 4n = n-fnH-n4-n 4ax = ax-^ax-\-ax-\-ax Coefficients are distinguished as numerical or literal, according as they are expressed in figures or letters. In the two terms above, 4 is the numerical coefficient. Any other factor of 4ax may be regarded as the coefficient of the product of the remaining factors. Observe that Aa-\-a = a-\-a-\-a-\-a-\-a = ba This shows that when no numerical coefficient is expressed, the numerical coefficient is considered to be 1. 35. Similar terms are terms which do not differ, or which differ only in their numerical factors, as, bxy, xy, and Sxy; Sab and 5a6; or 4ax, ax, and 7 ax 36. Dissimilar terms are terms that are not similar, as 4ab, ax, 36c; Sac, ixy; 2xy, xz, 3yz 37. Partly Similar Terms. Terms that have a common factor are said to be partly similar, or similar with respect to that factor. Thus, ax, ix, and bx are similar with respect to x; and 5xy, axy, bxy, and 4cxy are similar with respect to xy. 26 ELEMENTARY ALGEBRA 38. The value of an algebraic expression is the number it represents when some particular value is assigned to each letter in the expression. Substitute 1 for a, 2 for 6, 3 for c, 4 for d, in the following expression and simplify the result : 2ah + 36c + 5cd - 4hd = 2-l-2+3-2-3+5-3-4-4-2-4 = 4 +18+60-32 =50 Exercise 10 Find the value of each of the following expressions when , a = l, 6 = 2, c = 3, c? = 4, e = 0, m = |, n = J: 1. hcd—dn—^a-^-Qam 3. Qh — adm-{-5hc — Qn 5. hcd — 5e — 4:m-jrScn 7. Sa-\-Qmn-2b-\-bcd 9. 5cd — Sm-{-9a — Qcn 11. 4bc-\-7d-9n+7ab 13. cdm-{-Scn-\-ab — 7e 15. 9n-\-cdm-\-de — 2ab 2. 6am+9a+26c-3n 4. 4ad-6n+2d(m-26 6. Sb-4am+9bn-2a 8. Sad—7e — bm-\-Qdn 10. 7a-\-9bd-Sm-\-9an 12. 56d+ac— 46m+6n 14. 66n+5e — 6m+8ad 16. cd — acn-\-Sm-\-Sem CHAPTER III ADDITION ADDITION OF MONOMIALS 39. Addition is the process of uniting two or more numbers into one number. 40. The addends are the numbers to be added; the sum is the number obtained by addition. 41. To Add Similar Terms. In adding 5-6 and 3 '6 in arithmetic, the two products, which are 30 and 18, are found and then added. Since 5 times 6 plus 3 times 6 is 8 times 6, they may be added also by adding the coefficients of 6, thus 5.6+3-6 = 8-6 42. Adding Indicated Products. Algebraic terms, which are indicated products, can be united into one term only by the latter method. For example: 1. A school hall is I yards long. I go through it 6 times on Monday and 14 times on Tuesday. How many yards do I travel through the hall on both days? Monday, 6/ yards Tuesday, 14'^ yards Both days, 201 yards 2. The tickets for an entertainment were t cents each. George sold 34 and Mary 28 tickets. Find the total receipts from the sales of George and Mary. George, 34^ cents Mary, 28^ cents Both, Q2t cents 27 28 ELEMENTARY ALGEBRA ADDING SIMILAR TERMS 43. The sum of two similar terms is the sum of their coeffi- cients with the common letters affixed. Whether the terms have Uke or unhke signs, the sum of the coefficients is found by §§ 25 and 26. Exercise 11 Give at sight the sum of each of the following : 1. 4-3 2. 4a 3. 8a; 4. -76 5. -3c 5-3 -5a -3x b -5c 6. 8-5 7. 9a 8. -9x 9. 96 10. -6c 6-5 -3a 4a; -6 -4c 11. 6-7 12. 2x 13. -46 14. n 16. -3c 7-7 -7x 56 -4n -7c 16. 5a 17. -7x 18. 26 19. - n 20. -8c 6a 6a; —66 6n —6c 21. a 22. 9a; 23. -76 24. -8n 25. -4c 7a —3a; 86 n —9c 26. 5a 27. -9a; 28. 96 29. - n 30. -7c 9a 8a; -26 4n -5c 31. 7a 32. 47/ 33. -66 34. -7n 35. -6c 8a -9y 76 n -9c 44. Rule. — Find the algebraic sum of the coeffix^ients, and to that result affix the comm^on letters. ADDITION OF MONOMIALS 29 45. Fundamental Laws. There are two fundamental laws of addition which it will be well to notice here. They are known as the law of order, or the commutative law; and the law of grouping, or the associative law. 46. Law of Order. The sum of two or more numbers is the sam£ in whatever order they are added. It is evident that: 8+6+4=6+4+8=4+8+6 for each member of this equality is the same number. This law is represented as follows : a+b+c = b+c+a = c+a+b = a+c+b, etc. 47. Law of Grouping. The sum of several numbers is the same in whatever manner they are grouped. Thus, 8+6+4 denotes that 6 is to be added to 8, and 4 added to the result; that is, 8+6+4= (8+6)+4. By the law of order, 8+6+4=6+4+8=4+8+6 Therefore, 8+6+4= (8+6)+4= (6+4) +8= (4+8) +6 This law is represented as follows : a+b+c= (a+b)+c= (b+c)+a= (a+c)+b Adding Several Positive and Negative Terms. The addition of several similar terms with unlike signs is based on the associative law. By this law, the positive terms are grouped together and added, the negative terms are grouped together and added, and the two sums then united. Exercise 12 — Adding Similar Terms Give the sum of each of the following : 1. 2a 2. ~4:X 3. 56 4. -3/i 5. - c a 2x -lb 6/1 -7c 3a — X 36 — 4n • —9c 30 ELEMENTARY ALGEBRA 6. a 7. Qx 8. -96 9. 2n 10. -8c 8a -9x 36 -4n — c 2a X 76 Sn -9c 11. a 12. -7x 13. 56 14. Qn 15. -3c 5a 4x -66 -9n 9c a -Qx 96 2n -6c 16. 5a 17. Sx 18. - 6 19. Qn 20. -9c a -Sx - 6 — n 4c 7a 2x 86 Sn -8c ADDING DISSIMILAR TERMS 48. Dissimilar terms cannot be united into one term. The addition can only be indicated by writing them in suc- cession in any order, each preceded by its own sign, as here shown : 3ac 5a —36c — 6c —26 — 4ac 2bd -3a -26 dac-bc-{-2bd 2a -26 -4ac-36c-26 We write a positive term first, if there is one. If all the terms are negative, any one of them may be written first. Exercise 13 Give at sight the sum of each of the following : 1. 3a 2. 6 , 3. -2x 4. -2n 6. -5c 2x -2c y -Sx — c 6. 4a 7. 2a 8.-71 9. 2a 10. 5x Sx - 6 -3a: - 6 -Qy a — c 4n -5c -4x ADDITION OF MONOMIALS 31 11. 2a 12. ^x 13.- X 14:. -In 15. • 2a * 6 —4a; —32/ — n — c 5a; — 5x 4x 4n —5c 16. 4a 17. 4a; 18. -36 19. 3a 20. -7c 26 -2y 66 — c -9c a — X -46 -7n -8c 21. la 22. -5a; 23. 46 24. -Sn 25. -6c a 32/ n 7n 4c 4a a; -Qx -5n -5c 26. 3a; 27. -2a; 28. 76 29. -2n 30. -5a a 5a; -4c Sn -26 5a -3a; 6 -6n -42/ 31. a 32. -5a; 33. -3a 34. 3n 35. - a 3n 3a; 6 -9/1 -5c a; -7a; — a 7n — c 36. 4a 37. X 38. -66 39. -4n 40. c a - y - 6 2a 9c 2a 4z -76 n — c 41. y 42. -5a; 43. 6a 44. n 45. - a 7x 3a; - 6 -3n -3c x -7x 2n n -4a 46. 4a 47. 36 48. -7y 49. 5c 50. -7a; a -56 y 6 3a; 6a - 6 Qy -2c — X a 96 -42/ 26 6a; 32 ELEMENTARY ALGEBRA Simplify the following: 51. 4a+2a+a+5a 52. 3n+8n+n+2n+6n 53. 2x+x-lx+^x 54. 56-26-66+6+96 55. 5c-6c+c+4c 56. Qiy-y-\-^y-1y-{-by 57. 7a— 3a — a — 2a; 58. 7n+5yi — n— 4n+3a 59. 8a;-4a;-3a;-2/ 60. 46+96+76-86-6 61. 6i/+8x-9i/-5a: 62. 6a-76-4a+36+a ADDITION OF POLYNOMIALS 49. Addition of polynomials proceeds much as addition of monomials, as the two following illustrations show: 1. The stairway of a school has 3 flights, of a, 6, and c steps, respectively. A boy goes up and down the stairway 3 times on Monday, 5 times on Tuesday, 4 times on Wed- nesday, 6 times on Thursday, and 4 times on Friday. How many steps does he take on the stairs during the week? Monday, 6a + 66+ 6c steps Tuesday, 10a + 106 + 10c steps Wednesday, 8a + 86+ 8c steps Thursday, 12a + 126 + 12c steps Friday, 8a + 86+ 8c steps Sum, 44a +446+ 44c steps 2. At a money-changer's are offered for exchange: At one time, 52 marks, 35 francs, 12 pounds; At another, 18 marks, 26 francs, 24 pounds; At another, 22 marks, 15 francs, 18 pounds. The exchange value of a mark being m cents, of a franc / cents, and of a pound / cents, find the total exchange value of the foreign currency in cents. First time, 52m +35/+ 12i cents Second time, 18w+26/+24i cents Third time, 22m + 15/+18Z cents Sum, 92m+76/+54/ cents ADDITION OF POLYNOMIALS 33 50. To add poljmomials, ivrite similar terms in a column and add each column, beginning at the left. Thus, 5ah+Sac-2bc+Sbd-{- 5xy - 7xz 2ab— ac —5bd-{-2xy 5ac— be -\-7xz ab -\-3bc -4:xy +6 Sab+7ac -2bd+3xij +6 51. A check on algebraic work is another operation which tends to prove the first result correct. 52. Checking Addition by Substitution. Addition may be checked by substituting any number in place of the letters and determining whether the sum of the valv£s of the addends equals the value of the sum. The following shows how addition of polynomials may be checked by substituting 1 for each letter. Work Check 5a-96+7c = 5-9+7 = 3 a+Sb-Qc = 1+8-6 = 3 3a-46+3c = 3-4+3 = 2 9a-56+4c = 9-5+4 = 8 The sum of the values of the addends is 8, and the value of the sum of the polynomials is also 8. Observe that when 1 is substituted for each letter, the value of each term is the numerical coefficient. In checking or verifying algebraic processes, any number may be substituted for each letter. To avoid large num- bers, it is well to substitute small numbers; but substitu- ting 1 checks only the coefficients and should not, in general, be done. 34 ELEMENTARY ALGEBRA Exercise 14 1. Add4a-3n+2:r, 5n-4a;+5, -7a-4n+7x, 2a+6n4-6, n — X— 14, and 5a — 4n — 3x+4. 2. Add 5b-i-Sc-Qd, c-2b+Sd, -M-dc+Sb, 6d-4c-76, and 2c-\-d-\-4:b, and check. 3. Add4a-Sb+5c,2c-2b-\-d, -4d-8a+76, 3c+4a+3(i, and 26 — d— 7c, and check. 4. Add7x-5i/+32, 31/-8-52;, -4:y-\-Qz-5x, Qy-2x-Sz, and 42 +32/ +8, and check. 5. Add Sax-{-4iby — 2xy, 5by — 7xz-\-Qxy, 2ax — Sxy — dby, and 7xz-\-xy — by, and check. 6. Add 2x-4y-{-Sz, -bz+y-Qx, 3y+z-^5x, -Sy+Ax —42, and 7y+Sz—4iX. 7. Add 7ac — an-\-Snx, 5ax-{-4:an — Qnx, 2nx — San — 5ac, and an—5ax — nx, and check. 8. Add 5a+66-7c, 4c-36+5, -2c+5b-Sa, 4c-76-9, and -6+2c+6aH-5. 9. Add 5ax-\-3bx — 2cx, Zdx—4:ax-\-5cx, 4:cx — 7bx — 3dx, and 66a; — Sex + ax. 10. Add 8a6 — 66c+4ac, 5ad — 5ab — 7ac, 46c — ad+5ac, and 36c — 3ad— 3a6. 11. Add 4an — 76n+5a6, 46n — 7ac — 6an, 3anH-6ac — 9a6, and 4bn — ac-\-4ab, and check. 12. Add 6x-7y+5z, 4y-u-Sz, -2u-\-Qy-5x, Az-by +4:U — x, and —Qz-\-2x — Su. 13. Add 4a6 — 2ac+46c, — 5ac — 2a6+66c, and ab — 2ac. 14. Add 4a -76 -5c, Sc-7a-d, -56+3d-2c, 86-2cZ -5+c, and 46+2c+4a+5. 15. Add 5xy-ixz+dyz, 2xz — 2xy — 7yz, 3xz—xy-\-9yz, and —Syz+Qxy — xz, and check. CHAPTER IV SUBTRACTION. SYMBOLS OF AGGREGATION 53. Subtraction is the process of finding one of two num- bers when their sum and the other number are known. 54. The minuend is the number that represents the sum; the subtrahend is one of the addends of the minuend. 55. The difference, or remainder, is the number which added to the subtrahend gives the minuend. SUBTRACTION OF MONOMIALS 1. A thermometer reads +13°, and four hours previously it read —7°. Through how many degrees and in what direction had the top of the mercury changed meanwhile? Present reading, +13° Previous reading, — 7° The change, +20°, obtained by subtracting -7° from 13°. 2. Starting from a stair-landing a boy goes up 17 steps, and drops his pencil, which rolls down to the landing, across the landing, and on down to the 6th step below the landing, where it stops, The steps are a inches high. How far and in what direction must the boy go to get to the step where the pencil Hes? CaUing upward + and downward — , the boy- arriving at — 6a starting from +17a goes —23a, meaning 23a inches downward. In these cases we have been subtracting signed numbers. Let us now learn the general plan of subtracting such numbers. 35 36 ELEMENTARY ALGEBRA SUBTRACTING SIMILAR TERMS 56. The following examples represent all cases in addition with reference to signs and relative values of addends : 5a 3a -5a -3a -5a +5a -3a 3a 3a 5a -3a -5a 3a -3a 5a -5a 8a 8a -8a -8a -2a 2a 2a -2a Write examples in subtraction, using the above sums as minuends and one addend as subtrahend, as follows: 8a 8a -8a -8a -2a 2a 2a -2a 3a 5a -3a -5a 3a -3a 5a -5a 5a 3a -5a -3a -5a 5a -3a 3a By the definition of subtraction, the difference or remain- der in each case must be the other addend. Show that the correct result might have been obtained in each case by changing the sign of the subtrahend and adding. 67. Principle. — Subtracting any number is equivalent to adding a number of equal absolute value but opposite quality. 58. Rule. — Conceive the sign of the subtrahend changed from -\- to — or from — to -\- and proceed as in addition. The change of sign should always be made mentally. Exercise 15 — Subtracting Similar Terms Give remainders in the following orally: 1. 9a 2. 4a e. 5a 4a -4x Qx 3. 8. -35 -86 -4b - h 4. 7n 2n 9. n 6n 6. 10. -lie - 3c 3x -7x -10c 4c SUBTRACTION OF MONOMIALS 37 11. 5x 12. -46 13. - n 14. 9c 15. -12a 9a: 56 —5n c 8a 16. 66 17. 9n 18. -8c 19. a 20. - 7x 76 -2n - c 3a -lOx 21. 2n 22. -8c 23. - a . 24. Qx 25. -146 7n 7c —2a a; - 66 26. 8c 27. 3a 28. -6a; 29. 6 30. 7n 7c -4a - X 86 -13n 31. 6a 32. -7a; 33. - 6 34. 7n 35. -lie 9a 9a: —46 n 4c 36. 5x 37. 86 38. -3n 39. c 40. -13s 6a: —66 — n 5c — lis SUBTRACTING DISSIMILAR TERMS 59. A man had 5a6 acres of land and sold 2xy acres of it. How many acres had he left? The subtraction of dissimilar terms is indicated by writing one term after the other. Thus, 5a6 3ac — 2aa: 2xy —4xy —36c 5ab — 2xy 3acH-4x!/ 36c — 2aa: In indicating the subtraction of dissimilar terms, the sub- trahend must be written with its sign changed. 38 ELEMENTARY ALGEBRA Exercise 16 — Subtracting Monomials Give remainders in the following orally : .. 3a 2. -4x 3. -4a 4. -5n 6. 4x b -7x -2n 7n - 2/ :. 7a 7. -7a 8. -66 9. 2n 10. 7c 8a -2c h -3x -6c 11. a 12. -3a; 13. - a 14. -3n 15. 5x 2b -9x -Sx 6n -Sy 16. X 17. 3a 18. -26 19. -3n 20. 5c 9a; -26 76 -4/1 - c 21. a 22. — aj 23. — a 24. 4n 25. — c 56 -9x -46 -3n 8c 26. 3a; 27. 5a 28. -36 29. -5n 30. -2x Qx -2c -76 -6a -5y 31. 4a 32. x 33. -46 34. -2n 35. —5c 9a — 6i/ 76 -4a — c 36. a 37. -7a; 38. -2a 39. -4n 40. -8c 7a; 5a; —26 -9n c 41. 3a 42. - X 43. 66 44. -4n 45. 7c 5a 85/ -86 5c -9c SUBTRACTION OF POLYNOMIALS 39 SUBTRACTION OF POLYNOMIALS 60. 1. Subtract 7 dollars, 3 quarters, 8 dimes from 16 dollars, 7 quarters, 12 dimes. Letting c be the number of cents in a dollar, q the number of cents in a quarter, and d. the number of cents in a dime, we write: From lGc+7q+12d Take 7c+3g+ Sd Difference, 9c+4g+ 4d 2. From 5ah—4ac-\-Sbc bushels of grain, 4a6 — 6ac+2cc? bushels were sold. How many bushels remained? Minuend, 5ab — 4ac-\-Sbc Subtrahend, 4a6 — 6ac -\-2cd Difference, ah -\- 2ac + 36c — 2cd 61. Rule. — Write the polynomials, similar terms in a column. Beginning at the left, subtract as with monomials. Subtraction is checked by determining whether the difference between the values of minuend and subtrahend is equal to the value of the remainder. Observe the work below: Work Check 5a6-4ac-f36c = 10-12+18 =16 . 4ab-6ac -\-2cd = 8-18 +24 = 14 ah-\-2ac+3hc-2cd = 2+ 6+18-24= 2 The above example in subtraction has been checked by substituting 1 for a, 2 for h, 3 for c, and 4 for d. It is now plain that subtracting is finding what number must be added to the subtrahend to give the minuend. Hence, another good check on subtraction is to add the subtrahend and difference and see if the sum is the minuend. 40 ELEMENTARY ALGEBRA Exercise 17 — Subtracting Poljmomials Solve the following and check the first nine : 1. From 8a6-5c+4d-8 subtract 4(i+3a6-12-6c. 2. Subtract 5ay — z-\-9ax-{- 14: from 4axH-6ai/ — 2;+8. 3. From 66c-56H-8de+/ subtract 8de+10+56c-56. 4. Subtract 4ac+Sbd-2bc-She from 4ac+26rf- 106c. 5. From4:CX-{-7by — xy — 9s\ihtrsiCi7by — 10-{-3cx-{-xy. 6. Subtract Aax — Axy + lab from 4aa: — 2ac — Sxy -h 8a6. 7. From ax — 7ay-\-dxy — 2z subtract 4:xy — 7ay — 7-\-ax. 8. Subtract Sab-\-Q — 7ac — ax — am from 4a6 — 6ac — am. 9. From 6am— 4an4-4ar— 7rs subtract 12H-6am— 12an. 10. From the sum of 3a-\-2b — Sc+d and 2d-|-2a— 46 sub- tract 36-5+3d+4a+3c. 11. From 4:X — dy-{-2z — u subtract the sum of dz-\-2x — Q —4?/ and —2z — x-\ry — 2u-\-Q. 12. Subtract the sum of 2y-\-2b — 5x — da and 3a: — 66+3?/ +4a from 2a-36-2x+4i/. 13. From the sum of 3c-2d-5e+2/ and 8e-4d-3/-6c subtract 5e — 5c— /— 3d 14. Subtract 26 — 2c +d — 2a from the sum of 2a — 56 -f- 2c -2d and 46+3d-3a-3c. 16. From the sum of Sx-\-2y—z-{-2u+S and 32-4a:-10 ^by—^u subtract 2z — 3 — bx — '3u — 2y. 16. From 5a6+2ac — 36c subtract the sum of 26c-|-3ac+6(/, 4a6 — 26d— 46c, and bd—iac — ab. 17. From the sum of 4:X-{-y — 2z and 4iU-\-Sy—7x — 2z subtract 4w+42/ — 52!+5 — 3a;. SYMBOLS OF AGGREGATION 41 18. What number must be added to —4a +66 — 8c to give 0? To give 8a+46-4c? 19. From 4a6 — 3ac+26c subtract the sum of Shc-^-hd—ac, Sab — Shd — he, and bd — 2ac — ab. 20. From the sum of 3a — 2a: +5 and 4:X-\-2y—4: subtract the sum of 3a; — 2aH-3 and y — 2x — 2. 21. If a; = 5a-36+4c, y = Sa-2b-3c, z = a-\-b+Qc, find the value of x — y — z. 22. What number must be subtracted from 2a6 — 3ac— 56c to give ac - 56c + 2a6 - 36d? To give 0? 23. Subtract 2a-36+4 from 7, 26-3a+3 from unity,* a — 26+2 from zero, and add the three results. SYMBOLS OF AGGREGATION 62. The product 8X14 can be shown thus: 8(10+4), which means 8X10+8X4 = 80+32=112. This use of the symbol ( ), called a parenthesis, is of aid in learning rapid mental calculation, thus: 7X25 = 7(20+5) = 140+35 = 175 6X49 = 6(40+9) =240+54 = 294 9X68 = 9(60+8) = 540+72 = 612, etc. 63. A man walks north 5 miles an hour for 2 hours and then south along the same road 3 miles an hour for 2 hours. How far is he then from the starting-point? The answer to this problem may be written thus: 2(5-3)=2X2 = 4 Show that the perimeter of a rectangle x wide and y long may be written: 2(x+y) or 2x-\-2y, or x-\-y-\-x-^y and that 2{x-\-y) =2x-\-2y. *Unity means 1. Rectangle 42 ELEMENTARY ALGEBRA 64. In a series of the four operations, the multiplications and divisions are to be performed first. Thus, 8+7X3-6+16-^2+5-5X3-8^2 = 8+ 21 -6+ 8 +5- 15-4 =17 In such a series the terms are the parts separated by the signs + and — . The above example contains seven terms. When such expressions are to be simplified or reduced, each term should be first simplified or reduced. When it is desired to perform the operations of a series in any order other than the one mentioned above, it is necessary to use some symbol of aggregation. 65. The symbols of aggregation are the parenthesis ( ), the brace [ } , the bracket [], and the vinculum . These mean that the operations indicated within them are to be performed before the operations upon them; in other words, that the expressions within them are in each case to be regarded as one number. Every part within the symbol is affected by the operation indicated upon the symbol. Observe the following: 18-9-4 = 5 15X12-8 =172 18-(9-4) = 13 15X12-8 =60 216-(24-36-^4)X4- (4+6X3-35-8X4) = 137 216- 60 - 19 =137 Notice the use of the parenthesis in the following: 1. If the smaller of two numbers is a; — 7 and the larger X— 2, their difference is (x — 2) — (x — 7). 2. If a rectangle is x+8 in. long and x+3 in. wide, the area of the rectangle is (x+8)(x+3) square inches. 3. If J of the distance between two cities is x+10 miles, the whole distance is 3(x+10) or (x+10)3 miles. SYMBOLS OF AGGREGATION 43 Exercise 18 Remove the symbols of aggregation and then simplify : 1. 465+67X8- (9X24+144^4-45^5X6) 2. 764-(245-465-^5)-14X7+789-540-^9 3. 238-8X9- 108^9+754- (84-58) -47X8 4. 9X(48+65) + 128^4-(8Xl2+7-8X8)X7 66. Operations on Compound Expressions. Symbols of aggregation are much used in algebra to indicate operations on compound expressions. To indicate the subtraction or multiplication of a poly- nomial, a parenthesis is necessary. Thus, x{a-\-h) represents the product of x and a-\-h and is read x times a +6, or a+6 times x. Exercise 19 1. Indicate the subtraction of x — 5 from 3x+4. Indicate the product of two binomials. 2. If a man has 8a; sheep and sells 2x+35 of them, what will denote the number he has left? 3. What does (x-\-5){x — 2) represent, if X represents the nuniber of feet on each side "^ of a square? 4. What does x{x-^S) represent, if x stands for the number of rods on each side of a square? 5. Represent in two forms 4 times the sum of any two numbers. 5 times the difference of any two numbers. 6. Represent the product of two equal numbers each of which is 8 greater than x. 7. At 85^ a rod, express in two ways the cost of enclosing a rectangular farm x rods by y rods. (N X 5 44 ELEMENTARY ALGEBRA 8. If X is any positive integer greater than 5, is x — 5 greater or less than a: — 3? Show why. 9. What is the equation which tells that the difference between a; — 9 and a: — 4 is a? 10. If the difference between x— 12 and x — 8 is n, what is the value of nl 11. If X is any positive integer, when is ax greater than x1 When is ax less than x? 12. How many trees are there in an orchard, if there are 20 more trees in a row than there are rows? 13. Write 3a times the product of two binomials divided by the product of a + 6 and a — 6. 14. Indicate how many acres there are in a rectangular field a; — 8 rd. wide and x+lO rd. long. 15. What may represent the product of 4 numbers, if any 2 of them in order differ by the same number? 16. Write an expression of 3 terms, each term containing one or more compound factors. 17. At $40 an acre, what is the value of 3 farms containing X, a:+20, and x — 5 acres, respectively? 18. Represent the product of two unequal numbers, part of each number being x. 19. Wh^t is the area of 3 equal rectangles, the width of each being x in. and the length 6 in. greater? 20. The area of a square x in. long is the same as that of a rectangle x+6 in. by x— 4 in. Express as an equation. 21. How much does a boy earn, if the number of cents he gets per day exceeds the number of days he works by 20? SYMBOLS OF AGGREGATION 45 67. Extended Meaning of Term. It is necessary now to enlarge our idea of a term, especially when signs of aggregation are used. For example, the expression, 2a{x-\-y) - (3a+5) - {2a^\)x- (a+6) {a-h) contains only /our terms. In an expression involving symbols of aggregation, that part of the expression within the symbol of aggregation is to be regarded as a term, or as one of the factors of a term. Exercise 20 1. Write 3 times the sum of a and h, diminished by 5 times the product of a, &, and c. 2. If a rectangle is x inches long and y inches wide, what does 2{x-\-y) represent? 3. If 2n — 1 represents an odd number, what will represent the next larger odd number? 4. How many square inches are cut off in a strip 3 inches wide all around a square of paper x inches long and wide? 68. Removing Symbols of Aggregation. Symbols of aggregation preceded by — may be removed by changing the signs of the terms enclosed. Thus, 3a-26-(2a-56+c>= 3a-26-2a+56-c The reason for this change is evident from the principles of subtraction, as the number enclosed is to be subtracted. Symbols of aggregation that are preceded by + are removed without changing the signs of the terms enclosed. The minus sign before a symbol of aggregation being a sign of opera- tion, students should remember that if the first term of the number enclosed has no sign expressed, it is positive. 46 ELEMENTARY ALGEBRA Exercise 21 Remove the symbols of aggregation in the following and express the results in as few terms as possible : 1. 4a-6-(a-26+c) 2. 3x- (~2x-\-Sy)+2y 3. 3a-(6-c+2a)+6 4. 4x-{-Sy+i-Sx-4y) 6. 5a-b-4a+h-c 6. 5x- i-2x-4y)-Sy 7. 2a+h-c-{-{Sa-h) 8. 2x-Sy- {-2x-4y) 9. Sa-c-b-4a-b 10. 4n-3x+(-3n-4a;) When x = 2a-3b+4:c, y = Sa+2b-5c, z = 4a-5b-Sc, find the value of each of the following : 11. x-{-y-\-z 12. x+y — z 13. —x — y — z 14. x — y — z 16. X — 2/+2; 16. — x+i/ — 2 69. To remove two or more symbols of aggregation, one within another, begin with the outer one.* 3a-{a+2b-a-^p^^n) = 3a— a — 2b-\-a-{-b — c—n It should be noted that the — sign before the b belongs to the vin- culum, not to the 6. The sign of the 6 is +. Removing the outer symbol changes the sign before b—c to -\-, and these two terms are brought down with the same signs. *Many teachers prefer to begin with the innermost symbol of aggre- gation. Either way becomes easy after a little practice. It is just about as easy and it is even quicker, to remove all symbols of aggregation at once by beginning at the left and bringing each suc- cessive term down with its own or the opposite sign according as there is an even or an odd number of the antecedent minus signs affecting it. Any one of the three ways becomes easy and reliable with a little practice. SYMBOLS OF AGGREGATION 47 Exercise 22 Remove the symbols of aggregation in the following and simplify the results : 1. Qa-{b+5a-{-c)+h 2. 2y-3x- i-4:X-3y) 3. 2a-(36-a+6-c) 4. Qx- {-2y+3y-5x) 5. 4a-{2b-a-\-c-h) 6. 5x- {-2y-4:X-3y) 7. 5a-{h+a-2h-a) 8. 4n-i-Sx-\-3n-Qx) 9. 3a-(6-2a+6-c) 10. 7x- (-4?/-3a:+32/) 11. 4a+(6-a+26-c) 12. 3x+27/-(-2x-4?/) 13. 2a-(c+6+2a-6) 14. An-{-2x-{-Sn-\-2y) 15. 36-(a+36+a+c) 16. 2y-i-2x-3x-Sy) 70. It follows, that in order to enclose two or more terms of a polynomial in a symbol of aggregation preceded by the sign — , we must change the signs of the terms enclosed. Thus, ab — ac-{-hc — cd = ab— {ac—bc-{-cd) Exercise 23 Enclose the last three terms of each of these polynomials in a parenthesis preceded by a minus sign: 1. ac—ax-\-ab-{-bx 2. 2x-{-2y—xy — xz-{-yz 3. ab-{-bc — ac-^ax 4. ax — ay — 2x-\-xy — 2y 5. ax — bx — be— by 6. 3a+26+aa; — a6+6c 7. an-\-ab-\-ac—bc 8. 2a — ab — ax-\-bc — 2c 9. ac — ax — bc-\-bx 10. bc-{-2a-\-ac-\-2x — ac 48 ELEMENTARY ALGEBRA ADDITION OF TERMS PARTLY SIMILAR 71. Terms that are partly similar, i.e., similar as to part of the letters only, may be united into one term with a polynomial coefficient. Thus, ay ax an by X —2n {a+h)y {a-{-l)x {a-2)n 72. Rule. — Write the dissimilar parts in a parenthesis as the polynomial coefficient of the similar part. The above answers are read: "a plus h, times y^'; *'a plus 1, times a:"; and ^'a minus 2, times n, " a slight pause in the reading occurring where the last curve of the parenthesis stands. . Exercise 24 Read the sums of the following : 1. ax 2. by 3. an 4. ax 5. —by bx — y — 3n x cy ax 8. —an 9. ab 4:X n 2b 6. y bi 11. Sx ax 16. Sx ax X 21. ax X ax 12. an 13. —by 14. ar — en y cr 17. -Ay 18. xy - y 10. — n an 15. - 5x — ax 22. ay 23. - y -2y 4:X 19. bx 20. xy — cx X - y X bx 24. ax 25. xy Sx -xy -2x X y — nx bx -xy SUBTRACTION OF TERMS PARTLY SIMILAR 49 SUBTRACTION OF TERMS PARTLY SIMILAR 73. Terms partly similar, i.e., similar as to part of the literal factors, may be subtracted by indicating the sub- traction of the dissimilar parts. Thus, ax hy n hx —cy an (a — b)x (b-{-c)y • (l — a)n 74. Rule. — Write the indicated subtraction of the dissimilar parts in a parenthesis as a polynomial coefficient. Observe that the sign of the dissimilar part in the subtra- hend is changed from + to — , or from — to +. Exercise 26 Subtract and read the results of the following : 1. ay 2.—bx 3. 4a 4. ax 5. —an cy —ax —46 x ' 2n 6. ax 7. n 2x —an 11. b 12. - y nb -xy 16. xy 17. — X ax ax 21. bx 22. y ax -by 26. y 27. — an by nx 8. — ax xy 13. -2x 2y 18. nx -xy 23. -3a -3x 28. by -41/ 9. ac 10. -2c ex — ac 14. nx 16. ac an -be 19. c ac 20. -ay by 24. 3?/ by 26. — an — n 29. be 30. -by cy y CHAPTER V GRAPHING FUNCTIONS. SOLVING EQUATIONS IN ONE UNKNOWN GRAPHICALLY GRAPHING FUNCTIONS 75. Algebraic Numbers, or Functions. For the present it is convenient to call a number expressed by the aid of one or more letters an algebraic number or a function of the numbers denoted by the letters. Thus, 2a: +3, n^ — 2n — S, a-\-h, x — y, etc., are algebraic numbers or functions. The n^, in n^ — 2n— 8, means nXn and is read n-square, just as 5^ means 5X5 and is read 5-square. With every algebraic number or function, such as 3a: +5 (or n^ — 2n — S), two numbers must be thought about, viz. : the algebraic number or function itself and the number x, or n, that it depends on for its value. The number n^ — 2n — 8 tells us to form a compound number by squaring some simple number (n), subtracting twice the simple number, and then subtracting 8. The two numbers to be thought about are the value of n^ — 2n — 8 itself, and the value of n, and so for other compound numbers. The number x, n, t, or y, in terms ♦of which the compound number (the function) is expressed, may be called the independent number. In other words, the value of 3a; +5 depends on what x is, and the value of 71^ — 271 — 8 depends on the value of n. The x and the n are the independent numbers. For the reasons just stated, a number expressed in terms of X, such as 3a: 4- 5, is called a function of x, and is written f(x) and read : function of x. Similarly, n^ — 2n — 8 or any other number expressed in terms of n, may be denoted by f{n) and read : function of n. 50 GRAPHING FUNCTIONS 51 A function is a number that depends on some other number for its valu£. An algebraic function is a number whose dependence on another number is expressed in algebraic symbols, as 3a; +5, n^ — 2n — S, a-\-b, x — y, etc. In this book the word ' 'function" means algebraic function. A function that depends on two other numbers, as a-\-b, is denoted by /(a, b) and read: function of a and b. Thus, also a; — ^ is denoted by f{x, y) and read : function of x and y. The parenthesis, ( ), in the function symbol does not mean multipli- cation, but is a part of the symbol. If the letter within the ( ) is replaced by a positive or negative arithmetical number, as in /( — 2), the meaning is that the number, —2, is to be substituted for the letter in the function. Thus, If/(x)=3x+5, then /(-2) =3- -2+5= -6+5= -1, and if/(n)=n2-2n-8, then /(5) =52-2-5-8 = 25-10-8 = 7. Find/(3)if/(x)=8a:-3. Find/(-4) if /(n)=3n+15. 76. Two very important problems of algebra are: I. Knowing the value of the independent number, to find the value of the function; and II. Knowing the value of the function, to find the value of the independent number. 77. We already know how to solve Problem I. For example, to find the value of 3a; +5 for a; = 4, we have only to substitute 4 for x in 3a; +5, thus 3X4+5. Reducing, we find 3x+5 = 17 for a; = 4, and so also for any other value of X. To find the value of n^- 2n — 8 for some value of n, as 5, we substitute 5 for n, thus 5^-2X5-8 = 7, to see that 71^ — 2n — 8 = 7 for n = 5, and so for other values of n. 52 ELEMENTARY ALGEBRA Thus we know that to solve the first of the above problems, we have only to substitute the value of the independent number in the function and to simplify. 78. The second problem occurs very frequently in algebra, viz. : To find the value of the independent number when the valy£ of the function is known. This is Problem II above and it is the converse of Problem I. For example, it is often necessary to solve such problems as : Given 3x+5 = 8, to find the value of x, or Given n^ — 2n — S = 7, to find the value of n. Such expressions as 3a:+5 = 8 and n^ — 2n — 8 = 7 are equa- tions, and to solve them means to find what value or values, of X or of n will make 3a: +5 equal to 8, orn^ — 2n — 8 equal to 7. Consequently, to solve the second problem stated above (§ 76, II), requires a knowledge of the ways of solving equa- tions. We shall first show by means of pictures what it means to solve equations. Let it be kept in mind that alge- braic equations are made up of algebraic numbers. 79. Dependence of an Algebraic Number, or Function. Let us first try to understand the relation that exists between x and 3x+5. Draw a vertical and a horizontal algebraic scale (YY' and XX^so that they shall be at right angles, with their 0-points together, as shown in the figure. This is quickly done with cross-lined paper. Pupils should have some pages of cross- lined paper in their note-books. Graph of 3a: +5 h '■' ^-f \1 zL iL 4 n/^ it^ ??/ \lWf lOH ^y~ /I I_ -^ *^ 3x ' jf^ ^ -6-5-4-3-2/ ' V -'. /C? ' i - ,Gr -4 j" - AT Te -p -J GRAPHING FUNCTIONS 53 Now proceed thus: Assume x=l, 2, 3, 4, 0,-1,-2,-3,-4, and calculate 3a:+5 = 8, 11, 14, 17, 5, +2, -1, -4, -7. These rows of numbers mean that 1 and 8 go together, as also 2 and 11, 3 and 14, and so on to —4 and —7. They are number-pairs, that are paired through 3x4-5, and are usually written: (1, 8), (2, 11), (3, -14), (4, 17), (0, 5), (-1, +2), (-2, -1), (-3, -4), and (-4, -7). The x- value is always written first. Now picture the number-pair (1, 8)* by starting from the 0-point, measuring 1 unit-space to the right and then 8 units upward vertically, and marking the point reached, as A . This point. A, pictures the number-pair (1, 8), for it is I unit from the vertical scal§. and 8 units from the horizontal scale. Then picture the number-pair (2, 11) by starting from the 0-point, measuring 2 units horizontally to the right, and then II units vertically upward to B. The point B pictures the number-pair (2, 11). Similarly picture the number-pairs (3, 14), (4, 17), (0, 5), ( — 1, +2), measuring minus values of x from toward the left, ( — 2, —1), measuring minus values of 3x+5 downward from the horizontal, ( — 3, —4), and ( — 4, —7) as at C, D, E, F, G, H, and /. If your measuring and your work are correct, and you stretch a string tightly just over the points, you will find them to lie on a straight line. If you do so find them, draw the straight line through them. *Remember that if no signs are written before numbers, the plus- sign (+) is understood. Thus, (1, 8) means (-+-1, +8), and (2, —5) means (+2, —5), etc. 54 ELEMENTARY ALGEBRA If you should substitute any whole or fractional positive or nega- tive value in 3a: +5 for x and locate the point-picture of the resulting number-pair, you would always find that the point falls on this same line. Tryx = i li - 2h, etc. The conclusion is that 3x4-5 con- nects numbers into number-pairs, whose picturing points all lie along ^ the same straight line. Any number of pairs of values are given by 3a: +5. What we have been doing in this section is called graphing the func- tion 3x+5. 1 1 -"i OO W ' ©* 1 r - K] i- ™- .J r ^1 - J - - V P- - V n ^ Yr V \ \ y r fj { 1^ \ \ L G V A^l 1 -yATT 80. Picturing n2-2n- 8. n^-2n-S. Graph of n2-2n-8 Let us now make a picture of Assume n = 1, 2, 3, 4, 5, 6, then calculate n2-2n-8 = -9, -8, -5, 0, +7, +16 0, -1, -2, -3, -4, ■8,-5, 0, -f7, +16. The number pairs are here (1, —9), (2, —8), (3, —5), (4,0), (5,7), (6, 16), (0, -8), (-1, -5), (-2,0), (-3, +7), and (—4, +16), the n-value being the first number of each pair. Using again a pair of perpendicular algebraic scales on cross-lined paper, picture the number-pairs as in the figure. Draw carefully freehand a smooth curve through points A, B, C, and so on to F and then to L, as shown. In this case the number-pairs lie along a curve, called a parabola. The parabola iS an open curve. SOLVING EQUATIONS GRAPHICALLY 55 Any value you might take for n," substituted in n^ — 2n — 8, would give a number-pair whose point-picture would lie on this same curve. Try n = |, n= —\, n = l\, n= —l\, etc. The function n^ — 2n — 8, is then a number-law which pictures into a parabola. What we have just been doing in this section is called graphing f(n)=n2 — 2n — 8. 81. To make pictures of functions we merely assume values for x, or n, etc., substitute the assumed values in the functions (3x+5 or n^ — 2n — 8), and calculate the second numbers of the number-pairs. It then remains to picture the number-pairs on a pair of perpendicular algebraic scales, as above. Any number of number-pairs are given by either 3x+5 or n^ — 2n— 8, or by any other function. Every such function has some straight or curved line- picture. The particular number-pairs given by any function always picture into points all of which lie on the same straight or curved line. Hence, every function has its own particu- lar line-picture. The rising and falling of the line or curve picture the changes in the function that are produced by changing the independent number, as x or n. SOLVING EQUATIONS IN ONE UNKNOWN GRAPHICALLY 82. Solving 3xH-5 = 8, Graphically. Suppose now that we were required to solve the equation 3x+5 = 8. We would calculate some number-pairs of So; +5, locate the picturing points (see figure in § 79), and draw through the points the straight line. 56 ELEMENTARY ALGEBRA So soon as we know the line-picture to be a straight line, two rather widely separated points are sufficient to give the line-picture. Since we want to j&nd the value of x that makes 3x+5 = 8, we measure 8 units up on the vertical scale, and draw a horizontal out until it crosses the line of 3a: +5. The length of this line, or its equal measured along the horizontal scale, is the required value of x. The length is 1, and as it extends to the right, x=-\-l. This horizontal is called the graphical solution of 3x+5 = 8. Notice that while any number of number-pairs are given by 3a;+5, only one of these number-pairs will make 3a;+5 = 8. 83. Solving n2-2n-8 = 7, Graphically. Similarly, let it be required to solve n^ — 2n — 8 = 7, graphically. Calculate some number-pairs by substituting values of n as in § 80, and draw the parabola-picture, freehand, as in § 80. Since we are seeking the value of n that makes n2-2n-8 = 7, we draw a horizontal through a point 7 units up on the vertical scale, and prolong the horizontal both ways until it crosses the parabola. The line is KE in the figure of § 80. It will cut the parabola in two points. The lengths of the parts of the horizontal between the vertical scale and the curve are the two values of n that will make n2-2n-8 = 7. The two values are n= +5, and n= — 3. Substitute each of the two values in n^— 2n— 8 and see if they make it equal 7. This shows that there are two values that will give the one value 7 for the algebraic number, n2-2n-8. SOLVING EQUATIONS GRAPHICALLY 57 Notice then that while any number of number-pairs are given by n^ — 2n — 8, only two of these pairs make n2-2n-8 = 7. This means there are only two points on the graph of n^ — 2n — 8 where n^ — 2n — 8 = 7. They are the points K and E in the figure of § 80. 84, We have now shown how to make pictures of number- laws such as 3rc+5 and n^ — 2n — 8, and have also shown how to solve graphically such equations as 3a;+5 = 8 and n^ — 2n — 8 = 7. For any other algebraic numbers or equa- tions that contain only one letter, the method is the same. Exercise 26 Draw the Hne-pictures of the following functions of x : 1. 2x+5 2. x+5 3. a:+3 4. 2a;+3 6. 3a;+2 6. 3a;-f 1 7. 3a;- 1 8. 2x-l 9. a:^-}- 8a; -1-12 10. a;2-3a;-10 11. x^-2x-3 12. x^-l 13. x^-Qx-{-S 14. X"-6x+5 15. x~-4x Exercise 27 Solve the following equations graphically: 1. 2x+5 = 7 2. a;+5 = 9 3.^:4-3 = 5 4. 2a;+3=-l 5. 3a;-j-2 = 8 6. 3x4-1 = 7 7. 3a;-l = 5 8. 2a;-l=-5 9. a;2-f8x+12 = 21 10. x2_3a;_io = 11. x^-2x-3 = 5 12. rc2_i=8 58 ELEMENTARY ALGEBRA SUMMARY 85. The work of this chapter has taught the following facts: 1. Algebraic numbers, or functions, require us to keep in mind two numbers, the function itself and also some other number, as x or n, that it depends on for its value. 2. An algebraic number or function is a shorthand descrip- tion of the way to calculate its own value. 3. Algebraic numbers associate numbers into number- pairs. 4. The point-pictures of the number-pairs of an algebraic number give the line-pictures of the algebraic numbers, called the graphs of the algebraic numbers. 5. To find the value of an algebraic function when the value of the number it depends on is given, we substitute the given value and simplify. 6. To find the value of the independent number when the algebraic function is given equal to a number, we must solve an equation. 7. An equation is only a shorthand way of saying a function is to have a certain value. 