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MATHEMATICS The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN MAY <3 19/0 A h\ iF noePn MAY F ee L161— O-1096 SCHOOL ALGELRA FIRST COURSE > Leer BY Pewee RZ Pa. UNIVERSITY OF ILLINOIS A. R. CRATHORNHE, Pa.'D! UNIVERSITY OF ILLINOIS AND E. H. TAYLOR, Pu.D. EASTERN ILLINOIS STATE NORMAL SCHOOL NEW YORK HENRY HOLT AND COMPANY 1915 et ANAL, AVMAMMONATCHOVANU NAAT Da. | USOur. Rt MATHEMATICS LIBRARY PREFACE | Tuts book is the first volume of a two-book series. It con- | ains ample material for a full year’s work in the first year of the aa school, and covers the parts of algebra most likely to be of use to the student who goes no further in the subject. Tt will prepafe for Plane Geometry and Physics, which come in the later years of the high school. The second volume, the Advanced Course, supplies the additional material de- manded for entrance into the scientific and technical courses in our colleges and universities. The text represents a special effort at presentation of prin- ciples and definitions in clear, simple style, with wordy unes- sentials eliminated. The study of algebra is taken up as an ex- tomien of arithmetic. The laws of algebra are first suggested by induction from familiar rules of arithmetic, and throughout the book the close connection with arithmetic is kept in view. The pupil is led to see that new symbols are introduced into algebra, not arbitrarily, but because of their real advantages in representing numbers. Difficulties are taken one at a time at sufficient intervals to allow the mastery of each one before proceeding to the next. Thus the fundamental idea of representing numbers by letters is developed in the first two chapters; that of signed numbers in Chapter III; while the equation is not formally introduced until Eiiater IV, though it is used informally without special efinition from the beginning. In accordance with the same ‘beneral scheme, each of the several chapters on the equation takes up a single new difficulty at a time when the pupil is ready for it. oye | A fourth fundamental topic, the notion of a function, is gradu- ally approached through evaluation of expressions and rou! the graph, so that the pupil is prepared for the more formal treat, ment, with the use of the functional notation, in the advancec course. Graphical work is treated as simply as possible, not as an added difficulty, but when occasion for it arises in th algebraic work. While graphical solution is important, it is not the fundamental notion underlying graphical representa- tion. The idea of functionality, the change in a function as the independent variable changes, is the idea about which the work in graphs should center. The book is well supplied with carefully graded exercises and problems. In general, preference has been given to mis- cellaneous groups rather than to groups illustrating but one process. With regard to the question of correlation with other high school subjects the book follows a middle course. Many problems involving applications to other subjects have been introduced, but care has been taken to keep within the limits of the pupil’s experience, as well as to give data that may be depended upon to be correct. During the last decade many teachers’ associations have dis- cussed the arrangement and content of the high school course in algebra. Many outlines of courses have been proposed and some detailed syllabi have been published. In the preparation of this book we have availed ourselves of these discussions and printed syllabi, and have endeavored to incorporate into the work the views which prevail among progressive teachers. We take pleasure