8. While an algebraic function may furnish a great number of number-pairs, usually only one or a few of these pairs furnish a solution of the equation which gives the algebraic function a particular value. Although the graphical solutions of equations make the meaning of solutions clear and comprehensible, even in minute details, still they are more tedious and cumbersome than the algebraic solutions. When it is only the results of solutions that are wanted, and after it is learned that alge- braic solutions are shorter and easier ways of reaching these results, we shall use algebraic solutions. Algebraic solutions are treated in the next chapter. CHAPTER VI EQUATIONS. GENERAL REVIEW EQUATIONS 86. The equation is the backbone of algebra. Its value consists in its power as a tool for solving problems. Other algebraic topics are needed to give insight into and power over the equation. Algebraic skill means and always has meant nearly the same as skill in using the equation. In mathe- matical history the evolution of the equation means the evolution of algebra. The earliest algebraists were the Egyptians. Thirty-five hundred years ago they said such things as, '^A quantity, its half and third make 19. Find the quantity." They used no s5mibols or abbreviations, but the language of words only. About sixteen hundred years ago Diophantus, a Greek mathematician, wrote down the initial letters of the verbal sentence as his equation. It was simply a shortened sentence. A thousand years later calculators wrote down rules for calculating in symbols, much as a postal clerk of our day might write down rules for calculating the postage on parcels for various zones. For example, if for zone 3 the postal rule is ''6j^ for the first pound or fraction and 2f^ for each additional pound in the weight of the package," the postal clerk might write 2x+4, in which x is the weight in pounds, as a short form of the rule. On weighing the package he might do as 2x4-4 says, i.e., double the number of pounds and add 4 to get the number of cents to charge as postage. Now if at the other end of the route the persons receiving the package had no scales and desired to know the weight of the package, knowing the postage to be 12^, they might 60 ELEMENTARY ALGEBRA write down 2x+4 = 12, and find what x is, if they could solve the equation. Again, if a man starts 5 miles from his home and walks away from it x miles an hour for 2 hours, the rule for finding his distance from home would be 2x+5. Suppose he did not know his rate but did know how far he was from home, say 13 miles. To find his rate he might write 2x-\-5 = 13 and, if he knew how to solve the equation, he could find his rate, x, of walking. At a later date men came to regard such forms as 2x+4 and 2a: +5, not as shortened rules, but as the results of fol- lowing the rules, i.e., as numbers. Then they began to apply the laws of number to them, that is they began learning how to add, subtract, multiply, and divide them, and alge- bra was a reality. 87. Equations expressed partly or wholly in letters are either identities or conditional equations. 88. An identity is an equation with like members, or mem- bers which may be reduced to the same form. 89. The sign of identity is = . It is read, is identical with, or is identically equal to, or simply is. The sign of equality may also be used in an identity when there is no need to distinguish the nature of the equality. Thus, 5a-\-4a-\-2a = 8a+3a, and ax-\-c = c-\-ax are identities, and it is evident that they are true for any value of each letter in them. 90. Substitution is the process of putting one number symbol into an expression in place of another which has the same value. 91. Satisfying an Equation. An equation is said to be satisfied by any number which, when substituted in place of the unknown number, reduces the equation to an identity. The equation, 5x-\-Sx = 72, is satisfied by x = 9, for the substitution of 9 for X gives the identity, 45+27 = 72. EQUATIONS 61 In the equation, 5x — x = 3d, since 5x—x is 4x, 5x — x and 36 are equal only when x represents 9. Thus 5x— x = 36 is a conditional equation, because it is true only for a particular value of the letter in it. Any equation may be reduced to an identity by putting the value of each letter in place of that letter. The equations used in solving problems are equations of condition. The conditions of the problem, which are expressed in language, are translated into the language of an equation. 92. A root of an equation is any value of the unknown number that satisfies the equation. In solving equations, we shall often get an equation one or both members of which are negative, such as, -3a:=-12 -5i/ = 35 It will be explained later that in such cases the signs of both members may be changed. When -3x=-12, 3x=12, and x=4. When -5?/ = 35, 5?/= -35, and y= -7. When -4s = 27, s= -6f . Exercise 28 — Oral Work Solve the following equations: 1. -4a;=-15 2. -3^ = 18 3. 17 = 2a; 4. 42= -51/ 5. -5x=-24 6. -7?/ = 49 7. 20 = 3a; 8. 15= -6?/ 9. -8a;=-44 10. -5y = Q0 11. 30 = 4x 12. 62= -3?/ 93. In solving problems, it may be necessary to multiply or divide a term or a binomial by some number. To multiply 3a by 2, multiply the coefficient by 2; to divide 8a; by 2, divide the coefficient by 2. Thus, 4aX2 = 8a 9a-^3 = 3a nX7 = 7n Qx-7-Q = x To multiply 2a +36 by 2, multiply both terms by 2; to divide Qx—12y by 3, divide both terms by 3. 62 ELEMENTARY ALGEBRA Exercise 29 Perform the indicated operations and answer the questions in the following: 1. (3a+66)X2 {Sn-10)XS (8a; -20?/) ^4 2. If x+S is the present age of a man, how old is another man who is twice as old? 3. (5x-12)X3 (4a-86)-^4 (6x+120)^6 4. If Tom has x dollars and Frank 3x — 20, how many has Fred who has half as many as both the others? 5. (8x-92/)X4 (2n+15)X4 (5a-356)^5 6. If X is one number and 2a;— 10 another, what is a third number which is twice the sum of the other two? 94. In the statement of many problems, one or both mem- bers may contain a known and an unknown number. Thus, 7a;-4 = 8+5a; Before solving, it is necessary to have all unknown numbers in one member and all known numbers in the other member. If by the addition axiom, § 15, we add +4 and —5x to both members of the equation without uniting similar terms, we have 7a;-5a; = 8-f4 The same result might have been obtained by subtracting +5x and —4 from both members of the equation. 95. This process of changing a term from one member of an equation to the other without destroying the equality is called transposition. To avoid mechanical work and to impress upon themselves what axiom is involved in this change, students should always explain the work by telling what they add to or subtract from both members. EQUATIONS 63 Exercise 30 In like manner solve and check the following equations, applying the addition and the subtraction axioms alternately: 1. 5a;-32 = 3x-16 2. 14-4n = n+32-8/i 3. 13-6s = 25-9s 4. 96+12 = 66+40-6 5. 8i/+14=4!/+74 6. 15-3x = x+75-9x 7. 9n-19 = 44+2n 8. 3s-s-18 = 36-8s 9. 32-2a; = 72-6a; 10. 7a+6-15 = 79-4a 11. 66+16 = 36+26 12. 10+9n = 88+2n-8 13. 34-56 = 49-86 14. 6a:-14 = 56-2a:+2 16. 9s-13 = 4s+27 16. 16+4n+7 = 3n+30 17. 23-3a: = 71-7a; 18. 4a-15-a = 35-2a Exercise 31 — Oral Practice Do this entire list of 14 exercises in 26 minutes. 1. A has X sheep, and B has y. How many would C have, if he had twice as many as A and B? 2. Indicate by use of parentheses the product of the sum and difference of any two numbers, as m and n. 3. If there are x hundreds, y tens, and z units in a number, what will represent the number? 4. What will represent the sum of four consecutive odd numbers of which n is the largest? 5. How many square feet are there in the walls of a room X feet square and n feet high? 6. The sum of the ages of 4 men is lOx years. What was the sum of their ages 12 years ago? 64 ELEMENTARY ALGEBRA 7. If n represents an integer, does 2n-\-2 represent an even or an odd number? Show why. 8. From x dollars a man paid two debts, one of a dollars and the other of h dollars. How much did he have left? 9. A paid x dollars for a harness and Ax dollars for a horse. Represent the cost of both. 10. If one part of x is 16, what is the other part? If one part of y is 45, what is the other part? 11. A boy bought x oranges at m cents apiece and sold them at n cents apiece. If he lost, what was his loss? 12. The difference between two numbers is 25, and the smaller number is s. What is the larger number? 13. Represent the number of acres in a rectangle of land x rods long and x — 5 rods wide. 14. A house cost n dollars, a farm 5n dollars, and a store* 4n dollars. Express in two ways the cost of all. Exercise 32 — Review Problems and Equations Solve and check the following problems and equations: 1. The sum of two numbers is 128, and their difference is 34. Find the larger number. 2. 7a;-13 = x+12+5 3. 6s+17 = 45-2s+8 4. Divide the number 184 into two parts so that the greater shall exceed the less by 48. 6. 9n-80 = 26-n-f4 6. 3?/+12 = 16-5?/+4 7. The sum of two numbers is 270, and their difference is 4 times the smaller. Find the numbers. 8. 18-f3a: = 40-a:+7 9. 76-50 = 23-26-1 EQUATIONS 65 10. A and B own a farm worth $13,100. A has 3 times as large a share as B. How much is B's share? 11. 4n-15+n = 5-5n 12. 60-3s = 6s-8s+7 13. One automobile ran 3 times as fast as a second and 6 miles an hour faster than a third. The sum of their rates was 120. Find the rate of the third. 14. 16+5x-38 = 7-a; 15. 8a+30 = 35+7a-3 16. Three times a number diminished by 57, is equal to twice the number increased by 68. Find the number. 17. 82/-40 = 50-2/+6 18. 9n-15 = 37+2n-h4 19. A horse and carriage cost $385, the horse costing $95 more than the carriage. What did the horse cost? 20. A and B are 57 miles apart. They travel toward each other until they meet, A traveling twice as many miles as B. How many miles did A travel? 21. A has twice as many acres of land as B, and B has three times as many acres as C. If all of them have 2400 acres, how many acres have A and B together? Exercise 33 — Oral Practice Do this entire list in 28 minutes. 1. A merchant sold x yards of silk for $45. What will represent the cost per yard? 2. If a man has a half-dollars and b quarters, how many cents has he? How many dollars? 3. Indicate the sum of a and b, diminished by c. The sum of dx and x, diminished by y. 4. What will represent the sum of three consecutive numbers of which s is the smallest? Of which s is the middle number? 66 ELEMENTARY ALGEBRA 6. If there are x tens and y units in a number, what will represent the number? 6. How much butter, at h cents a pound, will pay for n pounds of tea at 60 cents a pound? 7. What will denote the number of square feet in a piece of paper I yards long and w feet wide? 4 8. A farmer received x dollars for sheep which he sold at y dollars a head. How many did he sell? 9. Find the value of a bushels of apples at m cents a peck and h bushels of pears at n cents a peck. 10. If a represents an integer, when does a+1 represent an even number? When an odd number? 11. If the difference between two numbers is 45 and the larger one is x, what is the smaller number? 12. What will represent the sum of three consecutive even numbers of which s is the smallest? s the largest? 13. The sum of two numbers is 175, and the difference between them is 5 times the smaller. Find the numbers. 14. The sum of the ages of 3 boys is 6x years. If they live, what will be the sum of their ages in 8 years? CLEARING EQUATIONS OF FRACTIONS 96. Clearing of Fractions. An equation containing frac- tions must be changed so as to remove the fractions before it can be solved. Observe that fX20=16 Multiplying this fraction by 20, a multiple of its denominator, the product is a whole number. Multiplying any fraction by a multiple of its denominator gives a whole number, for the denominator cancels with one factor of the multiplier. EQUATIONS 67 97. Principle. — // any fraction is multiplied by a multiple of its denominator, the product is a whole number. 98. Problem. — To clear of fractions, the equation 1-10+1+3 = 1-5+1 (1) Multiply both members of this equation by 12, the least common multiple of the denominators, by multiplying each term in it, applying cancellation to the fractional terms, and the result is 6x-120+3a:+36 = 4a;-60+2x (Mult. Axiom) (2) Every term in this equation is a whole number. This work is called clearing an equation of fractions. In describing this transformation of an equation, students should tell by what they multiply both members of the equation, rather than use the expression, clearing of fractions, i.e., they should say: **by the use of the multiplication axiom," etc. Solving equation (2), x = S Checking in (l): f-lO+f +3 = |-5+f 4-10+2+3 = %^— 5 or, -1 = -1 Exercise 34 Clear of fractions, solve, and check the following: .|-,+| = 3+| q--+S = S-J 3. 1+2+1=4+1 . |+.i = 38+| 68 ELEMENTARY ALGEBRA Exercise 35 — Problems and Equations Solve and check the following: 1. A woman bought silk at $2 a yard and had $14 left. Twice as many yards at $1.50 a yard would have cost $4 more than she had. Find the cost of the silk bought. Let n = the number of yards she bought; then 2n + 14— the number of dollars she had, and 3n — 4 = the number of dollars she had. 3w-4 = 2n + 14 n = 18, and 2n = 36 ($36, cost of silk.) Check: 3- 18-4 = 2- 18-|-14 50 = 50 2. A has twice as many sheep as B and 35 less than C. If all have 635, how many has A? ^ X . ^ X „ X 2s is , 3s+5 3. 4 + 6-3 = 7-- 4. _+2^+- = s+^3- 6. A boy has J as many 5-cent pieces as dimes. If he has $9 in all, how many coins has he? 8. A and B together earn $200 a month; A and C, $215; B and C, $235. How much do all earn? 11. Two horses cost $350, one costing 1 J times as much as the other. Find the cost of each. 12. Half of a number, diminished by 6, is equal to ^ of the number, increased by 2. Find the number. 13. The sum of the ages of mother and daughter is 48 years, and the difference between their ages is four times the daughter's age. Find the mother's age. EQUATIONS 69 14. A man bought 48 sheep and had $22 left. If he had bought 56 at the same price, he would have needed $14 more to pay for them. How much money did he have? 16. A horse and harness cost $260, the horse costing 5^ times as much as the harness. Find the cost of each. 16. 8x-12-(3a:-15) = 5x+33-(15+3a;) 17. A, B, and C together earn $3700. A earns $300 more than B and $400 less than C. How much does C earn? 18. 5n-(7n-fl6) + 10 = 12-(9n-30)-h3n 19. A has J as much money as B, and C has 2f times as much as A and B together. If C's money exceeds B's by $2800, how much have all? 20. 6s+14-(5s+20)=2s+13-(6s-16) 21. A father and son earn $126 a month. If the son's wages were doubled, he would receive only $18 less than his father. How much does the son receive? 22. 22/ -(32/+ 18) -12 =10 -(42/ -24) -5!/ 23. Three men raised 2040 bushels of corn. A raised three time» as many bushels as B and 165 bushels more than C. How many bushels did A and B together raise? 24.. Ix- (14+6a:- 18) =55- (5a;-48+3x) 25. A has \ as many sheep as B. If A should double his flock and B should sell 120 to C, A and B would then have the same number. How many sheep has B? 26. Frank has \ as many marbles as John. If John loses 186 and Frank loses 12, they will each have the same number left. How many marbles have both? 27. In a company of 112 persons, it was found that there were twice as many women as men and twice as many children as women. How many children were there? 70 ELEMENTARY ALGEBRA GENERAL REVIEW Exercise 36 — Oral Review Do this page in 15 minutes. 1. Express six times the product of a and h, increased by 3 times the sum of x and y. 2. What will represent the sum of 4 consecutive numbers of which X is the largest? 3. A man's capital doubled for 3 successive years, when it was $16,800. How much had he at first? 4. What is the age of a man who y years ago was a times the age of a boy whose age was x years? 6. How many square yards are there in the walls of a room 3x feet by 2x feet and y feet high? 6. What will represent the sum of 5 consecutive numbers of which m is the middle one? 7. A boy had a dollars. He earned h dollars and then spent c dollars. How much did he have left? 8. If one number is n and another number is 4 times as large, what is the sum of the numbers? 9. A farm cost 3 times as much as a house. If the*farm cost $6200 more than the house, what did both cost? 10. If a field is x rods square, how many rods of fence will be required to enclose it and divide it into 4 squares? ^ 11. A girl has x quarters, y dimes, and z nickels. Give an expression to denote how many dollars she has. 12. What will denote the number of feet in the perimeter of a rectangle 6x feet long and 3a: feet wide? 13. A man bought x sheep at a dollars a head and had b dollars left. How much money had he at first? 14. A house cost 3 times as much as the lot, one costing $5000 less than the other. What did both cost? GENERAL REVIEW 71 Exercise 37 — Written Review Solve all the problems of this page in 20 minutes. 1. A's age is to B's as 5 to 7, and the sum of their ages is 132 years. Find the age of each. Let 5n = the number of years in A's age, and 7n = the number of years in B's age. 5n+7n = 132 The pupil will understand that the number sought is not the value of n, but the numbers represented by 5w and 7n. 2. B's age is to A's as 4 to 7, and the difference between their ages is 27 years. Find A's age. 3. Seven boys and 12 men earn $275 a week. If each man earns 4 times as much as each boy, how much do the 7 boys earn per week? 4. A has 3 times as many cows as B; but if A should sell 6 to B, they would then have the same number. How many cows have both men? 5. Three men engage in business with a capital of $11,000. B invests half as much as A and $200 more than C. How much have A and B invested? 6. A, B, C, and D have 290 sheep. B has 15 more than A, C has 15 more than B, and D has 15 more than C. How many have A and B? 7. Three men raised 1684 bushels of oats. A raised 3 times as many bushels as C, and 185 bushels more than B. How many bushels did B and C raise? 8. A horse, carriage, and harness cost $350. The horse cost $95 more than the harness, and the carriage $35 less than the horse. Find the cost of the horse. 9. A boy bought oranges at 3^ apiece and had 20^ left. At bi apiece, he would have needed 16j^ more to pay for them. How many did he buy? 72 ELEMENTARY ALGEBRA Exercise 38 — Questions and Problems 1. Define algebraic expression; term; monomial; poly- nomial; similar terms; value of an algebraic expression. 2. From what expression must 9x-\-Qy — 5z be subtracted to give —4x — 3y-j-5z? 3. How are 8-9 and 6-9 added in arithmetic? Why cannot 9a and 8a be added in the same manner? 4. What are the factors of a number? Distinguish between the parts of a number and the factors of it. 6. How may numbers which are expressed by 2 factors be added, if they have a common factor? , 6. How is 4-8 subtracted from 9-8 in arithmetic? In what other way might it be subtracted? 7. What expression must be subtracted from 7a — 56+ 4c to give 9a +66 -2c? 8. Define identity; equation of condition. Give examples and show how they differ. 9. SimpUfy 9a- (2b-c)-\-2d- (5a+36)+4c-2c^, and find its value if a = 8, b= —4, c= — 5. 10. Add a(a+6)+2(6+c)+2(6-c), -3(6+c)+2(a+6) +6(6 — c), and a(6+c)— a(a+6)— 4(6 — c). 11. How is the correctness of subtraction proved in arith- metic? Is the same test applicable in algebra? 12. Subtract 2z-]-x — 2u-\-y-\-7 from the sum of 4:X — 2y-\- 5z—u and Sy+Q—4z — 2x. 13. Perform two different operations on an equation so that one term shall be transposed from each member. 14. Describe four operations which change the form and value of the members of an equation, but not their equality. GENERAL REVIEW 73 15. What expression must be added to 6a; — 5^+ 4^; to give 9x-\-4y-7z? 16. In the identity, 5x-{-Sx — 2x=10x—4:X, what number does X represent? 17. From 36 — 2cH-5d— 4e subtract the sum of 3d — 5e—4c +26 and c+e + 2d-4h, 18. How do you subtract one term from another, if the two terms are partly similar? 19. To what expression must 8a— 46+ 9c be added to give 5a+26-6c? To give 0? 20. Define root of an equation. How do you determine whether a number is a root of an equation? 21. Subtract 2a— 4b +5 from 0, and add the difference to the sum of 5a — 3c and unity. 22. Name the different steps in the solution of a problem by the use of an equation. Illustrate. 23. What must be true of two number expressions in order that we may place them equal to form an equation? 24. State the principle for enclosing two or more terms of a polynomial in a parenthesis. Illustrate. 25. How are terms that are partly similar added? Write 3 terms that are partly similar and add them. 26. From the sum of 2xy-\-3xz — yz and Sz — 2xz-^xy sub- tract the sum of xz — yz and 5z-\-Sxy — xz. 27. Add a(a-x)-2(a+a;)+a(a-2), 3(a+a;) + (a+3)- (a — 2)— a(a — a;), and a(a+3) — (a+x). 28. How do you prove whether the numbers found in solving a problem satisfy the conditions of the problem? CHAPTER VII GRAPHING DATA. SOLVING SIMULTANEOUS EQUATIONS GRAPHICALLY GRAPHING DATA 99. Graphing, as was illustrated in Chapter V, means representing by pictures and diagrams. 100. The diagrams and exercises below show how to picture laws that connect two sets of related numbers, such as prices and dates, temperatures and times, etc., when the laws cannot be expressed as equations, as well as when they can be so expressed. Exercise 39 1. In a newspaper of January, 1916, the prices of wheat from Jan. 10 to 15 on a Board of Trade were given as in the figure. The numbers along the horizontal are the dates, and those along the vertical, the prices per bushel. What was the price of wheat on Jan. 10? On Jan. 11? 12? 13? 14? 15? 2. On what date was the price highest? Lowest? Be- tween what dates did the price change most? 3. The average price per share, for dates Jan. 8-15, 1916, of 20 leading stocks of the New York Stock Exchange, was as shown in the figure. How much did the price fall from Jan. 8 to Jan. 10? Between what other dates did the price fall? Rise? What day was the rise greatest? The fall greatest? II 13 13 la JANUARY ! $93 55| r 93 00 r— 1 75 -A II 12 13 in JANUARV 74 GRAPHING DATA 75 4. What was the average price of these stocks on Jan. 11? On Jan. 14? On Jan. 15? JAN.1§ + 7° + 6° + 5° + 4° -4-3 = _2 = - 3° 6789 lO 11121 23 45 6 6. The hourly temperatures from 6 a. m. to 6 p. m. of Jan. 18, 1916, in Chicago, were as shown in the figure. Observe the degree-numbers along the vertical and the hour-numbers along the horizontal, and give the temperature at 6 a. m.; at 9 a. m.; at 12 m.; at 2 p. m.; at 6 p. m. 6. At what hour was the te:nperature lowest on Jan. 18? At what hours highest? When does the graph show the temperature stationary? In graphing temperatures, the lines connecting the points that repre- sent hourly readings do not represent the temperatures for the intermediate points. The temperature was probably not stationary at any time. But from the hourly readings it was apparently stationary. Nevertheless, the graphs give a good notion of the general trend of the temperature for the day. 7. The hourly thermometer readings from 6 a. m. to 6 p. m. on Jan. 17, 1916, in Chicago, were: A. M. M. p. M. Hours 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 Reading +2°, 2°, 0°,-l°,-l°,-2°,-2°,-2°,-2°,-2°,-2°,-2°,-r Show that the temperature line is as given in the figure. When was +-2 it coldest? Warmest? When grow- o ing colder? Warmer? When ^^ stationary by the graph? -a See note after problem 6. , n s \ A M Ppl 1 '< 4-i§s^ 01 1 1 2 2 : } 4 5 6 ^ ' \ >-C )-C )-6-< :tf' ^ ■■ 1 76 ELEMENTARY ALGEBRA + 2 JA^ (.1 7 JAN ie ■ P M A_^M °< L|.^°J V 2 2 : ^ s > € )-< / ] r ) k , .;^ f ' ~ ,, ^ y )-C N >- ^ s " 1 1 2's < 7 g 6 W 5 - 1 8. The hourly temperature curve from 6 p. m. Jan. 17 to 6 a. m. Jan. 18, 1916, was as shown in the figure. What was the ther- mometer reading at 7 p. m.? At 8, 9, 10, and 11 p. m.? At mid- night? At 3 a. m. of the 18th? At 5 a. m.? At 6 a. m.? 9. From Jan. 17, 6 p. m. to Jan. 18, 6 a. m. when was it growing warmer? Colder? When stationary by the graph? 10. A class studied the move- ment of a snail by having ft crawl along a foot-rule. The observing time, in minutes, was written along the horizontal, and the distances crawled, in inches, along a vertical, giving a picture of the snail's rate of crawling as in the figure. How far had the snail crawled the first minute? The first 2 min.? In 4 min.? In 6 min.? In 12 min.? What minute did it crawl most rapidly? Most slowly? 11. The daily growths of a tulip in inches were: Day 123 4. 56789 10 Height Ij 3 3| 6 8 8^ 9 lOj llj' 12 Mark off the days along a horizontal and the growths along verticals through 1, 2, 3, etc., using a scale of 1 short side to 1 inch, and draw a broken line connecting the points. What was the least growth on any day? The greatest growth? ,Jr^ i^ ii t ■y lU ^ ■-■ ^ U _ L_ to 12 14 16 18 20 13. At what age does the average boy grow most rapidly? 14. Graph these average heights of girls : Ages 2 4 6 8 10 12 14 16 18 20 years Heights.. 1.6 2.6 3 3.5 3.7 4.5 4.8 5.1 5.3 5.4 feet See the figure above. 15. At what age does the average girl grow most rapidly? 16. A varying rectangle 5 units wide, has the following lengths and areas: Lengths Areas 2 10 3 15 4 20 5 6, etc. 25 30, etc. Graph these lengths along a horizontal and areas along verticals, and connect the points. In this case we can show the law of the areas by the picture, and we can also express this law algebraically as an equation, thus, A=5L ''A / ■^ / / b^ A G u _ — 1 O 1 2 3 4 5 6 LENGTHS "Vertical Scale 1 space = 6 Pupils may find tables of values in books, daily papers, trade journals, etc., and graph them. 78 ELEMENTARY ALGEBRA 17. The rate per cent being 5, graph the following percent- ages and bases : Base $100, $200, $300, $400, $500, $600, $700, $800, $900, $1000 Percentage.. $5, $10, $15, $20, $25, $30, $35, $40, $45, $50 S50 45 M ■ ^ ^ -f^ y^ O <35 -30 -25 ,^ ^ .^^ >\ y^ ^ ■^ ^ .^ > )X- ->1 ^ >- ) _ SIOO 200 300 400 500 600 700 800 900 1 OOO Base Scale 1 horizontal space = $50 1 vertical space = $5 There is also an algebraic expression of this law, thus, P = 5.— orp: h_ 20 18. If now we take an algebraic law like y = x-\-\, we may substitute successive values of x, right and left from 0, and calculate the corresponding values of y, thus : x=l, 2, 3, 4, 5, 0, -1, -2, -3, -4, -5, etc. 2/ = 2, 3, 4, 5, 6, 1, 0, -1, -2, -3, -4, etc. Mark off the x-values along the horizontal, to the right if positive, and to the left if negative. Measure, to a convenient scale, the corre- sponding ^/-values on the verticals, upward if positive and downward if negative. Connect the points with a line. This line is the graph Graphof y=a;+l oiy = x-\-l. \ 1/ i / e r •>> ^ / o ■X- aa -: '/ -> 1/ / - /^ / ) GRAPHING DATA 79 19. Graph the algebraic law y^x^, by substituting successive values for x and calculating in y = x'^, the corresponding values of y. x = 0, 1, 2, 3, 4, 5, -1, -2, -3, -4, -5, etc. y = 0, I, 4, 9, 16, 25, 9, 16, 25, etc. k 1 1 i K rjo A > DO .1 \ 80 / \ J 1° SO \ ( \ Jn / \ ;'o ^ ^ ^f Jo } Y - y\ L 'o *^ Y ,._ - Y. lo 1 -y if 1 ' i> rd° r ^^ .^-^ 1 -1 0-9 -t 3-7-€ -1 -4- •?-? \ ?H = 6 7 8 9 1 o Graph of y=x^ Scale 1 horizontal space = 1 1 vertical space = 10 Mark the x- and i/-values off on horizontal and verticals. 20. Graph the algebraic \awx^-\-y^ = 25, or ?/ = ± \/25 — x^* by calculating values of y and plotting points as above : 5, etc. 0, etc. x= 1 :4.9 + 2 4.6+ 4 5-1 3 ±4.9 2 4.6 3 -4 4 ±3 ^ — r^ i^ 1^ f^ s^ — / '^ N -2 \ / >^ If -fe- 1- 3 - 2- 3 i I 1 V /^ v ^ J \ J ' ^V y k^ u ^ Graph of ?/ = ± V 25-2:2 or x2+z/2 = 25 *The expression V 25— a;^ means the square root of 25— x^. The sign ± means that the number calculated for V 25— x- may be either positive or negative. 80 ELEMENTARY ALGEBRA 101. From the above problems it is seen that a group of facts expressed by two different sets of connected numbers, like dates and prices, times and temperatures, ages and heights, a;-values and ^/-values in an equation, may be pictured, or graphed. This is generally done by measuring off the numbers of one set horizontally and of the other set vertically, locating points, and then connecting the points. 102. Problems 18, 19, and 20 have shown the following important facts: 1. A single equation in two unknowns is satisfied by many pairs of values of the unknowns. 2. By measuring off x-values horizontally and 2/- values vertically to suitable scales, locating points and connecting them, equations may give either straight or curved line graphs, or pictures. 3. Every pair of values of x and y that satisfies a given equation gives a point-picture that lies on one and the same line or curve. 4. It is easy to see that the x- and ^/-distance of any point on the curve from the chosen reference lines, would, if substituted, satisfy the equation that gave the graph. 103. In problem 18 the graph of y = x-\-l was found to be a straight line. This could be shown by stretching a string along the row of points. Any equation in two unknowns in which each unknown has the exponent* 1 (as 3x — 2t/= 1) gives a straight-line graph. Knowing this, it is easy to draw graphs of such equations by merely ch'oosing two values of x, calculating the corresponding two values for y, locating the two points, and drawing a straight line through the two points with a ruler. *With numbers like x, x^, y, y^, the small number written (or understood) at the right and above the letter is called an exponent. When no number is written, as with x, or n, or y, 1 is understood to be the exponent, just as though the written forms were, x^, or n^, or y^. GRAPHING DATA 81 ,Ji 1 ^ ^ 1 "y f f 5- 4- 3-?- J f / A / -- _ L"^ _ _ Y' A third point may well be calculated and located as a check on the work. It is best not to take the values of x too near together, as it is difficult to draw a line accurately through two very near points. 104. Linear Equations. Since equations in two unknowns both with exponent 1, have straight-line graphs, they are commonly called linear equations. Y 1. Graph the linear equation ?>x-2y = l. Take x = {), +3, -2, and compute y=-\, +4, -3|. The number-pairs for the points are written thus : - (0, -i), (3, 4), (-2, -3i), the first number in the parenthesis being the x- value. Graph the first two points (0, — ^) and (3, 4), as at A and 5, draw a hne through them with a ruler, and test whether the point ( — 2, —3^) lies on the line, as at C 2. In a similar way graph each of the following equations: i. y = x-2 2. y = x-^ 3. y = 2x 4. y = 4:-x 5. y = 2x-\ 6. y = 2x-\-?> 7. x^-2y = ^ 8. 2x-y = 4. 9. 3a;-4t/ = 4 105. We have just seen that one linear equation in two unknowns is satisfied by many pairs of values of x and y. But two linear equations in two unknowns, such as 2x--^y = 7 2y — x = 4: can both be satisfied at the same time by only one pair of values of x and y. Linear Equation Straight-Line Graph Graph of Sx-2y = l 82 ELEMENTARY ALGEBRA A" T^ ^K \ S- % D\- L^^ X<^B l.r-'^S- *^Ji^ V ^WG O El\ ^ 4--- T ^ it _SJ For example, graph 2x-\-y = 7, using a: =+1, +3, and —1, giving 2/= +5, +1, and +9, and graph 2?/ — x = 4, using x= 0, +4, and —3, giving y=-\-2, +4, and H-|. (See figure.) Now, we ask, can a point he so as to give X- and ^/-distances that will satisfy both equations? The answer is yes. The point of intersection, P, of the graphs sat- isfies the requirement. For the point, P, x=-\-2 and y=-\-S, and these values satisfy both equations. Hence, the x- and ^/-distances of the point of intersection of the graphs are the graphical solution of the two given linear equations. Since the graphs cross at only one point there is only one solution of the pair of equations. 106. Hence, two linear equations in two unknowns can be satisfied by only one pair of values of the unknowns. SOLVING SIMULTANEOUS EQUATIONS GRAPHICALLY 107. Simultaneous Equations. Equations that can be satisfied by the same values of the unknowns are called simultaneous equations. 108. It is now worth while to see that not all pairs of linear equations in two unknowns can be satisfied by even one pair of values of the unknowns. Two or more equations considered together are said to form a system. Y' Simultaneous Equations Intersecting Graphs SOLVING EQUATIONS GRAPHICALLY 83 1. Consider the system 1. 2y-x = 4: 2. Qy-Sx = Q The graphs of the equations are shown in the figure. Dividing 2 through by 3, gives 2y — x = 2, and the graph on which this is written is the graph of 6y — 3x = Q. The graphs are a pair of parallel lines. They y do not meet, and there is no point that Ues on both graphs. This means there is no pair of values of x and y that will satisfy both equations. "H n .M'l r^ ^"^1 X ^ ^ ^ ft (• i>< ^ ni> ii> >^ ^^^ ^ ^i^ r ^ Non-Simultaneous Equations Parallel Graphs 109. Inconsistent Equations. Equations which cannot be satis- fied by any pair of values of the unknowns are called non- simultaneous, or inconsistent equations. That the equations of § 108 are inconsistent can be seen without graphing, by dividing the second through by 3. This does not change the relation between x and y. Then one equation says that 2y—x = 4:, and the other that 2y — x is at the same time equal to 2. This is obviously absurd. The number, 2y—x, cannot at the same time be both 4 and 2. 110. For a system of two linear equations in two unknowns to be capable of solution, the equations must be simultaneous. 111. Dependent Equations. It is, however, not sufficient that the equations be simultaneous. We shall now see that two linear equations in two unknowns can fail to give a definite solution because they have too many solutions. 1. Consider the system. 1. 2y-x = 5' 2. Qy-3x=15 84 ELEMENTARY ALGEBRA 1 V ^ ?il>1 jX 6^ X <;, °> *r f^ Y^ ■ *L ^ o ^ 1 Both graphs are shown in the figure as a single line. They coincide. Every point that is on one is on the other also. Hence, any pair of values of x and y that satisfies one of the equations, satisfies the other also. Dividing the second equation through by 3, gives 2z/ — a: = 5, which is identical X with equation 1. One equation de- pends on the other in the sense that one can be derived from the other by simple division by an arithmeti- cal number. Such equations are called dependent equations. 112. Finally, for a system of two linear equations in two unknowns to be capable of solution, the equations must be both simultaneous and independent. Dependent Equations Coincident Graphs Exercise 40 — Graphical Solutions Solve the following systems graphically, or in case there is no definite solution, tell whether the system is inconsistent or dependent: 4. x-y = 2 Sx-2y = 9 x-\-y = 5 x-Sy=l 2x-5y=15 5y-2x=-15 2. 8. ( x+y=l \2x-\-5y = n I x-{-2y = Q \2x-\-4y=12 y = 2x-3 x-\-2y = 14: 3. 9. f x+y=2 \3x+Sy = Q y = x-S Sx-6y=ll 5x-Sy = S 2x+y=10 The graphical way of solving equations makes the mean- ing of solutions clear; but the algebraic way of the next chapter is shorter, and as it can be applied to equations in 3, 4, 5, and even n unknowns, it is also much more generally useful than the graphical way. CHAPTER VIII SIMULTANEOUS SIMPLE EQUATIONS. ELIMINA- TION BY ADDITION OR SUBTRACTION SIMULTANEOUS SIMPLE EQUATIONS 113. A determinate equation is an equation which has one root, or a limited number of roots, as, 114. An indeterminate equation is an equation which has an unlimited number of roots. Consider 2x-{-2y = 12 Any value may be assigned to x in this equation and a value of y found that will satisfy the equation. For example, when x = l, y = 5; when x = 2, y = 4:; when X = 3, 2/ = 3 ; and so on indefinitely. 115. Now consider 2x — 2y= —4: Any value may be assigned to x in this equation and a value of y found that will satisfy the equation. For example, when x=l, y = S; when x = 2, 2/ = 4; when x = S,y = 5; and so on indefinitely. It is evident that every simple equation containing two or more unknown numbers is indeterminate. But there is one set of values, and only one, that satisfies both equations, 2x-{-2y = 12 and 2x — 2y= —4, and these values are x = 2 and i/ = 4. 85 86 ELEMENTARY ALGEBRA 116. Independent equations are equations which cannot be derived one from the other by addition of, or multipH- cation or division by a positive or negative arithmetical number. The equations given above are independent, for one cannot be derived from the other by simple multiplication and division. So also are 4a:+3^ = 28 and 2x+3y = U. 117. A system of equations is two or more equations involving two or more unknown numbers, as, x+2y = d2 (2x-\-Sy = SQ x-2y = 12 \Qx-2y = 20 By a set of roots is meant the values of the unknown numbers in a system. As has been noted, each equation of a system, when taken by itself, is indeterminate. It was noted, also, that only one set of roots will satisfy two independent equations. In the two systems above, a: = 22 and y = 5 in the first and x — Q and y = S in the second, were the sets of roots. Simultaneous simple equations were solved graphicallj^ in Chapter VII. They will now be solved algebraically. To solve two simultaneous equations containing two unknown numbers, it is necessary to obtain from them a single equation containing but one unknown number. This can be done only in case the equations are indepen- dent as well as simultaneous; see § 112. 118. Elimination is the process of combining two or more simultaneous equations containing two or more unknown numbers in such a way as to obtain a single equation in which one of the unknown numbers does not appear. ELIMINATION BY ADDITION OR SUBTRACTION 87 ELIMINATION BY ADDITION OR SUBTRACTION 119. The following examples indicate the method of elimi- nation by addition and by subtraction. Solve the systems: x-\-y= 8 (1) f3a;+32/ = 9 (1) • ^ x-y= 6 (2) ; \3x+ y = b (2) 2x =14 22/ = 4 X =7 y=2 We add (2) to (1), member to We subtract (2) from (1) member, eliminating y, and then eliminating x, and then find the find the value of x. value of y. We then substitute these values in one of the equations of the sys- tem that gave it, and find the value of the other unknown number. From (1). y=l From (2). x=l checking 1^+^ = 7 + 1=8 (1) p+3^ = 3. 1 + 3- 2 = 9 (1) cneckmg \^_^^^_^^q (2) cneckmg ^^^^ ^ = 3.i_|- 2 =5 (2) In example 3, given below, we multiply both members of (2) by 2 and eliminate y by subtracting (3) from (1). 3. 9a:+4i/ = 43 (1) 4. Sx-\-2y = 2l (1) Sx-{-2y = 17 (2) 2x-\-3y = 19 (2) 9x+4y = 4S (1) Qx-\-4y = 42 (3) 6a;+4!/ = 34 (3) 6x+9y = 57 (4) Sx =9 5y = 15 In example 4, we multiply (1) by 2 and (2) by 3 and eliminate x by subtracting (3) from (4). 120. Rule. — Determine first which of the two unknown numbers it is more convenient to eliminate. By the multiplication axiom, §15, make the coefficients of that unknown number the same in both equations. If the signs of the terms to be eliminated are unlike, add the equations, member to member; if alike, subtract one equa- tion from the other, member from member. 88 ELEMENTARY ALGEBRA Exercise 41 Solve the following equations, checking some of them : \Sx-2y = 4: Ax — 5y=l 2x-2y = 2 5. 9. 11. (Sx-\-6y = Q \Qx-3y = 2 5x-3y = 23 7x-4y = SS (5x-^Sy = SS \9y-Sx=15 7x-Sy = 29 13. 9x-4y = S5 (9x-\-Sy= 12 \4y-Qx=-l 2. 8. 10. 12. 14. 4x-2y=-8 x-\-Sy=-9 (5x-\-'6y=-'i \2x-{- y=-l f2x+3i/=-4 \Sx+5y=-5 3x-Sy=-d 7x-Qy=-l \4:y-\-5x=—7 (5x-^Sy=-5 \Qy+9x=-Q f4x+6y=-8 \8!/+5x=-4 PROBLEMS 121. Solving Problems. In algebra many problems in which two or more numbers are to be found can be solved by the use of a single equation containing but one unknown number, but in many problems it is more convenient to introduce as many unknown numbers as there are numbers to be found. Such solutions involve a system of simultaneous equations, and to make a solution possible, there must be as many independent equations as there are unknown numbers used. ELIMINATION BY ADDITION OR SUBTRACTION 89 Exercise 42 — Problems in Two Unknowns 1. The larger of two numbers exceeds 4 times the smaller by 17, and twice the larger exceeds 7 times the smaller by 48. Find the numbers. Let X = the larger number, and y — the smaller number. x-4i/ = 17 2. If 7 pounds of tea and 5 pounds of coffee cost $6.50 and 6 pounds of tea and 10 pounds of coffee at the same prices cost $7, what are the prices per pound? Let X =the price of the tea in cents, and y = the price of the coffee in cents. 7a;+5i/ = 650 62: + 10^ = 700 3. The sum of two numbers is 121, and their difference is 25. Find the two numbers. 4. A boy has $2.00 in dimes and nickels, 28 coins in all. How many coins of each kind has he? 5. Eight sheep cost $12 more than 9 lambs, and 5 sheep and 3 lambs cost $42. Find the price of each. 6. B's age exceeds A's age by 8 years, and 3 times A's age exceeds twice B's age by 28 years. Find their ages. 7. Find two numbers such that if 4 is subtracted from the first and 8 added to the second, the results are equal; while if 2 is subtracted from the first and 6 from the second, the first remainder is twice the second. 8. Nine apples and 8 oranges cost 59^, and at the same prices 7 apples and 6 oranges cost 45^. Find the price of each. 90 ELEMENTARY ALGEBRA 9. A man sold 80 sheep for $390, selling some of them at $4 a head and the rest at $6 a head. How many sheep did he sell at each price? 10. The sum of the ages of A and B is 92 years. If B were twice as old as he is, his age would exceed A's age by 16 years. Find the age of each. 11. In an election 5163 men voted for two candidates, and the candidate elected had a majority of 567. How many votes did each candidate receive? 12. The sum of two numbers is 255, and f of the larger is equal to f of the smaller. By how much does the larger number exceed the smaller? 13. Twelve men and 6 boys earn $24 a day, and at the same daily wages, 7 men and 8 boys would earn $16.25 a day. How much does each man earn per day? 14. A miller mixes corn worth 80^ a bushel with oats worth 60^, making a mixture of 100 bushels worth 72^ a bushel. How many bushels of each does he use? 15. Eight years ago B was 3 times as old as A, but if both live 8 years, B will be only twice as old as A. What was the age of each 8 years ago? 16. A merchant sold 48 yards of silk for $89, selling part of it at $1.75 a yard and the rest at $2 a yard. How many yards of the better silk did he sell? 17. A has 160 sheep in two fields. If he takes 15 from the first field to the second, he has the same number in each field. How many are there in each field? CHAPTER IX MULTIPLICATION 122. Multiplication is the process of taking one number as an addend a certain number of times. 3X5 = 5+5+5 = 15 123. The multiplicand is the number taken as an addend. 124. The multiplier is the number which denotes how many times the multiplicand is taken. 125. The product is the result of multiplication. THE SIGN OF THE PRODUCT 126. Taking +5 twice as an addend, we have +10; three times, +15; four times, +20;. five times, +25. Thus, 3- (+5) = +15, 4- (+5) = +20, 6- (+8) = +48, which are the same as (+3)(+5) = +15, (+4)(+5) = +20, (+6)(+8) = +48. Taking —5 twice as an addend, we have —10; three times, — 15; four times, —20; five times, —25. Thus, 4-(-5) = -20, 7-(-5) = -35, • 9-(-5) = -45, which are the same as (+4)(-5) = -20, (+7)(-5) = -35, (+9)(-5) = -45. A negative multiplier means that the product is of the opposite quality from what it would be if the multiplier were positive. Therefore, (+5)(-4) = -20 (-7)(-6) = +42 (-8)(-5) = +40 From the foregoing examples, (+6)(+5) = +30 (+7)(-5) = -35 (-6)(-6) = +36 (-8)(+6)=-48 91 92 ELEMENTARY ALGEBRA From these results we may derive a law of signs for multiplying positive and negative numbers. 127. Sign Law of Multiplication. — Like signs of two numbers give a positive product, and unlike signs give a negative product. 128. The product of two or more numbers must contain as factors all the factors of each of the numbers. Thus, 2aX36 = 2-3-a-6 = 6a6 Exercise 43 Give the products of the following orally : 1. Sx 2y 2. -a6 3c 3. 4a -3n 4. -Qx yz 5. -5a -Sx 6. 5a 46 7. -xy 2s 8. -6a -6c 9. 46 -3c 10. -76 -3a When a term contains a twice as a factor, it is not written aa, but a^, and is read : a square. When a term contains x 3 times as a factor, it is not written xxx, but x^, and is read : x cube. 129. An exponent is a symbol of number written at the right and a little above another symbol of number to show how many times the latter is taken as a factor. 2ab^& = 2'a'b'b'C'C'C = 2abbccc This is its signification only when the exponent is a positive integer. It must be remembered that when no exponent is expressed the exponent 1 is always understood. Thus abx means a'b'x\ Observe that a^ = aXaXaXaXay while 5a = a-|-a-|-a4-a-|-a MULTIPLICATION 93 Students should note carefully the difference in meaning of exponent and of coefficient. 130. The sign of continuation is a series of dots . . . , and is read, and so on, or and so on to. THE EXPONENT IN THE PRODUCT 131. By § 128, a^Xa^ = aaa'aa = aaaaa = a^ In this particular example, the exponents of a in multi- plicand and multiplier are added. This illustrates a law of multiplication. The student should understand here that to prove any general law, general numbers must be used. To prove that this law of multiplication is general for any positive integral exponents, let a represent any number and m and n any positive integral exponents. Then, by § 129, a"' = a'a'a'a'a . . .torn factors; and a"" = a- a- a* a- a. . .ton factors. The product, a*" times a"", must contain a to m factors and a to n factors, or a to (m+n) factors. Therefore, 132. Law of Exponents for Multiplication. — The expo- nent of the product is the sum of the exponents of the factors. The exponents, m and n, used in this discussion are general numbers only in the sense that they denote any positive integers. MULTIPLYING ONE MONOMIAL BY ANOTHER 133. Rule. — Write the sign of the product, if negative {if positive no sign need be written), placing after it the product of the numerical factors and all the different letters, giving each letter an exponent which is the sum of the exponents of that letter in the factors. 94 ELEMENTARY ALGEBRA Exercise 44 Give the following products : l,Qa^x 2. SaH 3. Qxy^ 4.- xy^ 5. -7a^b - ax^ 7ifz -2¥c 8. -362c 9. 8aa;2 10. -^xhj — 4a2 h — \xHj — 2x^y 13. la¥ 14 bo^x 15. -%a?x -2a^b -9xhj -2hH Sx-'y -baH 6. 4a26 Sab^ 7. —iax^ 4:a^x 11.8ax2 5a'b 12. -7ax2 26x2 There are three important fundamental laws of multiplica- cation which it will be well to notice here. These are : law of order, or commutative law; law of group- ing, or associative law; and distributive law. 134. Law of Order. — The product of several numbers is the same in whatever order they are used. It is evident that 8-5-3 = 5-3'8 = 3-8-5 for each member of this equality is the same number. In general numbers, abc = b'C-a = c-a-b 135. Law of Grouping. — The product of several numbers is the same in whatever manner they are grouped. 8- 5- 3 denotes that 8 is to be multiplied by 5 and the product multiplied by 3 ; that is, 8 • 5 • 3 = (8 • 5) • 3. By the law of order, 8-5-3 = 5-3-8 = 3-8-5 Therefore, 8-5-3 = (8-5)-3 = (5-3)-8=(3-8)-5 In general numbers, a*b-c= (a-b)-c= (bc)*a= (a-c)-b MULTIPLICATION 95 136. Distributive Law. — The product of a 'polynomial and a monomial is the algebraic sum of the products obtained by multiplying each term of the polynomial by the monomial. (8+7)-6 = 8-6+7-6 In general numbers, (b+c)a = ab+ac This is called the distributive law, because the multiplier is distributed over the terms of the multiplicand. 137. A power is the product obtained by taking a number any number of times as a factor. 138. A square, or second power, is the product obtained by taking a number twice as a factor. Thus, 52 = 5.5 = 25 72 = 7-7 = 49 (6a)2 = 6a -60 = 3602 139. A cube, or third power, is the product obtained by taking a number three times as a factor. 53 = 5 . 5 . 5 = 125 (3a2)3 = 3a2 • 3a^ • Sa^ = 27a« The repeated factor is the root of the power, and the exponent indicating the power is the exponent of the power. The product is the power. Thus, exponent root — » 2^ ^ 8 '^— power POWERS OF MONOMIALS 140. To find a power of any number is simply to find the product of two or more equal factors. Thus, (2a62)4 = 2a¥ - 2ab'' • 2ab'' • 2ab^ = Ida'^b^ By the law of signs in multiplication, § 127, all powers of positive numbers and even powers of negative numbers are positive; odd powers of negative numbers are negative. 141. Rule. — (1) Raise the numerical coefficient to the re- quired power, (2) multiply the exponent of each letter by the exponent of the power, and (3) give the result the proper sign. 96 ELEMENTARY ' ALGEBRA Exercise 45 Gh ^e these indicated powers : 1. {2ay 2. (-2c2)3 3. i-^^y 4. (a^x^y 6. {sxy 6. (-4a'y 7. i-hr 8. {a^x'y 9. {2yy 10. i-zx^y 11. i-la^y 12. (a^x'y 13. {2ay 14. (-2a'y 16. i-Wy 16. (xvy 17. {7xy 18. i-5x'y 19. i-^x^y 20. {a'b^y 21. {Say 22. {-4a^y 23. {-h'y 24. {xvy MULTIPLYING A POLYNOMIAL BY A MONOMIAL 142. Observe carefully: 3a'¥-2a^b^+3a''b-2ab^ 2ab^ Qa^¥-4:a^b^-\-Qa^b^-4a^b^ 143. Rule. — Multiply each term of the multiplicand by the multiplier as in multiplication of monomials. Exercise 46 Multiply: 1. 3ax^+4:a^x by Sa^x^ 2. Sa'^¥-ab^-\-Sa^¥ by 4a^b^ 3. 5x'^y-3xy^ by 4a;y 4. 5ahi^-a^n-^a^n^ by 5aV 5. 3ac3-4a2c by 5aV 6. Qa^b^-a¥-\-Sa^b^ by Qa^¥ 7. Qa^x-7ax^ by 3a^x^ 8. 5¥c^+b^c-Wc'^ by Sb^c'^ MULTIPLICATION 97 MULTIPLYING A POLYNOMIAL BY A POLYNOMIAL 144. It follows from the distributive law, § 136, that {a-\-b) times any number is a times the number plus b times the number. 145. Rule. — Multiply the multiplicand by each term of the multiplier and add the products. Observe carefully: Work Check ^x'-2x''y-^Zxy^-y^ = 3 2x^-xy = 1 6x^ — 4:X^y-{-Qx^y^ — 2x'^y^ - Sx^y + 2xY - ^^V + xy"^ 6x^-7x^y-\-SxY-5xY-\-xy^ = ^ The work is checked by substituting x = l and y = l in the multipli- cand, multiplier, and product. It is plain that since any power of 1 is 1, i.e. 1^ = 1^ = 1^ = 1^ = 1, substituting 1 for x and ij does not check the exponents of x and ij. Exercise 47 Multiply: 1. 3c2+4c-6-by 2c^-c-\-S 2. 4a2+5a-3 by da^-a+5 3. Sx^-{-2x^-4x+l by 2x+4 4. a2-f3a6+62 ^y 2a^-Sab+b'' 6. x^y — x'^y^ — Sxy^ by 2x'^y—xy^ 146. A polynomial is arranged when the exponents of some letter increase or decrease with each succeeding term. Thus, 120^2— 4+ 2x is arranged when put in the form 1 20^24- 2x — 4, or -4+2x+12x2. As a convenience for the beginner, the multiplicand and the multi- plier should be arranged with reference to some letter; if possible, the same letter. 98 ELEMENTARY ALGEBRA Exercise 48 Multiply: 1. 12x2-4+2x by 4+5x2-3a; 2. 3a-2+4a2 by 4:a-\-3a^-2a' 3. 2x-{-3+x^ by Sx^-2x+S-x^ 4. 2a6-362+a2 by 2a''+3ab-b'' 5. 3a2-262-f3c2 by 4a2-262+4c2 6. 4ac-3a2+2c2 by 2a2-c2+3ac 7. a:^+2a;22/H-4x|/2 by 2?/+?/2 — 40^^ 8. 3a3-3a+2a2-4 by 5-3a2-3a 9. 2x?/2— x^+Sa;^?/ — ?/^ by i/ — dxy-\-x^ 10. 3a4+2a-3a2-a3+3 by 3a-a2+4 11. 4a:3-3x-4a;2+l by 3a:3+x-3a:2-5 Perform the following indicated multiplications and unite results into as few terms as possible:* 12. {a+c-{-x)2{a-\-c-x)-{2ac-x^y 13. 3(a-2c)(a+2c)2-5(a-3c)2+69c2 14. 2(3a+x)2-(3a-3a;)(3a+3x)-llx2 15. (a-2c)(c-3a)-(3a+c)(2c-a)-2ac 16. (2x+32/)2(2a;-32/)+2(2a;+32/)2-19a:2/ 17. 7{x+Sy){x-3y)-{x-5yy-6{x^-4y^) 18. 4(a:+3)(a;4-2) + (a:-6)(a;+4)-3a:(a;+7) *First decide how many terms there are in each of the given exercises. MULTIPLICATION 99 Exercise 49 — Special Products Perform the following indicated operations as rapidly as you can, using pencil only when necessary : 1. (a+l)(a-hl)= 2. (a+5)(a+5)= 3. ia-\-xy = 4. (2a: + l)2= 5. {7-\-xy= 6. {x+4){x+S) = 7. (:c+12)(x+3)= 8. (8+a)(5+a)= 9. {x-\-a){x+b) = 10. (m+r)(?n+s) = 12. (x^-\-y^)(x^-y^) = 14. (x+y){x'^-xy-\-y^) = 16. (3x+22/)2 = 18. {a-h)ia'-2ab-\-¥) = 20. (a:-l)(a:+l)(:c2+l) = 22. (a+6+a:)(a+b-x) = 24. (a-5)(a-5) = 26. (2a:- 1)2 = 28. {x+4){x-S) = 30. (8-a)(5-a) = 32. (m+r)(m — s) = 34. (x^-{-y^y = 36. (x-2/)(a:2+xi/+^2)=. 11. (a2+62)(a2-62) = 13. (x4-h2/4)2 = 15. (2a:+l)2 = 17. {a-b){a''+ah-\-h-) = 19. (x+2/)(a:2+a:2/+2/2) = 21. (a+6)3 = 23. (fl-l)(a-l) = 25. (a-a;)2 = 27. {7-xy = 29. (a;- 12) (a? -3) = 31. {x-a){x-b) = 33. (a2+62)2 = 35. (x^- 2/4)2 = 37. (2a:- 1)2 = CHAPTER X SIMPLE EQUATIONS 147. The degree of a term is indicated by the sum of the exponents of the literal factors. Thus, a^x^ is a term of the fourth degree. The degree of a term in any particular letter is indicated by the exponent of that letter in the term. Thus, a^x^ is of the second degree in x. 148. The degree of an equation in one unknown is the degree of the highest power of the unknown number. 5a; + 7 = 2x — a is an equation of the first degree. x—b = 4a:2 — 3 is an equation of the second degree, 149. A simple equation, or linear equation, is an equation which, when cleared and simplified, is of the ^rs^ degree. Whether or not a fractional equation is a simple equation cannot be determined until it is cleared of fractions and the resulting equation reduced to its simplest form. Also, rc2 + x-4 = x2 + 3 and 2x^-]-x+5 = x'^+x{x+2) are simple, or linear equations. These are simple equations, because when similar terms are united, the square of the unknown number disappears. 150. Checking or verifying a root of an equation is the process of proving that the root satisfies the equation. This is done by substituting the root found in the equation and ascertaining whether the result is an identity. 100 SIMPLE EQUATIONS 101 Solving the equation, 5a; — 6+3a;+7 = 6a;+19, we find the root of the equation to be 9. Substituting, 45-6+27+7 = 54+19 73 = 73, an identity In checking or verifying the root of an equation, the substitution should always be made in the original equation. When any term with the same sign is found in both members of an equation, it may by the subtraction axiom, § 15, be dropped from both. Thus, 2a:2+5a:-4 = 2a;2+2x+8 151. The directions for solving equations are generally summarized in a rule similar to the following: 1. Clear the equation of fractions by multiplying both members by the lowest common denominator (I. c. d.). 2. Transpose all unknown terms to the first member and all known terms to the second member. 3. Unite all terms containing the unknown number into one term, and unite similar terms in the second member. 4. Divide both members of the equation by the coefficient of the term that contains the unknown number. Exercise 60 Solve and verify these equations: 1. 3(a:+l)(8x-4) = (6a:-2)(4x+2) 2. 2(4a;-3)(2a:+2) = (4a:+7)(4x-4) 3. 5(2a:+3)(3a;-4) = (6x-3)(5a;-9) 4. 5(a:+3)+4(8-a:) = 15a:-6(8a;+3) 6. (a:-2)(9-x)-(a:+5)(2-x)-7 = 6. (a:-3)(3+a:)-(4+a:)(a;-4)-2a; = 7. 4(2a:-l)-2(2a:-6)+2(a:-8)-9 = 7 102 ELEMENTARY ALGEBRA 8. 5(a;+3)-4(x-2)-3(2+a:)-7 = 9. 5x-3(x-6) = 2(8-x)-4(9+x)+6 4+^ ^8 5- ^^-^ 8~ "^ ^3~4 12. 43-x+-g- = - 13. --x-3H-3^ In checking or verifying the solution of a problem, the substitution should be made in the problem itself. 14. A has twice as much money as B, and B has twice as much as C. If all have S595, how much has C? 15. The sum of the third, fourth, and eighth parts of a number is 68. Find the number. 16. The length of a rectangle is twice its width, and the perimeter is 144 feet. Find the dimensions. 17. A man gave $125 to his 5 sons, each of 4 of them receiving S5 more than his next younger brother. How much did the oldest son receive? 18. James has J as many marbles as Frank. If James buys 120 and Frank loses 23, James will then have 7 more than Frank. How many has each? 19. The sum of two numbers is 85, and 3 times the smaller exceeds twice the larger by 20. Find the larger number. 20. Seven men agreed to share equally in buying a boat, but, as 3 of them were unable to pay, each of the others had to pay $30 more than his original share. Find the cost of the boat. 21. Three men invested $9400 in business. A put in $600 more than B, and C invested $200 less than A. How much did A and C together invest in the business? 22. A farmer sold 30 lambs and 60 sheep for $300. He received twice as much per head for the sheep as for the lambs. How much did he receive for the 60 sheep? SIMPLE EQUATIONS 103 152. In clearing an equation of fractions, if a fraction is preceded by the minus sign, the sign of each term of the numerator must he changed, for the fraction-line is a vinculum for the numerator, gives, 12x-8a;-96 = 6-3a; Exercise 61 — Equations and Problems in One Unknown Solve the following, checking some of them : 3±2x x-S ^_4x-\-5 x-5 „ 2a:+2 2rc-3 , ^i x , 4.T-3 ^ x-H , ^ 5x-12 2x-6 , 2x-{ 3-— +2^—4-=-^ + ^- 5a;+15 _2a-5_^2_ 3a:+8 _2 a:-7 *• 2 ^~ ^^~~3~ "6^ «.^_5(^+2(x-3) = 6f+?^ . 31+8 „2 2(x-3)_4(x-2) 3(3:+2) 6. -^ H 2— 7. 3^-7H2(x+3)=^+3(W) 8. Nine boys and 16 men earn $365 a week. If each man earns 4 times as much as each boy, how much do the 9 boys earn per week? 9. A boy has $3.60 in dimes and 5-cent pieces, and he has 4 times as many 5-cent pieces as dimes. How many coins has he and what is the value of each kind? 104 ELEMENTARY ALGEBRA 10. A walked 95 miles in 3 days, going 4 miles more the second day than the first and 3 miles more the third day than the second. How far did he go the third day? 11. A is 3 times as old as B. Ten years ago A was 5 times as old as B. Find A's age now. Let x = the number of years in B's age now. and 3a: = the number of years in A's age now. X — 10 = the number of years in B's age 10 years ago, 3x — 10 = the number of years in A's age 10 years ago. 5>{x-10)=3x-lO 12. A is 4 times as old as his son, and 5 years ago he was 7 times as old. Find the father's age. 13. A man is 24 years older than his son. Fourteen years ago he was 3 times as old. Find the age of each. 14. A farmer sold corn, wheat, and oats. For his corn and wheat he received $800. For his corn and oats he received $720, and for his wheat and oats $840. How much did he receive for all his grain? 16. A man spent J of his money for a suit of clothes, J of it for a watch, and had $115 left. How much did he spend? 16. The sum of two numbers is 82, and if the greater is divided by the less, the quotient is 5 and the remainder 4. Find the two numbers. 17. D is 6 years older than C; C is 4 years older than B; B is 3 years older than A. If they live 5 years, the sum of their ages will be 135 years. Find D's age. 18. A grocer mixed tea worth 70^ a pound with tea worth 50^ a pound in such proportions that the mixture weighing 100 pounds was worth $58. How many pounds of each kind were in the mixture? SIMPLE EQUATIONS 105 19. A man paid a bill of $12.95 in quarters, dimes, and 5-cent pieces, giving in payment 3 times as many dimes as 5-cent pieces, and twice as many quarters as dimes. How many coins were there in the whole amount? 20. Find two numbers differing by 96, the sum of which is equal to twice their difference. . 21. Divide 32 into two parts such that the sum of twice the less and 5 times the greater shall be 118. 22. A man divided $3500 among his 5 sons so that each one received $100 more than his next younger brother. How much did the youngest son receive? 23. A farmer sold |^ of his potatoes and had left 665 bushels less than he sold. Find the value of his whole crop at 55^ a bushel. 24. The sum of three numbers is 170. The second exceeds the first by 8, and the third is 14 less than the second. Find the sum of the second and third numbers. 26. A lady bought a hat and a dress for $72, and the differ- ence in the cost was 4 times the cost of the hat. How much did she pay for the dress? 26. A man bequeathed his property, which amounted to $30,300, to his wife, son, and daughter. The son received $1200 more than the daughter and $3000 less than the wife. How much did the wife receive? 27. A man paid $12,800 for two houses and a farm, paying the same sum for each house. If he had paid twice as much for each house, the two houses would have cost $1600 more than the farm. Find the cost of the farm. 28. There are 3 times as many pupils in one school as in another. If 120 pupils were taken from the larger school to the smaller, the larger would still have twice as many as the smaller. How many are there in both schools? 106 ELEMENTARY ALGEBRA Exercise 62 — Problems in Simultaneous Equations Solve the following problems in simultaneous simple equations : 1. The sum of two numbers is 85, and their difference exceeds J of the smaller by 8. Find the numbers. Let X = the larger number, and i/ = the smaller number. y x-y-8=- 5 The second equation contains a fraction. Clear this of fractions and then with the other equation, ehminate. 2. If 5 is added to the numerator of a certain fraction, its value is f ; and if 1 is subtracted from the denominator, its value is J. Find the fraction. Let n = the numerator, and d = the denominator. 3. Three times the larger of two numbers exceeds ^ of the smaller by 66, and 3 times the smaller exceeds ^ of the larger by 46. Find the numbers. 4. If 3 is added to both terms of a certain fraction, its value is f ; and if 4 is subtracted from both terms, its value is f . Find the fraction. 5. A miller bought 50 bushels of corn and 40 bushels of oats for $64. At another time he bought at the same prices 38 bushels of oats and 70 bushels of corn for $78.80. How much did he pay for all of the corn? 6. A dealer bought oranges, some at 2 for 5^ and some at 3 for 5^, paying $12 for all. Three dozen were unsalable, but he sold the remainder at 30^ a dozen, making a profit of $2.10. How many oranges did he buy? CHAPTER XI DIVISION 153. Division is the process of finding one of two numbers when their product and the other number are known. 154. The dividend is the number to be divided and repre- sents the product of the two numbers. 155. The divisor is the number by which we divide and represents one factor of the dividend. 156. The quotient is the number obtained by division and represents the other factor of the dividend. Since division is the reverse of multiplication, the rule for division is derived from the process of multiplication. Three things must be determined : The sign of the quotient, the coefficient J the exponent of each letter. DIVIDING A MONOMIAL BY A MONOMIAL 157. The Sign of the Quotient. (+7)(+5) = +35, therefore (+35)^(+5) = +7 (+7)(-5) = -35, therefore (-35)^(-5) = -f 7 (-7)(-5) = +35, therefore (+35)-^(-5) = -7 (-7)(+5) = -35, therefore (-35)^(+5) = -7 158. Sign Law of Division. — Like signs of dividend and divisor give a positive quotient; unlike signs, a negative quotient. 107 108 ELEMENTARY ALGEBRA Give the following quotients : (+56)-^(+7) (-64)-(-8) (96)-f-(-8) (-63)^(+9) (+84)^(-7) (-72)-(-9) (68)-^(-4) (+75)-^(-5) Since 5aX3x=loax, therefore 15ax-^3x = 5a The coefficient of the quotient is the coefficient of the dividend divided by the coefficient of the divisor. 159. The Exponent in the Quotient. Since the dividend is a product, one factor of which is the divisor, the exponent of the dividend is the sum of the exponents of divisor and quotient. To find the exponent of the quotient, subtract the exponent of the divisor from that of the dividend. 160. Law of Exponents for Division. — Each exponent in the divisor is subtracted from the exponent of the same letter in the dividend. Since a^Xa'^ = a^, therefore a^-^a^ = a^ In general numbers, Observe the following: 2ab)W^ -2a^b )-M¥ -Za^h )l2a^¥c 4a^b Aa'^b'^ — 4a6c By the law of exponents for division, a^-i-a^ = a^. But any number, except 0, divided by itself also equals 1. Therefore, a« = l. 161. Meaning of Exponent 0. Since a may represent any number, it follows that any number with a zero-exponent is equal to 1. Thus, 2abcP = 2ab-l = 2ab From this equation it is evident that any letter with a zero-expo- nent may be omitted from a term, because its presence only multiplies the rest of the term by 1. DIVISION 109 Exercise 53 — Dividing Monomials Find the following quotients : 1. 46c3)166V 2. Qx^y)-SOxY 3. -xyz)-2x^yz^ 4. Qxy'')lSx^y^ 5. 7a'x)-2Sa''x^ 6. -acx)-5ac^x^ 7. Scx^)l5cH^ 8. 9b^c)-lS¥(^ 9. -hxy)-db^xy^ 10. 962)18a53c2 11. 3a:)-21a2xy 12. -axy)-4axY 13. 2ac3)14a3c3 14. 8a;?/)-24xy 15. -bcx)-Wcx^ 16. 8xi/^)16a^Y 17. Qa)-lSa^¥c'' 18. - a6a^) - Sab^a;^ 19. 5?/2)15a;?/322 20. 5a^x)-S5a^x'^ 21. -xyz)-7x^yz^ DIVIDING A POLYNOMIAL BY A MONOMIAL 162. Since division is the inverse of multiplication, (see §§ 142, 143), to divide a polynomial by a monomial, we divide each term of the polynomial by the monomial divisor. Thus, 5a62 c )20a''¥c-15a^b^c^+25ab^(^ 4a62 - Sa^c + Sc^ Exercise 54 — Dividing a. Polynomial by a Monomial Divide : 1. 32a63c2-l66Vc?+1663c*-864c2 by Wc'' 2. 14a6*c2+286V(^-186V+26V by 2b^c^ 3. 10b^x^y-25¥cx^+15b*x^-5¥x^ by 5¥x^ 4. 15aV-12a36c2(i+18aV-3a4c2 by Sa'c^ 5. 18a;V2+24axy-12x4i/3+6a:y by Ga^V 6. 36x^3+ 18xV- 27 6a;4!/3z+9a:Y by 9x¥ 7. 16a^a:3-24a36cx4+32a3a;3-8a2a:^ by Sa^x^ 8. 21a363c+14a464rf-28a26^-7a263 by. 7a^b^ no ELEMENTARY ALGEBRA DIVIDING A POLYNOMIAL BY A POLYNOMIAL 163. The rule for dividing a polynomial by a polynomial is deduced from the process of multiplication. Study this example carefully: . a4+2a36-6a2fe2+26a63-156^|a2+4afe-362 a^+4a^6-3a262 a'-2ab-\-5b^ -2a36-3a262+26a63 ' 5a262+20a63-1564 ba'b^ + 20ab^-15b'^ Arrange the dividend and divisor with reference to the descending powers of a, writing the divisor at the right of the dividend. Since the dividend is the product of the divisor and quotient, it is the algebraic sum of the products obtained by multiplying the divisor by the several terms of the quotient. Hence, when dividend, divisor, and quotient are arranged with reference to the descending powers of some letter, the first term of the dividend is the product of the first terms of the divisor and quotient, whence the first term of the quotient is the quotient of the first term of the dividend divided by the first term of the divisor. Dividing the first term of the dividend by the first term of the divisor, we have a^ for the first term of the quotient. Since the dividend is the algebraic sum of the products obtained by multiplying the divisor by the several terms of the quotient, if the product of the divisor and first term of the quotient is subtracted from the dividend, the remainder, which is a new dividend, is the product of the divisor and the other terms of the quotient, and the next term of the quotient is the quotient of the first term of the remainder divided by the first term of the divisor. Dividing the first term of the remainder by the first term of the divisor, we have —2ab for the second term of the quotient. Repeating this process until there is no remainder, we obtain the quotient a^ — 2ab-\-5b^. Each remainder must be arranged in the same manner as the divi- dend and divisor. DIVISION 111 Observe the following solutions: (I) a4-a262+2a63 - b^ \ a^-ab+b^ a^-a'b + d'b' a^^ab-b^ a^b -2a262+2a63 a^b - a^b^^ a¥ - a262+ ab^-¥ - a^62+ ab^-b^ (II). x'+4y' \x'-2xy+2y' Divide: x'-2x'y+2x^y^ x^+2xy-^2y^ 2x^y-2xY+^y'' 2x^y — 4x'^y'^ + 4:xy^ 2xhf-4xy^+4:y* Exercise 65 — Dividing Polynomials 1. a2-a-42by a+6 2. x^-x-SO by x-6 3. a2+a- 72 by a+9 4. x^-\-x-5Qhy x-7 6. a2-6a-16by a+2 6. a:2+8x-33by a-3 7. a2-9a-52by a+4 8. x2+7a:-98by a;-7 9. a2+32a+60 by a+2 10. a:2-17a;-18by l-\-x 11. a2+25a-54by a-2 112 ELEMENTARY ALGEBRA 12. a;2+20x-f75bya;+15 13. a*-15a2+56by a2-7 14. 3a6-8a3-28by a3+2 15. 5x8+42x^+85 by x*+5 16. a^— a(^ — 3a^c-\-c^ by a — c 17. ¥-\-b^x — hx^ — oir^ by h — x 18. a^+ax^+a^x+o^ by a+x 19. a4+64+a2fe2bya2+fe2_a6 20. ax''^+G6x+ 6x^+62 by ax+?> 21. x^+3x?/^+3x2|/+i/^ by x+i/ 22. x^ — 2/^+2;^ — 2x2; by X — 2/ — 2; 23. a^-\-2xy — y^ — x'^ by a+x — ?/ 24. 32x3-6x-l by 8x2-2x-l 25. 4a4+6a2+8a3-24 by 2a+4 26. a3+27 by a+3 27. 36x2-812/2 by 6x+9i/ 28. a^-ie by a-2 29. 27x3+642/^ by 3x+42/ 30. a3-64bya-4 31. 25x2-162/2 by 5x-4?/ 32. 0^-81 by a+3 33. Sx^-125y^ by 2x-5y 34. a6+27bya2+3 35. 16x2-64^/2 by 4x+Sy^ 36. a9-64 by a3-4 37. IQx'-Sly* by 2x+3t7 38. a8-16 by a2+2 39. 25x^-49y^ by 5x-7y CHAPTER XII APPLICATIONS OF SIMPLE EQUATIONS. ELIMINATION BY SUBSTITUTION SUGGESTIONS ON PROBLEM-SOLVING 164. Read the following suggestions carefully: I. Solving problems in algebra is finding one or more unknown numbers by the use of equations. II. Bear in mind that in all problem-solving the general plan is to find two different expressions to represent the same number, place them equal to form an equation, and solve the equation. III. The absolutely necessary condition of success in this work is the power to focus every faculty of the mind on the task in hand. For the time being every thought of other things must he banished from the mind: IV. First of all, you should read the problem attentively and thoughtfully several times before you attempt to form the equation. The purpose of this careful reading is to see clearly what facts are given and what is to be found. V. Very few advanced pupils can see through a problem at a glance and determine the equation, and of course a beginner cannot do it. You must not allow this partial and hazy grasp of the problem to discourage you. Never permit yourself even to think that you cannot conquer the problem. VI. You must advance by short steps at first. Here is a most important suggestion for you: Do not at the outset try to see every number in the problem represented in sym- bols and even to see the equation to be used, for very few 113 114 ELEMENTARY ALGEBRA can do that; but express in symbols all you can of the con- ditions of the problem, no matter how useless this may seem. From these expressions you will see what numbers are equal, and the formation of the equation will become a simple matter. VII. It is much easier to reason about small numbers than about large ones. If the numbers in a problem are large, or complicated, or are general numbers, simplify the problem by replacing them with simple arithmetical numbers; then reread the problem using the simple numbers, and try again to sense the meaning. To form the habit of doing this will help you greatly. VIII. School work that requires little or no effort on your part will not increase your power to do harder things. You should welcome some tasks that test you to the limit; and if you would grow stronger, you must always rely upon yourself. It is ruinous to your progress to rely on others to assist you in solving your problems. IX. Appeal to your teacher for assistance only after you have really done your best, and then ask only for one or two hints to start you right. Exercise 56 — Problems Requiring Simple Equations Solve the following equations and problems: ^ x+5 x-\-2.x-d_. ^ s+3, s+1 s+8_. ^* "3 6~+^ ^ ^* ~r"+"3 5~~^ , 7n+4 , „ 3n-9 „, ^ . 5x-S ,„ 8:r+5 3. ^-tan— 3- = 31 4. 40.-^—17 = ^- ^ 22/-5 , 35 _ 62/-h3 ^ n+9 n+5_ n+9 6. -6-+^-52/ ^ 6. — __n— ^ ^ S+5.S-1 s+7 APPLICATIONS OF SIMPLE EQUATIONS 115 8. The sum of two numbers is 94, and their difference is 38. Find the numbers. 9. A boy has 3 times as many dimes as quarters, and he has $11 in all. How many coins has he? 10. Seven times a certain number is 176 more than 3 times the number. Find the number. 11. A man bought 50 sheep, some at $3.75 a head and the others at $4.50 a head. The average cost was $4.05. How many did he buy at the lower price? 12. A boy earns $1.25 a day less than his father, and in 14 days the father earns $15 more than the son earns in 16 days. How much do both earn per day? 13. A clerk spends J of his annual salary for board, | for clothes, J for other expenses, and saves $1100. How much are his annual expenses? 14. At what rate per annum will $8000 yield $540 interest in 1 year and 6 months? Let X = the rate per annum. 8000 Xj^x| = 540 16. A man invested a certain sum at 5% and twice as much at 6%. His annual income from both investments was $680. How much did he invest? 16. A is 64 years old, and B is f as old. How many years have passed since B was J as old as A? 17. The sum of two numbers is 84, and 7 times the less exceeds 5 times the greater by 12. Find the numbers. 18. A had 8 acres of land less than B, but A sold 24 acres to B. A then had left only J as many acres as B. How many acres did each have at first? 116 ELEMENTARY ALGEBRA 19. A woman bought 36 yards of silk for $31, paying 75^^ a yard for part of it and $1 a yard for the rest. How rnan}^ yards of each kind did she buy? 20. A grocer has tea worth 40^ a pound and some worth GOjzf a pound. How many pounds of each must he take to mix 60 pounds worth 54^ a pound? Solve the 20th with one and then with two unknown numbers. 21. A boy bought a number of apples at the rate of 7 for 10|^ and sold them at the rate of 10^ for 3, gaining $2. How many apples did he buy? 22. If it costs the same at $1 a yard to enclose a square court with a fence as to pave it at 10^ a square yard, what are the dimensions of the court? 23. A mason received $3.60 a day for his labor and paid 85^ a day for his board. At the end of 44 days he had saved $92.20. How many days did he work? 24. A, B, and C together earn $5000. A's salary is f of B's and $450 less than C's. Find C's salary. 26. The sum of J and J of a number exceeds 5 times the difference between ^ and ^ of the number by 29. Find the number. 26. If f of a certain principal is invested at 4% and the remainder at 5%, the annual income is $690. Find the whole sum invested. 27. A bought sheep at $4 a head and had $33 left. If he had bought them at $4.75 a head, he would have needed 75^ more to pay for them. How many did he buy? 28. The length of a rectangle exceeds its width by 13 inches. If the length were diminished 7 inches and the width increased 5 inches, the area would remain the same. What are the dimensions of the rectangle? APPLICATIONS OF SIMPLE EQUATIONS 117 29. What is the distance between two cities, if an express train which runs 60 miles an hour can go from one city to the other in 6 hours less time than a freight train which runs 20 miles an hour? 30. A man owed $140. He sold wheat at $1 a bushel and corn at 75^ a bushel, selUng the same number of bushels of each, and received just money enough to pay the debt. How many bushels of grain did he sell? 31. A man was employed for 56 days at the rate of $3.25 a day and his board, and for every day he might be idle he was to pay $1 for his board. At the end of the time he received $148. How many days did he work? 32. Two wheelmen are 144 miles apart. They ride toward each other, A riding 8 miles an hour and B 6 miles an hour. B sets out 3 hours before A. How many miles w411 A have I'idden when they meet? 33. At what time between 3 and 4 o'clock are the hands of a clock together? Let m = the number of minute-spaces passed over by the minute-hand before the hands are together. Since the minute-hand goes 12 times as fast as the hour-hand, m divided by 12 = the number of minute-spaces passed over by the hour-hand in the same time. The hour-hand must pass over 15 spaces from 12 to 3, and, in addition, as many as the hour-hand passes over in the meantime. Hence the equation is m Solving, the number of spaces passed over by the minute- hand is 16^^, and the time is 16yy minutes past 3. 118 ELEMENTARY ALGEBRA 34. At what times between 5 and 6 o'clock are the hands of a clock at right angles to each other? The hands are at right angles twice between 5 and 6, once before the minute-hand passes the hour-hand, and once after. In the first case, the minute-hand must pass over 25 spaces, pkis the number of spaces passed over by the hour-hand, minus 15 spaces. In the second case, the minute-hand must pass over 25 spaces, plus m divided by 12, plus 15 spaces. The two equations are m = 25H -15 12 m = 25+- + 15 36. At what time between 3 and 4 o'clock are the hands of a clock opposite «ach other? m m = 15 + -4-30 1 z 36. At what time between 8 and 9 o'clock are the hands of a clock to- p;ether? 37. At what time between 2 and 3 o'clock are the hands of a clock at right angles to each other? 38. A is 54 years old, and B is \ as old. In how many- years will B be \ as old as A? 39. A, B, and C together earn $3650. A's salary is J of B's and $650 less than C's. Find C's salary. 40. A boy has $11 in half-dollars and 5-cent pieces, in all 58 coins. How many has he of each kind? APPLICATIONS OF SIMPLE EQUATIONS 119 41. Find the number whose double dhninished by 23 is as much greater than 53 as 68 is greater than the number. 42. A is 28 years older than his son, but 5 years ago he was 3 times as old. Find the father's age. 43. A man bought some cows at $40 a head. If he had bought 2 less for the same money, each would have cost $10 more. How many did he buy? 44. A had twice as many sheep as B. Each sold half his flock to C, and A sold 30 to B, whereupon A and B had the same number. How many had each at first? 46. One of two numbers is 4 times the other. If 24 is sub- tracted from the greater, and the less is subtracted from 66, the remainders are equal. Find the numbers. 46. A woman bought 12 yards of silk, but if she had bought 8 yards more for the same money, it would have cost 60<;^ a yard less. How much did it cost? 47. A father and two sons earn $222 a month, the two sons receiving the same wages. If the sons' wages were doubled, they would together receive only $6 less than their father. How much does the father earn per month? 48. A man bought land at $90 an acre and had $1000 left. At $105 an acre, he would have needed $200 more to pay for it. How many acres did he buy? 49. A fruit dealer bought some oranges at the rate of 3 for 5^ and twice as many others at the rate of 2 for 5fj. He sold them all at 36cf a dozen and made a profit of $5.60. How many oranges did he buy? 50. A pedestrian walked a certain distance at the rate of if miles an hour. He rested 2 hours at the end of his journey and returned at the rate of 2^ miles an hour. If he was out 9 hours, how many miles did he walk? 120 ELEMENTARY ALGEBRA ELIMINATION BY SUBSTITUTION 165. The following example illustrates the method of elimination by substitution: Sx+2y = Q5 Transposing 2?/ in (1), Dividing (3) by 3, Substituting in (2), 4x-Sy = S0 Sx = Q5-2y Q5-2y x = 260-8?/ -dy = SO (1) (2) (3) (4) (5) Solving (5), we have the value of y, and substituting this value in (1) or (2), we find the value of x. 166. Rule. — Determine first which of the two unknown num- bers it is more convenient to eliminate. From either equation, find the valu^ of that unknown number in terms of the other. Substitute this value for the same un- known number in the other equation. Exercise 57 Eliminate by substitution and solve: 1. 3. 7. f 4x — 6?/ = 6 \2a;+3i/ = 9 3a;-3i/ = 9 4a;- 52/ = 7 f4x+2?/ = 5 \5a;+3^ = 7 bx ^y 4^8 f5x+4!/= —4 \4x+37/=-2 4. 6. 8. 4a;-5?/=-2 3a:-4i/=-3 f 2x4-2!/ =-5 \6a;+9?/=-6 3^_% y y 9x 14 3i/=- ELIMINATION BY SUBSTITUTION 121 Exercise 58 — Problems — Eliminate by Substitution 1. The sum of the two digits that express a number is 14; and if 18 is added to the number, the digits are interchanged. Find the number. Let i = the digit in tens' place, and u = the digit in units' place. t-\-u = U iot+ii+m = iOu-\-t 2. The sum of the two digits of a number is 12; and if the number is divided by the sum of the digits, the quotient is 7. Find the number. 3. The sum of the two digits of a number is 12; and if 18 is subtracted from the number, the digits are interchanged. Find the number. 4. In 6 hours A rides 9 miles more than B does in 5 hours, and in 10 hours B rides 2 miles more than A does in 7 hours. How manj^ miles does each ride per houi? 6. The sum of the two digits of a number is 14; and if the digits are interchanged, the resulting number exceeds the given number by 18. Find the number. 6. A number exceeds 4 times the sum of its two digits by 6. If the number is divided by the tens' digit, the quotient is 10 and the remainder 4. Find the number. 7. A man invested $28,000, partly in 5% bonds and partly in 6% bonds. The annual income from the 5% bonds ex- ceeds the annual income from the 6% bonds by $80. How much did he invest at each rate? 8. A dealer bought 60 barrels of apples and 10 barrels of pears for $195. He sold the apples at a profit of 40% and the pears at a profit of 20%, receiving $264 for all. How much per barrel did he receive for each kind of fruit? 122 ELEMENTARY ALGEBRA 9. In 4 years a sum of money at simple interest amounts to $768, and in 5 years at the same rate it amounts to $800. Find the sum invested and the rate. 10. A pound of tea and 5 pounds of coffee cost $2. At prices 20% higher, 3 pounds of tea and 11 pounds of coffee would cost $6. Find the price of each. 11. If 7 is added to the sum of the two digits of a certain number, the result is 5 times the tens' digit, and if 45 is added to the number itself, the digits are interchanged. Find the number. 12. If the sum of two numbers is divided by 5, the quotient is 21 and the remainder 4; and if the difference of the numbers is divided by 10, the quotient is 6 and the remainder 3. Find the numbers. 13. A man paid $14 for oranges, buying some of them at 12 for 25^ and the rest at 14 for 25^. He sold them all at 30cf a dozen and made a profit of $4.30. How many did he buy of each kind? 14. If the larger of two numbers is divided by the smaller, the quotient is 6 and the remainder 8; but if 7 times the smaller is divided by the larger, the quotient is 1 and the remainder 9. Find the numbers. 