in expressing to many teachers our appreciation for helpful suggestions and criticisms. We are especially in- debted to Prof. E. J. Townsend and Dr. E. B. Lytle of the Uni- versity of Illinois and to Miss Jessie D. Brakensiek of Quincy High School, Quincy, IIl., for careful and critical reading of the manuscript and for suggestions as to exercises and problems; to Mr. J. L. Dunn of the Lewis and Clark Higt School, Spokane, Wash., Mr. C. H. Fullerton and Mr. W. B.: lv PREFACE Tt a mg a PREFACE Vv Skimming of the East High School, Columbus, O., Mr. E. A. ook of the Commercial High School, Brooklyn, N.Y., Mr. . C. Irwin of the Joliet Township High School, Joliet, IIl., nd Mr. R. L. Modesitt of the Eastern Illinois State Normal chool, Charleston, Ill., for reading the proof and seeing the ook through the press. H. L. RIETZ A. R. CRATHORNE Hi LAN LO \ { | CONTENTS NOD ONNN ORR He CHAPTER I INTRODUCTION PAGE . Numbers and Language of Arithmetic. .........2.. IIEIPERIGOOTA 0 re te kk kk Smuprmmriana@r UNeETALION. . ... 6.1. 6 ee ee m. Use of Letters in Solving Problems. .......2.2.2.2.~. . Use of Letters to Abbreviate Statements ...... Sete 2 EOS ee a Serermerierenra of @ Circle... ww kk kk Seenren Ora wircle =... .. 1... . AR ce BM PUTS oe kw. Lyne (tk CRE aes ee ere oo Se ee eae re SE 3 CMUMEPIEIEIIIREOWETH 9.0. ec. c e te ee Mertmmrerasotoonpymbols ... . . . 1 1 wk ew we 1 CHAPTER II ALGEBRAIC EXPRESSIONS MnMMtRIPREROIFOMRIONS, .. . . wk tt ke tk ke le 14 RIVIERA OMA ok eh oe ee ke ee 14 SS ee 15 . Evaluation of Expressions... . . RG My Se coats aed 16 pereapreseions « ontaining one Letter . . 2... 0.2... .8.. 17 8B. Graphical Representation. ...... Ee A ek + at aes 19 CHAPTER III POSITIVE AND NEGATIVE NUMBERS ®. The Use of a Scale to Represent the Numbers of Arithmetic. . 23 p. Addition and Subtraction on the Scale... .... . Si tte pabosmive and Negative Numbers .....:... . Mare ee | vill CONTENTS 22. Illustrations. . ..0 0 6% 3s 5 23. Numerical or Absolute Value. . .. . Pee 24. Greater and Less. ..: 2. 0. 4 2 0) 2 a 25. Addition of Signed Numbers. . . . <2) pige enn 26. Subtraction of Signed Numbers . . . 0) 3 27. Subtraction on a Seale... . . . . soe eee 28. Rule for Subtraction . - .. . . 2 . Se 29. Addition and Subtraction of Several Numbers. ....... 30. Multiplication in Arithmetic. - 2 . 293 S52 ee 31. Multiplication of Signed Numbers . ~ 790) eee 32. Division of Signed Numbers - . © >.> 25.) ee 33. Fractions 2... 2... 1s 6 ee se CHAPTER IV EQUALITIES 34, Members of an Equality. . . . . . 2 . 290) ye 35. Identities. 2... 2... 2. eh. Ds 3G, puquatlons YU - meicdseeeeeh) eee eS . do ke, 37. Solution of Equations. . . .. .. . . . 29a 38. Principles Used in Solving Equations : .. 3.) Jee 39. Verification of Solutions by Substitution. . . 7) 2) ieee 40. Transposition . . 2... . . ...5 + « 9) 0 41. Translation of English Expressions into Algebraic Expressions . CHAPTER V ADDITION 42. Terms of an Expression. . ... . . « = 2) 0) 43. Monomials and Polynomials... . . . ) 3 44, Similar Terms . .-. . . .. . 3-4 « 1) < 45. Addition of Monomials . ... . . . 9 3933 46. Simplifying Polynomials. . . . >... + sue 47. Arrangement of Terms ina Polynomial. .......... 48. Addition of Polynomials. ... ... . . J 2 CHAPTER VI SUBTRACTION 49. Subtraction of Monomials. . .. .. . . ss gene 60... Subtraction of Polynomials... . . issue ke aaa CONTENTS CHAPTER VII PARENTHESES CHAPTER VIII MULTIPLICATION #. Multiplication of a Product by any Number... ...... fetrouuc, Oo. &tolynomial by a Monomial .......... merit !wo Polynomials... 2 .. . . 2. 2 ee ee CHAPTER IX EQUATIONS AND PROBLEMS pebeuations lrvolving Parentheses... . i... ...:. 66656 e-e mrouemone involwing Fractions... . . ... 2 6. «se ew CHAPTER X DIVISION SRE OOMUAI 3 ec ee ke Division of a Polynomial by a Monomial .......... erry me OrvPOUNaL 5 6 kk kk ee SR co CHAPTER XI LINEAR EQUATIONS NMEETIREUQUIS Eo ge a sk i emf eee ee el CHAPTER XII IMPORTANT TYPE FORMS EN ee) es ee wei. @ gale ale MMII ICAL cg my ge se ee es 1X 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89." 90. CONTENTS . Product of the Sum and Difference of Two Numbers. ... . . Product of Two Binomials Having a Common Term .... . Cube of a Binomial... 3... 3 Le - Square of a Trinomial, © . <>... > 7) Ree CHAPTER XIII FACTORING Prime Factors in Arithmetic. . 5...) Prime Factors in Algebra . . .°.. . 40 2 ee Factors of Monomials. . Sa Monomial Factors in Polynomials Factors Found by Grouping. Terms. . . © .. 2m Difference of Two Squares Trinomial Squares .. . « at! he ah ee Trinomials of the Form 2? a ay e ate x x ae . igo ae os General Quadratic Trinomial. . . . . Mer Sum and Difference of Two Cubes... . . 2) Se Summary of Factoring ...-. .. ..% . - ne CHAPTER XIV EQUATIONS SOLVED BY FACTORING Quadratic Equations . Factoring Applied to the Bolitian Ef Guanrane Equationes CHAPTER XV HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE Greatest Common Divisor in Arithmetic. . . . . jae ae Common Factor oe Sk WS Ak CHAPTER XVI FRACTIONS Fractions in Arithmetic... ... ... + . ee Fractions in Algebra ..... . ere Division by Zero... 6.0. «6 6 se 8 } CONTENTS xi { PAGE Po Oe STS Sl Pe 138 ( Reduction to Lowest ane Pe aati 0 Fe Oe | ee ere 140 Je Gancellation...... . Oo Loe! ee pes TAl : « Reduction to Common Dencnistor SOL Se ea (. Addition and Subtraction of Fractions... ......... 148 i Beoneamomorrractions 2... wk ee eh 1G PMIMOSEOUETAGUONS 6 ee ee 1D , I SE Gr a ee ks Sa ee a 151 ih i| CHAPTER XVII FRACTIONAL AND LITERAL EQUATIONS ieeeaearing quations Of Fractions. . ........ . '.. + 159 |0. Unknowns in the Denominator... ........... 160 SESS ee a a 164 Sema I MOS Ss pe tk A he ew we ws 165 6 CHAPTER XVIII [ RATIO, PROPORTION, AND VARIATION eee Tatag)). ao Ys een est eee ee. F218 WEIRUECMOTO OAM ww. kk kk _ OEE Sree oy rack dt | Mean Proportional”. ..... ae ot oF ae SY Ga eee 1b | Perera our troportional. . . ... .. . ...... <1 Debra Ve UCrnation fos 4... ee ew ee ew NT re GrvoMnvernsion ©. ee ee on NTS DtrruOnre A OMpodilioOn. - . . . 5. we ee... 8 . Proportion by Division. . . . eres, * haaiaermmrmemte eats WE | . Proportion by Composition and Divison: Oey. | abery > einen be fo DI SONRtANIS ws ee et ee eS TE fa SE, Oe a a TG Wy 5 i CHAPTER XIX | | GRAPHICAL REPRESENTATION OF THE RELATION i BETWEEN TWO VARIABLES 4, I 2 a es we ete 180 | i Axes, Codrdinates. . .. . Pea Pe A Pee) Aye nek st LO SMIIPRIRMEL Ce hy es ks ces eee Loe, ES RR RD, Cee an gee aire Wiki 183 Ke MOTTO ANCE cin a te ere we ew 182 ( Ua INCION ee eG el aw ke 188 ( x CONTENTS ; PA 68. Product of the Sum and Difference of Two Numbers... . . , 69. Product of Two Binomials Having a Common Tle@Rior icra 70. Cube of a Binomial... 9. .-. . 2) 71... Square of a Trinomial. . . ..°. | 1p CHAPTER XIII FACTORING 72. Prime Factors in Arithmetic. . . >.) Geen 73. Prime Factors in Algebra . . . \. ) .) a) 74. Factors of Monomials. . .. . 1 8 6 a or 75. Monomial Factors in EAlvromiale on ee al 76. “ie 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89." 90. Factors Found by Grouping. Terms : <> =) 3) eee Difference of Two Squares ......... Trinomial Squares .). .°s 5. . .0 5 ee Trinomials of the Form a? + Bae + b) x + ab General Quadratic Trinomial. . ... » 2 ee Sum and Difference of Two Cubes . . . . 3) ene Summary of Factoring . .-. . ©. +. Jens CHAPTER XIV EQUATIONS SOLVED BY FACTORING Quadratic Equations... . : 3... 