16. If the numerator of a certain fraction is doubled and 3 added to the denominator, its value is f ; if the denominator is doubled and 2 added to the numerator, its value is y. Find the fraction. 16. If a rectangular plot of land were 20 feet longer and 10 feet wider, the area would be increased 3000 square feet; but if the length were 10 feet more and the width 30 feet less, the area would be diminished 2400 square feet. How many square feet are there in the plot? CHAPTER XIII GENERAL NUMBERS. FORMULAS. TYPE-FORMS GENERAL NUMBERS 167. Representing Numbers. B}^ common usage, the Arabic numerals of arithmetic and the letters used in algebra are called numbers. It must be remembered, however, that all number symbols are used simply to represent numbers. Since letters are used in algebra to represent any numbers, these letters are called general numbers. 168. A general number is a letter or other number symbol that may represent any number. To be able to read algebraic expressions in concise English and to express mathematical statements in algebraic symbols is of great importctfice. For example, Sab, dax, or Sxy represents three times the product of any two numbers. Also, 2{a — b) or 2{x — y) may represent twice the difference of any two numbers. Since a and b may represent any two unequal numbers, the equality — ia-\-b)-{a-b)=2b expresses the following principle: The sum of any two unequal numbers exceeds their difference by twice the smaller number. If a and b are any two numbers of which b is the smaller, what principle does this equality express 2(a+6)-2(a-5)=46? What principles do the following identities express (a-f-6) + (a-6)=2a ia+iy-a^ = 2a-\-l? 123 124 ELEMENTARY ALGEBRA FORMULAS 169. A formula is an expression of a general principle, or rule in general number symbols and in the form of an equality. The expression of a formula in words is a principle, and the expression of it as a direction is a rule. The ability to express general principles as formulas, and to read formulas accurately as principles and rules is of the greatest value to students of algebra, physics, etc. The truth of the following algebraic statement, called a formula, may be verified by performing the indicated oper- ations : ix+yy-{x-yy = 4:xy Supposing that x and y are any two numbers, what principle does the formula express? Exercise 59 1. Verify the truth of this formula: {a-\-xy — {a-{-x) {a — x) =2x{a-\-x) 2. Having verified the truth of this algebraic statement, tell what general principle it expresses. Since a fornmla expresses a general principle, it applies to all particular examples of that type. 3. By how much does 687 + 125 exceed 687-125? By how much does 2(759+45) exceed 2(759-45)? 4. How much does the square of 50+3 exceed the square of 50 — 3? Give result without squaring. 6. Without squaring the binomial, give the difference be- tween (20+6)2 and (20+6) (20 -6). 6. By how much does 569+350 exceed 569 - 350? By how much does 3(476 + 150) exceed 3(476-150)? FORMULAS 125 7. How much does the square of 40+5 exceed the square of 40 — 5? Give the result without squaring. 170. Deriving Formulas. The use of general numbers enables us to derive formulas for solving whole classes of problems. General numbers may be used to represent any units of measure as well as to represent abstract numbers. We have learned that the area of any rectangle is equal to the product of the length and width. area = length X width length X width = area Using the initial letters of these words, this principle may be expressed in the following formulas: a = Iw or Iw = a 171. Solving Formulas. To solve a formula completely is to find the value of each general number in terms of the others. Dividing both members of Iw = a, first by I and then by w, we obtain the two new formulas : w = T and 1 = — 1 w This formula may be stated in words, thus, Either dimension of any rectangle is equal to the area divided by the other dimension. This holds only when the area and the given dimension are expressed in the same units of measure. Exercise 60 1. If a rectangular lawn 48 feet long contains 1728 square feet, what is its width? 1728 126 ELEMENTARY ALGEBRA 2. When a rectangle 18 feet wide contains 150 square yards, what is its length? 9X150 18 3. A rectangle of land 64 rods long contains 18 acres. Find its width in rods. 160X18 w = 64 4. Express in general numbers two rules for finding the perimeter of any rectangle. 5. Using any general numbers, write three formulas for finding the area of any triangle. 6. Solve one of the three formulas of problem 5 and give the rule which each of the derived formulas expresses. 7. If a triangle whose altitude is 24 feet contains 52 square yards, how long is its base? 8. If x is the age of a boy now, make the problem of which this equation is the statement: x+3 = 3(.T-7) 9. Using any general numbers, write the formula for find- ing the volume of any rectangular prism. 10. Solve the formula of problem 9 and give the principle which each of the three derived formulas expresses. 11. Express in a formula the relation of dividend, divisor, quotient, and remainder, in division. 12. A has X acres of land and B 3a: acres. Make the problem of which the statement is 3x — 20 = 2(x+20). 13. Without squaring the binomial, give the difference between (75+3)' and (75+3)(75-3). 14. Give a formula for finding one dimension of a rectangle when the perimeter and the other dimension are given. FORMULAS 127 15. If a rectangle 64 feet long has a perimeter of 226 feet, what is the width? 16. Represent the number of cubic yards in any box- shaped excavation when the dimensions are given in feet. 172. The formula as a compact shorthand of number laws is perhaps the most practical part of algebra. The following list of problems will give practice in formulating arithmetical, practical, and scientific laws. Exercise 61 — Stating and Formulating Laws 1. Denoting the minuend, subtrahend, and difference by m, s, and d, respectively, show by a formula the relation of these numbers. 2. Add s to both sides of m — s = d and state what the resulting formula means. 3. Show by a formula the relation of the product, p, multiplicand, M, and multiplier, m. 4. Divide both sides of p = M'm by m, and state the meaning of the resulting formula. 6. State as a formula: "The product of a fraction, H - , by a whole number, a, is the product of the whole number by the numerator, divided by the denominator." 6. Show by a formula the principle for multiplying a a c fraction, -, by a fraction -, caUing the product p. a 7. State by a formula the relation of the percentage, p, the rate, r, and the base, h, and translate the formula into words. 8. Divide both sides of p = br, by r, and tell the meaning of the resulting formula. 128 ELEMENTARY ALGEBRA 9. State and give meaning of the formula for the interest, i, in terms of the principle, p, rate, ?-, and time, t (in years). 10. Divide both sides of i = prt by rt, and tell what the resulting formula means. 11. Divide both sides of i = prt by pt, and tell what the resulting formula means. 12. State as a formula the law for subtracting two fractions. 13. State as a formula the law for multiplying two fractions. 14. Show by a formula the law of area, A, of a square of side, s. 15. State by a formula the volume, V, of a cube whose edge is s. 16. State by a formula the value, J, of a decimal fraction having t units in tenths' place and h units in hundredths' place. / h Ans. f = — -\ . • 10 100 17. Solve the formula in the answer of problem 16 for t; for/j. 18. State as a formula the cost-law, in which c is the total cost, n the number of articles, and p the price of each. Solve the formula for n;ior p. 19. Calling d the total distance, r the rate of movement, and t the time, state the distance-law for uniform motion, as a formula. 20. Solve the formula of problem 19 for r, and tell the meaning of the result. Solve for t. 21. The velocity, v, of a freely falling body is the product of the gravity-constant, g, by the time, t, of fall. Formulate this law. Solve it for g; for t. 22. Solve the formula, A = 2Tr{h+r) for 7r; for zrr; for /i+r; for /i. FORMULAS 129 23. Formulate the principle: ''The reciprocal, i?, of a number, n, shows how many times the number goes into 1." 24. The area, A, of an equilateral triangle of side a is given by A=— v3- State this formula as a rule. 25. State as a formula: "The value of a fraction is not changed by multiplying both numerator and denominator by the same number, m. " 26. Write as a formula: "The value of a fraction is not changed by dividing both terms by the same number." 27. Write as a formula: "The commission equals the product of the rate and the principal." 28. Formulate: "The area of a parallelogram equals the product of the base and altitude." 29. Give the meaning of the formula : A = \/s{s — a){s — b) (s — c), in which A is the area of a triangle, a, 6, and c the lengths of the sides, and s is J the sum of the sides. 30. Find by the formula of problem 29 the area of a tri- angle whose sides are 6, 8, and 10. 31. Give the meaning of the formula E= , where E is the energy of a moving mass, Af , of velocity, Y . 32. Solve the formula of problem 31 for M\ For y^. 33. Solve -i-4 = l for ^; forP; for^. ti r i^ 34. The law of the see-saw board, balanced by two boys is: di and c^ being the distances from the support of the weights, Wi and W2, of the boys. Translate the law into words. 130 ELEMENTARY ALGEBRA FORMS AND TYPE-FORMS OF ALGEBRAIC NUMBERS 173. Meaning of Type-Forms. A very important thing to learn in algebra is the meaning and use of forms and type- forms of algebraic numbers. By the form of a number is meant how, from its written appearance, it looks as though it were made up out of simpler numbers. A bit of valuable advice, often given, but seldom appreciated by the beginner, is always to look carefully into a problem- or exercise before putting pencil to paper. ''Look before you leap" is a good motto for the young algebraist. Make it a habit. The habit is particularly valuable in factoring. The amount of useless labor it will save you will compensate many-fold for the effort. The way to start the practice is to learn what number-forms mean and how to use them. This is not an entirely new thing, for number-forms are used early in arithmetic. For example, when you learned to tell, without dividing, whether 5 is a factor of a number, by noticing whether it ended in or 5, you were using the form of the number to lighten your work. Likewise, you have probably learned to use the form of a number to decide, without dividing, whether the number is divisible by 10, 100, 2, 4, 8, etc. In algebra, an acquaintance with number-forms is much more useful than in arithmetic. If we were asked to indicate the sum or the difference of two different numbers in some suggestive form, we might write : ( ) + (J and ( )-( ), the empty curves suggesting that any numbers whatsoever might be written within them. But while these forms show sum and difference, they do not suggest that the two numbers FORMS AND TYPE-FORMS 131 ill question are to be different numbers. To obviate this objection we might suggest these forms: ( ) + [ ] and ( )-[ ], with the understanding that the curved and the square- cornered symbols are to suggest that different numbers are to be written inside the differently-shaped symbols. If we had been ingenious enough to see what it took mathematicians hundreds of years to discover, that by simply calling one number x and the other y, and writing, x-\-y and x — y, we have everything shown easily and fully, then our problem would have been solved. We merely remember that the different letters are in general to denote different numbers. 174. Examples of Type-Forms. Any other letters, as a and 6, might as well have been used as x and y in the last sec- tion. But X and y are easily written, and serve just as well as any other letters, so algebraists fall into the habit of using them more than others. We say then that x-\-y and x — y are respectively the forms for the sum and the difference of any two different numbers. Since x-\-y may stand for (typify) the sum of any two num- bers^ it may be called a type-form for the sum. Similarly, X — y is called the type-form for the difference of two numbers. The type-form for the sum of two products is ax-\-hy; for the difference of two products, ax — by. The type-form for the sum of two products having one factor common to both products is ax-\-ay, and for the differ- ence of such products, ax — ay. The type-form for the sum of two squares is x^+l/^, and for the difference of two squares, x^ — y'^. Observe that x^+i/^ means that a number is made by taking two different num- bers, squaring both, and adding the squares, while x^ — y^ directs us to form a number by choosing two different num- 132 ELEMENTARY ALGEBRA bers, squaring both, and subtracting. Clearly then, sucli short forms as x^-\-i/ and x'^ — y^ are very compact ways of saying a great deal. Such a number-form as x^-\-ax-\-b is the type-form for num- bers to be built up by choosing a number, squaring it, adding the product of it and some second number, and then adding a third number. As x^-\-ax-\-h has three terms, it is a tri- nomial. But is made up of three different numbers, x, a, and b. Since one of these numbers, x, is squared, the tri- nomial is called a quadratic (square-like) trinomial. The form, x^-\-ax-\-h, is then a type-form for quadratic trinomials. 175. Tjrpe-Forms Interpreted. Since x-{-y stands for the sum of any two numbers, if we multiply it by itself we get the square of the sum of any two numbers. Multiplying x-\-y by x-\-y gives us x^-\-2xy-hy^ 1. Hence the type-form for the square of the sum of two numbers is x^-i-2xy-\-y^. As a type-form, this x^-\-2xy-\-y^ tells us much. 1. It tells us that the square of the sum of two different numbers is a trinomial. 2. It tells us that two of the three terms of the trinomial are made by squaring the numbers to be added separately. . 3. It tells us that the remaining term of the trinomial is made by doubling the product of the two numbers that were added to give the original sum. 4. It tells y^ that a shoi^t way of getting a square of the sum of two numbers is to square each of the two numbers, to form their product and double it, and then to add the three results. FORMS AND TYPE-FORMS 133 Thus, to square the sum 10 -f- 5, or 15, calculate 10^, 5^, and 2X5X 10, getting 100, 25, and 100, and then add 100, 25, and 100, getting 225. All this can be done mentally. 2. Similarly, x—y multiplied by itself, gives x'^ — 2xy-\-y'^ which is the type-form for the square of the difference of any two numbers. Thus, 38 = 40-2, hence 38- = (40 -2)2 = 40^-2x2x40 + 22 = 1600 — 160+4= 1444. Most of this calculating can be done mentally. 3. Since any binomial is either a sum or a difference, x=i=y is the type-form for any binomial. 4. The type-form for the square of any binomial is then x'=t=2xy-\-i/ the upper or lower sign being used according as the binomial is a sum or a difference. 6. The type-form for the difference of two cubes is x^ — y^. 6. The type-form for the sum of two cubes is x^-{-y^. 7. The type-form for the difference of two like powers is .T" — ?/". 8. The type-form for the sum of two like powers is x"-f-?/". 9. The type-form for the product of the sum and difference of any two numbers is (x-\-y){x — y). 10. Give in words the meanings of the type-forms 5 to 9. CHAPTER XIV FACTORING 176. The factors of a number are the numbers whose product is that number. Factors of a number are the makers of the number, by multiplication. 177. From the law of the algebraic notation and the mear - ing of integral exponents, the factors of a monomial are the factors of the coefficient and each letter as many times as there are units in its exponent. Thus, 6a^6-c = 3 • 2 • aaa • 66 • c = 3 • 2aaahbc MONOMIAL FACTORS Type-form: ax-f-ay-faz 178. Polynomials having a common factor in every term are the product of a polynomial and a monomial. By definition of factors, since 3a(2a — 36)=6a2 — 9a6, 3a and 2a — 3fc are the factors of 6a^ — 9a6. The monomial factor is the greatest common factor of the coefficients multiplied by the lowest power of all the common letters. Thus, 14a2 + 21rt = 7a(2a+3) and mx^-\2x^ = (Sx\{Zx^2). When the monomial factor is one term of the polynomial, the corresponding term in the polynomial factor is 1. Thus, lox=' + 10x2-5a: = 5a:(3:c2^2x- 1) 179. Rule. — Divide the 'polynomial by the monomial factor and write the divisor and the quotient for the factors. Factors may always be checked by multiplying them together and comparing the product with the number to be factored. 134 FACTORING 135 Exercise 82 Factor the following and check the last four: 1. da^-lOa" 2. a^x^+a^x^ 3. ia^b-lOab^ 4. 6x^+15x^ 6. xV-^y . 6. Uab^+7a^h 7. QaH^-\-Sa^x^-2ax^y 8. 'da^x^+2a^x^-4:a'xY 9. 46V-8a6c2-4a26c3 10. aa;y - Sa^x?/ - a^x Y 11. 6a2c2+9a26c2d-3ac2 12. a^b^d-Sab^c'+a^bh^ COMMON COMPOUND FACTOR Type-form: ax+ay+bx+by 180. The terms of a polynomial may sometimes be so grouped as to show a common compound factor. Consider ax-\-ay-\-bx-\-b2j The first and second terms of this polynomial contain the common factor a, and the third and fourth terms contain the common factor 6. Grouping the terms in this manner and factoring each group, we have : a{x-\-y)-\-b{x-\-y) By the use of parentheses, the polynomial is thus reduced to two terms, which are similar with reference to the com- pound factor, x-\-y. Combining the terms according to the rule for addition of terms partly similar, §72, we have: {a+b){x-\-y) The first term is not always grouped with the second. It may be grouped with the third term, or the fourth. Factor ax-\-bx-\-2a-\-2b, grouping the first with the third term, and the second term with the fourth. Thus, a{x-{-2)-hb{x-\-2) 130 ELEMENTARY ALGEBRA Exercise 63 Write the factors of the following and check : 1. ac — ad-\-cn — dn 2. ax— cy-\-cx — ay 3. ax-\-2x-\-ay-\-2y 4. an-\-hn — ax — hx 5. a^-\-ahi-\-an'^-\-tf 6. x^ — 'Mj — xy-\-Zx 7. a^—mn — an-\-am 8. af'-\-aH-{-a'^x^-\-x^ 181. In the preceding examples, a positive monomial factor is taken out of each group. Observe the following: ax-{-ay — bx— by = (a — b) {x-\-y) ax—ay—bx-\- by = {a — b){x — y) Convince yourself that the equations are correct by multi- plying a — bhyx-\-y and a — bhyx — y. A polynomial cannot be factored in this manner unless the compound factor is the same in each group. To get the same compound factor in each group, —6 is taken out of the second group in each of the two examples above. Exercise 64 Factor the following polynomials and check: 1. an—b7i — ax-\rbx 2. bx—by-\-y'^ — xy 3. ax—by-\-ay — bx 4. ab-{-xy — ay — bx 6. 71^ — nx-\-ny — xy 6. ax^ — by-\-axy — bx 7. a}^-\-7n-x — ani- — aH 8. abx — bc-\-ai — anx 182. In some cases the compound factor in one group is like the remaining terms of the polynomial, or like those terms with their signs changed. In such examples the monomial factor taken out of one group is +1 or —1, as, for example, ax-ay-\-x-y={a-\-l){x-y) ax~ay-x-\-y^{a-l)(x-ij) FACTORING Exercise 65 Factor and check the following : 1. ax+2x-\-a+2 3. 3-c2+3c-c3 5. r'^+4-c2+4c 7. o — a' — r>o-{-a^ 9. rt^-ea—as^-e 11. l-7x3-x+7a;2 13. am-\-cn — an— cni ■ 15. a(x-y)-b(y-x) 17. {a—c)a—(c—a)b 137 2. ah — an-\-n — b 4. a^ — a-\-ay — y 6. aa;H-6 — a — 6a' 8. a6 — 6cH-a— c 10. ax- — 6.T- — a-f 6 12. rt-3a:2-|-3-aa:- 14. 263+3-362-26 16. a3-12-2a4-6a2 18. a^ — x*-\-a'^x — ax^ 183. Some polynomials may be separated into three or more groups that contain a common compound factor, as with ax — ay—hx-^hy-{-x — y-{a — h-{-l){x—ij). Exercise 66 Factor and check the following: 1; ax — a — hx-\-h—nx-\-n 2. a;3-5x2-4a;+20 3. a3-3a-f2a2-6 4. ax—hx—x — ay+by-\-y 5. 4a6+c — 6— 4ac 6. a^ — a—a%n-{-bn SQUARE OF THE SUM OF TWO NUMBERS Type-form: a2+2ab+b2 184. Since a and 6 are any two numbers, (a-\-by is the square of the sum of any two numbers. The square of a-f-6 is found by multiplication to be a^-\-2ab-^b^, or the square of a, plus twice the product of a and 6, plus the square of 6, or (a+b)2 = a2-f2ab4-b2 138 ELEMENTARY ALGEBRA 185. The square of the sum of two numbers is the square of the first number, plus twice the product of the first and second, plus the square of the second. Exercise 67 Give the results of the following: 1. (6+c)2 2. (x-hl)(x-hl) 3. iax+h)~ 4. If a man lives 8 years, he will be n years old. How old was he 8 years ago? 5. (a+cY 6. (n+3)(n+3) 7. {a+byY 8. What will represent the sum of 3 consecutive odd num- bers of which s is the smallest? 9. {b+xY 10. Cr-f2)(x+2) 11. {2a-\-by 12. A man was x years old a years ago. If he lives, how old will he be in b years? 13. (x+yY 14. (n+4)(n+4) 16. {x+SijY SQUARE OF THE DIFFERENCE OF TWO NUMBERS Type-form: a--2ab+b2 186. Since a and b are any two numbers, {a — by is the square of the difference of any two numbers. The square of a — b is found by multiplication to be a^ — 2ab+¥, or the square of a, minus twice the product of a and b, plus the square of b, or (a-b)2 = a2-2ab+b2 187. The square of the difference of two numbers is the square of the first number, minus twice the product of the first and second, plus the square of the second. FACTORING 139 Exercise 68 Give the results of the following, without multiplying : 1. (b-cy 2. (n-l)(n-l) 3. {ax- by 4. {a-cy 6. {x-S)(x-3) 6. (a-btjy 7. {b-xy 8. (n-2){?i-2) 9. (3x-4y 10. {b-yy 11. (.x-4)(a;-4) 12. {x-Syy 13. (a-a;)2 14. (n-6)(?i-6) 16. (4a-5)2 188. An arithmetical number may be squared mentally by considering it to be the sum or the difference of two numbers. Thus, 462 = (40+6)2 = 1600+480+36 = 21 16 462 = (50-4)2 = 2500-400+16 = 2116 Exercise 69 , Express the squares of these numbers, first as the sum, then as the difference of two numbers : 1. 382 2. 472 3. 652 4. 542 6. 732 6. 582 7, 642 8. 762 9, 352 ^q 952 189. A trinomial may be squared by grouping two terms to make a binomial of it. Thus, (a+6+c)2 = a2+2a6+62+2c) {a'-¥y (a+8)(a+2) 9. The perimeter of a square is 12x feet. What will denote the number of square feet in its area? •10. (6-5)(5-f6) (3a+26)2 (h-9)ib-Q) 11. At m dollars a week for men and b dollars a week for boys, how much will 6 of each earn in 4 weeks? 12. (x+7)(x+4) {4x-xyy (2/+8)(i/-3) 13. How many square yards are there in the ceiling and walls of a room 4x ft. by 3a; ft. and 2/ ft. high? 14. (7+a;)(a:-7) (5a4-36)2 ^a-S){5+a) 15. What may represent the area of any rectangle the length of which is 8 inches greater than its width? 16. At a cents a square yard, what will it cost in dollars to plaster a ceiling I feet long and w feet wide? REVIEW AND PRACTICE 165 Exercise 93 Solve the following problems and equations : 1. The difference of two numbers is 9, and their product is 630. Find the numbers. 2. 6a:2-5x-5=-4.T 3. 6^/2 + 2?/ + 12 = 31/+ 14 4. The sum of the squares of three consecutive numbers is 245. What are the numbers? 5. 8.T2+3a--9=-3x 6. Gj/^+Qy-f 15 = 2i/+13 7. The sum of the squares of three consecutive even numbers is 308. Find the numbers. 8. 2a;2-2a:+3=-7a;' , 9. 3y''-2y-{-17 = 9y+n 10. One number is f of another, and the difference of their squares is 80. Find the numbers. 11. (a;+4)2-9 = 3(3x+9) 12. ax''-a = bx^-b 13. The square of a number exceeds the square of f of it by 567. What is the number? 14. {x-\-S){x+7)=S(x-h5){x-2)+Q 15. A rectangle of land 5 times as long as it is wide con- tains 8 acres. Find the dimensions. 16. There are 48 sq. yd. in a floor which is 6 feet longer than it is wide. Find the dimensions. 17. The square of a number increased by the square of half the number equals 980. Find the number. 18. Four equal squares of paper contain 208 square inches less than one square 28 inches on each side. Find the length of each of the four squares. 19. A man bought land for $1280, paying | as many dollars per acre as there were acres in the piece. At what price per acre did he buy the land? 166 ELEMENTARY ALGEBRA Exercise 94 — Oral Practice Formulate the odd numbered exercises and give the products in the even numbered exercises. 1. If a rectangle is 6 in. longer than wide, what are the dimensions, if each is increased 8 in. ? 2. {x+7){x-{-7) (a+3)(a-l) {x+4){x-}-2) 3. What is the area of a square formed by adding 3 feet on all sides of a square x feet long? 4. (a:-6)(x+5) (a-2)(a-l) {x-\-4:){x-S) 6. What may represent the perimeters of the first and the enlarged rectangles in the first problem? 6. (a;-7)(x-5) (a-8)(a-8) {x-4){x-\-l) 7. What will represent the sum of four consecutive even numbers of which n is the largest? 8. (x+6)(x-f 5) (a-8)(a-f3) ix-\-Q){x-\-Q) 9. What restriction is placed on the exponents used in proving the law of exponents for multiplication? (§ 132.) 10. {x-H)ix-Q) (a-f8)(a-7) (?i+8)(n+3) 11. What does {x-{-2y represent, if x in the expression represents the side of a square? 12. {s-7){s-7) (6-9)(6+3) {x-9){x-4) 13. Write 5 times the square of a — b, diminished by the product of the binomials, x — 7 and re — 9. 14. (n+5)(n-h2) (a+8)(a+8) (x-|-7)(x-l) 16. What does (x— 4)(a; — 3) represent, if x in the expres- sion represents the side of a square? 16. (s-9)(.s-9) (6-6)(6+4) (?/+9)(2/+7) 17. What will represent the quotient of a number of three figures divided by 3 times the sum of the digits? REVIEW AND PRACTICE 167 Exercise 95 — Problems for Review Solve the following problems and exercises: ' x-4:'^x+4: 2 '48 3. The sum of two numbers is 24, and their product is 128. Find the numbers. 4. The sum of the squares of three consecutive odd num- bers is 37 1 . Find the numbers. 6. The sum of two even numbers is 18, and the sum of their squares is 164. Find the numbers. 6. Find two numbers whose difference is 8 and whose sum multipUed by the smaller number is 280. 7. Find two consecutive numbers the sum of whose squares exceeds 10 times the smaller number by 155. 8. Find the side of a square whose area is doubled by increasing its length 6 in. and its width 4 in. 9. The square of the sum of two consecutive numbers exceeds the sum of their squares by 112. Find the numbers. 10. A man worked 17 times as many days as he received dollars per day and earned $272. How many days did he work and how much did he receive per day? 11. At 20(f a square foot, it cost $56 to lay a parquet floor in a room whose length is 6 feet more than its width. Find the dimensions of the floor. 12. A mason worked 32 days more than he received dollars per day for his labor and earned $105. How many days did he work and how much did he receive per day? 13. An aeroplane flew 50 more miles an hour than the number of hours it flew. It flew 399 miles on the trip in question. How long was it in making the trip? 168 ELEMENTARY ALGEBRA Exercise 96 — Oral Review Answer the questions and perfonn indicated operations: 1. What does (x+2)(x — 2) represent, if x in the expres- sion represents the side of a square? 2. (a;-6)(x-2) (5a -56)2 (^4_7)(^i_|_5) 3. At X cents a rod, how many dollars will it cost to enclose a rectangular field I rods by w rods? 4. {x-\-S){x+2) (a+7)(a-5) (n-4)(n+2) 6. What is the area of a square formed by cutting off a strip 2 yards wide from all sides of a square x yards long? 6. (a;+9)(x-7) (a-9)(a-f-5) (n-8)(n-3) 7. What will represent the quotient of a number of x hundreds, y tens, and z units, divided by 8? 8. (x+8)(x+4) (a+8)(a-4) (n-9)(n-7) 9. What is received for x sheep bought at a dollars a head and sold at a profit of b dollars a head? 10. (x+6)(x-h3) (a-}-5)(a-2) (/i-9)(?i-f-4) 11. A man worked 8 days of n hours each at x cents an hour. He spent b dollars. How much had he left? 12. (x-3)(x-l) (a-5)(a+2) (?i+8)(n+7) 13. What is received for y horses bought at p dollars a head and sold at a loss of q dollars a head? 14. {x+Q){x-2) (a-8)(a-f 1) (n-7)(n-6) 16. A rectangular field 5x rods long has a perimeter of 18a: rods. What will denote the area in acres? 16. (x+9)(a:+3) (a+9)(a-5) (n-8)(w-l) 17. If the quotient is represented by q, the divisor by d, and the remainder by r, what is the dividend? REVIEW AND PRACTICE 169 Exercise 97 — Test Questions Answer, solve, and perform indicated operations: 1. What is the last step in finding the root of an equation? What axiom is involved? 2. (204-2)2 (30-1)2 (40+5)(40-5) 3. Define transposition. State the principles that are involved in transposing a term. 4. (40+5)2 (50-1)2 (20+3)(20-f2) 5. Indicate the product of three binomials without using the sign of multiplication. 6. (60+5)2 (40-1)2 (30-6)(30-4) 7. Read the sum of a(x+i/) and b(x-^y). Of a(b—l) and (h—1). Of a(m — n) and 2(m — n) . 8. (20+8)2 (30-5)2 (40+5)(40-4) 9. Define: power; square; cube. How do you find the square of a number? The cube? 10. (30+6)2 (50-4)2 (50-7)(50+5) 11. State the law of signs to be observed in raising a mono- mial to any power. 12. Write two identities that express in algebraic symbols the rules for squaring any binomial. 13. When is the value of a+6 a negative number? When is the value of a — 6 a positive number? 14. Indicate the product of two binomials and two mono- mials without using the sign of multiplication. 16. Write an identity which tells how to find the product of the sum and difference of any two numbers. 170 ELEMENTARY ALGEBRA 16. Read the sum of (a-\-b){x-\-y) and {a—b){x-\-y). Read the sum of (a+ c) (n — 1 ) and ( c — a) (n — 1) . 17. Write an expression that represents 5 times the square of the sum of any two numbers. 18. From what law do we obtain the rule for multiplying a polynomial by a monomial? 19. Show that the difference of the squares of two consecu- tive integers is an odd number. 20. Represent 3 times the sum of the squares of any two numbers multiplied by their difference. 21. Show when the product of several negative numbers is positive and when it is negative. 22. From 4ab — 3ac-\-2bc subtract the sum of Sbc+bd—ac, Sab — 2bd—bc, and bd — 2ac — ab. 23. What does a^+b^ represent? What does x~ — y- represent? What does 2(a+l)(a— 1) represent? 24* Define coefficient; exponent; and show the difference in their meaning or signification. 26. What does 2{a-\-hy represent? What does 3(a-6)2 represent? What does (a + 6) (a — b) represent? 26. Subtract 7x — 5y-\-3z from 3x — Sy-{-Qz, subtract result from zero, and add to 4x — Sy-}-2z. 27. Simplify 12a-(26- c)+4c-(5a-h36) and find its value when a = 7, 6=— 3, c= —4. 28. State the sign law of multiplication. State the index law of multiplication. Prove both laws. 29. Represent 5 times the sum of the squares of any two numbers multiplied by the square of their sum. 30. How much does the square of 70+3 exceed the product of (70 4- 3) (70 -3)? Give result without squaring. REVIEW AND PRACTICE 171 31. Subtract the sum of 5m—a — 9n and 56+5n+a— 4m from a-|-46-|-6m — 4w. 32. Represent the product of any three numbers, the last two of which differ by 2. 33. How is the dividend found, when the divisor, quotient, and remainder are known. 34. State the sigii law of division. State the index law of division. Prove both laws. 36. Without squaring the binomial, give the difference between (60+4)^ and (60+4)(60-4). 36. Find the value of (a-6)2+(6- c)2+(a-6)-f2c2 when a=l, 6 = 3, and c= —4. 37. Write the product of 51 and 49 by expressing them as the sum and difference of two numbers. 38. How do you determine whether a trinomial of the form of x'^-\-bx-{-c is the product of two binomials? 39. Represent 4 times the sum of the cubes of any two numbers multiplied by the sum of their squares. 40. Show that the difference of the squares of two consecu- tive odd numbers is twice the sum of the numbers. 41. Add (a-f c)-h2a(6 + c), b(b - c) -\-a{a-\-c)-{b-\-c), {a-\-c) — {b — c)—a{b-\-c), and 4(6 — c) + (a+c). 42. From the sum of 2ab — ac-\-2bc and 2ac — bc — Sab sub- tract the sum of 3ac — 46c — ab and 2a6c — 2a6 — 2ac. 43. Find the cost of x books at a^ apiece, x+5 books at b^ apiece, and x — 3 books at n^ apiece. CHAPTER XVI HIGHEST COMMON FACTOR. LOWEST COMMON MULTIPLE HIGHEST COMMON FACTOR 217. A common divisor, or common factor, of two or more numbers is an exact divisor of each of them. Thus, a- is a comtnon factor of 2a^, 3a*b, and a^bc. 218. The highest common factor (h.c.f.) of two or more numbers is the product of all their common factors. Thus, x^ is the h.c.f. of x^, x'^y, and 2x^y^z. The term greatest common divisor is used in arithmetic, but it is not appUcable in algebra. For example, x^ above may or may not be greater than x. Thus, if x = ^, x^=^, and x^ is therefore less than x. In algebra the term highest common factor is used. That is, x^ is higher than X (meaning x^) in the sense that its exponent is higher than that of x. fflGHEST COMMON FACTOR OF MONOMIALS 219. The highest common factor (h, c. f.) of two or more monomials may be determined by inspection. Consider: Sa^c\ 4a26c^, IGa^ft^c^, 12aV The h. c. f. of the coefficients is 4. The highest common literal factors are a^ and c^. The h. c. f. is 4aV. Observe that the power of each letter in the h. c. f. is the lowest power of that letter found in any of the monomials. 220. Rule. — To the h.c.f. of the coefficients, annex the highest power of each letter conimon to all. 172 HIGHEST COMMON FACTOR 173 Exercise 98 Give the h. c. f. of each of the following sets of numbers: 1. a', a\ 2a^b 2. ^xY, ^^f, SxHf, Qx^ 3. (jn\ 97^^ San^ 4. Sa^b\ 9a^b^, Qa'^b^ 12a^b^ 6. 4x% 2x\ Sx^ij 6. 5aV, Sa^x\ 2a''x', lOa^x^ 7. W, 3a^ 12a^b 8. 2x''y', SxH/, Qx^y^, 14xV 9. 5x^ 6^6, lOax^ 10. Qa^n\ 3aV, 9aV, ISa^n^ fflGHEST COMMON FACTOR OF POLYNOMIALS BY FACTORING 221. To find the highest common factor of compound expressions by factoring, proceed as follows: 4a;2-24x+36 = 4(a:-3)(x-3) 6a;2-42a;+72 = 6(a:-3)(a:-4) 2x^+12x-5'i = 2ix-S){x+9) The common prime factors of these numbers are 2 and x — 3, and the h.c.f. is their product, or 2.t — 6. 222. Rule. — Resolve the nutnbers into their prime factors, and find the product of all the common factors. Observe that each factor is taken the least number of times it is found in any of the given expressions. Exercise 99 Find the highest common factor of the following expres- sions, by inspection as far as possible : 1. 3x^+3y^, x-\-y, and x* — y^ 2. 2x2-14x+24 and x^-Qx+9 3. 5a'^ — ob* and ac-\-ad—bc—bd 4. 5x^+40, ^2-4, and x^+Gx+S 6. 3x2-llx-20 and Sx^-12x-15 174 ELEMENTARY ALGEBRA 6. x^-Qx+O, x''+2x-15, and a:3-27 7. 2ax — 2af Qax^ — Qax, and 2abx — 2ab 8. lHax'^-\-Qa, lSax^ — 2a, and 54ax+2a 9. 27a'-U, 9a2-16, and Sa'-2Sa-\-32 10. 40^2-20x4-25, 8x3-125, and 4x^-25 11. x3+27, x2-9, 8ax2+24ax, and x^-81 12. 24x^-81x, 12x2- 18x, and 48x3 -108x 13. a2+3a6-1862, a''-27a¥, and (a -36)2 14. a'»+a2_2^ Sa^b-Sab, and 4-8a2+4a^ 16. a2-4a6+462, a^-S¥, and a'-ab-2b^ 16. 9x2-6x+l, 6x2+10x-4, and Oax^-ax 17. \^a^bc-Wbc, 16-a^ 8-a«, and 4-a^ 18. a4-4a262+36S a'-b\ and 6^-2a2624-a4 19. x2-10x+16, \2xy-Zxhj, and x2-4xH-4 20. 63a2-36a, 49a2-16, and 16-56a+49a2 21. 24x2+18x-15, 1-4x4-4x2, and 8x3-2x 22. x2+12x+36, x2-2x-48, and x2-3x-54 23. 9x2-12ax+4a2, and 2a6+2ai/-36x-3xi/ 24. a2_[-2ac+c2, a^ — a?c — a&-\-(?, and 4a2— 4c2 26. ay^-\-ax^, bo?x^-\-ba?xy, and ax^-\-ay'^-\-2axy 26. 64a-32ax+4ax2, 5(x-4)2, and x2+2x-24 27. ba^-\-ba¥, aH — a}y — abx-\-aby, and a^ — abx 28. x^ — x^y—xy^-\-y^, (x — yY, and x* — 2x^y^-{-y^ 29. Sa^-Sb^, a^-^-a'b-ab^-b^, and b^-2ab-\-a' 30. a^-f-a^c— ac2 — c^, ax — ay — cx-\-cy, and a2_c2 31. 16x3+4x2-2x, 2x- 16x2+32x3, and IGx^-x LOWEST COMMON MULTIPLE 175 LOWEST COMMON MULTIPLE 223. A multiple of a number is a number that is exactly divisible by that number. For example, 4a6, 8ac, and 2ax are multiples of 2a. 224. A common multiple of two or more numbers is a number that is exactly divisible by each of them. Thus, \2a%c is a common multiple of 2a, 36, and 2c. 225. The lowest common multiple (1. c. m.) of two or more numbers is the product of all their different factors. Thus, 18a^ is the 1. c. m. of 3a, Qa^, and 6a^. >. Principles. — Every multiple of a number contains all the factors of that number. The lowest common multiple of two or more numbers con- tains only the factors of all the numbers. If two or more numbers have no common factor, their lowest common multiple is their product. LOWEST COMMON MULTIPLE OF MONOMIALS 227. The lowest common multiple of two or more mono- mials is determined by inspection. Consider : 2abc, Qab\ Sa%X 4a¥, a%c The 1. c. m. of the coefficients is 12. The lowest common multiple of the literal parts is a%^c. Hence 120^6^0 is the lowest common multiple. Observe that the exponent of each letter is the highest exponent that letter has in any one of the monomials. 228. Rule. — To the lowest common multiple of the coefficients, annex all the letters of each monomial, giving each letter the highest exponent it has in any monomial. 176 ELEMENTARY ALGEBRA Exercise 100 Give the lowest common multiple of the following: 1. 2a2, Sa\ 5a^b 2. ^ax"", 2a'x, 5ay\ lOa^x 3. Sx\ 6i/2, 9x''y 4. 9a^b, 4ab\ Sa^a, 12¥c 6. 6a^ 5x^ 3a^x 6. 4x1/, Sx'^y, oxy^, 15x^z 7. 07i\ 2n^ S¥n 8. 8a*6, Sfe^o:, 4aa:3, l&x^^ 9. 4a^, 5 c-, la^h 10. 5a:^?/, Ixy^, 2x^y, \4yz^ LOWEST COMMON MULTIPLE OF POLYNOMIALS BY FACTORING 229. The lowest common multiple of polynomials is found by resolving them into their prime factors, and finding the product of all the different factors. For example : a2+7a+12=(a-f3)(a+4) o2+8a+16=(a4-4)(a+4) a2-4a-32=(a+4)(a-8) The 1. c. m. is (a+3) (a - 8) (a+4)2. 230. Rule. — Find the product of all the different prime factors of the numbers, taking each factor as many times as it is found in any of the given numbers. The factors of the lowest common multiple may often be determined without writing the factors of the expressions. Consider: 2x-\-y, 2xy — y'^, 4x^ — y- The different factors in these expressions are y, 2x-\-y, and 2x—y, and the lowest common multiple is y{4x^—y-). Exercise 101 Find the 1. c. m. of each of the following exercises, deter- mining it without writing the factors, as far as possible: 1. x2-3a:-4 and x^-1 2. 6a-66 and 4a^-4b^ LOWEST COMMON MULTIPLE 177 3. x^-\-4:X-\-4 and x'^ — A 4. x^ — 2ax-\-a^ and a^ — x^ 6. 2(a2-|-x2) and bia*-x*) 6. 0:^-8 and x^-lOo^+lG 7. x'^-\-2xy-{-y^ and x'^ — xf- 8. a2-62 and a2-2a6+62 9. a;^-!, x^-fl, and a;^-! 10. a^-fl, a^-\-a, and a^— 1 11. a;2+14x+40 and x^-lQ 12. 3c(c-a)2 and 2a{a^-c^) 13. 3— 4a;4-a:^ and a:2+4 — 5.x 14. 27+a;3, 64-2a:, and x^-O 16. a2+a- 12 and a2-a-20 16. a^ — 2a^b-\-ab^ and ax—bx 17. a — a;, a^ — x^, and a^ — x^ 18. a2_5a-f-4 and a^-2a-\-l 19. a'* — x^ a — x, and a^—x^ 20. a^ — x^ a^ — a;2, and (a — x)^ 21. a2+6a+8 and a^+5a+Q 22. x2+a;-20 and 12-7a:+x2 23. l+4a;2, 4a;2-l, and 2x-l 24. d- — ax-\-x'^, a^-|-r\ and a+x 25. a — x, ar-\-x^ — 2ax, and x^ — a^ 26. a;2-llx+24 and x'^-Gx-lG 27. a2+2a-15 and 21 -\2a-\-a'- 178 ELEMENTARY ALGEBRA 28. 8^3-64, 4^2-16, and 6a:- 12 29. 2a+66, 3a-96, and Sa'-27b^ 30. a2-4, a2-4a+4, and a^-f 2a3 31. a^ — b^, a—b, b-\-a, and b — c 32. x'^-5ax-24:a^ and x2+8aa: + 15a2 33. x^ — xy, x^— 2/^ and x^-\-xij-]-7f 34. a2_i^ 2a+2, 3a-3, and 5a-5 35. a^ — Sab — ^b^ and ax— 4a+&x — 46 36. ac{x — y), 2a{x-\-7j), and 3c(x+?/) 37. ^2-1, l-2a;+x2, and H-2x+a;2 38. 20a -5, IGa^-l, 2a, and 12a2+3a 39. Aa?c-Wc, 2a2+2a6, and 3a6-36'' 40. l-h2x H-a;2, l-2x''+x\ and (l-a;)^ 41. x^-A:, x^-f-4x2+4, and 4-4x2+a;^ 42. a2H-8a4-16, a^-ie, and a2-8a+16 43. a;3+2a;2-4a:-8 and x3-2.r2+4x-8 44. x2+?/2, .T7/-7/2, x?/4-?/2, and x^-f-x?/ 46. l+x2+x^ 1-X+X2, and l+x+x^ 46. 12x2+12, 2x2-2, 8x+8, and 4x-4 47. 2a4(a^+x2), 5a3(a2-x), and 3a2(a2+x) 48. x^ — x^y-\-xy^ — y^ and x^+x'^y—xy^ — y^ 49. a2-a-6, a2-lla+24, and a2-6a-16 50. x2-2x-3, x2+2x-hl, and 9-6x+x2 51. a3-3a2-4a+12, a2-4, and a2-a-6 52. x2+7x+10, x2-4x-45, and x2-7x-18 CHAPTER XVII FRACTIONS 231. An algebraic fraction is the indicated division in fractional form of one number by another (see § 7). As examples, observe: a-\-b x-\-y a^—¥ 232. The numerator is the number above the line. The denominator is the number below the line. The numerator of a fraction represents the dividend, and the denominator represents the divisor. The numerator and denominator of any fraction taken together are called the terms of the fraction. Recall that the dividing line is a symbol of aggregation as well as one of division. See § 152. 