02 Factoring Applied to the Solin ot Quadratic pes CHAPTER XV HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE Greatest Common Divisor in Arithmetic. . . ....4.5.5., Highest Common Factor Pee Lge CHAPTER XVI FRACTIONS Fractions in Arithmetic. ... . . . . . ee Fractions in Algebra . . . . . . = ws) wen atne nnn Division. by Zero... 2. 605 se se CONTENTS . Signsin Fractions ....... Sipl<_ + .. 7 sera ee ea tetcrias Gwent LeIm0s . . . 4... ... « 6 8 bbl eoa a @eeaneenation. .. .- .. . fot 3. Se Reduction to Common eh arniiator Stee mniiog an sultraction Of Fractions. .. . . . ... 9s © ws . Multiplication of Fractions . Division of Fractions . . Complex Fractions . . CHAPTER XVII FRACTIONAL AND LITERAL EQUATIONS . Clearing Equations of Fractions . . Unknowns in the Denominator . . BRUCPPNAGI ek ee ke ee si SI IICEIE ST MUOM GS rcs ss ts, se Ne bape vee CHAPTER XVIII RATIO, PROPORTION, AND VARIATION RS ge >) See nee : Se a RS EM TMOPUIG GRIER Se ee ee eerrraricerouren troportional .. . . 3... 2s ek te ee . Proportion by Alternation ..,.. selene re” kOe ome Proportion by Inversion . . Proportion by Composition. . . Proportion by Division. : ee: . Proportion by Composition and ivisions het ee eae . Variables and Constants . . DITA EMVN ST ek ke CHAPTER XIX GRAPHICAL REPRESENTATION OF THE RELATION BETWEEN TWO VARIABLES SG eal TS RR Axes, Codrdinates. . .. . ria 85 5 8 ue POOL OImts’.. 5... - - GM arene case eereeuooriimate Paper... 5 6. 8 ee we ke OEMS ee Mee OTA ies, eek Xl CONTENTS 120. Graph of a Function. . . .°. 9... 2) ee 121. Graph of an Equation : . . . . . 12 122. Locus of a Linear Equation. . . . “4 5) een 123. Graphic Solution of Equations: . . . .) 73a neuen m. 124. Graphical Representation of Scientific Data ee ee CHAPTER XX SYSTEMS OF LINEAR EQUATIONS 125. Solution of Equations in Two or More Unknowns. .... . 126. Simultaneous Equations . ... . 2 -, . 9 127. Independent Equations. . . . . =. = = Secsnneeneenuene 128. Dependent or Equivalent Equations. ........... 129. Inconsistent Equations. . .-. .. 72 .) sees 130. Elimination. . . . ) ... 2. . 4 . 0) Bes 131. Elimination by Addition ae Siibtracian oe SE a ales 132. Elimination by.Substitution®’ | . . . 7 | ee 133. Standard Form az + by +c=0 . . . |. =e 134. Literal Equations Containing Two Unknowns. ....... 135. Linear Systems in Three or More Unknowns ........ CHAPTER XXI SQUARE ROOT AND APPLICATIONS 246. Definition of a Square Root... . . . 2). . eee Sage Hadical Sign wa. . 6 i oe eg eh yee 138. Square Root of Monomials . . . . . . . sss 139. Equations Solved by Finding chee are Roots’... [2 aan 140... Square Roots of Trmomials. . . : ./. 2) 141. Process of Finding the Square Root . . . . . Jes 142. Square Roots of Numbers Expressed in Arabic Figures . ; 143. Explanation of Process of Finding Square Root in Arithmetic : 144. Number with More than Two Periods. .........°. 145. Squaré Roots of Decimals ... .. . . 3) 2 0) eee 146. Approximate Square Root... . : Se) 2) se CHAPTER XXII RADICALS 147, Radicals ... . . . 1.05. 6 5 202. Wr 148. Rational and Irrational Numbers . . <2) 2) eee 2 70. TA: 72. 73. 74. CONTENTS I ee ek ee eae tg Square Root of a Fraction ~ LD MU ice: ur eek Bremineniomor Radicals. .-. 6 6 6 ee ee ee Meaning of Simplification of a Radical... ... 1... . Addition and Subtraction of Radicals *. ......... Multiplication of Quadratic Surds .“ . 2 2. 1 ee ee es Mivision of Quadratic Surds. . .“.. 3’... se we Rationalization of Denominators “: ........... MUERTE CtOY fo ee i of ce ee et Solution of Equations Involving Reticnls. fo. Ts oh CHAPTER XXIII QUADRATIC EQUATIONS Quadratic Equations Solved by Factoring ........-. IEE QUaATC. 