233. An integer, or integral number, is a number no part of which is a fraction, as 5, 11, 16. A fraction of anything is defined in arithmetic as one or more of the equal parts of it; but since the terms of an algebraic fraction may be any numbers, positive or negative, integral or fractional, it is quite evident that the arithmetical definition does not accurately describe an algebraic fraction. The value of any arithmetical fraction is the quotient of the numerator divided by the denominator. This is true of any algebraic fraction, and for this reason it is defined as in § 231 above. A fraction whose numerator is a-\-h and whose denominator is a — h, is read: a +6 over a—h, or a-{-h divided hy a — h. 179 180 ELEMENTARY ALGEBRA 234. The sign of a fraction is the sign written before the line that separates the terms. 235. Since a fraction is an indicated division, by the hiw of signs in division, § 158, the following is true: -{-3 3 -3 3 +9.^9 -9__9 -3 3 +3 3 Changing the signs of both numerator and denominator does not change the sign of the fraction. Changing the sign of either numerator or denominator changes the sign of the fraction. If either term of a fraction is a polynomial, its sign is changed hy changing the sign of every term. a—b_—a-^b_b — a x-y —x-\-y y — x 236. Two principles are to be observed when the terms of a fraction are expressed by their factors, viz. 1 . Changing the sign of one factor in numerator or denomi- nator changes the sign of the fraction. For: {a-b){b-c) _ _ {a-b){b-c) _ _ {a-b){c-b) ix-y)(y-z) {x-y)(z-y) {x-y){y-z) This is evident, for changing the sign of one factor changes the sign of that term of the fraction. 2. Changing the sign of two factors in numerator or denomi- nator does not change the sign of the fraction. For: (a-b)ib-c) _ {a-b)ib-c) _ {b-a)ic-b) (x-y)(y-z) (y-x)(z^y) (x-y)(y-z) This is true, for changing the signs of two factors does not change the sign of that term of the fraction. FRACTIONS 181 237. Reduction of fractions is the process of changing their form without changing their value. Let a and h denote any two numbers, and m the quotient of a divided by h. Expressing this in an equation, a - = m h Since m is the quotient of a divided by 6, and since the dividend equals the product of the divisor and quotient, a = hm, and by the multipHcation axiom, §15, a'n = bm'n. Dividing both members of the last equation by 6«n, and indicating the division in the first member, we have: a-n b-n By the comparison axiom, §15, a a-n . a , - = , smce m=— also. b b-n b a a-n We multiply both terms of the first member of - =- — by n to get the b-n second member, and divide both terms of the second member by n to get the first member. This being an equation, by the multipHcation and division axioms, §15, the value of the fraction is not changed. Multiplying or dividing both terms of a fraction by the same number does not change the value of the fraction. Exercise 102 a~\~x 1. Change to an equivalent fraction whose numerator „ , a—x IS a^ — x^. 2. Change • to an equivalent fraction whose denomi- y-x natorisx— 2/. X 3. Change — — ^ to an equivalent fraction whose numerator is x^ —2x. 182 ELEMENTARY ALGEBRA x-\-2 4. Change to an equivalent fraction whose denomi- x — 2 nator is {x — 2y, 'V 6. Change to an equivalent fraction whose denomi- y-x hator is a;2 — 1/2. 238. A fraction is in its lowest terms when the numerator and denominator have no common factor except 1. To reduce a fraction to its lowest terms, we must remove all factors found in both numerator and denominator. This is done by canceling the common factors, which is equivalent to dividing both numerator and denominator by them, thus, 15aH'' _Sa x^-3x+2 _ Cx^^)ix-l) _ x-l 239. Rule. — Resolve numerator and denominator into their prime factors and cancel {divide out) all factors common to both. When the numerator of a fraction is a factor of the denomi- nator, the numerator of the result is 1 . For example, a-\-x _ 1 a^—x^ a — x It is often advisable to change the sign of a factor in one term to make it like a factor in the other. Thus : (x+7){x-4) _ {x+7)(x-4) _ x+7 ^ 5i4-x) 5(x-4) 5 We change the sign of the factor, 4— x, in the denominator and also the sign of the fraction, and then cancel the common factor. Exercise 103 Reduce the following fractions to their lowest terms, giving results at sight as far as possible : , 2a3 ^ 3x2 4^^. 6ct2 ^ 12xhj 1. — 2. — 3. 4. — 6. 8a2 6X3 g^y 3ct4 lQ^y2 6. 9x' n. '-^ 7. — 12. 8x^ FRACTIONS 8. 13. 9yz 7a6 Ihc 14. 8x3 4x^ 9n5 10. 16. 183 lOo^fe \ba¥ 2Sxy^ Reduce to lowest possible : ^ a*x-}-Max Exercise 104 terms, giving results at sight as far as a2-4a+16 4. a^+a 3b-\-Sab 7. 9x^y-Sx^y x2-8x+15 10. X2-1 5xy+5y 13. x^-y' x'-2xy+y' 16. 2a2-4a Sab-Qb 19. a^-¥ a2+2a6+62 22. 4xyH-4 5xY-5 26. ax" — a^ a2-2ax+x2 28. 9a3-6a6 6a26-462 <»i a^-x^ 2. ¥-1 ¥-1 6. c?-x^ a^-x^ 8. a — x a^-x^ 11. x'-y' x'-y' 14. a2+x2 a6+x6 17. a2-62 (a+6)2 20. x'-y' {x-yf 23. a3+x3 (a+x)3 26. x'-y' {x'-y'f 29. {a+by (a'-b^y 90 {x'-y'Y (f-2ax-{-x^ (x+yy 3. 6. 9. 12. 16. 18. 21. 24. 27. 30. 33. ^2-9 x2-6x+9 n-1 n' a^-\-4 — 4a a2+6-5a a^ 1 r2_ 3a+2 n2+l .2_ 2n+l y'-i a^+a — 6 a2+6a+9 n2-l x^-m x^+2x-S 184 ELEMENTARY ALGEBRA ' Sxy'-Qy' ' {a'-x^y ' a'-^l „ aH-x^ {a-hy a'+2+Sa ' a^-\-2ax+x^ ' (a'^-b^y ' a'+3+Qa 240. A mixed number is a number one part of which is integral and the other part fractional, as x—y x—3 241. A proper fraction is a fraction which cannot be reduced to a whole or a mixed number, as x-\-y abc x—S a+6 xyz 2/— 4 242. An improper fraction is a fraction which can be reduced to a whole or a mixed number, as a^-¥ x^-5x-\-9 x^-Y a2+62 x-2 ' x^-f REDUCTION OF IMPROPER FRACTIONS 243. An improper fraction is reduced to a whole or a mixed number by performing the indicated division. Thus, to reduce to a mixed number, proceed as follows: a+2 a^+x 1 .7-1-2 c^'+2q' a2-2a-f4 -2a2+x -2a2-4a 4a+a; 4a+8 x-% Therefore, ^5!+^ = a^- 2a+4-f ^ a-\-2 a+2 FRACTIONS 185 We continue the division until the remainder is of a lower degree in the leading letter than the denominator. When the sign of the first term of the remainder is plus, we write it over the divisor at the right of the integral part, connecting the integral and the fractional parts with the plus sign. An improper fraction reduces to an integral expression when the numerator is exactly divisible by the denominator. 244. Again, reduce to a mixed number: W-^a''-(Sa-\-n 3a2-2 Observe carefully: 6a3-9a2-6a+ll|3a2-2 6a^ -4a 2a-3 -9a2-2a+ll -9a2___f_6 -2a+ 5 Therefore 6a3-9a^-6a+ll _ 2a-5 iheretore, ^^^_^ -la 6 ^^^_^ When the sign before the first term of the remainder is minus, change the sign of each term of it, write it over the divisor, and annex it to the integral part, connecting parts of the quotient with the minus sign. Why connect them with the minus sign? Exercise 106 Reduce to whole, or mixed numbers : 1. o3+l a A x^-l x^ 7. a2-l 2. ?^ 3. " x+1 a — b a^ — 4 ^ x^ — y^ 5. 6. a — 2 x-{-y ^2-4 a4+.T^ 8. 9. a — 1 x+2 a — x 186 ELEMENTARY ALGEBRA ,o.V^ n.^!±i 12.^^+^^ n+1 a — 2 x-\-y 13.^!^ ii.'f±l 16. ?!z^ a — 1 n+2 a; — 2/ 16 ^'+^ 17 ^'-^ 18 ^'-^' . ''' 5-x '°--T+2- „^ 12a2-4a+5 „„ fe2-76+12 21. 22. 2a 6-3 REDUCTION OF MIXED EXPRESSIONS 245. Mixed expressions are reduced to improper fractions as in arithmetic, except that when the fractional part is minus, the numerator of it is subtracted. Observe : a2+9 (a-3)(a+2)= a^-a-Q ^ ^~^a+2 Adding a^ +9 Hence, a — 34 2a2-a+3 a2+9 2a2-a-f3 a+2 a+2 ., a^-^-x^ {a — x){a — x)=a'^ — 2ax-\-x'^ a — x Subtracting a^ -{-x^ — 2ax Tx a^+x^ —2ax 2ax Hence, a — x— = or a — x a — x a—x Exercise 106 Reduce to improper fractions : , ^ , 2aH-5 ^ 8a — 2a; „ , ^ 1. a+lH — r— 2. — - — -2a+3j 4a 3 FRACTIONS 187 3. x-3- 5. a-4 + Sx-4: 2x 4a+3 7. a+5- 9. x-2 ba 7a+4 a+5 4x — 5 11. a-44 13. x+6- 16. a+6- a-16 a+4 a;-36 aH-262 a+6 4. ?4^-3x-2. 6. 5a-36 8a -46 8. 'S^-2.-Zy Sx-2ij 10. 2a -4a;- Sa'-dx^ 4a+3a; 12. 6^^!z9^-3x-4, 14. 2x-Sy 4x^+92/^ 3a;-2?/ 16. 3a -2a;- •4X+32/ 8a^-7a;^ 2a -3x LOWEST COMMON DENOMINATOR 246. Two or more fractions have a common denominator when their denominators are the same numbers. The lowest common denominator (Led.) of two or more fractions is tlie l.c.m. of their denominators. Consider: a a(a-\-x) a — x {a—x){a-\-x) a a(a — x) and a-\-x {a-{-x){a—x) 247. Rule. — Find the lowest common multiple of the denom- inators for the lowest common denominator. Divide this denominator by the denominator of each fraction and multiply both terms of the given fraction by the quotient. 188 ELEMENTARY ALGEBRA Exercise 107 Reduce the following fractions to equivalent fractions having the lowest common denominator : 1. 3a2 3' 2ax 2 ' 4:xy 6 3. 2a c h ' 4^2 a bax c 6. 4 2ah' 3 6a^' b ^ax 7. 3a2 3' 4a6 X 2xy c 9. a 2bx 4 5a2 Sab ' 2 11. 5 ' a 4bx a 13. 3 Sax a 662' c 4ab 2. x+1 a x-\ b ' X2-1 c 4. 2 a-1' 3 a+l' 4 a2-l 6. a2+4 a aH-2 6 ' a-5 2 8. b c x-S' a x+S 10. 5 4-^2' 3 2-:.' 4 2+x 12. 0:2-4 i a a-1 b ' 6-2 c 14. a2+4 a2-4' a-2 a+2' a+2 a-2 ADDITION AND SUBTRACTION OF FRACTIONS 248. Similar fractions are added or subtracted by perform- ing those operations upon the numerators and writing the result over the common denominator. We have learned in division that: a-{-c-\-e—n — x_acc n x 6 6'^6"^6~6~6 Interchanging the members of this equation, we observe the rule for addition and subtraction of fractions. 249. Rule. — Reduce the fractions to the Led., change the signs of all the terms of numerators of fractions that are pre- FRACTIONS 189 ceded hy the minus sign, find the algebraic sum of the numer- ators, and write it over the least common denominator. Observe the following : a a a'^ — axa^-\-ax 2a^ a-\-x a—x a-—X" a'^—x'^ a^—x'^ In many examples in addition and subtraction it is best to express the lowest common denominator in its factored form. Exercise 108 Perform the indicated operations : 2a-{-x a — 3x 2a-\-h h-{-c a-^c 1. — 2. -\- 3 4 3 2 2 da-\-b a-2b ^ Zx-{-ij _ x-y x-j-y a b '428 Aa — x 2a-\-x Aa-\-x a — x x — a 5. 6. -f- 3 5 2 6 3 ' \-\-2x l-3a; '3 2 4 9.-A._^ 10.^-^ + -^ 2x+3 2a:+8 a+x a'-x' a^x 11. 3a+4 ^ 5-3ff ^^ X ^ y _x-y 4 5 • x^ — 2/2 x+y x-\-y 2x+3 2-f4a; x+lO a:+9 a;+4 13. 14. \~ 3 6 2 4 3 _ 2a 2a ..6 8 , 2a; 15. — 16. — h 2a-\-b 2a— b a-\-x a — x a^ — x .„ 3x+?/ 3x+?/ a^x a , 2a^ 17. — lo. a—x a-\-x a^—x^ 190 ELEMENTARY ALGEBRA 19. -^— - -^ 20. 2a- 6 2a+6 21. ^IZI-'^J^ 22. 23. 3M:_^+_A_ 24. a — 6 3a+6 „^ 4a-l , 6a+2 26. 1 26. 2a+2 3+3a' a-h a+6 h'-o? x-\-y X x{x'^ — y) X --] y h ^ xy X 1 '^ f x-y a;+?/ x'-f 5 , 3 9x x+4 'x-4 x'^-lQ a 6 62 27. 28. 2/ a:+?/ y(x^-y^) b ¥ d'b a+6 (a+6)2 (a+6)3 29. -i^-+-^4- ^^ (a— x)^ a — x {a — xY 2 4 2 (a;-i/)2 7/2-x2 {x^-ij)' 3 4,2 31. -T ;^ ^-^-1- a{a-2) a2-4 a(a+2) „^ 4a6 , 2 a+6 32. 7^ A W-a^ a-h a2H-a6+62 rt+4 7t"+31 n-2 ^^* ii^ n2_2^_i5"^n+3 34. y^-\-x^ x-\-y x^ — xy-^-y^ x-y {x-zY ^x-z 35. — 7 TT -i x—z {x—z){x — y) x-y a+4 a2_^4a-2 1_ • a2+a+l l-a» a-1 FRACTIONS 191 2 3n+6 , n2+3n+5 gy _L_ n+1 n^— n+1 n^ + l ^^ a2+4a-f9_ 2 a+4 39. 40. a3+27 a+3 a^-Sa+Q n2H-2w+28 , 3 n+6 8-n3 'w-2 n2+2n+4 4a;^ , 1 , x-2y x^+S?/^ a:+2|/ x^ — 2aj?/+42/^ MULTIPLICATION OF FRACTIONS 250. The product of two fractions is the product of the numerators over the product of the denominators. a c a c From T = ^ and -:= n, we have YX-; = mw. Why? a a From the first equations, a=bm arid c = dn. Multiplying a = 6m by c = dn, member by member, we have ac=bdmn. Dividing both members of ac = bdmn by hd, we have — = mn, and by the comparison axiom, a c_ac b d~hd This method is applicable also when either factor is integral, for integers may be expressed in fractional form. Since the product of the numerators is divided by the product of the denominators, cancellation may be employed. Exercise 109 SimpHfy the following: ^ U2b^bac ^ a2-25^a2-9 !• ": — X";r~X-r — r 2. —z — ^r— X- Ax 5a 6a6 a'^ — 3a a'^-\-5a Sb Sa 3cy a-1 4a^+a 9a 2x 4ab ' 2a-\-Sa^ a-2 192 . ELEMENTARY ALGEBRA 6. ^-1 y}+y Q x^-^y\ , x-y 1 — y^ x—1 ' x^ — xy x'^+2xy x^-4: x-l ' y^-\-xy x'^-Qz'^ 9. ^'-y' y^^+^ 10 x'+8y' ^^ (x+yy x^-\-xz x — y ' (a;2 — 2/2)2 x-\-2y (a+6)2 a-b ' (x+d)^ yz^+Sy'' bc-{-bx Sax c — x a^ — x"^ 13. r— X"ri — X X a^-{-ax 4:by a — x c^ — x^ 6a2 a — x 5by {a-\-xy a-\-b v,3ac a — b ab—b 1^» ~/ TTTiX -. Xt — : — ttitX (a-^;)2'' 6 (a+^)' a6+a2 Ig^ {x-yy ^^ x-4: ^^ iy+xy ^^ x'+27 d-\-x 0:2 — 1/2 0:^ — 64 x^ — y^ ^„ a2_^a6^^ a6-6\ 4a;+rr\ a2_52 17. 7 ttttXt — r-^X — ^ X (a- 6)2 ^{a+by a-\-x a¥x a^-5a-\-Q a''+2a a^-^S-6a * a2-2a-8 a2-4a+4 a2+3-4a n2+2n+l n-5 n^-Sn 19' ";; :; ;:: — X „ . — : — r~7: — X n^— n2 — 2n n^-\-4:-\-2n n^ — 4n — 5 ax-\-a'^^a^-\-x'^ — ax^^Zab 4a2 — Qa;^ 20. X X X 4a — 6a; 2a^ — Sa^x 2xy x^-\-a^ b aj \ aj \ \ — aJ\ \-\-a y xj \ xj \ 1— x/\ 1+x 26. (x-?^^xf:.-?^^ 26. ("2+ 2y Y, a.-j/\ yj \ x-yj\ x-\-yJ FRACTIONS - 193 x-\-zy\ xj \ n-\-xJ\ n — x 6c/ \ 3c/ \ 6+c/V 6 31. (,,^\x(^- + l.) 32. r .■^''-^^, Yl+i+^ \2x2+i/v \2/ 2x/ * \xz-^xy-{-yz/\x y z. DIVISION OF FRACTIONS 251. The reciprocal of a is 1 divided by a, or -. The reciprocal of a fraction is the fraction inverted. The reciprocal of y is -. a 252. The quotient of two fractions is the product of the dividend and the reciprocal of the divisor. T^ a , c , a c m , J. From -r=w and -:=n, we have t"^-7 = — > by div. axiom. b d b d n . From the first equations, a = b7n and c = dn. Dividing a = bm by c = dn, member by member, we have: a _bm c dn' Multiplying both members by -, we have : ad m , , ^, 7- =— , and by the comparison axiom, be n a ^ c _ad b ' d be 253. Write integral expressions in the fractional form, invert the divisor, and multiply. Exercise 110 Perform the indicated divisions: 2+a_^4-a2 4x'^-\-x ^ 2x-{-Sx'^ ' x-S ' x-3 x-S x-2 194 ELEMENTARY ALGEBRA a-\-S^a'^ — 9 a-\-x _^ a^-{-ax ' ' ■ ' * ~ ~ ' a^-Sax ' a2-9x2 a^-W ' 2ab-\-a^ a—2x ' a^ — 8x^ (x2-25)2 , (x-5)2 7. 6+1* 62-1 ^+?/ . x^^xy a+x a'-x^ a-h (a -by a+6 * a^-b^ a2-25 1 a2-5a • 6. 9. -^ 10. a2+6a a2_36 2/ 2// \ X 17. Il+-Vfl+?) 18. Sa^+ax^ * a:+2a x2+4-2x .x2+8 x-2 'a;2-4 a^ — x^ a — a: x^-Zx-^ 'x^+o; n^ — 6— n . 2/1 +n2 n^-2-n ' n2-2n a^+x^ a+x x/ V 3/ a2+6a+8 4+2a x/ \ 2// a^+a;^ ar — ax-j-x^ 21. ( 1--Vfl+-^ 22. ^'+2a;2/+?/^ . x^-i/^ aV \ a;/ xy-^rlf 'f 23. a H- 1+- 24. a/ \ a/ a^ — 9 a+3 26. (I-i-V^^+ll 26. ^'-^' -^'-^^ 2/2 xv \x J x^-\-2xy-\-\f- x-{-y 27. U'-^ H- 1-- 28. ^^^ xj\xj a^ — 25 5a-\-a'^ 29. ( ,.+_J^^l+-j 30. -^-^-.-^^^^ FRACTIONS 195 31. (^-4)-.(l-M^ 32.^4- 6 ^V.,8-2a y^ x^J \ xj \ a+1/ \ o?—\ Exercise 111 — Test Questions and Review Answer all you can orally : 1. Show that a common divisor of two numbers is a divisor of any multiple of either of them. 2. Write an algebraic expression that is exactly divisible by aH-6 and 2a — 36. 3. Give some algebraic expressions of which the lowest common multiple is 2a^ — 2x^. 4. How do you determine whether a binomial is the pro- duct of the sum and difference of two numbers? 6. Show that a common divisor of two numbers is a divisor of their sum and also of their difference. 6.- Recalling the solution of equations by factoring, what are the roots of the equation, a;^ — 5x + 6 = 0? 7. Show how much the square of the sum of two numbers exceeds the product of their sum and difference. 8. Without squaring the binomial, give the difference between (30+7)2 ^^d (30 +7) (30 -7). 9. How much does the square of 40 +5 exceed the product of (40 +5) (40 -5)? Find the value of the following expressions when a = l, 6 = 2, c = 3, d = 4, e = 0, m = \^n = \. 10. cd2m-8a263-j-d2m2+9c2X262+6a3(/m2n-62c2(i2n 11. m3X(i'+76cm-h9a3c3-a2cV+c2(i2^64m+a^Xc^3 196 ELEMENTARY ALGEBRA Find the value of the following expressions when a = J, b = i, c=l,d = 4,x = ^^y = S. 12. Qa^d^-5(^y''XS¥x 13. 86y+a(2d2-2d)+5x2 14. 9¥d^+Sa^d'^5c'x^ 15. 2c'd'-{2y'-5d)x-4y^ 16. Show how much the square of the sum of two numbers exceeds the square of their difference. 17. Give the difference between (20+4)2 and (20-4)2 without squaring either binomial. 18. What is meant when it is said that a certain number satisfies an equation? 19. By what niust a fraction be multiplied to give the smallest possible integral product? 20. Show to what the sum of any two numbers divided by the sum of their reciprocals is equal. 21. Find the h. c.f. and the 1. cm. of x*-{-x'^y--\-y^, x^-\-xy-\-y^, and x'^ — xy-\-y^. 22. How much does the square of 50+4 exceed the square of 50—4? Give result without squaring. 23. What is the result in multiplication of fractions, when all factors in numerator and denominator cancel? Find the value of the following expressions when a = 2, b = l^c = 4:,d = S,n = 5,x = l,y = ^. d+n b^+y 27. '+1 c-b y" U^ on 2 2 30. — - X y 31 ^'-^' 32 ^"'4-^'^ 33. a+2 b- -y 36. X y a X FRACTIONS 197 a^+n' 3n2_8fe2 ' a^+h^ . • (Px 2y^ 39. Find the h.c.f. and the l.c.m. of x^ — x^ — x-\-l, 2x^-x^-x^, and 2x''+x-S. 40. Write an expression of 3 unlike terms of the third degree, the terms involving x and y. 41. Define elimination. What axioms are involved in elimination by addition and subtraction? 42. By what is the sum of the same odd powers of two numbers divisible? The difference? - 43. Give the difference between (60+2)2 ^nd (60-2)2 without squaring either binomial. 44. State the law for changing the signs of one or more factors in numerator or denominator. 45. How do you determine whether a trinomial of the form ax^-\-bx-\-c is the product of two binomials? 46. Define determinate equation; indeterminate equation; independent equations; simultaneous equations. 47. If two equal fractions have the same numerator or denominator, except 0, how do the other terms compare? 48. How much does the square of 80+2 exceed the square of 80 — 2? Give result without squaring. 49. If one of the factors of Qa'^x'^—4:aH — 4:ax^-\-x*-{-a'^ is x2+a2 — 2ax, what is the other factor? 50. Give the two formulas that express in general numbers the index laws of multiplication and division. CHAPTER XVIII LITERAL AND FRACTIONAL EQUATIONS. SOLUTION OF FORMULAS LITERAL AND FRACTIONAL EQUATIONS 254. A literal equation is an equation in which there are two or more general numbers. In solving such equations, the value of any letter may be found, but only in terms of the other letters. Solve for x, ax — ar = bx — 6^ Adding a^ and —bxto both members of this equation, and uniting the terms containing x, we have {a — h)x = o? — ¥ By the division axiom, x = a-\-h Checking: a{a-\-h) — a^ = h{a-\-h) —h"^ or ah = ah To solve a literal equation for any letter in it is to find the value of that letter in terms of the others. Exercise 112 Solve the following equations in the left column for x, and those in the right column for y and check : 1. 4a-x = 4b-bx 2. 2b-\-6y = Sc+ay 3. 5n—x = 4n-\-nx 4. ay — ab = 3y — 3b 5. 3a — x = 2a—bx 6. 5a — by = ay — 5b 7. n^—nx = ax—a^ 8. 2a — Qy = ay— IS 9. _-4a = 46-^ 10. --n+- = y---- LITERAL AND FRACTIONAL EQUATIONS 199 255. Special Devices. It will be well to note here some special devices for clearing equations of fractions. Thus, clear of fractions — - — + = — — -— 5 'Sx—1 10 Multiplying both members by 10, the lowest common multiple of the monomial denominators, we have 3a; — 1 Subtract 4x in each member, unite other monomials, clear of fractions, and complete the solution. In some examples it simplifies the solution to combine fractions before clearing of fractions. Thus, from 1 , a—c 1 .a-\-c . . we obtain a—c X a-\-c X 1 1 _a+c a—c ' a—c a-hc x x Combining the fractions in each member, we have 2c _2c If two equal fractions have the same numerator, not 0, their denominators are equal. Hence, x = a^—c^ Check by substituting in equation (1) Exercise 113 Solve the following equations: 3x+8 _ 4a;-3 x 4y+S y-2_y+2 ' 12 3x+4 4 * 2/'-4 2+2/ y-2 2x-5 x_ 5x+S . 3j/-f4 _i/+3_l-y * 3x-2 3 15 • 1-1/2 i_y 2/+1 5x—4:_Sx-^S X a-\-l _5x — a a — 1 • ~10 2x+5 2 • a^~~oF^~^a+l 200 7. — = x — a-\— a a ELEMENTARY ALGEBRA 3x . X «• ro+r^o-3 „ 3a; 2a; 9. ^-^ = x+3 11. .13. 15. 17. 19. 21. 23. 2+l£5 1^+2 a6 _ ca; ex a6 5a^+4_5a:— 4_a; 2x+2~2 10. 12. x—c (2x-— c)' X — a {2x — ay a-}-c x — 2a . : ac 10 2x+l 14. ?^+26-x^^ 2x+a 46H-X 3x-2 6x-l 5 6x4-3 15 3x+2 5x4-6 , X 1 x-6 "^3 4 5 X — 5 x4-l X — 2 5x+4 _ 6x4-4 _x4-6 6 3x4-2 5~ 3x4-8 4x4-8 2x4-5 6 3x4-6 4 2x4-1 16. 18. 20. 22. 24. X.-2 c x — 2ac 2x ^ , 6c — = 54- — a a l_ x-Sb X bx X acx 8x4-3 x4-2 x+3 34-x x-3 9 8 x2-9 6 ^ x4-2 2x4-4 8x+3 2-x 2x.24-2x x+2 26. 26. 27. 28. 29. 3x2 x4-2 2x-l 2-x 2x-l 4x2-1 14- 2x 12x4-ll_9x4-7_10x-5 8 6 46x4-8 x4-4a4-c , 4x4-a4-2c _ x4-a4-c 2 3 x-\-a— c 4 5 X— 2 X— 3 X— 4 1 X — 5 1 (x-7)(x4-2) (x-4)(x-3) LITERAL AND FRACTIONAL EQUATIONS 201 SPECIAL METHODS 256. Observe the form of the following equation and the method of solving it. x-\-2 x — 5_x — Q x-\-S ,^v ^+3 x^~x^ x+i ^ ^ Transpose one fraction from each side to make each mem- ber the difference of two fractions. Thus, x-\-2 x-\-3 _x — Q x — 5 x-\-3 x-\-4: x — 5 x—4: Performing the indicated subtraction in each member and simplifying the numerators, we have -1 _ -1 x2-h7x+12~a:2-9:c+20 Since the numerators of these equal fractions are equal, and not 0, the denominators are equal. x^-9x-\-20 = x^+7x-\-12 X = -2 Check by substituting in (1) 257. In solving equations like the following, it is best to reduce the fractions to mixed expressions: 5x-7 2a;-17 _ 4x-l 3a;-21 . . ~^^ x-1 ~ x-i x-Q» ■ y) x-2 x-1 x-l a;-6 Subtracting the sum of the integers from both members, dividing by 3, and combining the fractions in each member, we have -5 _ -5 x2-9x+14 a;2-7a:+6 The denominators are equal, hence, a;2-7a:+6 = a:2-9a;4-14 a: = 4 Check by substituting in (1) 202 ELEMENTARY ALGEBRA Exercise 114 Solve the following equations : x-i-2 x+4_ 2x^-Sx-^2 ' x-2 x-S~ x^-5x-\-6 x — 4iX — 8_x — 7x — 5 x—5 x—9 x—S X— 6 x-{-l x+4: _ x-\-2 x-\-S x—1 x—4: x — 2 x — S x-\-5_x — Q_x — 4_x — 15 x+4: X — 1 x — b a;— 16 o?-\-x a'^ — x_4acx-\-2a^ — 2c^ c^ — x c^-\-x c'^ — x^ x-{-3a x-\-2a_x-{-a x-\-2a x+a x — a x+3a x-\-ba 5a;-8 6x-44 10a;-8 _ a:-8 ' x-2 x-1 x-l ~'x^ 5x-64 , rc-6 Ax-bb , 2x-ll a;— 13 x — 1 X— 14 x — Qi 9a;+4 3x+2 i_ 3a;+3 2a;-5 15 3x-4~^ ^" 5 3x-4 _ 8a:-h5 4 Sx-a 4x-2 2x-2a 10. — X = 9 5 2x-a 45 2x-a 2/^ _ 3 , . 4 6. 6. ^^' 2{y-\) '^'y-l y-1 " S{y-l) y _ 2y^ _ l/-2 __p__ 9 ^^- ^^ 3to^ 3~ 3(2/-3) ^ 1/-2 97/-1 gi_2-|-?y 7?y+86 ^^' 2+2/ 2(2/-2) 2 ^_2 2(7/-4) LITERAL AND FRACTIONAL EQUATIONS 203 ^^ x-\-2 . x-7 x+3 x-Q 14. -f 16. 16. X X— 5 x+l X— 4 x-\-S x — Q _x-\-4: x — 5 x-\-l x-4:~ x-\-2 x^ x — 5 x—4: X— 10 x — 9 x+5 x+4 x+lO x+9 2c , _ 2x+3c , 3x+6c ic+4c x+c x-\-2c Exercise 115 — Problems in Simple Equations Solve the following problems: 1. Separate 59 into two such parts that 4 times the smaller shall exceed twice the larger by 26. 2. From what number must 135 be subtracted to get 273? 3. Find the number to which if 329 be added, the sum will be 642. 4. What number must be multiplied by 37 to obtain 999? 5. A is 3 times as old as B, but in 20 years he will be only twice as old. Find the age of each. 6. What number must one divide by 23 to obtain 163? 7. Divide 220 so that the quotient of one part divided by the other is 4 and the remainder 20. 8. What number must be added to .378 to give .65? 9. A is 53 years old and B is 33. How many years have elapsed since A was if times as old as B? 10. What number must one subtract from 3f to get 2 J? 11. Divide $15 into two parts so that there are twice as many dimes in the first part as there are 5-cent pieces in the second part. 204 ELEMENTARY ALGEBRA 12. By what number must one multiply 3^ to obtain 7y? 13. The difference between two numbers is 17; and if 4 is added to the larger number, the sum is 4 times the smaller number. Find the numbers. 14. Divide $9000 into two parts such that the interest on the greater part for 2 years at 6% shall be equal to the interest on the other part for 3 years at 5%. 15. What number subtracted from 164 gives the same result as 92 added to the number? 16. Of what number is 5 J the three-tenths part? 17. The difference between two numbers is 32; and if the greater is divided by the less, the quotient is 5 and the re- mainder 4. Find the numbers. 18. By what number must one divide 3 J to get 5 J? 19. Three men earned a certain sum of money. A and B earned $180; A and C earned $190; and B and C earned $200. How much did they all earn? 20. What number is as much under 7^ as it is over 5 J? 21. The length of a rectangle is if times its width. If each dimension were 3 inches less, the area would be dimin- ished 279 square inches. Find the length. 22. What number lies midway between 3j and 7^ ? 23. A man bought a coat for $36, paying for it in 2-dollar bills and 50-cent pieces, giving twice as many bills as coins. How many bills did he give? 24. Of what number does the double exceed by 9 its J? 26. A man invested a certain sum at 5% and twice as much at 6%. His annual income from both investments was $765. How much did he invest? LITERAL AND FRACTIONAL EQUATIONS 205 26. Of what number is the 9th part 3 less than its ^? 27. A had $50 more than B. A bought land at $18 an acre and had $140 left. B bought 10 acres less at $24 an acre and had $30 left. How many acres did each buy? 28. A and B are 106 miles apart. They travel toward each other, A at the rate of 13 miles in 3 hours, and B at the rate of 9 miles in 2 hours. How far will each have traveled when they meet? 29. A can do a piece of work in 8 (3^, 7n)* days, and B can do it in 12 (4f , n)* days. In how many days can both doit? A can do |- of it in a day; B can do -j^ of it in a day. Let X = the number of days in which both can do it. XX 111 --] = lor- = -H 8 12 X 8 12 30. A can do a piece of work in 12 (8^, a) days, B can do it in 15 (12^, h) days, and C can do it in 20 (18f , c) days. In how many days can all do it working together? 31. A and B can do a piece of work in 8 (2, d) days, and A can do it in 20 (4f , a) days. In how many days can B do it? . B can do ^--^q of it in a day. 32. A boy spent part of 78^ and had left 12 times as much as he spent. How much did he spend? 33. Find the number whose third, fourth, sixth, and eighth parts are together 5 less than the number. 34. John's father gave him $3 yesterday. He spent 50^ today, and he still has ^i more than he had day before yester- day. How much has he left? *The numbers inside the parenth^es may be used instead of those outside, if preferred. 206 ELEMENTARY ALGEBRA 35. A can do a piece of work in 12 days, which B can do in 18 days, and with C's help they can do it in 4 days. In how many days can C do the work? 36. If A can do half of a piece of work in 10 days and B can do the whole of it in 15 days, in how many days can both of them do it working together? 37. A speculator bought two pieces of land at the same price. He sold one piece at a profit of $1700 and the other at a loss of $900, receiving twice as much for one piece as for the other. How much did each piece cost him? 38. At what rate per annum will $3600 give $270 interest in one year 8 months? Let r = the rate % per annum. 39. What sum must be invested at 5% to give a quarterly income of $105? 40. What sum put at interest at 5% per annum will amount to $6000 in 1 year 9 months? 41. A father is 42 years old, and his son is y as old. If both live, in how many years will the son be f as old as his father? 42. Separate the number 145 into two parts so that the excess of the greater over 50 shall be 4 times the excess of 50 over the smaller. 43. If f of a certain principal is invested at 5% and the remainder at 4%, the annual income from both investments is $660. Find the whole sum invested. 44. The width of a room is f of its length. If the length were 4 feet less and the width 4 feet more, the room would be square. Find the dimensions of the room. LITERAL AND FRACTIONAL EQUATIONS 207 45. A man invested $20,000, part of it at 5% and tiie remainder at 6%. The interest on the former for 2 years is the same as the interest on the latter for 2 years 6 months. How much was invested at each rate? 46. A man invested J of his money in 4% bonds, f of it in 5% bonds, and the remainder in 6% bonds, buying them all at par. His annual income from the whole investment amounts to $2550. Find his whole investment. GENERAL PROBLEMS 258. A general problem is a problem all of the numbers in which are general numbers. It is therefore evident that the solution of a general prob- lem involves a literal equation. For example: The sum of two numbers is w, and the larger number is n times the smaller. Find the numbers. Let X = the smaller number, and nx = the larger number. x-{-nx = m m mn Solvmg, x== and nx= l-\-n l-\-n The result obtained in solving a general problem is a form- ula for solving all problems of that type. To find the smaller number, divide the sum of the numbers by 1 plus the ratio of the two numbers. To find the larger number, divide the product of the sum and ratio by 1 plus the ratio of the numbers. These are the rules for finding any two numbers when their sum and their ratio are known. 259. Generalization in algebra is the process of solving a general problem and interpreting the formula obtained as a rule for solving all problems of that type. , 208 ELEMENTARY ALGEBRA Exercise 116 1. The larger of two numbers is 7 times the smaller, and their sum is 1488. Find the numbers. m mn = 1488-^8 = (1488X7) -^8 l-\-n 1+n 2. The smaller of two numbers is f of the larger, and their sum is 21. ' Find the numbers. 3. If two numbers are added, the result is 2769, and one is 8f times the other. Find the two numbers. 4. The sum of two numbers is s, and the difference of the same numbers is d. Find the numbers. Let X = the larger number, and s— a; = the smaller number. x — {s—x)=d Solvmg, x = and s—x= 6. Read these formulas as rules for finding two numbers when their sum and their difference are known. 6. The sum of two numbers is 768, and their difference is 116. Find the numbers. s+d ' 768 + 116 s-d 768-116 2 " 2 ^" 2 ~ 2 7. A man sold a piece of land for $6800 and gained the same sum he would have lost, if he had sold it for $5200. How much did he pay for the -land? 8. The sum of two numbers is a, and m times the smaller is equal to n times the larger. Find the numbers. Let L = the larger number, and a — L = the smaller number. am—mL=nL Solvmg, L = and a— L = m-f-n m+n LITERAL AND FRACTIONAL EQUATIONS 209 9. Read these formulas as rules for finding the two num- bers in any problem of this type. 10. The sum of two numbers is 472, and 3 times the larger is equal to 5 times the smaller. Find the numbers. am _ 472X5 an _ 472X3 m+n 8 m-\-n 8 11. Divide the number c into two parts so that a times the larger part equals b times the smaller. 12. A boy bought oranges at a cents apiece and had b cents left. At m cents apiece, he would have needed n cents more to pay for them. How many did he buy? 13. According to the conditions of the preceding problem, is m greater or less than a? Show why. 14. Make a particular problem which may be solved by the formula obtained in problem 12. 15. The length of a rectangle is m times its width, and the perimeter is n feet. Find the dimensions. 16. Make a particular problem which may be solved by the formula obtained in problem 15. 17. The sum of two numbers is s. m times their sum equals n times their difference. Find the numbers. 18. To make the preceding problem true, is m greater or less than n? Give your reason. 19. A rectangle is a feet longer than it is wide, and the perimeter is p feet. Find the dimensions. 20. Make a particular problem which may be solved by the formula obtained in problem 19. 21. Find two parts of a such that the quotient of the greater divided by the less shall be m divided by n. 22. A can do a piece of work in a days, B in 6 days, and C in c days. In how many days can all working together do it? 210 ELEMENTARY ALGEBRA SOLUTION OF FORMULAS 260. The student of physics and higher mathematics will often find it necessary to solve formulas. For example: The distance passed over by any body moving with a uniform velocity in any number of units of time is the product of the velocity and the time. This law expressed in a formula is — , d = vt Solving this equation for v and t, we have v — d-T-t and t = d-i-v What is the average velocity of a train, if it runs 448 miles in 16 hours? (^-^^ = 448-^16 261. The interest is the product of the principal, the rate expressed as hundredths, and the time. i=prt It must be remembered that r in this formula represents the rate per annum and t the number of years. Solving this formula, or literal equation, for p, r, and t, we have the following formulas : p = i-i-rt r = i-^pt t = i-^pr 1. What sum put at interest at 6% for 1 year 4 months will yield $60 in interest? ^^r^ = 60^yfo•i = 750 2. At what rate per annum will $1300 amount to $1391 in 1 year 4 months 24 days? z> p^ = 91 -T- 1300 • If^ = .05 3. In how many years, months, and days will $2200 amount to $2345 at 5% per annum? ^>pr=143-^2200•lfo=ltf =1 yr. 3 mo. 18 da. SOLUTION OF FORMULAS 211 262. The ratio of the circumference of any circle to its diameter is approximately 3.1416. The exact value is repre- sented by TT. The formulas for the circumference of a circle are c = 7rd and c = 27rr, in which c is the circumference, d the diameter, and r the radius. Solve c = Tvd for d^ and c = 2irr for r, and read results as rules for finding d and r. 263. Denoting the area by A, the base by h, and the alti- tude by h, the formulas for the area of a triangle are : A = b.| A = h| A = bh.^ The area of any triangle is the product of the base and half the altitude, the altitude and half the base, or half the product of the base and altitude. Solve each of the above formulas for b and h, and read the results as rules for finding those dimensions. 264. Primes and Subscripts. Different but related num- bers in a formula are often denoted by the same letter with different primes or subscripts. Primes are accent marks written at the right and above the number; subscripts are small figures written at the right and below the number. For example, a', a", a'", no, n^, n2, n^. These are read a prime, a second, a third, n sub zero, n sub one, n sub two, n sub three, respectively. In the formula for the area of a trapezoid we shall find the two parallel bases denoted by bi and 62. 265. Denoting the area by A, the two parallel sides or bases by 61 and 62, and the altitude by h, the formula for the area of any trapezoid is : A=*^^h 2 212 ELEMENTARY ALGEBRA The area of a trapezoid is the product of half the sum of the two bases and the altitude. Solve the above formula for 61, 62, and h, and read the results as rules for finding those dimensions. Exercise 117 — General Formtilas Ir — a 1. Solve the formula s = for a, r, and I. r— 1 2. A man sold a piece of land for n dollars and gained a per cent. How much did he pay for it? 3. Solve the formula s = — - — for a, I, and n. 4. What sum must be invested at n% per annum to yield a quarterly income of a dollars? 5. Solve dxW\ = d2W2 for each general number. 6. By selling silk at m cents a yard, a merchant lost b%. Find the cost per yard. 7. Solve V2t = Vit-\-n for Vi, V2, and t. 8. The length of a rectangular field is m times its width. Increasing its length a rods and its width b rods would increase its area n square rods. Find the dimensions. 9. What sum put at interest at r per cent per annum will amount to m dollars in n years? 10. Solve the formula -—- = - for q, v, and/. q P f 11. At what rate per annum will a dollars yield b dollars interest in c years? 12. Solve the formula F = ~+S2 for C. 5 13. In how many years will the interest on a dollars amount to 7n dollars at r% per annum? 14. Solve the f ormula - = T-77 for a, g, h, and L Q n-\-i CHAPTER XIX SIMULTANEOUS SIMPLE EQUATIONS ELIMINATION BY COMPARISON 266. Elimination by comparison is accomplished by solving each equation for the same unknown number and forming an equation of the two values obtained. Solve the system: (Qx-5y = 15 (1) {3x-h2y = 21 (2) Trans- '(l)6rc-5i/ = 15 (2) 3a:+2t/ = 21 posing: (3) 6a; =15+5?/ (4) Sx = 2l-2y (5) .= 1^ (6) x = 21-2^ By comparison (7) axiom, § 15: Solving (7), Substituting Ans. Exercise 118 Solve the following, eliminating by comparison: Ux-8y=^12 (Sx+3y=-25 \Qy+Sx = SQ \42/+6x=-28 1. 3. 4x-5y = 27 \Sx-4y=-lS Sx-3y = 24: \4x-5y=-21 213 214 ELEMENTARY ALGEBRA 5. 7. 11. 13. 15. 17. (Sx+5y = 2S \2y+4x==17 (Sx+2y = 27 \2x+Sy = 2S (Sx+5y = 45 \4x+3y = 25 (2x+Sy = 5S \7x-2y = 2S 6y+6x = 47 Sx-Sy = 2Q 7y-2x = S4: 7x-2y = lQ (Sy+Qx = 50 \9x-4y = 24: 6. 8. 10. 12. 14. 16. 18. (Qx-\-7y=-70 \2y+2x=-25 Sx-7y=-40 ix-5y=-2S 2y-3x=-25 2x-\-5y=-40 (5y-4:X=-19 \sx+2y=-2Q (Qy-Qx=-Ql \4:X+9y=-20 Uy-9x=-52 \Qx+Sy=-m Sy-7x=-lH 4x-7y=-S2 267. In eliminating one unknown number from a system of fractional equations, it is often best to proceed without clearing the equations of fractions. 2x 5y ^+^ = 33 5 4 (1) (2) Multiplying (1) by 3 and (2) by 2 and subtracting the second result from the first, we have "3""T-^^ From this equation the value of y is found to be 12. Sub- stituting this value in (1), the value of x is 40. ELIMINATION BY COMPARISON 215 268. Systems of fractional equations having the unknown numbers in the denominators, though not simple equations, may be solved as such for some of their roots. In solving such equations, one of the unknown numbers should be eliminated without clearing of fractions. Thus, (1) (2) Multiplying (1) by 3 and (2) by 2 and subtracting the second result from the first, we have 2 X y = 22 3+L X y = 30 y = 6 from which y = ^; and substituting in (1), Check by substituting in (1) and (2) Exercise 119 Solve the following and check the first six: 1. < 6. < \Wr = 25 1 1_ ,^ y~ = 15 X y = 2a 1_1_ .^ y = 26 X y --4c a_b_ .X y = M 2. V = 27 X y §-1=36 X y 4. < ?+3 X y 5_4 X y 28 = 24 6. 4 2 --- = 22 X y «-? = 15 y X 216 ELEMENTARY ALGEBRA In the following systems, first multiply each equation through by the 1. c. m. of the known factors in the denomi- nators : 2 11a; ]__ Ey 1 lly 1 _1 l0i~2 1 5x~^Sy ^ 1+^ = 5^ 'Sx^y 2 8. < 10. 7x^Uy Sx 2y 2 = 3 3a: 5y '-+- = 6 [Qx by Exercise 120 Solve and check, eliminating by any method (see §§ 119, 165, and 266): 4i/-2x=16 5x — 3i/ = 44 \5x+62/ = 28 2x-|2/ = 36 fx+2i/ = 56 7. < 9. < 3 5 2x 5?/_ 2a 26^4(^ a; ?/ c 2a 26 4 c 11. a; 2/ ^ f5i/H-6x = 47 \4a:+32/ = 35 2. f5x+4?/ = \6?/+7a; = • \9a;+32/ = rfx+42/= 6. 25 45 -20 •27 44 24 8. < 10. < 12. 3x % 4"^2 = -30 4^ +3a:=-48 2x 3?/ ,32/"^ 3x 6?/-2a; = 27 31 -34 30 ELIMINATION BY COMPARISON 217 13. 15. < \lx-\-Sy = 35 ^+^ = 27 4 5 17. < 19. 21, 24. 27. --f =10 ax by 1+^ = 24 ^ax by (4:X+2y = a \4x-3y=b Sx — 2y = a 2x-3y = a ax-\-by = b cx — dy = d 26. 14. 16. < 2x+6y=-29 3y+Qx=-SQ = -35 18. < 7y_2x 6 5 i— ?-=-18 Sx 4y 4y 3x /2p+18 = 5 \ p+ 9= -5 ( 14:171— l = 2n \ n — 6m = 22. (t+u =13 \lO^+w-f27 = 107i+^ 25. f3n + 15s = 7 \5s+12=-r x{a-\-b)—y(a — b)=4 x{a — b)-\-y{a—b) =4 2.4a;- 1.5?/ =12 1.2a:H-.15i/ = 42 r+l= —4s 2s-13=-5r 29. 28. a(x-y)-{-b{x+y)=2c a{x-\-y) + b{x — y) = 2d 30. (mx+ .sy=.in \ .8x-.24y = m2 31. (2-p)4 = 3g (2-2?)2 = 2g-4 f (l+a)x+(a-2)2/ = 4a • \ (3-ha)x+(a-4)y = (ja 218 33. ELEMENTARY ALGEBRA (ay-\-hx = 2ab \by+ax = a^-\- 34. 35. a-\-b = x+y ax-\-ar = by-\-b'^ y{a-\-n) = x{a—n) -\-an x{a-\-n)=y{a—n)-\-^an 36. 38. iay-{-bx = a^-\-b^ \ax — by= —a? — X\ X2 1__1^ Xi X2 = 25 40. < rii ^2 i+?-=3i rii n.2 42. < .44. < ^+^ = 12 2/1 2/2 12/1 2/2 2^+1= ^« 1+1 = 21 4a; 62/ 46. { x-[-y-l x-y+l y-x^-l x-y-\-\ = 10 10 48. { ^.4x .9?/ 37. a? — b'^ = ax — by {a+by = ax-\-by . 3xH-42/+6 2a;-32/+l ^4a;+5?/-2 = 20 L3a:;-3i/-8 a+3 . n—5 41. < 2 6 71+7 a-\-9 = 17 29 20 18 43. < P+l q+l |-^r-^ = 45 45. 47. tp+1 q+1 n+S s— 4 12 2 n+5 s— 4 '_6 3_ x—1 y—l 5 1 49. < a:— 1 y—l _y I X a—d a—c y X b-d b- = 15 = 10 = 39 = 37 = 20 = 20 ELIMINATION BY COMPARISON 219 50. < — ?-*=180 .02a; My 61. y ^ X _ 1 [Mx my n — s n-{-s n — s X y 1 n-{-s n — s n+s PROBLEMS IN SIMULTANEOUS SIMPLE EQUATIONS 269. Many problems, which really contain two or more unknown numbers, are easily solved by the use of a single equation containing but one unknown number. This method is advisable only when the relations between the unknown numbers are so simple that all of them can be expressed in terms of a single unknown. In other problems it is better to introduce as many equa- tions as there are unknown numbers. When using a system of two or more equations to solve problems, enough conditions must be expressed in the prob- lem to furnish as many independent equations as there are unknown numbers to be found. Exercise 121 — Problems in Two Unknowns Solve the following problems using equations involving two or more unknown numbers : 1. The sum of two numbers is 148, and their difference is 38. Find the numbers. 2. The larger of two numbers is 3^ times the smaller, and their sum is 324. Find the numbers. 3. A man changed $7 into dimes and nickels, receiving 111 coins. How many of each did he have? 4. Of two consecutive numbers, f of the smaller number exceeds ^ of the larger by 6. Find the numbers. 5. Divide 118 into two parts so that 7 times the smaller part shall exceed 3 times the larger by 100. 220 ELEMENTARY ALGEBRA 6. If the pupils of a class are seated 3 on each bench, 5 pupils must stand. If 4 are put on each bench, one seat is not occupied. How many pupils are in the class? 7. Half the sum of two numbers is 73, and 4 times their difference is 128. Find the numbers. 8. The length of a rectangle exceeds its width by 14, and its perimeter is 116. Find the dimensions. 9. Find three numbers whose sum is 50, the first being 20 greater and the second 15 greater than the third. 10. In the equation ax-{- by = 32, find a and b, if, when x=4t, y = 2; and if, when x=10, y= —3. 11. There are 4 more spokes in each front than in each rear wheel of a wagon, and in the 4 wheels there are 112 spokes. How many spokes are in each wheel? 12. The sum of three numbers is 59. The second is 8 greater then the first, and the third is 7 greater than the second. Find the numbers. 13. If 3 carpenters and 7 masons together receive a daily wage of $61.20 and a mason receives 20 cents a day more than a carpenter, what is the daily wage of each? 14. Three tons of hard coal and 2 tons of soft coal cost $32. At the same prices, 2 tons of hard coal and 6 tons of soft cost $43.50. Find the price per ton of each. 15. The first of three numbers is twice the third, the second is 5 less than the first, and the sum of the three numbers is 55. Find the numbers. 16. A man invests part of $3200 at 6% and the rest at 5%. If the annual income from the two amounts is $180, what is the amount of each investment? 17. One dimension of a rectangle is 8, and one dimension ELIMINATION BY COMPARISON 221 of a smaller rectangle is 6. The sum of the areas is 144 and the difference is 48. Find the unknown dimensions. 18. There is a number which is expressed by two figures. The digit in tens' place exceeds the digit in units' place by 4 ; and if 31 is added to the number, the result is nine times .the sum of the digits. Find the number. Let ^ = the tens' digit, and w = the units' digit. 19. A number of three digits is equal to 18 times the sum of its digits. The digit in tens' place is 3 times the digit in units' place; and if 99 is added to the number, the digits are interchanged. Find the number. Let h = the digit in hundreds' place, and ^ = the digit in tens' place, and u = the digit in units' place. lOOh-{-mt+u = lS{h+t-{-u) t = Su lOOh+10t-\-u+99 = 100u-\-10t+h 20. A number of two digits is 9 less than 7 times the sum of its digits. If 18 is subtracted from the number, the digits are interchanged. What is the number? 21. A is 2 years older, and C is 2§ years younger than B. The sum of their ages is 73. Find the age of each. 22. In a picnic party of 38 persons there were -f as many men as women, and f as many children as women. How many of each were there in the party? 23. A boy said to his playmate, ''Give me 5 of your marbles and then we will have the same number." His playmate replied, "Give me 10 of yours and I will then have twice as many as you." How many marbles did each have? 24. Find three numbers whose sum is 168, iff of the first 222 ELEMENTARY ALGEBRA plus f of the second plus ^ of the thh'd is 92; and if when 21 is added to the first, the sum is twice the third. Let / = the first number, and s = the second number, and t = the third number. /+s+< = 168 2f 2s t —+-+- = 92 3 5 2 f+21=2t 25. The average weight of 3 persons is 164 lb. The aver- age weight of the first and second is 159 lb., and of the second and third 165 lb. Find the weight of each. 26. In a company of 29 persons there were 15 more adults than children and 4 more men than women. How many- persons of each kind were there in the company? 27. How many bushels each of new wheat at $1.05 a bushel and old wheat at 85^ a bushel may be mixed to make a mixture of 200 bushels worth 90^ a bushel? 28. The numerator of the larger of two fractions is 8, and the numerator of the smaller fraction is 5. The sum of the fractions is lyf; and if the numerators are interchanged, their sum is l|^. Find the fractions. 29. The sum of the three angles of a triangle is 180°. The sum of twice the first and the second exceeds the third by 90°; and the sum of the first and twice the third exceeds twice the second by 70°. Find the three angles of the triangle. 30. If a rectangle of paper were 4 in. shorter and 3 in. wider, the area would be 2 sq. in. less than it is. If a strip 2 in. wide is cut off on all sides, the area is diminished 184 sq. in. Find the dimensions. 31. In the equation ax — by = 20, find x and y if when a = 7, 6 = 5; and if when a = 8, 6 = 3^. ELIMINATION BY COMPARISON 223 32. The sum of the three digits of a number is 15. The digit in tens' place is half the sum of the other two; and if 198 is subtracted from the number, the first and last digits are interchanged. Find the number. 33. A man has $49 in dollar bills, half-dollars, and quarters. Half of the dollars and J of the half-dollars are worth $15.50; J of the half-dollars and J of the quarters are worth $5. How many coins has he? 34. A and B are 8 miles apart. If they set out at the same time and travel in the same direction, A will overtake B in 4 hours. If they travel toward each other, they will meet in 1^ hours. At what rate does each travel? 35. A man bought a piece of land. At $5 less per acre, he could have bought 40 acres more for the money; at $4 more per acre, he could have bought 20 acres less for the money. Find the number of acres bought and the price per acre. 36. One woman paid $2.75 for 7 lb. of cofifee and 5 lb. of sugar; another paid $2.05 for 3 lb. of coffee and 10 lb. of rice; another paid $1.02 for 7 lb. of sugar and 6 lb. of rice. Find the uniform price of each per pound. 37. A harvest hand engaged to work two months, July and August, for his board and $2.50 for each work-day. For each week-day he did not work he forfeited 50^ for his board. The term of service contained 8 Sundays. At settlement he received $123. How many days did he work? 38. The sums of the three pairs of sides of a triangle are 14, 15, and 17. How long is each side? 39. A classroom has 36 desks, some single and some double. The seating capacity of the room is 42. How many desks of each kind are there? 224 ELEMENTARY ALGEBRA 40. A sold 35 sheep to B and 25 to C. They each then had the same number. Before A made these sales, he had 10 more than B and C together. How many did each have at first? 41. A boy bought some peaches at the rate of 2 for 5^ and some others at 3 for 5^, paying $6 for all of them. He sold them all at 40^ a dozen and made a profit of $4. How many did he buy at each price? 42. A has his money invested at 4%, B at 5%, and C at 6%. A's and B's annual interest together is $398; B's and C's together is $441.50; and A's and C's together is $409.50. How much money has each one invested? 43. The width of a rectangular sheet of paper is 6 inches greater than half its length. If a strip 3 inches wide were cut off on the four sides, it would contain 360 square inches. Find the dimensions of the paper. 44. If the sum of f of the first of three numbers and f of the second is 118, the sum of f of the second and f of the third is 93, and the sum of f of the third and f of the first is 112, what are the numbers? 45. A street car has 12 short and 4 long seats. When the seats are all occupied, 56 persons are seated, each long seat holding 6 more passengers than each short one. How many passengers does each kind of seat accommodate? 46. The sum of two sides of a triangle is 58 feet, and the difference is 14 feet. The perimeter of the triangle is 103 feet. Find the length of each side. 47. In 8 months a sum of money at simple interest amounts to $780. At the same rate, in 14 months it amounts to $802.50. Find the sum invested and the rate. ELIMINATION BY COMPARISON 225 48. Angle A of a triangle is 14° less than angle B, and angle B is 10° larger than angle C. How many degrees are there in each angle? 49. The perimeter of a triangle is 80 feet. Two of its sides are equal, and the other side is 8 feet longer than either of the others. Find each side. 50. In a factory where 600 men and women are employed, the average daily wage for men is $3.25 and for women $1.75. If the sum paid daily for labor is $1650, how many men and how many women are employed? 51. How must a man invest $42,000, partly at 4|%, partly at 5%, and partly at 6%, so that he may receive an annual income of $2200, if he invests f as much at 4|% as he invests at the two higher rates? 52. Of three fractions the sum of the reciprocals of the first and second is 3y^; the sum of the reciprocals of the first and third is 2^; the sum of the reciprocals of the second and third is 3^^. Find the fractions. 53. A miller has corn worth 80^ a bushel, rye worth 70^ a bushel, and oats worth 60^ a bushel. He wishes to make a mixture of 200 bushels of the three kinds worth 72^^ a bushel, and use 40 bushels of rj^e. How many bushels of corn and oats must he use? 64. In a public school the number of pupils in the third and fourth grades is 225; in the third and fifth grades, 200; in the fourth and fifth grades, 185. How many more pupils are in the third grade than in the fifth? 55. If a fruit dealer had bought and paid for 90 lemons at a certain price he would have had 75^ remaining. If he had bought 150 lemons at the same price he would have had 75 oi*, in words: A mean proportional between two numbers is the square root of their product. 283. A Third Proportional. In the proportion a :b = b : c, the number c is a third proportional to a and b. Thus in ^ = f ^, 80 is a third proportional to 5 and 20. 284. A Fourth Proportional. A fourth proportional to the three numbers, a, b, and c, is the number d in the proportion a c - = -. It is the number, which with the three given numbers, completes a four-termed proportion. Thus in i^ = f^, 39 is the fourth proportional to 7, 13, and 21. , « Exercise 127 Find mean proportionals between : 1. 3 and 27 2. 4 and 16 3. 1 and 81 4. a and b 6. J and ^^3- 6. 1 and x^ 4 9,Pi 7. Z and —^ 6. a+b and dia+by a ax^ PROPORTION Find third proportionals to the following: 9. 2 and 6 10. 4 and 9 11. 2 and 22 12. J and 2 13. f and —8 lA. x — y and x+y 16. x-\-y and x^ — y"^ 235 16. - and - a X Find fourth proportionals to the following : 17. 4, 8, and 12 19. 5, 6, and 12^ 21. m, n, and p 23. m-\-n, m — n, and w? — n'^ 1 18. 12, 3, and 1 20. a, X, and ?/ 22. a, a^, and a'' 24. x^ x^, and a;^ 25. a:+l, , and x—1 x-l PRINCIPLES OF PROPORTION 285. Since each of the following products is 24, we may write 2- 12 = 3-8. Using only the four numbers of these two products, we may write the two columns below, the first being proportions and the second, not proportions. Test by § 281 the expressions of both columns and show that the expressions of the first column meet the test, while those in the second column do not. PROPORTIONS EXPRESSIONS NOT PROPORTIONS 1. 2 :3 = 8 :12 1. 2 :12 = 3 :8 2. 2 :8=12 :3 2. 2 :8=12 :3 3. 12 :3 = 8 :2 3. 12 :2 = 3 :8 4. 12 :8 = 3 :2 4. 12 :8 = 2 :3 5. 3 :12 = 2 :8 5. 3 :8=12 :2 6. 3 2 = 8 :12 7. 8 3 = 12 :2 8. 8 12 = 3 :2 236 ELEMENTARY ALGEBRA 6. 3 :2 = 12 :8 7. 8 :12 = 2 :3 8. 8 :2 = 12 :3 Notice that in the first column the proportions are made by using both factors of one of the products as means, and both factors of the other product as extremes. In the second column notice that this plan is not observed, and that the expressions obtained are not proportions. 286. Principle. If the product of two numbers equals the product of two other numbers, the factors of either product may be made the means and those of the other product the extremes of a proportion. Suppose a'd=b' c a c , a b To prove -r = -, and - = -,, etc. a c a Proof. Divide both members of a'd = b'C by bd, a c and simplify, obtaining -r=-. b a Also divide both sides of a'd=b' c by cd, and obtain - = -, etc. Other proportions are proved similarly, c a See how many of the 8 possible proportions you can write from the equation a'd=b' c You should be able to write two, beginning with any one of the 4 letters. Exercise 128 Write all the proportions you can from the following equations : 1. 3-12 = 4-9 3. 3-7 = 2M 2. 2-25 = 5-10 4. m-q^U'p 287. Just as equations may be derived from other equa- tions so may proportions be derived from other proportions. The principles for deriving proportions from proportions are now to be established. PROPORTION 237 288. Proportion by Alternation. // four numbers are in proportion, they will be in proportion by alternation, or the means of the proportion may be interchanged. _^, , . .- a c ,, a b That IS, if T = -i, then -=- b a c a a c From T=-iJ we have, by § 281, a ad = bc From which by § 284 we obtain : a _b c d' This expresses the principle that if the means of a pro- portion be interchanged, the result will be a proportion. Deriving a proportion in this way is said to be taking the given proportion by alternation. 289. Proportion by Inversion. // four numbers are in proportion they will be in proportion by inversion or the two ratios may be inverted. That is, if - = -, then- = -. a a c From the proportion, a_ c b~d By § 281, we obtain: b- c = a'd. From which by § 284, we have -=-. a c That is, the two ratios of a proportion may be inverted without destroying proportionality. Deriving a proportion in this way is called taking the given proportion by inversion. 290. Proportion by Addition. If four numbers are in proportion, they will be in proportion by addition, that is the sums of the two terms of the ratios will form a proportion with either the antecedents or the consequents. 238 ELEMENTARY ALGEBRA a a c a To prove (1) and (2), proceed by analysis, thus: ANALYSIS Assume (1) = , or (2) -—— = —1-. a c a Reduce the improper fractions to mixed numbers thus : 1+-=1+- or -+1=3+1 a c b a -ni7L b d a c Whence, - = - or 7- = - a c b d PROOF We may now construct the proof, by reversing the steps just given. We know that - = - if ? = -^. Why? a c b d Add 1 to both sides of the equations : a c b d Reducing to improper fractions we have: a+6 c-\-d , a+6 c-\-d = and — r— = — —. a c b d When either of the last two proportions is inferred directly a c a c from Y = -;, the proportion, 7 = -;, is said to be taken by addi- d d tion. Proportion by addition is often called proportion by composition. 291. Proportion by Subtraction. // four numbers are in proportion, they will be in proportion by subtraction. That is, the difference of the terms of each ratio form a proportion with either the antecedents or the consequents of the ratios. PROPORTION 239 T- a c ^, a—h c — d a — b c — d If- = -, then = , or — — = _— d a c .0 d Use the method of analysis just as it was used above. Proportion by subtraction is often called proportion by division. 292. Proportion by Addition and Subtraction. If four numbers are in proportion they will be in proportion by addi- tion and subtraction. ^p a c ^, a+6 c-\-d If - = -, then r = -. a a—b c—d Combine the results of the principles of §§ 290 and 291. Proportion by addition and subtraction is often called proportion by composition and division. Exercise 129 1. From each of the following, write a proportion (1) by alternation, (2) by inversion, (3) by addition, (4) by sub- traction, and (5) by addition and subtraction: 1 3_9 o 2_i4 o 6_-42 8_ m+n 1-2-6 ^.9-63 ^. zrn 77 ^-3";^^^ 2. From each of the following equations write a proportion, commencing with each of the four factors; then take each proportion (1) by alternation, (2) by inversion, and (3) by addition and subtraction: 1. 2-9 = 3-6 2. 3-8 = 2-12 3. f •7 = 3-i- . 4. 2.7-3 = 9-0.9 3. Find values of x in the following proportions: a;-7_6 a:+5_5 • 7 4 * x-Z 1 x^^]_ 3a;+5 _ll • x+2 11 * 5a;-5 5 240 ELEMENTARY ALGEBRA * x-3 8 . ' a:-6 5 4. Divide 91 into two parts that are to each other as 5. Divide m into two parts that are to each other as a : b. 6. The difference between two numbers that are to each other as a : 6, is d. Find them. 7. What number must be added to each term of 3 : 6 = 4 : 8 to give another proportion? 8. By what number must each factor of the products 25-51 and 31 -40 be reduced that the products may be equal? 9. By what number must each factor of the product 30-147 be reduced and each factor of 14-62 be increased, to make the products equal? 10. What number must be added to both m and n to give sums which are to each other as a : 6? 11. What number added to m and subtracted from n gives numbers to each other as a : 6? 12. The value of a fraction is f . Increasing numerator and denominator by 2 gives a fraction whose value is f . What is the fraction? 13. The denominator of a fraction is 6 greater than the numerator. Reducing both terms by 1 gives a fraction whose value is J. Find the fraction. 14. If the denominator of a fraction whose value is f , is increased and the numerator decreased by 3, the value of the resulting fraction is ^. Find the fraction. 15. By what number must both terms of |-| be increased to give a fraction whose value is ^? 16. The value of a fraction is f . If 7 is added to the numerator and 2 to the denominator, the reciprocal value of the original fraction is obtained. Find the original fraction. VARIATION 241 VARIATION 293. Direct Variation. Suppose water to be flowing through a tube into a pail. If lo denotes the weight of water in the pail at any time t minutes after starting, then w and t have different values at different times. They are therefore called variables. If the flow is uniform and q denotes the weight of water flowing into the pail in one unit of time, (1 min.), w = qt. The number q differs from w and t in that q is constant for a given flow. The height, h, of a growing tree and its age y, the price, P, of a load of corn and the price, p, per bushel, are other examples of variables. Give other examples. 294. If in a given discussion or problem a number may have many different values it is a variable number, or a variable. All numbers that are not variables are constants. Thus, if y varies as x we may always write : -=k, a constant, or y = kx. X Show that 2/ is a function of x. Since the distance, d, that a train runs during the time, t hours, varies as t, we may write : d = kt. If, now, the train runs 30 miles an hour. A; = 30, and we may write : d = 30i . Show that d is a function of t. 295. When a variable, as y, is so related to another, as x, that as they change, their ratio, -, remains constant, the X one variable is said to vary directly as, or to vary as the other. In symbols this is written : yocx, and read : 2/ varies as a;. _. , 242 ELEMENTARY ALGEBRA Exercise 130 1. Assume the amount, w, of water in a barrel to vary as the tune, t, since the in-flow began. Write the general law for the amount of water in the barrel at time, t. Ans. w = qt 2. Suppose that after 3 minutes of flow there are 36 qt. of water in the barrel. Find q and state the law definitely. Substitute it; = 36, and t = 3 in the general law, w=qi, obtaining 36 = 9*3, or 9 = 12, and the definite form of the law is then w = 12t 3. After 2 minutes of flow how many quarts will have passed into the barrel? Substitute t = 2 in w = 12t, obtaining w = 12'2 = 2A. 4. The amount of water in a cistern is assumed to vary as the square of the time, t, since the in-flow through a tube began. Express the general law connecting w and t. General law : w = qt^ 6. Suppose that after 5 minutes there are 225 qt. in the cistern. Find q and state the law definitely. In w =qt'^, put w = 225 and t = 5, obtaining: 225=5-25, or q = 9, Definite law, w = 9t^. 6. Find the quantity of water in the cistern after 3 minutes of flow. Inw== 9P, substitute t = 3, giving w = 9'9 = Sl. (81 qt. in cistern) 7. After how long will there be 900 qt. in the cistern? 900 = 9- f^ or <^ = 100, and t = 10 (after 10 min.) 8. li y cc X and y=lO when x = 5, what is the law con- necting X and y? We have, first, y = kx. Making y = lO and x = 5, 10 = 5A: Therefore, ' k = 2 Hence, y = 2x. VARIATION 243 9. When a spring is stretched a()^II||][^^ by a force, F, the amount of stretch, { — s — . s, varies as the strength of the ^ |)CfiMlMMMMl^ force, F. Express the general law The stretch, s, varies as of stretch. the force, F. How is this law shown by the graduation marks of an ordinary spring balance? 10. When the force is 20 lb., the stretch is 5 inches. Find k, and express the law definitely. 11. How much would a force of 32 lb. stretch the spring? 12. How many pounds of force would have to be exerted to give a stretch of 10 inches? 13. The area, A, of a square varies as the square of a side, s. When s = 5, A = 25. Find /c, and express the law connecting A and s in definite form. Have you met this law before? 14. If the altitude of a rectangle is constant, the area. A, of the rectangle varies as the base, x. Write the general law. 16. If the base is 12, the area is 96. Express the law in definite form. 16. The area. A, of an equilateral triangle varies as the square of a side, s. Express the law connecting A and s in general form. 17. When the side of the triangle is 6 the area is 3 v|^. Find k, and express the law in definite form. 18. The work, w, of a machine varies as the number of hours, h, that it runs. Write the general law of work for the machine. 19. Working 3 hours, the machine does 59,400 foot-tons of work. Express the law of the machine in definite form. 20. How much work would the machine do in 1 minute, or ^^0^ of an hour? CHAPTER XXI POWERS. ROOTS INVOLUTION 296. In §§ 140, 183, 185, and 187 we learned how to raise monomials to any power, also how to square binomials and polynomials. Those sections should be reviewed here. 297. Involution is the process of raising a number to a power whose exponent is a positive integer. Involution is indicated by an exponent, and the exponent which indicates how many times the number is taken as a factor is called the exponent of the power. Thus, 298. The base of a power in involution is the number which is raised to a power. It has been shown that to multiply any power of a base by any power of the same base, the exponents are added. Thus, The expression of this law in general numbers is a'"Xa" = a*"+". 299. It has been shown that to divide any power of a base by any lower power of the same base, the exponent of the divisor is subtracted from the exponent of the dividend. Thus, 244 INVOLUTION 245 The expression of this law in general numbers is At this point it is necessary to prove three general laws for the involution of monomials. 300. The sign of continuation is a series of dots ... It is read and so on, (See § 130.) POWER OF A POWER 301. Let a represent any number, m any positive integral exponent of a, and n any positive integer. Then {a"'y represents any power of any power. By definition of a power: (^gjm)n=^m.^m.^m.^m ^ tO 71 f actOrS, -_ Qm+m+m+m ... to n terms, The nth power of the mth power of any number is the mnth power of the number. The expression of this law in general numbers is Exercise 131 Give the result of each of the following : 1. (a^y 2. (23)3 3^ (a;2)« 4. (c")"» 5. (x")^ 6. (a^y 7. (52)5 8. (x^y 9. (c"')" 10. (x'^y 11. What power of 3 is (27)^? What power of 2 is (16)^? What power of 5 is (125)' ? 12. Express (81)^ as a power of 3; of 9. 13. Express (64)^ as a power of 4; of 2; of 8. 14. Express (9)^ as a power of 81 ; of 3; of 27. 246 ELEMENTARY ALGEBRA POWER OF A PRODUCT 302. Let a and h represent any two numbers and n any positive integer. Then (aby will represent any power of the product of any two numbers. By definition of a power : {ahY = ah-ah'ah'ah'ah • • • to n factors, = {aaa • • • to w factors) (666 • • • ton factors), = a"6" The nth power of the product of two or more numbers is the product of the nth powers of the numbers. The expression of this law in general numbers is (ab)" = a"b". In a similar manner it may be shown that the law holds for the product of any number of factors. Thus, (2a26"c)3 = 23a«63"c3 = Sa^^^'c^ Exercise 132 Write the power of each of the following: 1. (2a2)3 2. (22-32)2 3. (a-6")2 4. {Sab^cy 6. {Sx^y 6. (43-5^)2 7. (xH/y 8. (2ac2.T)" POWER OF A FRACTION 303. We have seen that: 'a\'' a a a © T ) —T'T'T ' ' ' to n factors, _a'a-a ' • • to ?i factors b'h'b • • • to n factors ~b- The nth power of a fraction is the nth power of the numerator divided by the nth power of the denominator. The expression of this law in general numbers is a\"_a^ iJ ~b^ INVOLU' noN 247 Exercise 133 Give the power of each of the following: ■• (iJ 2. r-^Y \ xyj ^- W) 4. (- "26; •■©■ •■ (-ST '■ (I?)' 8. (- 3?// '■ (iJ ■«■ (-5)' ■■■ (:?)■ 12. (^ 2«y ■'■ (IJ "• (-S)' ... (=?)- 16. (- 3 2iy POWERS OF BINOMIALS 304. By multiplication, the following powers of a-\-b and a—b may be obtained : (a-fb)=^ = a=^-}-3a2b+3ab2-hb3 (a-b)3 = a^-3a2b-|-3ab2-b3 (a+b)^ = a^+4a^b-f-6a2b2+4ab3+b^ (a - b)4 = a^ - 4a^b +6a^b ' - 4ab3+b^ (a4-b)^ = a^H-5a4b + 10a"^b2+10a2b=^+5ab4+b5 (a - b)^ = a^ - 5a% + lOa^b^ - lOa^b^ + Sab^ - b^ 305. From an examination of these powers, or expansions, considering n to represent the exponent of the power, the fol- lowing laws hold in each expansion : 1. Evei^y term of the expansion, except the last, contains a; and every term, except the first, contains h. 2. The number of terms in the expansion is n-fl; that is, it is 1 greater than the exponent of the power. 248 ELEMENTARY ALGEBRA 3. // both terms of the binomial are positive j all the terms of the expansion are positive. 4. If the second term is negative, the odd terms of the expan- sion are positive, the even terms negative. 5. The exponent of a in the first term of the expansion is n, and it diminishes by 1 in each succeeding term. 6. The exponent of b in the second term of the expansion is 1, and it increases by 1 in each succeeding term. 7. The coefficient of the first term of the expansion is 1; the coefficient of the second term is n ; and the coefficient of any succeeding term is found by multiplying the coefficient of the preceding term by the exponent of a in that term, and dividing the product by a number 1 greater than the exponent of b in that term. The statement of these laws constitutes what is called the binomial theorem. The theorem is true of all the examples given. We shall take it for granted that it is true for any positive integral power of a binomial, but a general proof lies beyond the scope of this book. Students will find it helpful to memorize the coefficients of the 1st, 2d, 3d, 4th, 5th, and 6th powers. 306. These coefficients may be arranged in a table forming what is known as Pascal's Triangle, as follows: Coefficients of 1st power: 1 1 Coefficients of 2d power: 2 1 Coefficients of 3d power: 3 3 1 Coefficients of 4th power: 4 6 4 1 Coefficients of 5th power: 5 10 10 5 1 Coefficients of 6th power: 6 15 20 15 6 Each coefficient is the sum of the number above it and the number to the left of the latter. The coefficients of two terms equally distant from the first and last terms of the expansion are equal. INVOLUTION 249 Exercise 134 Expand the following binomials to the powers indicated, reading the powers at sight, if possible: 1. {x+yy 2. {a-^xY 3. {b-cY 4. {a+yY 6. (x-aY 6. (c-bY 7. {a-\-yy 8. {b-{-xY 9. {a- cY 10. {b-\-xY 11, iy-xY 12. (a- cY 13. (a-w)' 14. (6-a)' 15. (a:-c)'^ 16. (n+xY 17. (6-a:)5 18. (a-xY 19. (a;-2/)^ 20. (a+x)« 21. {b-xY 22. (a+x)8 23. {a-xY 24. (a-?/)" 307. When a or b is 1, that term of the binomial appears only in the first or last term of the power. Thus, (a+l)' = a'+5a4+10a3+10a2+5a+l (l-a)«=l-6a+15a2-20a3+15a4-6a-'+aV Exercise 136 Give the following powers : 1. {x+lY 2. (l-aY 3. {b-lY 4. (l+xY 6. (a-lY 6. {l-xY It must be remembered that a and b in the binomial theorem of § 305 represent any terms whatever. Observe: (2a2+4)3 = (2a2)3-|-3(2a2)24-f 3(2a^)42+43 = 8a«-h48«4+96a2-f64 Exercise 136 Give the expansions of the following: 1. {b-2Y 2. (3-x)^ 3. {a-4Y 4. {2-xY 5. (a-3)^ 6. (6-2)7 7. (a;2+a:)^ 8. {a+a'Y 9. (t^-t)^ 250 • ELEMENTARY ALGEBRA EVOLUTION 308. A root of a number is one of the equal factors whose product is the number. Thus, 2 is a root of 8, 16, 32, 64, etc. 3 is a root of 9, 27, 81, 243, etc. 5 is a root of 25, 125, 625, etc Roots are named from the number of equal factors that make the number. See two definitions § 190. What root of 16 is 2? What root of 16 is 4? What root of 64 is 2? What root of 64 is 4? What root of 81 is 3? 309. Evolution is the process .of finding a root, or one of the equal factors, of a number. Evolution is indicated by the radical sign \/~which is placed before the number. The radical sign alone indicates the square root. If any other root is required, it is indicated by a small figure called the index of the root, written in the\/ of the radical sign, thus: \/l6, v^, \^, V^, V^ A symbol of aggregation with the radical sign indicates the part of the expression that is affected by the sign. Thus, \/254-24 means the sum of \/25 and 24, while \/25+24 means the square root of the sum of 25 and 24. The long bar above is a vinculum. See § 65. Any root of a number indicated by the radical sign is called a radical. Since evolution is the reverse of involution, the nth root of a is a number the nth power of which is a. EVOLUTION 251 ROOT OF A POWER 310. Since (a"^)" = a'"'S V a»»« = a"*, by extracting the nth root of both members, The nth root of a power is obtained by dividing the exponent of the power by n. Exercise 137 1. How would you find the square root of a power? The cube root? The fourth root? The fifth root? 2. Give the indicated root of each of the following: 1. \/a^ 2. \/^ 3. V P 4. v^ 6. S/Jc^^ ROOT OF A PRODUCT 311. Since {ab)" = a''b", then Va^- = ab. Why? The nth root of the product of two or more factors is the product of the nth root of the factors. Exercise 138 Find the indicated root of each of the following : 1. \/a2^ 2. -v/2W 3. \/lQa^ 4. \/a^^ 6. \/x^' 6. \/59a^ 7. v^81^4 g. \/¥^i'^ \/l«X25X36 = 4X5X6 = 120 9. \/25X 49X121 10. Vl6X 25X36X144 11. V27X 64X125 12. a/8X 64X216X348 By the same principle, any root of a number may be found by resolving the number into its prime factors. Observe the following: V'99225= V34-52.7'^ = 9-o-7 = 315 252 ELEMENTARY ALGEBRA In like manner, solve: • 13. \/30625 14. V86436 15. \^2T9E2 16. ^54872 Observe, also: V^45-60-80 = a/(32-5)- (22.3-^) •(24-5) = V2'-33-53 = 60 Solve the following: 17. \/l4X2lX42X63 18. VT5a^Fx2lF?><35^V 19. >^36X63X72X98 20. \/l2a'b' X bWc^ X 72a' c^ 21. V(^'+a:-2)(a:2-x-6)(x2-4x+3) • ROOT OF A FRACTION 312. From the law, -j— =t — , we have — b^ The nth root of a fraction is the nth root of the numerator divided by the nth root of the denominator. Exercise 139 Give the following indicated roots : 256a^a;8 '625xV 313. A root is called an odd root, if its index is an odd num- ber; an even root, if its index is an even number. NUMBER OF ROOTS 314. Since 8X8 = 64, the square root of 64 is 8, and since ( - 8) X ( - 8) = 64, the square root of 64 is also - 8. EVOLUTION 253 It is evident that every positive number has two square roots, one positive and the other negative. It may also be shown that every number has three cube roots, four fourth roots, and so on. IMAGINARY ROOTS 315. The square root of —25 is not 5, for 5^= +25; neither is the square root —5, for ( — 5)^= 4-25. The square root of —25 is therefore impossible, as is the square root of any other negative number. We can only indicate the square roots of a negative num- ber. The square roots of — 25 may be written \/-25 and - \/^^ 316. An imaginary number is an indicated even root of a negative number. 317. Since no even power is negative, all even roots of negative numbers are imaginary. We shall learn later that imaginary numbers are as real as any other numbers, but the old name imaginary still clings to mathematical literature. 318. The system of numbers as presented in arithmetic consisted of integers and fractions. Early in our study of algebra the number system was extended to include negative numbei*s. Now the number system is further extended to include imaginary numbers. These will be studied later. 319. A real number is a number that does not involve an even root of a negative number. SIGNS OF REAL ROOTS 320. Since even powers are positive, even roots of positive numbers are either positive or negative. 254 ELEMENTARY ALGEBRA To indicate that a root is positive or negative, the double sign, read plus or minus, is generally used: V^=±a2 ^^=r±a;2 ^"64= ±8 \/8l=±3 321. Since odd powers have the same sign as the number involved, odd roots have the same sign as the number. Thus, ^ySa^ = 2a^, \/^^^=-x', ■ a-\/5; av^, 6\/.r; and 3-^, 5v^, 9-^, etc. Two or more surds can be united into one by addition or subtraction only when they are similar, as is shown here: 2V45+4\/20+5\/80-3Vl25 = 6\/5+8V5+20-\/5 - 15\/5 = 19 V5, and 5^yTQ^-^yMx^-V2x^ = lOv^^-S v''2^- v^2i2 = 6^2x2 Exercise 163 Simplify the following : 1. 3V300+2V243 2. 2 ^yiSx^ - \/M^ -\- ^y^Ox^ 3. 4\/45+2V48-4\/27-3.\/20+2-\/l2 4. 7-v/T75-5\/Il2 5. 4\/l6^-2\/25a3+2\/36^ 6. 3Vn2+6\/45-3V28-f V80+3\/63 7. 3^^375-2^/192 8. 3V24a^- \/96^+2\/54^ 9. 2\/360-4Vl0-fV90+3'v/40-|\/250 10. 4Vl28-3Vl62 11. 2v^81x^-3v^l6^4-v/80i^ 12. 5>/l6-f-f >/128-5v/54+4v^250-2aJ^'486 TO REDUCE SURDS TO THE SAME ORDER 360. Surds of different orders are changed to the same order by expressing the radicals as fractional exponents, and reduc- ing the fractional exponents to equivalent fractions having a common denominator, and then expressing the surds unth radical signs. Thus, '\/2 = 2^ andAy3 = 3^ 272 ELEMENTARY ALGEBRA The lowest common denominator of the exponents is 6. Then, 2^ = 2"^ = y/2. This principle enables us to compare radicals of different orders as to relative magnitude. The signs of inequality are > and < . The sign, > , means greater than] < means less than. Exercise 154 Compare the following pairs of radicals: 1. \/5 and \/7 2. \/5 and \/2 3. 2 V3 and 3 v^2 4. v^ and v^ 5. ^ and i/6 6. 2\/5 and 3 ^y^ 7. Arrange in order of value, \/7, \/6, and V^- MULTIPLICATION OF SURDS 361. The product of two or more surds of the same order is found by law 5, § 335. For fractional exponents this law takes the form: Notice that this applies only when the surds are of the same order. Thus, V5 • \/35 = a/175 = 5\/7, and 2v^6. 3^/18 = 6^^108 = 18^ Exercise 166 Multiply as indicated : 1. 4\/3-3V5 2. 2\/7-3-n/7 3. 2\/5-3\/l5 4. 5^-2v^ 6. 4\/5-5V5 6. 5\/2'^^yS2 7. 3^y4'4^yS 8. 3V6-2\/8 9. 4^yQ'2\/l2 RADICALS 273 362. To find the product of surds of different orders, first reduce them to the same order. For example : \/2- ^V^= \/8- \/400 = 2-v/50 When numbers are large they can be better managed by the use of exponents. Thus, V21- \/9S= y/Tl'-i/T^= x/7^3. ^71:22= 7\/756 Exercise 166 Perform the following multiphcations : 1. 4V2.2>^ 2. 2Vi->/| 4. 3v^-4v^9 5. 3v^-\/| 7. 2\/6-5\/6 8. 5V^|-V| 12. f Vf.f Vf.sVI 3. 3\/2.5a/4 6. av^-a^v^ 9. 2a/5-4\/2 11. v^-\/3a2- V^6a2^ 13. v^9?^- V3x-\/2^ 14, I Vf -f V|"-tVj 15. V2a- V^4a2- ^Sa*-^^ 363. Observe that the square of the square root of a num- ber is the number itself. For example, Va-\/a = a, \/5-\/5 = 5, and VlO* VlO = 10 Multiplication of surds may be extended to radical poly- nomials, as shown here: 5+ \/3 3+ 2\/3 15+ 3\/3 IOV3+6 2H-13V3 Multiply the following: 3V2+2V6 \/2-3\/6 2-v/7-3v^ 3\/7H-2Vx 42-9\/7x 4V7x-6x 42-5-\/7x-6x 3\/^ — Sa/?/ 274 ELEMENTARY ALGEBRA Exercise 167 Multiply the following : 1. ^-^/ahy4+2\/a 2. 5-f4V^by 3-3\/2 3. 12+3\/5by 4-2\/5 4. 3a-3 Va by 2a+2 V« 6. V3-2\/6 by V3+5\/6 6. 2V«+3Vcby\/a+6A/c 7. v^-f-VS- VS by\/2- V3H-2V5 8. \/S- V5+2 V? by 2 V3+2V5- V^ 9. 3\/7-2\/3+4\/5by4\/7-3\/3-\/5 Multiply by inspection : 10. (-v/7-h2)(\/7-2) 11. (V«+V^)(V^-fV^) 12. (V3+5)(V3-f 5) IS, {Vi-Vy)(V^-Vy) 14. {x-VS){x-\/^) 16. (V8-V3)(V8-f-V3) 16. {^/a-hx){^/a-x) 17. (Va;+ V^)(V^-|- V^) 18. (\/T0-5)(Vl0-5) 19. (V^+\/3)(V8+\/3) DIVISION OF SURDS 364. The quotient of two surds of the same order is found by the principle of evolution as stated in the formula: Vi^ n /X Vi Vy RADICALS 275 As in multiplication, so here, this principle applies only to surds of the same order. Thus, 2\/60-^V5 = 2\/l2 = 4-v/3, •^/54-^3^/3 = | VT8= V2, also 6\/T5^2-v/T8 = 3\/f=J\/30. Exercise 168 Give these quotients by inspection : 1. 4V24-^-\/3 2. V32 4. SVSI-^V^ 5. V54 7. 3\/40^-\/2 8. \/56 2\/2 3. 6\/45-^2V3 3V2 6. 2\/90^9\/5 2\/7 9. 3V75-T-5V3 RATIONALIZING SURDS 365. Rationalizing is the process of multiplying a surd by a number that gives a rational product. Observe the following: V^--v/5 = 5, ^-^ = 3, V8-V2 = 4, and ^^^•->^ = 5. The rationalizing factor is the factor by which a surd is multiplied to give a rational product. When the product of two surds is rational, either surd is the rationalizing factor of the other. Name a rationahzing factor of each of the following surds and give the products : 1. \/Q 2. 2\/l2 3. 5v^ 4. 2V27a6 5. \/8 6. 5v^ 7. 3v^ 8. 4:^ymxy 366. A binomial surd is a binomial one or both of whose terms are surds. Thus, 4+ \/5, \/3 - 2, and \/6-f \/7. 276 ELEMENTARY ALGEBRA 367. A binomial quadratic surd is a binomial surd whose surd, or surds, are of the second order. 368. Conjugate surds are two binomial quadratic surds that differ only in the sign of one of the terms. For example a-\- y/h and a— \^b, as also \/7-V5andV7+V5 are conjugate surds. Since conjugate surds are of forms a-{-b and a — 6, the product of any two conjugate surds is rational. Hence it follows that any binomial quadratic surd may be rationalized by multiplying it by its conjugate. Thus, (4+\/7)(4-\/7)=9, and (\/IO-V2)(\/lO+\/2)=8 Exercise 169 Name a rationalizing factor of each of the following surds and give the products : 1. 8-Vl4 2. 2\/5-3a/2 3. y/a^+y/a 4. a+2'\/6 6. 2\/T5+V7 6. \/^-\/a-b 7.\/7d-5 8. \/70-3a/6 B.\/a^-\-\/a Exercise 160 Rationalize the denominators of the following: J 4 2 V^-V^ 3 V^+1+2 ' 3-\/5 'VS-hV^ *\/«+l-2 3— \/2 ■\/a-{-\/x a— -x/x+l '3+^2 " '\/a-\G ' a-^y/x'Ti a ^ \/a-\/x ^ \/a^b-\-c a—\/b y/x—y/a ^Ja-^-b c RADICALS 277 Rationalize and simplify by inspection: 13. -^ 14. -— 16. -^ 16. -^-= 17. — ^ 18. -^ 19. ;:^— ^ 20. — ^ V7 V« 3v^ aV^ 369. Any power of a monomial surd is found by the rule for multiplication of surds of the same order, § 361. SQUARE ROOT OF BINOMIAL SURDS 370. The square of a binotnial quadratic surd is a binomial quadratic surd one term of which is rational. Observe the examples : (\/5+\/3)' = 8+2\/l5, and (\/7- \/2)2 = 9-2\/T4. It follows then that some binomial surds are squares, and the square root of them may be found. Observe, first, that the rational term of the square is the Slim of the radicands of the two given surds; and second, that the irrational term of the square is twice the product of the two given surds. 371. Rule. — Reduce the binomial surd to the form a =±= 2 -y/S- Separate a into two parts whose product is b. Extract the square roots of the two parts of a, and connect the roots with the sign of the irrational term. Thus, Vl.54-V200= Vl5+2\/50= VlO+V^, V20+4-\/24= V20-h2V96= \/l2-f \/8, V20-5\/T2= V20-2V75= Vl5- \/5. Exercise 161 Find the square root of each of the following: 1. 9+2 V20 2. l-2-\/30 3. 13+4\/l0 4. 5-10\/2 5. 10+4\/6 6. 10-4^12 278 ELEMENTARY ALGEBRA 7. 12+4\/5 8. 15+3V6 9. 30-6\/20 10. 2x-f-32/-2\/6x^ 11. 2x+2V^'-2/' 12. 2a + h-2V^Tab 13. a2+6+2aV6 APPROXIMATE VALUES OF SURDS " 372. The approximate value of a surd is found by extract- ing the indicated root to the required degree of accuracy. It is frequently necessary to find the value of a fraction with a radical denominator. In such work much labor is saved by first rationalizing the divisor, or denominator. Thus, 3 _ 3a/5 3' 2.23607 \/5 5 5 Simplify each of the following divisions, finding the numer- ical value correct to 5 decimal places, having given that, V2 = 1.41421, \/3= 1.73205, and \/5 = 2.23607. Exercise 162 1. lO-^VlS 2. 13^\/27 3. 25^V20 4. 3VT5-^2\/3 5. 18-3\/8 6. 7-^2^/75 IRRATIONAL EQUATIONS IN ONE UNKNOWN 373. An irrational, or radical equation is an equation con- taining an irrational root of the unknown number. Thus, V^ = 3, V^-4 = 5, V3aJ-5=\/^+35. To solve an irrational equation the first step is to free the equation of radicals. This is done by raising both members of the equation to the same power. Power Axiom. — The same powers of equal numbers are equal. RADICALS 279 To solve, \/2x-6 = 3, or V^a;-? = \/2^+17 . Squaring, 2a; — 5 = 9, 5a; — 7= 2a; +17. The results of squaring in these two examples are simple equations, and are solved as such. Radical equations containing more than one radical may have to be squared more than once. Thus, to solve: V^.T — 5+ V''^' = 5 Subtracting ■\/x, \/.t — 5 = 5— \/x Squaring, x — 5 = 25 — 10 \/lc-\rX Uniting terms, 10v^a; = 30 Dividing by 10, ■\/x = S Squaring, x = 9. 374. With radical equations it is agreed that the radical sign shall denote only principal roots. Verifying, \/9 — 5 -f \/9 = 5 2+3 = 5 5 = 5 Since the substitution of 9 for x in the original equation gives an identity, 9 satisfies the equation. Exercise 163 Solve and verify the following: 1. \/^-H = 9 2. \^x — a = a 3. \/.^c+5 = 4 4. v^x+6 = 2 5.\/x-\-b = a 6. v^a; — 3 — 3 7. \^-\/x-^=V^ 8. \/^T2+\/x = 2 9. \/x-7+\/7=\G 10. V^+5-\/x=l 11. \/^+9-\/^-7 = 2 12. y/x-4:+\/x = 2 13. 3v^-\/4a;-9= V3 14. \/x+ V^+8 = 4 ^^ V3^+9+5\/^^ ^ ^^ V^+9 V^T2 15. . =o 16. , = , \/3.'r + 9~5\/-'^-5 ; \/^+3 \/^p-2 280 ELEMENTARY ALGEBRA 19. \/x-{-\^a+x = —== 20.\^a — \/x-{-\/x-{-a = \/x ■yx^a 21. y/x+d+\/x-2=\/5-h4x 375. A statement may be in the forrn of an irrational equation which, under the assumption that \/ shall mean the positive square root, cannot be satisfied. Thus, solving by the usual method, '\/x — 5= y/x-^^ we obtain a; = 9. Attempting to verify we have 2 = 3-f5 which is not an identity. Setting aside the assumption and recalling that \/x may be either the positive or negative root, as the conditions of the problem require, and retaining both signs in verifying, we have, ±2= ±34-5. Of these possibihties as to sign, we can get an identity by using +2 for \^x — 5 and —3 for \/x. It is worth noting that this state of things would not have been found if verify- ing had been omitted. In squaring a radical equation, a root is sometimes i7itrO' duced which the given equation did not contain. Thus, \/4xTl=S-V^^ freed of radicals and solved by the usual process, leads to x = 2 and x = 6 Verifying for 2, 3 = 3-0 This checks. Verifying for 6, 5 = 3 — 2 This does not check. Hence, 2 satisfies the equation under the assumption that \/~indicates only the positive square root, while 6 does not RADICALS 281 satisfy the equation. Removing this assumption, however, and choosing the -j- or — sign for the symbol, V^, according to the requirement that substitution must lead to an identity, 6 also will satisfy. For substituting 6, ±5=:3-(±2) From the possibilities as to sign here, and one sign has as good a right as another to be chosen, can we make an identity out of this? By taking -y/ to indicate + on the left side, and — on the right, we have : 5 = 3+2 376. The point to be noted is that without verifying we should not have found precisely which root would satisfy under the assumption of § 375 as to the sign of ■\/~. In solving radical equations verify all results and reject those which do not satisfy the original equation, on the assumption that \/~always denotes the positive value of the root. Under this agreement, which is convenient, but arbitrary, some radical equations have no solution. For example. Solve Vx^-m = 7 + Vx' - 9 (1) Squaring and simplifying, a;2 - 37 = V(a:2-lGHx2-9) (2) Squaring again and simplifying a:^ = 25, or x = =*= 5 Substituting in (1), vnder the arbitrary agreement, we have 3=7-|-4, which is absurd. No solution is possible under Ike agreement. Setting aside the agreement and substituting, ±3=7±4_ Choosing the positive value ol the V on the left, and the negative value on the right, we have 3 = 7 — 4, an identity. Hence, without the agreement, a solution is possible, though the radicals must be given whatever algebraic sign will lead to an identity. CHAPTER XXIII QUADRATIC EQUATIONS 377. A quadratic equation is an equation of the second degree in the unknown number. For example, x^+5x = 20, 4:r2 = 36, and 2x'--4x = 5a, are all quadratic equations. In determining the degree of an equation it is assumed that the equation is first reduced to its simplest form. 378. The constant term in a quadratic equation is the term that does not contain the unknown number. Some quadratic equations contain only the square of the unknown number; others contain both the square and the first power of it. Hence there are two kinds of quadratic equations. 379. A pure quadratic equation is an equation that does not contain the first power of the unknown number. Thus, 3a;2=108, x^-\-2x = 2x''+2x-m, 4x2 = 36a. 380. An affected quadratic equation is an equation that contains both the first and second powers of the unknown number. Thus, 3x2-f5a; = 15, x^-ix = S, x^-ax=b. Pure quadratics are also called incomplete quadratics, and affected quadratics are called complete quadratics. THE GRAPfflCAL METHOD OF SOLUTION 381. The Graphical Solution. The normal form of the pure, or incomplete quadratic, is a;^ — a = 0. Exercise 164 — Graphing We shall now graph x^ — a, for a = 9, a = 4, a = 0, and a = — 4. 282 QUADRATIC EQUATIONS 1. Graphing x' — a for a = 9, or graphing x' calculate and locate the points : x= 0, 1, 2, 3, 4, 5, -1, -2, - a;2_9=_9^ _8^ _5^ 0, 7, 16, -8, -5, 283 9, we first 3, -4, - 5 0, +7, + 16 Draw a smooth curve (1) through these points. Recall that ^2_9_Q asks: ''What is x where .T2-9is0?" or ''What is x where the curve crosses the horizontal V The answer is readily seen from the figure to be +3 and —3. Hence, the roots of x^ — 9 = are +3 and —3. These substi- tuted in x'^ — 9 = are seen to ^" satisfy it. 2. Graphing x- — a for a = 4, or graphing .r'- — 4, we calculate and plot the points : 2=1 vertical space x= 0, 1,2, 3, 4, 5,-1,-2,-3, -4, -5 x2-4= -4, -3, 0, +5, +12, +21, -3, 0, +5, +12, +21 and draw a smooth curve, like curve (2), through the points. This curve is of the same form as curve (1), but is simply raised upward 5 units. The a;-values of the crossing points are here +2 and —2, which are the roots of x^ — 4 = 0. 3. Similarly, graphing curve (3) for x- — a, for a = 0, or graphing the curve for x^, the required curve is drawn through the following calculated and plotted points : x = 0, 1, 2, 3, 4, 5, -1, -2, -3, -4, -5 _j _j \ J h viAt t7$'7 ^^|i\i«. Hz'} fi f% o> ^'m J jlrj IJUi \i\\ n t' ) ®/ n \ i^i^ ^Y Ywl f \l\ ^Ja A/i V i \ g5 Jf J ''' \ xfvt^i Ahl > \ \%^ jjf'i//" ^ .-^-'i q^ + /2 -y^ ^ \ |<4 rX 1/ ~\ -■^ ~o~ \rf J/_ ~TSp^ 1 '" Scale horizontal epact 25 x2 = 0, 1, 4, 9, 16, 25, 1, 4, 9, 16, Here there is but one x-value of the crossing-, or rather touching-point with the horizontal, viz. : 0. 284 ELEMENTARY ALGEBRA Because there were two crossing-points as the curve moved upward so long as it crossed the horizontal, we say there are two equal O's here. In reahty there is only the root 0, because -fO and —0 are the same point. 4. Graphing x^ — a for a= — 4, or graphing x^+A, we cal- culate and plot the points: x= 0, 1, 2, 3, 4, 0,-1,-2,-3,-4,-5 0:2-1-4= +4, +5, +8, +13, +20, +29, + 5, + 8, + 13, + 2(), + 29, and draw the smooth curve (4) through them. The curve being 4 units higher than curve (3) does not touch the horizontal at all. There are no crossing-points and the algebraic way of saying this is to say the roots are imaginary. We shall see later that the roots are +2\/— 1 and — 2\/ — 1. 382. We see then that a pure quadratic in general has two roots that are numerically equal but of opposite signs, but that if the graph of the first member just touches the horizon- tal there is but one root, viz., 0. If the graph does not cut the horizontal, there are no real roots. But since two results are found by solving x'^= —a i.e., x= \/ — a, and x= — ^—a, we say that if the graph lies entirely above the horizontal, there are two roots, one positive and the other negative, and both imaginary. SOLVING QUADRATICS BY FACTORING 383. The solution of quadratic equations by factoring, given in § 215 and on page 164, should be reviewed here. This is not a general method, for it is limited to those equations the first members of which are readily factored. A pure quadratic equation which is reducible to the form a:^ — a = is readily solved by factoring. When reduced to this form it is evident that the first QUADRATIC EQUATIONS 285 member is the difference of two squares. For example, {x-2a) {x-\-2a)=Q (.c- Vs) (.c+ V5)=0 x = 2a, and -2a x=V^, and - V5 Verify that the values found for x are solutions by sub- stituting them in the original equations. Exercise 166 Solve the following by factoring and verify: 1. {x-iy = 5-2x 2. (.'r+3)--6(x+3)=9 3. 4x'-\-9 = x^-\-m 4. x-\-Vx^+2'\/T^=l 6. 2a;2-« = x2+3a 6. Vx-{- y/¥^^^= Va+x ^^±n x-n^ \/2x'~-\-l-\/2x^-l ' ^^ x-\-n~ ' ^2x^-\-l-\-\/2x''-l^^ ^ x+2 . x-2 „i ^^ / 6 a:-2 ' a:+2 -* v x -r-ii-.t.- y^2_ii ^^ x-t-4 , a:-4 ,7 ^„ /» , /^ — r-^ 11. :ri-5+z — o = l8 12. ^ ^^ _ =x4- v^ — 13 5 5 1 a-H-y^'' ^ x4-cr , .g-ft 3 — X 3+x ^ ' x^— n^ x-\-n x—n 384. Some affected quadratics may be solved in a similar manner by factoring. For example, x2-4a:-12 = 10x2-llx+3=0 - (a:-(5)(x+2)=0 (5a:-3)(2a;-l) =0 a: = 6, and —2 ^=f, and ^ Substitute these values of x in the given equations and verify that they are the correct roots. 286 ELEMENTARY ALGEBRA Exercise 166 Solve the following by factoring and verify : 1. a;2+llx-26 = 2. 4x2-12a;=-9 3. 2a^2_5^_i2 = 4. 6x2+lla;=-4 . 6. 3a;2-7a;-20 = 6. x2-20a:=-51 385. Some equations of a higher degree than the second may be solved by factoring. Observe the following: x^—x- = l2x x^— X- — 4x-f 4 = x(x-4) (x+3)=0 (x-2) (x+2) (a;-l)=0 x = 0, 4, and -3 x = 2, -2, and 1. Substitute these values of the unknown in the given equations from which they were found, and verify that they are the correct roots. Exercise 167 Solve the following by factoring and verify: 1. a;3+8x2-9x = 2. x^-{-dx^-x = 5 3. x'^+ab-ax-bx = 4. x^-5x^-\-4: = 6. x^-{-x^-4.2x = 6. x^+ax+bx-\-ab = 7. x^-\-5x-Qx^ = 8. 6.T--49x=-8 9. x{x''-l)-2{x-\-l)=0 10. x^-\-x'-'SOx = 11. x^+7x'-7 = x 12. x{x''-4)-S(x-2)=0 13. 6x2-f3x-18 = 14. x^-x^-{-9 = 9x 16. (x-2)2-4(x-2)4-3 = 16. x^-{-5x^-Qx = 17. 6x2+17x=-5 18. {x^-x-2){Sx~--x-2)=0 19. 6x2-5^-21=0 20: .T^-17.'c2+16 = QUADRATIC EQUATIONS 287 SQUARE ROOT METHOD OF SOLUTION 386. A pure quadratic is solved by i^educing it to tlie normal form, x- = a, and taking the square root of both members. Root Axiom. Equal principal 7'oots of equal number's are equal. Extracting the square root of both members, "we have: x= =t: -y/a The double sign belongs to the unknown number as well as to the second member, but x = =*= \^a is the same as — a; = =1= \/a. For this reason the double sign is used before the second member only. A pure quadratic equation has two roots numerically equal, one positive and the other negative. For example, X- = 2o, x2 = 8, x^= — 5, have the roots : Since the square root of a negative number is imaginary, we observe that when a is negative, both roots are imaginary. All this was shown more clearly in § 381 by the aid of the graphs. Exercise 168 Solve by the square root method : 1 ^_ . _i_=5 2 _1 3_^i 5 . a 3. \/^+5= ' 4. ^.^ =\/x-a ■\/x — o \/x-\-a 387. Any complete quadratic equation may be reduced to the normal form, ax2+bx+c = 0, a, b, and c denoting any real numbers, positive or negative, integral or fractional, though a may not be 0. 288 ELEMENTARY ALGEBRA Since any complete quadratic may be reduced to this form, it is called the general quadratic. To apply the square root method of solution, the first member must be made a square. For this purpose the form of the equation is changed to : ax~-\-hx= —c 388. The process of making the first member of a quadratic equation a square is called completing the square. The value of a in the general quadratic, ax--{-hx-\-c, may be 1 , or it may be any number greater than 1 . TO COMPLETE THE SQUARE WHEN a IS 1 389. Consider the arranged trinomial -square, Two of the terms are squares and the other term is the product of three factors, viz.: The factor 2, the square root of the^rs^ term, and the square root of the last term. The binomial x^+2cx represents the sum of the first and second terms of any arranged trinomial square. Dividing the second term, 2cx, by twice the square root of the first term, i.e., by 2x, the quotient is c, which is the square root of the missing term. Adding c^ to x^-\-2cx will therefore complete the square. 390. Rule. — Reduce the equation to the general form and add to both members the square of half the coefficient of x. To make the first member of x^ — 6x = 7 a square, we must add 9 to both members, thus obtaining: a-2-6x+9 = 16 By the root axiom, .r — 3 = =*= 4 Whence, x — T, and —1 Substitute these in the given equation and verify. Carefully observe the following important truth : QUADRATIC EQUATIONS 289 391. The sum of the two roots is the coefficient of x with reversed sign. The product of the two roots is the constant term of the equation in the general form. To the teacher: Require pupils to test or verify the roots of all quadratic equations by reference to the foregoing principle. For example, solve a:2-3x-18=0. x'--3x = 18, x2-3x+f = 8i . ^-f=-f x = 6, and 3 The sum of the roots is 3, the coefficient of x with reversed sign; and their product is —18, which is the constant term. Again, solve x- — 6a: + 1 2 = a:2_6x=-t2 x2-6x+9=-3 x-3=±V^ x = 3±V^ The sum of the roots is 6, the coefficient of x with reversed sign; the product is 12, which is the constant term. Exercise 169 Solve by completing the square and verify : 1. x2-f-10x=-21 2. ?/-4?/-117 = 3. n2-14n=-24 4. .?/-6?/-160 = 6. x2-12a:=-32 . 6. 7/-2?/-143 = 7. /i2+lln=-24 8. x2-3a:-180 = TO COMPLETE THE SQUARE WHEN a IS NOT 1 392. Observe the following solution of 2x2 — 6x — 5 = 0, jj^ which the coefficient of x^ is greater than 1. Dividing through by 2, x^ - 3x - f = ( 1 ) Transposing the |^, x^ — 3a: = f Completing the square, x- — 3a:-}-f =f -f f By the root axiom, x — f = ± i\/l9 Hence, j;=f=*=i\/lO 290 ELEMENTARY ALGEBRA The sum of the roots must be the negative coefficient of x in the equation in which the coefficient of x- is 1 [i.e., in (1)], and the product of the roots must be the constant term in the same equation. The sum of the roots is 3, the coefficient of x with reversed sign; the product is — -|, which is the constant term. Exercise 170 Solve the following and verify : 1. x2-168=-2x 2. 2x^+Sx-U = 3. Zx^-10x=-S 4. 3a;2-h4x-39 = 5. 2/2-120= -2?/ 6. 2x2+7a:-39 = 7. 8a:-a;2=-180 8. n2-lln-60 = 9. a:2-16x=-60 10. if -\-loy-d^ = Q 11. 3x2-33= -2a; 12. x2-13a;-30 = 13. w2-lln=-30 14. 3a;2-f-x- 200 = 16. 3x2-95= -Ix 16. ?/2-ll?/-|-28 = 393. To avoid fractions, first multiply both members of the equation by four times the coefficient of x^. For example, to solve : 2x- — 7x — 1 5 = Multiply by 8, 16x2-56x = 120, Dividing b^x by twice the square root of \^x-, the quotient is 7. Squaring 7 and adding, 16x2-56rc+49 = 169 By the root axiom, 4a: —7 = =*= 13, Whence, x = b and — f. If the coefficient of x- in the given equation is made 1, the coefficient of X is — 1^ and the constant term is — V^. The sum of the roots is -g-, the coefficient of x with reversed sign; the product is — V", which is the constant term. This checks the work. Observe that the number added to complete the square is the square of the coefficient of x in the given equation. QUADRATIC EQUATIONS 291 Exercise 171 Complete the square, solve and verify : 1. 3a:2-7x=-2 2. 2x2~5a;-42 = 3. x'^-12=-4x 4. 3a;2-2a;-40 = 5. 4x'--7x=-3 6. 5r2-14r+8 = 7. 7w'+Qm=-n 8. 31*2 -f-9ii- 30 = 9. 2n2-5=-3n . 10. 3?/2- 101/4-3 = 11. x2+6a;=-25 12. ?/-10i/+21=0 13. 3<2-2=-5^ 14. 2s2+7s-22 = 15. rt2-j_8a=-21 16. 52-125-45 = SOLUTION BY FORMULA 394. The equation ax2H-bx+c = may be taken to repre- sent, or typify, any quadratic equation, in which all terms have been transposed to the first member, the a:2-terms being combined into a single term, as also the x-terms, and the constant terms. The solution of ax^-]-hx-\-c — gives a formula, or short- hand law for writing the roots of any equation of that form. Completing the square and solving, -b=bVb2-4ac x = 2^ 2a This is the formula for writing the roots directly without completing the square. It is the final result that is always arrived at by completing the square, and it may always be written down at once. 202 ELEMENTARY ALGEBRA Notice there are tvw roots, viz.: -b + \/b'-4ac b , \/b2-4ac Xi = 2a 2a 2a _ -b-\/b^-4ac _ b \/b^-4ac ^^ 2a 2a 2a Solve the following by the formula: 1. a:2-10a:-24 = a; = 5±\/25+24 a: = 12 and -2 2. 2x2 -*13a;+ 15 = X X — 13.7 a: = o and 1^ By the use of this formula write by inspection the roots of the equations at the end of Exercise 171. TO FIND APPROXIMATE VALUES OF ROOTS OF QUADRATIC EQUATIONS 395. Observe the following process for calculating approx- imate roots: (1) (2) :2-9x+16 = x2- -12a:+25 = -9a:4-(|)2=-V— 16=\'- 0:2- -12a:+62 = 62-25 = ll x-^ = ^\y/\7 • a:-6=+JVTi x- = 4.5±2.062- a: = 6=±=l.658+. a: = 6.562 - a; = 7.658 + and 2.438 4- and 4.342- Observe in each case whether the sum of the roots equals the coefficient of x with reversed sign. Exercise 172 Find the approximate roots to two places of decimals, of the following: 1. a:2-3:r-8 = 2. a:2-5x+3 = QUADRATIC EQUATIONS 293 3. x2+7x-22 = 4. a;2-8x-38 = 5. a:2-f lla:+27 = 6. a;2-6a:-35 = 7. a;2-10.T+23 = 8. x^-{-4:X-U,92==0 9. x2_2x-5.76 = 10. a:2+2x- 20.78 = 11. .T2-5.2a; + 5.76 = 12. .t^- 9.65a; +10.5 = 13. j2- 11. 05a; -96.6 = 14. a:2-22.55x+96.6 = EQUATIONS IN QUADRATIC FORM 396. An equation is in the quadratic form when it contains but two powers of the unknown number, the exponent of one power being twice that of the other. Show that the following are in the quadratic form: 2x'-3x'' = S, 2a;+4a-- = 13, and 3a;' -2a:' =4 These equations which are said to be m quadratic form may be reduced to the form ax2"+bx"+c = 0, and they may be solved by any of the methods for solving complete quadratics. The first solutions, however, are the values of x", that is, the values of x with half the larger exponent. Evolution and involution must then be applied to both members of the equation to find the values of x. Solve the following equations that are in quadratic form: (1) .T^- 13x2+36 = (2) 2a:+3V7=27 (.r^-4)(.r2-9)=0 16j;+24x' =216 .t2 = 4 and 9 1 6.r + 24x' +9 = 225 .r==t2and ±3 4x-i+3=±15 a;' =3 and -f .T = 9 and -V" In verifying these values, remember that in this particular example the square root of V" i^ ~f > because — f, not +f , was squared to give -\^ . 294 ELEMENTARY ALGEBRA Exercise 173 Solve the following equations: 1. x4+4a;2-45 = 2. a;^+x*-30 = 3. x'-5x'-2A = 4. 2v^4-3\/x = 6 6. rc+6V^-20 = 6. a;^+4a:^-5 = 7. x4-5x2-36 = 8. \/x-3v^ = 28 9. a:'+2x^-8 = 10. 2x3H-5-v/x3 = 7 11. x^+4x2-32 = 12. 0^6+2x3-80 = 13. x-5V^-14 = 14. 4x^+x«-39 = 397. Some expressions are in quadratic form with reference to a compound expression, such for example as, (x-\-2y-{x-\-2) = 12 and a:+3+2\/^+3-3 = These equations may be solved by factoring, the first one for {x-{-2) and the second one for \/^H~3. Exercise 174 Solve the following by factoring: 1. x-8-V^^ = 20 2. (x-2)2-3(a:-2) = 10 3. a;+6-2\/i+6 = 8 4. (0:2-5)2-4x2+20 = 77 6. x+4+(x+4)' = 20 6. (x2+8)2-5x2-40 = 84 398. Some equations may be put in the quadratic form by adding a number to both members. For example, x2-4x+\/^'-4x+12 = 8 may be put in quadratic form by adding 12, thus: x2-4x+12+\/^'-4x+12 = 20 This is in the quadratic form with reference to .r^— 4^+1 2. By factoring, ^ vx^ — 4a; + 12= —5 and 4 Squaring, .t- — 4a; + 1 2 = 25 and 1 6. andx2-4a:-4 = The last two equations are ordinary quadratic equations. QUADRATIC EQUATIONS Exercise 175 Solve the following equations in quadratic form : 1. 3x^+50:2-8 = 295 3. x^+5x^+6 = 6. V^-3\/^=21 7. 3^-5x--2^ = 9. a:-^-5a:-^+4 = 11. 2x-^-a;-^-45 = 13. (x-l)^ + (x-l)^ = 2 15. (x-5)2-x+5 = 110 2. x2-7a;-V^2_ 4. x^ — Qx— \/4x2 7x+l = 5 24x = 8 6. x2+V^2_5^_|_3^5^^3 8. x''-2x-\/9x'--lSx = 4: 10. (x2+3)2-5(a:2+3) = 14 12. r'-6x-3 = 2\/^^-6x 14. x^-5x+ \/4x^ - 20x = 48 16.-\/2x2+14x+2 = x2+7x-3 GRAPHICAL SOLUTION OF QUADRATICS 399. The graphical solution of quadratic equations makes the meaning of the roots, and the possibility of solutions, somewhat clearer. To solve graphically the equation x2-6x+8 = 0. First graph the function x^ — 6x+8 for the values: x=-l, 0, 1, 2, 3, 4, 5, 6, 7, etc., x2-6x+8=+15, +8, +3, 0, -1, 0, +3, +8, +15, etc.. Plotting these points and connecting them as in the figure we have the graph of .t^ — 6x+8. To ask for the values of x that give x^ — 6a:+8 = 0, is to ask what are the x-values of the crossing -points of the graph over the horizontal. Clearly these values are x=+2 and x=+4. The curve of the figure is called a parabola and any quadratic like x^-{-px-\-q always gives a 'parabola for its graph. — i: V — :::^:::i:;^i:: :i:i^:ii:^Ei:: :i::Vi|iii: ±::iffi:iii:'' Y' Scale 1 •■ 1 horizontal space 2 "1 vertical space Graph of rc2-6a;+8 296 ELEMENTARY ALGEBRA Graph of x'+ax + 12 for a =+8 (I) a = - S (2) a = -i-7 (3) a^-7 (4) Scale 2—1 vertical space 400. This figure gives the graphs of four quadratics ob- tained by keeping the constant term equal to + 12 and chang- ing the coefficient of the x-term only. The quadratics x^ + 7.r + 1 2 and ^2 — 7x+ 12 give the same shape of curve; either being turned over the vertical axis gives the other. The same is true of the graphs of x"-\-Sx -\-l2'dndx--Sx+l2. This may be expressed by saying that reversing the sign of the coefficient of x in the quadratic, turns the graph over around the vertical axis. The roots of such pairs of quadratics are numerically equal, but of opposite signs. Give the roots from the figure for quadratic equations made by putting each of the four quadratic trinomials equal to 0. All four of the graphs go through the point + 12 on the vertical. 401. This figure shows ^' the graphs of quadratics all of which have the constant term —12. Compare the graphs of the pairs: Y \ i R~r N / J t i\ I rrV" JLJiLA- 1 lit -T/fr p- "^trW"" i" "l^r "fi~ A jA V-ju _j y I'^Ih /y s \ <^^\ \w\^ / / / /-v |_ "^ 1 1 \ \1 \ 1/ Mf 7^ "$Il \ \ f ! //'^^ ^ / "^ ^ Y \ Y A /[/ / / 3 ? A Lri-'' t- \ W\ t/\ / >i' V \ \i\ y/iZ^ J. ^v-^^^^j: \ 42a_ 2 it " V.^f Vvf 1 x2+a:-12 Graph of x'-¥ aa> - 12 a;2-x-12, for o = +^ a = -4 a= +i x2+4a:-12 a= -; x2-4x-12. Scale 2 -= 1 vertical space QUADRATIC EQUATIONS 297 Reversing the sign of the x-term again is seen to turn the graphs over. Through what point do all these graphs go? Read from its graph the roots of — ^ a;2-4x-12 = x2-a;-12 = 402. The figure shows tliat the effect of changing only the constant term is to hold the curve of the same shape and to raise it by just as much as the constant term is increased. The curve crosses the horizontal in two points, giving two real roots, ^ until it just touches the horizontal. Then the two roots coalesce into one. As the curve rises, it ceases to touch the horizontal and the roots become imaginary, as the following algebraic solution of x^ — 2x4-5 = will show. Solving x^ — 2x+5 = 0, which is the same as — X' r n _i (^' A T % //! ^v\v )l jv^ \\iv yj// n]\V\ JjIL'^ 'i\i^\ tiff \''W Sj'j \\ \l\ 1 ih 1 ~^Wv~ -4ffi-r — WM W" ""■rVr ^Xxj \-k ^^ 1 4^ y ^ J r^ .■^ Graph of x- for 6= - 8, 6= -5. 6= + 5. - 2x 4- 6 curve (I) curve {2) curve (S) curve d) Scale 2=1 vertical space x2-2a:+l=-4 (a:-l)2=-4 x—l= =*=\/^, or =t2-\/^, we obtain, x=l=i=2^/— 1 These roots are imaginary, since they contain \/ — 1. Thus, failure of the graph either to cut or to touch the horizontal indicates that imaginary roots are present. When the graph touches, or cuts, the horizontal how could the factors of the first number of its equation be read from the graph? 298 ELEMENTARY ALGEBRA Exercise 176 Solve the following quadratic equations graphically : 1. a:2-3x~10 = 2, a;2+3a;-10 = 3. x2-5x-6 = 4. x2+a:-20 = 6. x2-x-20 = 6. x2+5x = CHARACTER OF THE ROOTS OF QUADRATIC EQUATIONS 403. The character of the roots of any complete quadratic equation is determined by examining the solutions of: ax2+bx+c = In this discussion it is assumed that a, h, and c are real numbers, a is greater than zero, and b and c are either posi- tive or negative. Denoting the roots by n and ro, we have the values: — b + \/b2— 4ac — b — \/b2 — 4ac 2a 2a The nature of the two roots, as real or imaginary, rational or irrational, depends on the value of 6^ — 4ac. The expression h^—Aac is called the discriminant of the roots. 404. Observing the formulas for n and r2, it is evident that: 1. When the discriminant is a square the roots are real, rational, and unequal. 2. When the discriminant is equal to zero the roots are real, rational, and equal. 3. When the discriminant is a positive number not a square the roots are real and conjugate surds, 4. When the discriminant is a negative number the roots are conjugate complex numbers. A complex number is a number of the form a-\-b\^ — 1, a and b denoting real numbers. The numbers a+^v — 1 and a— 6V — 1, are conjugate complex numbers. QUADRATIC EQUATIONS 299 405. It follows that we can determine the nature of the roots of any quadratic equation without solving it. For example: 3x2-7x-h2 = In this equation 6^— 4ac = 25. Since 25 is a square, the roots are real, rational, and unequal. But take the equation In this equation b^—4ac= —24. Since —24 is a negative number, the roots are conjugate complex numbers. Exercise 177 By the use of the discriminant determine the nature of the roots of each of the following equations : 1. 4a:2-7x+3 = 2. a:2-7a;-8 = 3. 2x2-4a^+2 = 4. x^+6x-\-^ = 6. 5x2-f-8x-2 = 6. x2-3x-h5 = 7. 7x2-5x+l=0 8. x^-\-Zx+5 = 9. 4x2-4x+l=0 10. x^-dx-9 = 11. 4a;2+6x-4 = 12. a:2-5a:+8 = 13. For what values of n will 2x^-\-nx-\-S = have equal roots? Irrational roots? 14. For what value of a will ax' — l2x+Q = have equal roots? Imaginary roots? 15. For what values of n will Sx^-{-2nx-\-S = have equal roots? Imaginary roots? 16. For what values of c will Sa;^ — 10xH-c = have equal roots? Real roots? Imaginary roots? 17. For what values of n will 9x'^-{-nx-{-x-\-l=0 have equal roots? Find the corresponding values of x. 300 ELEMENTARY ALGEBRA 406. By dividing both members of the general quadratic equation, ax^-\-bx-\-c = by the coefficient of x^, the equation becomes of the form: x2-f2px+q = in which p and q are positive or negative," integral or frac- tional, and 2p is any coefficient of x. The solutions of this equation are, by § 394 or § 403. ri= — p+Vp^ — q r2=-p-\/p'-q The sum of the two roots of JC--|-2^x+^ = is — 2p, the coefficient of x with reversed sign. The product of the two roots of X"+2px-\-q = is q, the constant term of the equxition. 407. The two foregoing principles enable us to form quadratic equations with given roots. If the roots of a quadratic equation are —9 and 5, the coefficient of x is 4, and the constant term is —45. The equation then is x2+4a;-45 = It has already been proved §§ 215, 384-5, that if (xH-9) (x — 5)=0, the roots are —9 and 5. Observe that the known numbers in (xH-9)(a; — 5) =0^, are the roots of the equation with their signs reversed. TO FORM A QUADRATIC EQUATION WITH GIVEN ROOTS 408. Rule. — Subtract each of the roots from x and place the product of the two remainders equal to zero. The equation whose roots are 6 and —7 is (a;-6)(x+7)=0, or a;2-f-x-42 = 0. QUADRATIC EQUATIONS > 301 Exercise 178 Give at sight the equations whose roots are : 1. 5 and 3 2. 2 and —5 3. —5 and —4 4. 7 and 2 5. Sand -3 6. -3 and -8 7. 6 and 5 8. 3 and -7 9. -7 and -5 10. 8 and 9 11. 9 and -4 12. -4 and -7 13. f andf 14. I and -| 15. -| and -| 16. a-\-n and a—n 17. —a-\-\/7iand—a — \/n 18. a — 2 and a+2 19. -2+\/3and -2-\/3. 20. 2a+l and 2a-: L 21. -3-\/5and -3-f-A/5 22. 3-2?:and3-h2i 23. -H-\/7and -1-V7 24. What is the sum of the roots of x^^- a; - 6 = 0? What is the sum of the roots of 2a;^+12a:-f 1 =0? 26. If a in the general quadratic equation is 5, what part of b is the sum of the roots? 26. For what value of c will 4^2— 16x4- c = have equal roots? Conjugate surd roots? Imaginary roots? 27. For what value of m will 3x2 — mx— 48 = h^ve equal roots? Conjugate surd roots? Imaginary roots? FACTORING BY PRINCIPLES OF QUADRATICS 409. The method of factoring quadratic trinomials, whose factors are rational, has already been explained. (See p. 163.) By the principles of quadratics, quadratic expressions whose factors are irrational may be factored. For example, To factor x2-8x+ 11. We place: x'-Sx + U^O By § 390, x = Aj-V5 and A-Vl. Hence, by § 408 {x-4-V5)(x-i-\-V5) =-x^-Sx-\-n._ The factors of a:- — 8x + 1 1 are a- — 4 — V 5 and x — 4 + v 5. 302 ELEMENTARY ALGEBRA Exercise 179 Factor the following: 1. a2-4a+l 4. a2+6a-3 7. a2-2a+4 10. a-+5a— 1 2. x^-Hx-2 5. a;2+4a;-4 8. a:2+8a;-8 11. x'^-Sx+l 3. n2-6n+ll 6. n2-6n+13 12. /i2+9n+23 PROBLEMS IN QUADRATIC EQUATIONS 410. Since quadratic equations have two roots, a problem whose solution involves such an equation apparently has two values of the unknown number, or two roots. Both roots may satisfy the equation, but only one of them may satisfy the conditions of the problem. Especially is this true when the roots are surds or imaginary. In solving problems that involve quadratics, we should examine the roots of the equation and reject any root that does not satisfy the requirements of the problem. Exercise 180 — Problems in Quadratics Solve the following problems: 1. The sum of two numbers is 42, and their product is 416. Find the two numbers. 2. The sum of the squares of three consecutive numbers is 590. Find the three numbers. 3. A rectangular field of 4 acres is 12 rods longer than it is wide. What are the dimensions? 4. The quotient of one number divided by another is 7, and their product is 2800. Find the numbers. 5. If the sum of the squares of three consecutive even numbers is 980, what are the numbers? QUADRATIC EQUATIONS 303 6. What is the price of eggs per dozen when 5 less for 50^ increases the price 6^ a dozen? 7. Find two consecutive odd numbers the sum of whose squares exceeds 20 times the larger number by 94. 8. The perimeter of a rectangular field is 84 rods, and the area is 432 square rods. Find the dimen'sions. 9. The sum of two numbers is 24, and their product is 139. Find the numbers and prove your answer. 10. The difference between two numbers is 16, and their product is 1380. Find the numbers. 11. The perimeter of a rectangular field is 114 rods, and the area is 5 acres. Find the dimensions. 12. Solve the formula d = ^gf for t and g. 13. The sum of two even numbers is 48, and the sum of their squares is 1224. Find the numbers. 14. The sum of two numbers is 40, and their product is 398 J. Find the numbers. Prove your answer. 16. The sum of two numbers is 96, and their product is 18 times as much. Find the numbers. 16. Solve the formula a^ = ¥+c'^ for b and c. 17. The hypotenuse of a right triangle is 9 feet longer than one leg and 2 feet longer than the other leg. Find the three sides of the triangle. 18. At 15^ a square foot, it cost $99 to lay a parquet floor in a room whose length is 8 feet more than its width. Find the dimensions of the floor. 19. The dimensions of a certain rectangle and its diagonal are represented by three consecutive even numbers. What are the dimensions of the rectangle? 304 ELEMENTARY ALGEBRA 20. A carpenter worked 30 days more than lie received dollars per day for his labor and earned $175. How many days did he work and how much did he receive per day? 21. Two numbers differ by 1. The square of their sum exceeds the sum of their squares by 220. Find the numbers. 22. There are 32 sq. yd. in a rectangle whose length is 18 times the width. Find the length in feet. 23. Find two numbers whose difference is 6, and whose sum multiplied by the smaller number is 756. 24. Find the side of a square whose area is doubled by increasing its length 9 yd. and its width 6 yd. 26. One square field is 10 rd. longer than another, and the area of both is 1 108 sq. rd. Find the length of each. 26. Find the numbers the sum of whose two digits is 13 and the sum of the squares of whose digits is 89. 27. The number of square inches in the surface of a cube exceeds the number of inches in the sum of its edges by 1170. Find the volume of the cube, 28. A man bought a piece of land for $4050. He sold it at $53 an acre, making a profit equal to the cost of 16 acres. How many acres did he buy? 29. A merchant sold some damaged goods for $24 and lost a per cent equal to the number of dollars he paid for the goods. Find the cost of the goods. 30. The length of a rectangle exceeds its width by 7 rd. If the dimensions were increased 5 rd., it would contain 5 acres. Find the dimensions of the rectangle. 31. A merchant bought lace for $100. He kept 30 yards and sold the remainder for as much as it all cost, gaining 75<^ a yard. How many yards did he buy? CHAPTER XXIV SIMULTANEOUS SYSTEMS SOLVED BY QUADRATICS 411. A quadratic equation in two variables (unknowns) is an equation of the second degree in the variables (un- knowns). Thus, X and y denoting variables, 3a:'--2i/2+x-4?/+3 = 0, x+Si/-y = 9, and xy-lQ = x-y, arc quadratic equations in two variables. 412. Two or more such equations in the same variables are called a system of quadratic equations. If all equations of the system can be satisfied by the same pair, or pairs, of val- ues of the variables, it is called a simultaneous system. Not all simultaneous systems of quadratic equations can be solved by elementary algebra. In fact the solution generally leads to biquadratic, or fourth degree equations. We shall consider here only systems containing one quad- I'atic and one linear equation. 413. Let us now examine the meaning of the solutions of such equations, beginning with the system, y = 4:X — x~—l (1) y = 2x-l (2) Equation (1) is the same sls x'^—4x-\-y-\-l=0. Calculating from equation (1), the ^/-values for the x- values of the first line, x=-l,-^, 0, fi,+l,+2, +3, -|-3i+4, -h4i,+5,etc., y= -6, -3i, -1, +i +2, -f 3, +2, +f , -1, -3^ -6, etc., plotting the number-pairs and connecting the points, gives the curT^e in the figure. 305 306 ELEMENTARY ALGEBRA ~JI ^ ^ffyf\ r yaw \ q »5=l=^ fi---#i Y' Scale 1 horizontal t>pace 1 vertical e^pacu On the same reference lines, graphing (2) for 'a:=+2, -2 i/=H-3, -5 x' \ I rj^K/?."\| I ' \ x gives the straight hne marked y = 2x—l in the figure. The solutions sought are the x- and ^/'distances of the crossing-points of the graphs of (1) and (2). The X- and y-values must he so paired that both numbers of each pair belong to the same crossing-point. The solutions are : x = 0, y= —1, and x = + 2, y=-\-S. 414. The graph of (1) is a parabola and any two-letter equation of the second degree with only one variable raised to the second power and without an xy-term, gives a parabola for its graph. 415. Suppose a line to start from the position marked y = 2x—l, moving across the parabola parallel to the starting position to the line y = 2x. In every position there would be two crossing-points until the position y = 2x is reached. At this position the two crossing-points blend into one, the line becoming tangent to the parabola. Beyond the position y — 2x there would be no crossing- point of the line and the parabola. Starting from the line y = 2x—l and moving parallel to itself toward the right, there would always be two crossing- points. Recalling that every crossing-point gives a value of x and of y, we observe that: /. There are in general two solutions of a system made up of a parabolic and a linear equation. II. When the line is tangent to the parabola there is but one solution, or since the tioo crossing-points coalesce, we may say two equal solutions. SIMULTANEOUS SYSTEMS 307 III. For an equation representing a Ime beyond the tangent position there is no real solution. Algebra shows that there are two solutions eve7i here, hut that they are imaginary. 416. The solution just given is the graphical solution of the system. We now give the algebraic solution of the same system. Writing the equations thus : x^-4x-\-y+l=0 (1) y = 2x-l (2) ?ubstitute the value of y from (2) in (1), simplify, and find: Whence, x = 0, and +2 Substituting these values of x in (2) , we find : y= —1, and -h3 The solutions are the number pairs : x = 0, x=-\-2, and ?/=-l,2/ = +3 These values agree with those of the graphical solution. Exercise 181 Solve the following systems algebraically: fx^-\-Sx-y = lS /2.T2-6x-f2/ = 8 \ y — 2 = 2x ' \ y — 4x=—4: y'--2y-\-x = 5 fy^-oy-\-3x = Q x-2y = S ' \ 2y-^x = 4 /a;2-?/ = 5 fSx^-9x-y = 2 \Zx~y=-5 \ 3x-y = 2 308 ELEMENTARY ALGEBRA 417. Solve next the system x^-\-i/ = 25 i/ = =fc \/25-xH]) or U^+y'- = 25, or ?/ = ± V^S - x^ (1) .(^ x-y=l,ory = x-l (2) Graphing (1) y= =t \/25 — x'^ using a;= +6, +5, +4, +3, +2, +1, , -1 , -2, -3, -4, -5, -6, etc., and calculating y from ?/ = ± -\/25 — x*. find y=imag. 0, =±=3, ±4, ±4.0, ±4.9, ±5, ±4.9, ± 4.6, ±4, ±3, 0, imog., etc. Graphing these pairs, laying off the values with double sign both upward and downward, obtain the circle of the figure. Graphing now the line y = x—l, obtain the straight Hne of the figure. The crossing-points give the following solutions : x=+4, x=-3, t/=+3, y=-4. This is the graphical solution. Suppose a line should start from the position x — y = l and move upward across the circle, keeping parallel to x — y = \j through the positions x — y = 0,x — y=—Z,iox — y=—iS \/2, or downward through the position, x — y = ^ to x — y = b\^2. In every position the line gives two crossing-points with the circle, until the tangent positions are reached, where the two crossing-points become one point of contact. For a line beyond the tangent positions the system would give two imaginary solutions. For the tangent positions of the line we might again say there are two equal solutions. For the upper tangent-point x= —■^\/2, y=-j-^\/2 and for the lower tangont-point, x= -\-^\^2, y= — ■§-\/2. T l¥^ z / X M^ ^^z zz r^^' yK. // -j^ 7 -.^t ^ *^1Z ,7 LjZ o T7 7 / '^^ \y / z zz * '^jA- ^zv:z^_. ^-¥^ 7 M/ 7i\2'^ /TH^y Qf^ zz z uz zz z VL y Scale =s horizontal ^pace ^ vertical space SIMULTANEOUS SYSTEMS 309 418. The algebraic solution consists in substituting the value of y from (2) in (1), obtaining: Or, or, Whence, x = J±-|-= +4, or -3, Substituting these values of x in (2) then gives: 2/= +3, or -4. These solutions agree with those of the graphical metho'd. Exercise 182 Solve the following systems algebraically : 4. 7. 2. 5. 8. a;2 4-|/2^58 x — 4iy= —5 a;2+i/2 = 29 2a;-6?/ = 2 9a:-2/ = 2 3. 6. 9. 6a;-5^=-26 r^2_^^2=74 \2a:+y = 19 x2+7/2 = 29 a;2+7/2 = 52 3a;-4^j = 2 419. Solve the system : r 4a;-5i/ = 20 (1), or j/ = |(x -5) ^ \l6x2+25i/2 = 400 (2), ori/ = f\/25-x2. Calculate the ^/-values for (2) from these aj-values: a:=+6, +5, +4, +3, +2, +1, 0, -1, -2, -3, -4,-5, ij=imag. 0, ±2.4,±3.2, Y 6, 3.6, .X ij ^ r^ ,^ ^ V I f ^ V \ / \ ,, ] / A \ &Y •- \ \d > 'U ^ ■>~< \^ k rvn-\ - Y' Scal6 1 ™ 1 horizontal space 1 = 1 vertical dpace 3.9, ±4, ±3.9, ±3.6, =^3.2, ±2.4, 0,imag. and graph (1) using x=+S, 2/=— 1.6, and x = 0, ^=—4, obtaining the figure here. The number-pairs of the crossing-points are x=+5, and 0, and i/ = and —4, which are the solutions. Show from the figure that — x=-f5 goes with ^ = 0, and x = with y=—4. 310 ELEMENTARY ALGEBRA 420. The algebraic solution is obtained by substituting the value of y from (1) in (2), obtaining 16x2+25[t(x-5)P = 400 Reducing, 32^2 - 160a:+400 = 400 Or, a:^ — 5x = Whence, x = 0, and +5, and from (1) i/= — 4, and 0. The graph shows that the 0-value of x must be paired with the — 4-value of y, and that the +5 and also belong to- gether. The graph for the equation 16x^+251/2 = 400, is an ellipse. Moving this line 4a: — 5i/ = 20 parallel to itself across the ellipse shows there are always two crossing-points, and hence two pairs of values of x and y, save for the tangent positions, where there would be only one pair or, as we prefer to say, two equal pairs. An algebraic solution would show that when the line does not touch the ellipse there would be two imaginary values of X and y. 421. A quadratic equation with no xy-term. but containing the square-terms of both variables, the coefficients of these terms being of the same sign, gives a graph that is 'an ellipse. Exercise 183 Solve the following systems algebraically: f 2x-Zy = ( x-3y = 2 ( Sx-5y = S \4a;2+92/2 = 36 \4x^+9y'' = SQ \x'+25y^ = 25 ( 7x-4y = 10 (5x-.3y = S (l0x-Sy = 5 , \x^+lQy^ = m \9x2+y2 = 9 \49a;2+^2 = 49 422. Solve the system: -1/2=16 ' (1), or2/==*=V^^^^ -2/ = 2 {2),ory = x-2 SIMULTANEOUS SYSTEMS 311 In equation (1) for all values of x between —4 and +4 the values of y are imaginary. Calculate y for the given .re- values, find: x=+10, +8, +5, +4, -4, -5, -8, -10, etc. ^=±9.2, ±6.9, ±3, 0, 0, ±3, ±6.9, =^9.2, etc. Plotting these points, drawing the graph, and graphing equation (2) for x = 0, y=—2, and x= —4, y=—Q, obtain the picture of the figure shown. The graph of equation (1) is a hyperbola. It has two disconnected parts, or branches. There is but one crossing- 1 9v .-i- ^ .^^ s .^^ Nk ^^ Y " 5c o X^' J n -^f' ^ y^ "v* ^v ^ >v ^^ '^ \^ ^ ^q^ s ifc -=^ ^ ^^ ± Scale ] := 1 horizontal space 2=1 vertical space point of the line and curve. The figure shows why. The graph shows the x- and ^/-values for this crossing-point to be x= -}-5, and y= +3. 423. The algebraic solution gives by substituting the value of 2/ from (2) in (1) x^-{x-2Y=\^ Reducing, we find, 4a: = +20, or, x=+5. This value of x, substituted in equation (2), gives 2/=+3. These values of x and y agree with the graphical solution. 312 ELEMENTARY ALGEBRA 424. A quadratic equation having both x-- and y^-terms with opposite signs, no xy-term being present, always gives a hyperbola for its graph. Could the straight line be turned around so that it would cut both branches of the hyperbola? How many values of x and of y would thece be? Exercise 184 Solve the following systems algebraically: (x^-y^ = 7 (x^-y^ = lS (x'~-y^ = ^5 ' \ x-y=l * \ x-y=l ' \ x-y = 5 *• ^ x-y = 3 \ x-y = 5 \x-3y = Q 425. Solve the system : xy = 12, or y = ^ (1) 12 X y-x = l, or t/ = x+l (2) In equation (1) calculate y for the following assumed values of x, a;=+12,-f6,+4,-f3,+2, +1,- 1,-2,-3,-4,-6,-12 y= +l,+2,+3,+4,+6,+12, -12, -6,-4,-3,-2,, -1 Plot these points, and draw the graph, obtaining a curve for xy = 12. Show both branches of the curve. Both branches together are spoken of as a single curve, the hyperbola. Graphing equation (2) on the same axes, using the follow- ing points, a:=+3, 0, -1, -4 y=+4, +1, 0, -3, the straight line graph for y = x-\-l is obtained. SIMULTANEOUS SYSTEMS 313 426. The roots are the x- and i/-values of the crossing- points of the two graphs, viz. : x=+3 and (x= —4 y=+4 12/= -3 Check: Substitute the number-pair (+3, +4) in equations (1) and (2) thus, in(l), +4 = 4^ and in (2), +4= +3 + 1. + 3 Then, substitute the other number-pair ( — 4, —3) in (1) and (2) thus, in (1), -3 = — . and in (2), -3= -4 + 1. — 4 Hence the pairs (+3, +4) and ( — 4, —3) are the root- pairs of the given system. 427. Equations Uke those of the system of § 425 have a hyperbola and a straight-line for graphs. The solutions, or roots, of the system are the x- and ^/-values of all the crossing- points. Such graphs in general have two crossing-points, and hence, two a;-values and two ^/-values, and these x- and i/-values must be so paired that the two numbers of a pair shall belong to the same crossing-point of the graphs. 428. The algebraic solution of the system of § 425, xy = l2 (1) _y-x = l (2) is as follows: From (2) we have y^x-1 (3) Substitute — (3) in (1), obtaining x(x-l) = 12, or a;2-x = 12 Whence, x=+3, or —4 Substituting these values of x in (1), we obtain x=+4, or -3 These are the values given by the graphical solution of §425. 314 ELEMENTARY ALGEBRA Exercise 185 xy = 12 2 f xy = 36 ^ ( xy = 20 y-x = 4: '\x—y=—5 '\x-4:y = 2 3xy = 21 g I 5xy = 150 ^ ( 7xy = 9S x — 8y=—l ' \x-y=—l ' \x-5y=-3 429. The main use of the graphical solution of equations to pupils is to enable them to see the meaning of solutions, and to understand why roots are paired in a certain way. For practical work of solving equations the algebraic solution, as given in §§ 416, 418, 420, 423 and 428, should always be used. In the exercises that follow the algebraic method is to be employed. Exercise 186 Solve the following systems and pair the roots properly 3. X -y = 3 ' [ x-i-y = 10 y-x^-\-x=l ^ jx^+y^ = 20 x = y—4: . [ x-\-y = Q x2+i/2 = 26 ^ fx^+y^ = 7S 6. x-y = 6 [y-2x = lS 3x2+81/2=147 ^ /x2+2i/2 = 89 8> x — y = 2 [ a:+^ = ll (5x'-\-y' = 4:5 I xy = 10 • \ x+2y = 12 ^^' \y-x = S 11. < ^^=^^ 12 ^ ^'^^^^ x-\-y = 9 [x — y = 2 f2?/ = 10 ^^ (Sx^-y = 7 x—y = 7 ' \ y — 5x = 5 SIMULTANEOUS SYSTEMS 315 16. < ^'=12-2/ 16./"+" = ^ ^^ 'm-n = S .- /a;2+a;?/+i/2 = 61 mn=18 \ x+7/ = 9 a+c = 14 \ y = ll—x 21. < „ 22. ?yi-n = 3 [ 3i/-2a;=l fm?-\-n?+mn = S9 (xy+y^ = 40 ^^' \ m-n = 3 , \i/-3a:=-4 a2-3i/=l3 \ mx = 85 ^ V4-62-a-6=18 „„ /a^2_^23=65 27. < , , r 28. a+6=-5 ]t/-2a:=-14 2^ a6+a2 = 40 , 30. J ^^"^—^ _ 15 ~ 4 6_3a=-4 *"" \ x-i/ = f ;c2-5d2 = 76 ,„ f2m-3n = 9 31. < . . , ^^ 32. 4c-5rf = 29 \ mn = Q gg (Sxy-hx'-2y'=^52 ^^ j 3m-2n = 2S \ 2x+Sy = SQ ' \m''-27nn = 45 cd = 57 ' ' \ y-z=lQ 37. ;^^-^^ = 16 33^ / xy = 4 n+2m=13 {x-y = ;4m2-9n2=19 ,^ /p2+4^ = 76 39. < „ . ^ .^ 40. ^"^ ^ 3n+2m=19 ' \Sp-q = 21 7nn = 30 ' \ a;-3t/=-5 ,m2+n2-m-n = 50 ^„ j3a;2-2/2 = 275 41. N «,x 42. 316 ELEMENTARY ALGEBRA 430. special Methods. Some systems may be conven- iently solved by special methods as well as by substitution. 431. One of these special methods is to divide the given equations, member by member, obtaining a derived equation which, with one of the given equations, furnishes a system of equations equivalent to the given system, and then to solve the derived system. (a) Observe carefully the following solution of the system : '2-1/2 = 33 (1) x-i-y = n (2) Dividing (1) by (2), x-y^S (3) The system consisting of" (2) and (3) is simpler than the given system and the simpler system gives x = 7 and t/ = 4. These are all the roots, for (1) represents a hyperbola and (2) a straight line, and they cross in only one point. (b) Solve the system : /36m?-p2 = 819 (1) \ 67n-p=-39 (2) Dividing (1) by (2), Q7n-{-p=-21 (3) The system (2) and (3) is equivalent to the given system and its roots are : m= — 5 and p= +9 Exercise 187 Solve the following systems, first dividing when possible and pairing results properly: '93.2 _ 4^2 ^ 308 /m2 - n2 = 64 ^' ' Sx-2y = 14: " 1 m+n = 16 3. < / 4. < 3p — = 8 Q m m -1~I2 = ^^ or If m m — -=n a SIMULTANEOUS SYSTEMS 317 5. 7. 3a;2=16-25?/2 7?i = f -|-9p 6. < 8. < 1 1 ^ ?" --2 = 5 1 1 -H- - = 5 .^ 2/ f ^ 32 = 6 Ri' i22^ 1 2 ■+-B- = 1.5 .^1 /^2 432. Another special method, or device, is to form systems equivalent to the given system by so combining the given equations as to obtain squares in both members and then to derive simpler systems by extracting the square roots of both members. (a) Observe carefully the following solution of the system : x2+?/2+a:2/ = 52 (1) x+y = S- (2) Subtracting the first equation from the square of the second, xy = 12 (3) The system made up of (2) and (3) is equivalent to the system (1) and (2). The system (2) and (3), solved as in § 428, gives x = 6 and 2 and y = 2 and 6. (6) Solve the system: x'^-{-xy=10 2/2-hxi/=15 Adding the equations we obtain : {x-\-yy = 25, or x-\-y= ±5 Subtracting (1) from (2), y^ — x'^ = 5 Equation (3) is really two equations, viz. : x-\-y = 5, Siud x — y= —5. (1) (2) (3) (4) 318 ELEMENTARY ALGEBRA The given system is then equivalent to the two systems: ^'-y'-=-^ and J^'-f=-5 x+y=-{-5 { x-]-y= -5 Dividing the first equations by the second, obtain: x — y=—l and x — y=-\-l Combining these with the second equations of the derived system, we have: x = 2 and y = S, and x= —2 and y=—d (c) Solve the system: x2+2/2 = 40 (1) xy = l2 (2) Multiplying (2) by 2 and adding to (1), obtain: x2+2x?/+2/2 = 64 Or, x+y==^S (3) Subtracting 2xy = 24: from (1) x^-2xy-{-y^=m, Or, x-y= ±4 (4) Now from (3) and (4) we form the four systems which are together equivalent to the given system, viz. : J x+y=+S jj (x+y=-\-S x — y=-}-4: [x — y=—4: III. ^+^=7^ IV. l^+y-'l . [x-y=-]r4: [x — y=—4: System I gives x = Q, y = 2, II, gives x = 2, y = Q, III, gives x= —2, y= —6, and IV gives x= —6, y= —2. Hence, the solutions of the given system are : x=+6, +2, -2, and -6, y=-\-2, +6, -6, and -2. The system (1) and (2) are both quadratic equations, so that this problem lies a little beyond the limits set for this book. But the method in most of its parts is so like that for systems made up of one quadratic SIMULTANEOUS SYSTEMS 319 and one linear as to bring it within the pupil's comprehension. The reason there are so many solutions lies in the fact that the graph of (1) is a circle and of (2) a hyperbola, since a circle and a hyperbola, in general, cross each other in four points. 433. In the following list of exercises we shall include a few systems in two quadratics of the type of the last. Exercise 188 Solve the following systems of equations: Qxy=lS ' \ rs= 12 ^' \ Smn=m ^ *• I 11 = 3 x''-xy = 22 (x''+4:xy+3Qif = 224 ^xy-y^=lS ■ \ I2xy = m Exercise 189 1. The sum of two numbers is 7 (or a), and the sum of their squares is 21 (or b) . What are the numbers ? 2. Find two numbers the difference of whose squares is 33 (or m), and the product of whose squares is 784 (or n). 3. The combined area of two square fields is 8| acres, and the sum of their perimeters is 200 rods. What is the area of each field ? 4. The sum of the squares of two numbers is 91 (or p),'and the difference of the numbers is 5 (or q). Find the numbers. 5. The difference of two numbers is 28, and half their product is equal to the cube of the smaller number. What are the numbers ? 320 ELEMENTARY ALGEBRA 6. The area of the ceiUng of a hall is 700 square feet, and its length is six feet less than four times the width. Find the dimensions. 7. The sum of two numbers is 13 (or s), and their product is 210 (or p). Find the numbers. 8. If the dimensions of a rectangle were each increased 1 foot, the area would be 99 square feet; if they were each diminished 1 foot, the area would be 63 square feet. What are the dimensions? 9. A number is expressed by two figures the sum of which is 14, and the sum of the squares of the digits exceeds the number by 11. Find the number. 10. The combined area of two adjoining square fields is 900 square rods, and it requires 150 rods of fence to inclose them. If they are so situated as to require the least amount of fence, what is the dimension of each ? 11. The area of a rectangle is 192 square inches, and its diagonal is 20 inches. Find the dimensions. 12. A rectangular field contains 270 square rods. If it were two rods longer and one rod wider, it would contain 50 square rods more. Find the dimensions of the field. 13. A farmer bought 12 sheep and 4 calves for $60. At the same prices, he could buy 3 more sheep for $24 than calves for $30. Find the price of each. 14. The perimeter of a rectangular piece of ground is 200 rods, and its area is 15 acres. Find the dimensions of the field. 15. The hypotenuse of a right triangle is 30 feet, and its area is 216 square feet. Find the length of the other two sides. SIMULTANEOUS SYSTEMS 321 16. The sum of the squares of two numbers is 74, and the difference of their squares is 24. What are the numbers ? 17. A merchant bought two kinds of silk, paying $63 for each piece, and buying 8 yards more of one kind than the other. The difference in price was 50 cents a yard. How many yards of each kind did he buy ? 18. A rectangular piece of paper contains 1350 square inches; but if the dimensions were each 5 inches less, it would contain 1000 square inches. Find the dimensions. 19. If the sum of two numbers is added to their product, the result is 31; and the sum of their squares exceeds their sum by 48. What are the numbers ? 20. A man bought sheep for $136. He kept 22 of them, and sold the remainder at a profit of $1 a head, receiving for them $2 more than they all cost. At what price per head did he buy them ? 21. The square of the sum of two numbers exceeds 6 times the sum of the numbers by 16. The difference of the num- bers is 2. Find the numbers. 22. The opposite sides of a parallelogram are equal. One pair of opposite sides of the parallelogram are denoted by m^—mn and 19 — n^, and the other pair are denoted by 7n and n+3. Find m and n and the length of the sides. SUMMARY OF DEFINITIONS FOR REFERENCE AND REVIEW (Definitions without page numbers are on page last indicated.) CHAPTER I The factors of a number are its makers by multiplication. (Page 8.) An equation is an expression of equality between two equal numbers. (Page 11.) The value of any letter in a number expression is the number or numbers it represents. (Page 12.) An unknown number is a letter whose value in an equation is to be found. (Page 13.) Solving an equation is finding the value of the unknown number, or numbers in it. An axiom is a statement so evidently true that it may be accepted without proof. In problem-solving the notation is the representation in algebraic symbols of the unknown numbers of the problem. (Page 15.) The statement is the expression of the conditions of the problem in one or more equations. CHAPTER II Directed numbers or signed numbers are numbers whose units are positive or negative. (Page 21.) The absolute value of a number is the number of units in it, regard- less of sign. (Page 22.) The + and — signs may denote either operations or opposing qual- ities of numbers. (Page 23.) Algebraic notation is a method of expressing . numbers by figures and letters. (Page 24.) An algebraic expression is the representation of any number in algebraic notation. A term is a number expression whose parts are not separated by the + or — sign. A monomial is an expression of one term. (Page 25.) A polynomial is an expression of two or more terms. 322 SUMMARY OF DEFINITIONS 323 A binomial is a polynomial of two terms. A trinomial is a polynomial of three terms. A coefficient of a term is any factor of the term which shows how many times the other factor is taken as an addend. Similar terms are terms which do not differ, or which differ only in their numerical factors. Dissimilar terms are terms that are not similar. Partly similar terms are terms that have a common factor. The value of an algebraic expression is the number it represents when some particular value is assigned to each letter in the expression. (Page 26.) CHAPTER III Addition is the process of uniting two or more numbers into one number. (Page 27.) The addends are the numbers to be added. The sum is the number obtained by addition. The fundamental laws of addition are the law of order, (the com- mutative law), and the law of grouping (the associative law). (Page 29.) The law of order states that numbers may be added in any order. The law of grouping states that addends may be grouped in any way. CHAPTER IV Subtraction is the process of finding one of two numbers when their sum and the other number are known. (Page 35.) The minuend is the number that represents the sum. The subtrahend is the given addend. The difference or remainder is the number which added to the sub- trahend gives the minuend. The symbols of aggregation are the parenthesis ( ), the brace [ } , the bracket [ ], and the vinculum ~". (Page 42.) CHAPTER V Algebraic number and function have the same meaning. (Page 50.) The independent number is the number on which the function depends. 324 ELEMENTARY ALGEBRA A function is a number that depends on some other number for its value. (Page 5L) An algebraic function is a number whose dependence on another number is expressed in algebraic symbols. CHAPTER VI Equations are of two kinds, identities and conditional equations. (Page 60.) An identity is an equation with like members, or members which may be reduced to the same form. Substitution is putting a number symbol into a number expression in place of another which has the same value. An equation is satisfied by any number which, when substituted for the unknown number, reduces the equation to an identity. A conditional equation is an equation that can be satisfied by only one or by a definite number of values of the letters in it. (Page 6L) A root of an equation is any value of the unknown number that satisfies the equation. Transposition is the process of changing a term from one member of an equation to the other, by adding or subtracting the same number in both members. (Page 62.) CHAPTER VII Graphing means representing number-pairs, related sets of numbers, and number laws by pictures and diagrams. (Page 74.) A linear equation is an equation in two unknowns both with exponent 1. (Page 8L) The graphical solution of two linear equations is the point of inter- section of the graphs of the equations. (Page 82.) Simultaneous equations are equations that can be satisfied by the same values of x and y. A system of equations is two or more equations considered together. (Pages 82 and 86.) Non-simultaneous or inconsistent equations are equations which cannot be satisfied by any values of the unknowns. (Page 83.) Dependent equations are equations in which one or more can be derived from another or others by some simple arithmetical operation. (Page 84.) SUMMARY OF DEFINITIONS 325 CHAPTER VIII A determinate equation is an equation which has one root, or a limited number of roots. (Page 85.) An indeterminate equation is an equation which has an unlimited number of roots. Independent equations are equations which cannot be derived one from another by a simple arithmetical operation. (Page 86.) A set of roots of a system of equations means the values of the un- known numbers of the system. Elimination is a process of deriving a single equation in one unknown from a system of two or more simultaneous equations in two or more unknowns. CHAPTER IX Multiplication is the process of taking one number as an addend a certain number of times. (Page 91.) The multiplicand is the number taken as an addend. The multiplier is the number denoting how many times the multi- plicand is taken. The product is the result of the multiphc.ation. A negative multiplier means that the product is of the opposite quality from what it would be if the multiplier were positive. An exponent is a symbol of number written at the right and a httle above another symbol of number to show how many times the latter is taken as a factor. (Page 92.) The three fundamental laws of multiplication are the law of order (commutative law), the law of grouping (associative law), and the distributive law. (Page 94.) The law of order is: The prod^ict of several numbers is the same in whatever order they are u^ed. The law of grouping is: The product of several numbers is the same in whatever manner they are grouped. The distributive law is: The product of a polynomial and a monomial is the algebraic su7n of the products obtained by multiplying each term of the polynomial by the monomial. (Page 95.) A power is the product obtained by taking a number any number of times as a factor. 326 ELEMENTARY ALGEBRA A polynomial is arranged when the exponents of some letter increase or decrease with each succeeding term. (Page 97.) CHAPTER X The degree of a term is indicated by the sum of the exponents of the literal factors. (Page 100.) The degree of an equation in one unknown is the degree of the highest power of the imknown number. A simple equation, or linear equation, is an equation which, when cleared and simpUfied, is of the first degree. Checking or verifying a root of an equation is the process of proving that the root satisfies the equation. CHAPTER XI Division is the process of finding one of two numbers when their product and the other number are known. (Page 107.) The dividend is the number to be divided and represents the product of the two numbers. The divisor is the number by which we divide and represents one factor of the dividend. The quotient is the result of division. Any number with a zero-exponent equals 1. (Page 108.) CHAPTER XIII A general number is a letter or other number symbol that may repre- sent any number. (Page 123.) A formula is an expression of a general principle, or rule, in general number symbols and in the form of an equality. (Page 124.) To solve a formula completely is to find the value of each general number in terms of the others. (Page 125.) CHAPTER XIV A root of a number is one of its equal factors. (Page 139.) The square root of a number is one of the two equal factors whose product is the number. (Page 140.) The cube root of a number is one of the three equal factors whose product is the number. SUMMARY OF DEFINITIONS 327 CHAPTER XVI A common divisor, or common factor, of two or more numbers is an exact divisor of eacii of them. (Page 172.) The highest common factor (h.c.f.) of two or more numbers is the product of all their common factors. A multiple of a number is a number that is exactly divisible by it. (Page 175.) A common multiple of two or more numbers is a number that is exactly divisible by each of them. The lowest common multiple (l.c.m.) of two or more numbers is the product of all their different factors. CHAPTER XVII An algebraic fraction is the indicated division in fractional form of one number by another. (Page 179.) The numerator is the number above the line. The denominator is the number below the hne. The terms of a fraction are the numerator and denominator together. An integer, or integral number, is a number no part of which is a fraction. The sign of a fraction is the sign written before the line that separates the terms. (Page 180.) Reduction of fractions is the process of changing their form without changing their f a/ wes. (Page 181.) A mixed number is a number one part of which is integral and the other part fractional. (Page 184.) A proper fraction is a fraction which cannot be reduced to a whole or a mixed number. An improper fraction is a fraction which can be reduced to a whole or a mixed number. The lowest common denominator (Led.) of two or more fractions is the l.c.m. of their denominators. (Page 187.) The reciprocal of a fraction is the fraction inverted. (Page 193.) CHAPTER XVIII A literal equation is an equation in which there are two or more general numbers. (Page 198.) 328 ELEMENTARY ALGEBRA A general problem is a problem all of the numbers in which are general numbers, (Page 207.) CHAPTER XX The ratio of one number to another is the quotient of the first number divided by the second. (Page 229.) The antecedent is the first number of a ratio, and the consequent is the second number. The terms of a ratio are the antecedent and consequent. The value of a ratio is the quotient expressed in its lowest terms. A ratio of greater inequality is a ratio in which the antecedent is greater than the consequent. (Page 231.) A ratio of less inequality is a ratio in which the antecedent is less than the consequent. A proportion is an equation of ratios. (Page 232.) The terms of a proportion are the terms of the ratios. The extremes of a proportion are the first and fourth terms; the means are the second and third terms. A mean proportional is the second of three numbers which form a continued proportion, as x in a: a: = x:b. (Page 234.) A third proportional is the third of three numbers that form a con- tinued proportion. A fourth proportional is the fourth of four numbers that form a proportion. A variable number, or a variable, is a number which in a given problem, or discussion, may have different values. (Page 241.) A constant number, or a constant, is a number that is not a variabl \ One variable varies as another if, as they vary, their ratio remains constant. CHAPTER XXI Involution is the process of raising a number to a power whose exponent is a positive integer. (Page 244.) The exponent of the power is the number which indicates how many times the number (the root or base) is taken as a factor. The base of a power is the number which is raised to a power. Evolution is the process of finding a root of a number. (Page 250.) SUMMARY OF DEFINITIONS 329 The index of a root is a number symbol written or understood in the opening of the sign V to denote what root is intended. A radical is any root of a number indicated by the radical sign, \/~> or V , or by a fractional exponent. (Pages 250 and 264.) An odd root is a root whose index is an odd number. (Page 252.) An even root is a root whose index is an even number. An imaginary number is an indicated even root of a negative num- ber. (Page 253.) A real number is a number that does not involve an even root of a negative number. The principal root of a number is the real root which has the same sign as the number itself. (Page 254.) The radicand is the number whose indicated root is to be found. (Page 264.) The order, or degree, of a radical is determined by the index of the root.' A rational number is a positive or negative integer or a fraction whose terms are integers. An irrational number is a number which cannot be expressed wholly in rational form. (Page 265.) A surd is an indicated root of a rational number which cannot be exactly obtained. An arithmetic surd is a surd whose radicand is an arithmetical number. An algebraic surd is a surd whose radicand is an algebraic expression. The coefficient of a radical is the rational factor before the radical. A pure surd, or an entire surd, is a surd having no coefficient ex- pressed. A mixed surd is a surd having a coefficient expressed. A quadratic surd is a surd of the second order. Similar surds are surds which in their simplest form are of the same degree and have the same radicand. (Page 271.) Rationalizing a surd is the process of multiplying the surd by a number that gives a rational product. (Page 275.) The rationalizing factor is the factor by which a surd is multiplied to give a rational product. A binomial surd is a binomial one or hath of whose terms are surds. 330 ELEMENTARY ALGEBRA A binomial quadratic surd is a binomial surd whose surd term or terms are of the second order. (Page 276.) Conjugate surds are two binomial quadratic surds that differ only in the sign of one of the terms. An irrational, or radical, equation is an equation containing an irrational root of the unknown number. (Page 278.) CHAPTER XXIII A quadratic equation is an equation of the second degree in the unknown number. (Page 282.) The constant term in a quadratic equation is the term that does not contain the unknown number. A pure quadratic equation is an equation that does not contain the first power of the unknown number. An affected quadratic equation is an equation that contains both the first and second powers of the unknown number. Pure quadratics are often called incomplete quadratics, and affected quadratics are also often called complete quadratics. The discriminant of the roots oi ax"^-\-hx+c = is b^ — iac. (Page 298.) A complex number is a number of the form a+6 v — 1, a and b denoting real numbers. Conjugate complex numbers are complex numbers which differ in the sign of the imaginary term. CHAPTER XXIV A quadratic equation in two variables is an equation in two variables, one or both of which are of the second degree. (Page 305.) A system of quadratic equations is two or more quadratic equations considered together. A simultaneous system is a system in which all the equations can be satisfied by the same values of the variables. INDEX PAGE Absolute value of a number . 22 Addends 27 Adding indicated products . 27 several positive and nega- tive terms 29 similar terms 27-28 Addition and subtraction of fractions 188 of surds 270 Addition defined 27 fundamental laws of . . . 29 law of grouping for ... 29 law of order for 29 of dissimilar terms .... 30 of monomials 27 of polynomials 32 of terms partly similar . . 48 proportion by 237 analysis of 238 Affected quadratic equation 282 solved by factoring . . . 285 Aggregation, symbols of . 41-43 Algebra defined 7 reasons for studying . . . 1-6 Algebraic expression .... 24 value of an 26 fraction 179 function defined 51 functions 50 language 8 notation 24 numbers 21, 50 signs 9 Alternation, proportion by . 237 Antecedent 229 Approximate values of surds 278 Arranged polynomials ... 97 PAGE Associative law of addition . 29 of multiplication .... 94 Assumption for irrational equations 280 Axiom, power 278 root 287 Axioms 13 Balance of values 12 Base of a power 244 Binomial defined 25 quadratic surd 276 surd 275 theorem 248 Binomials, powers of ... . 247 Brace 42 Bracket 42 Check or test 14 Check on algebraic work defined 33 Checking 16 addition by substitution . 33 a problem 101 or verifying a root . . . 100 Clearing equations of fractions 66 principle of 67 application of 103 Clock problems 117 Coefficient 25 of a radical 265 Common compound factors . 135 Common divisor 172 fraction, square root of . . 262 multiple 175 331 332 INDEX PAGE Comparison, elimination by . 213 Complete divisor 256 quadratic equation . . . 287 approximate values of roots .292 normal form 287 roots of the 292 quadratics 282 Completing the square . . . 288 a = l 288 a not 1 289 Complex number 298 Composition, proportion by . 238 Compound expressions, oper- ations on 43 Conditional equation ... 61 Conjugate surds 276 Conjugate complex numbers 298 Consequent 229 Constant 241 term of a quadratic . . . 282 Cube defined 95 root 140 Definition of a*^ . .. . 108, 263 of a^ 263 of a-" 264 Definitions, summary of . . 322 Degree of an equation . . . 100 Denominator defined . . . .179 Dependence of a function . . 52 Dependent equations ... 83 Deriving formulas 125 Determinate equations . . 85 Difference defined 35 of same odd powers ... 153 of two squares 143 Digits, Arabic 8 Directed numbers 21 Directions for solving equa- tions 101 PAGE Discriminant of roots . . . 298 Dissimilar terms ..... 25 Distributive law 95 Dividend defined 107 Dividing a monomial by a monomial 107 a polynomial by a mono- mial 109 a polynomial by a poly- nomial 110 Division defined 107 indicated 9 of fractions 193 proportion by 239 sign law of 107 Divisor, common 172 complete 256 defined 107 partial 256 Double meaning of + and — 23 EHmination, defined .... 86 by addition or subtraction 87 by comparison 213 by substitution 120 Ellipse 310 Equation defined 11 degree of 100 determinate 85 history of 59, 60 indeterminate 85 linear 100 literal and fractional ... 198 quadratic 282 simple or linear 100 in quadratic form .... 293 members of an 11 root of 61 solving an 13 Equations, dependent . . .83 inconsistent 83 INDEX 333 PAGE Equations, graphing . . . 77-81 linear 81 non-simultaneous .... 83 simple 100 simultaneous 82 Equality, sign of • 11 Even powers 95 roots 252 Evolution 250 principle of 274 Examples of type-forms . . 131 Expansions 247 Exponent in multiplication . 92 in product 93 in quotient 108 law of, for division . . . 108 law of, for multiplication . 93 of the power 244 zero, meaning of 108, 263 Exponents, fractional . . . 272 fundamental laws of . . . 263 theory of 263 Expression, algebraic 24 Extended meaning of term . 45 Extremes of a proportion . . 232 Factor, common 172 defined 8 highest common 172 rationalizing 275 Factoring 134 by principles of quadratics 301 Factors, common compound 135 defined 8, 134 monomial 134 First member 11 Forming quadratics with given roots 300 Formula defined 124 Formulas derived 125 solved . 125 PAGE Formulas, solution of ... 210 Fourth proportional .... 234 Fraction in lowest terms . .182 improper 184 proper 184 Fractions, addition and sub- traction of 188 division of 193 multiphcation of .... 191 Fractional exponents .... 272 Function defined 51 dependence of 52 Function of a:, n, etc. . . . 50 fix), fin), etc 50 Generalization in algebra . . 207 General number defined . .123 General numbers 123 quadratic 288 quadratic trinomial ... 148 Graphical solution of one- letter quadratics . . . 159-60, 282, 295 of quadratic systems . 305-313 Graphing data 74 functions 50, 54 equations 77-81 Graph of a;2-a 283 Higher degree equations by factoring 286 Highest common factor . . 172 of monomials 172 of polynomials 173 Hyperbola 311,312 Identity defined 60 sign of 60 Imaginary number .... 253 roots 253 Improper fraction 184 334 INDEX PAGE Incomplete quadratic equations 282 trinomial squares .... 151 Inconsistent equations ... 83 Independent equations . . 84, 86 Independent number .... 50 Indeterminate equation . . 85 Index of the root 250 Indicating division .... -9 multiplication ....;. 9 Inequality, ratio of greater . 231 ratio of less 231 signs of 272 Inversion, proportion by . . 237 Involution .244 Irrational equations, assump- tion for 280 equations in one unknown - 278 number 265 Language, using algebraic . 8 Letters representing numbers 17 Law of exponents for division 108 for multiplication .... 93 Law of grouping for addition 29 for multiplication .... 94 Law of order for addition . . 29 for multiplication .... 94 Linear equations 81, 100 Literal and fractional equations 198 Lowest common denominator 187 Lowest common multiple . .175 of monomials 175 of polynomials Meaning of exponent . 108, 263 type-forms 130 Mean proportional .... 234 Means of a proportion . . . 232 Measuring is ratioing . . . 230 Members, first and second . 11 176 PAGE Mixed number 184 surd 265 Mixed surd to an entire surd 270 Monomial defined 25 Multiple, common 175 lowest common 175 Multiplicand defined .... 91 Multiplication defined ... 91 indicated 9 law of exponents for . . . 93 of fractions 191 of surds 272 sign law of 92 Multiplier defined 91 negative 91 Multiplying monomials . . 93 a polynomial by a mono- mial 96 a polynomial by a poly- nomial 97 Nature of roots of quadratic 298 Negative multiplier .... 91 Non-simultaneous equations 83 Notation 7 algebraic 24 in problem-solving . . . 15, 16 system of 24 Number 13 imaginary 253 independent 50 irrational 264 mixed 184 of roots 252 rational 264 real- 253 unknown 13 Numbers, directed 21 general 123 of arithmetic 20 INDEX 335 PAGE Numbers, positive and nega- tive 21 represented 8 Numerator defined .... 179 Odd powers 95 root 252 Operations on compound ex- pressions 43 Opposite qualities of alge- braic numbers .... 21 Order of a radical 264 second and third .... 264 Parabola 54, 306 Parenthesis 41 defined 42 Partial divisor 256 Partly similar terms .... 25 Pascal's triangle 248 Picturing functions .... 54 Polynomial, arranged ... 97 defined 25 square root of a 254 Polynomials factored by grouping 137 Positive and negative num- bers 21 problems in 23 Power defined 95 base of the 244 of a fraction 246 of a monomial surd . . . 277 of a product 246 axiom 278 second 95 third 95 Powers and roots 244 of binomials 247 Primes and subscripts . . .211 Principal root 254 PAGE Principle of evolution . . . 274 Principles of proportion . . 235 Problem, general 207 solving a 15 quadratics 302 Problems in sumultaneous equations 106 three or more unknowns . 226 two unknowns ...... 89 Problem-solving, suggestions on 113 Product defined 91 of sum and difference of two numbers 142 of two binomials with a common term 147 of two numbers equal to o 158 Product, sign of the .... 91 Products, how written ... 8 Proper fraction 184 Proportion defined .... 232 by addition 237 by addition and sub- traction 239 by alternation 237 by composition 238 by division 239 by inversion 237 by subtraction 238 extremes and means of . . 232 principles of 235 Proportional, mean .... 234 fourth 234 third 234 Proportionality, test of . . . 233 Pure quadratic equation . . 282 normal form of 282 solved by factoring . . . 284 Quadratic equation, affected . 282 nature of roots of ... . 298 336 INDEX PAGE Quadratics, pure 282 Quadratic equations .... 282 solved by formula . . . .291 Quadratic surd 265 binomial 276 trinomial 132 Quality of number 21 Quotient defined 107 Radical, coefficient of . . . 265 defined 250, 264 degree or order of ... . 264 reduction of 267 sign ! ... 250 Radicand 264 Ratio, antecedant of ... 229 consequent of 229 defined 229 of greater and less inequali- ty 231 Rational number 264 Real number 253 roots 253 Reasons for studying algebra 1-6 Reciprocal of a number . .193 Reduction of fractions . . . 181 of improper fractions . . 184 of mixed expressions . . . 186 of radicals 267 of surds to same order . .271 Remainder in subtraction . 35 Removing symbols of aggre- gation 45 Review of factoring . . . .156 Root of a fraction 252 an equation 61 a number 139, 250 a power 251 a product 251 cube and square 140 index of the 250 PAGE Root, principal 254 square, of a decimal . . .261 square, of numbers . . . 259 Roots, imaginary 253 of complete quadratic . . 292 sets of 86 Satisfying an equation ... 60 Second number 11 power 95 Sets of roots 86 Signed numbers 21 Sign law of division .... 107 of multiplication .... 92 Sign of a fraction 180 continuation . . . .93, 245 negative numbers .... 22 positive numbers . . . .22 product 91 quotient 107 real root 253 Signs of inequality .... 272 Similar terms 25 Similar with respect to a fac- tor 25 Simultaneous equations de- fined 82 system of 82 Simultaneous simple equations 85 Solution of equations by fac- toring 15? formulas 210 Solving an equation .... 13 a problem 15 equations linear in - and - 215 problems 88 the equation 15 Solving one-letter equations graphically 55 INDEX 337 PAGE Solving formulas 125 quadratics by factoring 284-285 simultaneous equations graphically 82-84 Special methods for systems of quadratics . . . 316-317 Special products 99 Special quadratic trinomials . 148 Square defined 95 of difference of two numbers 138 of sum of two numbers . .137 Square root 140 of a binomial surd . . . 277 of a common fraction . . 262 of a decimal 261 of a polynomial ..... 254 of numbers 259 Statement in problem-solv- ing 15-16 Stating and formulating laws 127 Subscripts, primes and . . . 211 Substitution defined .... 60 elimination by 120 Subtraction defined .... 35 of monomials 35 of polynomials 39 proportion by 238 Subtrahend defined .... 35 Subtracting, defined for alge- bra 39 dissimilar terms 37 monomials 38 polynomials 39 similar terms 36 terms partly similar ... 49 Suggestions on problem-solv- ing 113 Summary of factoring ... 155 work on graphing .... 58 Sum of the same odd powers 154 Surd 265 PAGE Surd, pure or entire .... 265 mixed 265 quadratic 265 Surds, addition and sub- traction of 270 binomial 275 binomial quadratic . . . 276 conjugate 276 division of 274 multipUcation of .... 272 of different orders . . . .271 similar 271 Symbols of aggregation . 41-43 removing 45 System of equations .... 86 System of notation .... 24 of quadratic equations . . 305 Systems solved by quadratics 305 Term, constant 282 defined 24 extended meaning of . . . 45 Terms, dissimilar and similar 25 of a fraction 179 of a ratio 229 partly similar 25 Test of proportionality . . . 233 of roots of quadratic . .. . 289 Theorem, binomial .... 248 Third power 95 proportional 234 Three or more unknowns . . 226 To check 16 Transposition applied ... 63 defined 62 Triangle, Pascal's 248 Trinomial, defined 25 general quadratic . . . .149 quadratic 132 squares 140 squares, incomplete . . 151 338 INDEX PAGE Type-forms, meaning of . . 130 Type-forms, interpreted . .132 examples of 131 Unknown number 13 Value of an algebraic ex- pression 26 of any letter 12 of a quadratic surd . , . 265 of a ratio 229 PAGE Values, approximate, of roots of quadratics 292 Values of surds 210 Variables 241 Variation 241 direct 241 sign of 241 Varies as, or directly as . . 241 Vinculum 42, 250 Zero-exponent, meaning of 108, 263 VB 35929 UNIVERSITY OF CAUFORNIA LIBRARY