2... we ee ee ee es Equations Solved by Completing Square. . .......- . Solution by Hindu Method of Completing Square... . . . Type Form of a Quadratic Equation... .... . . Quadratic Solved by Formula. ........- a ee ais . The Special Quadratic az? +c=0 .... 2.2. + ee ees . Graphs of Quadratic Functions... . . MIIPIGREVONUIMDEIS. . . 6 ee es . Graphical Meaning of Imaginary Roots ...... . Historical Note on Quadratics. . ... . 22 6 1 se ee ee CHAPTER XXIV SYSTEMS OF EQUATIONS INVOLVING QUADRATICS ye yh ESS SS Sa a PRA Solution of Simultaneous Quadratics. . . . - 1. - ee. One Equation Linear and One Quadratic. . . ...... . Equations Containing x? and y2only. .......-.---: ES de i Xill PAGE 226 227 227 228 229 231 232 233 234 235 238 239 239 240 242 242 245 247 249 250 255 SCHOOL ALGEBRA ‘FIRST COURSE CHAPTER I INTRODUCTION _ 1. Numbers and language of arithmetic. In counting, the child learns numbers which are called integers. The written language of arithmetic uses the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, to represent numbers and the signs +, —, &, + to denote operations. In problems where two or more integers are added or are multiplied together, or when the smaller of two integers is sub- tracted from the greater, the answer is always an integer. In he use of the sign + however, another kind of numbers, called ractions, is obtained. These two kinds of numbers, integers and fractions, have been studied in arithmetic. | @, Language of algebra. Algebra is a continuation of arith- metic. ‘In it we use not only the numbers and symbols of arithmetic, but we also introduce new kinds of numbers and new symbols. The written language of algebra makes much use of letters to represent numbers. This use of letters is not entirely new to the student, for it is customary in arithmetic to represent certain numbers by letters. Thus, the radius of a circle is often represented by r, the diameter by d, or by 2xr. The altitude and the base of a rectangle may be repre- sented by a and b, the area by a Xb, and the distance around by at+ta+b+b, Bee 2 xb. 3. Symbols of operation. The signs +, —, X, + are used ‘in algebra as in arithmetic, but the sign of aot is I | ] 2 INTRODUCTION [Cuap. I usually omitted. For example, ax 6 is usually written ab | and 2x ris written 2r. If a sign of multiplication is used, it i customary to use a dot, written a little above the position for a period to distinguish it from the decimal point, instead of \the sign x. For example, 2 x 3 is written 2:3. The sign of division is not much used in algebra. Thus, a+ 6 is usually written : The use of letters to represent numbers enables us to write many statements in very brief form. Thus, if A is the area, b the base, and a the altitude of a rectangle, the brief statement A =ab gives the rule that the area of a rectangle is equal to the prod uct of the base and altitude. It is important to be able to translate English sentences into such algebraic statements. : EXERCISES 1. If / stands for the length of a running track in yards what stands for the length of a track 50 yards longer? aa 1+ 50 yards. 2. If l stands for the length of a track, what is the leratif of a track 100 yards longer? What is the length of a tracl twice as long? 3. What is the cost of 5 railway. tickets at a2 dollars a ticket? 4. What is the wale of 3a when a is 1? When a is 2? When a is 13? 5. How many pecks are there in 6 bushels? Inz bushels? 6. How many inches are there in y feet? : 7. If n represents a number, what represents a number three times as large? 8. Write the sum of a, b, c, and d, using the sign of addition. 9. Write the product of a and 2 as it is expressed in algebra , { | Ane, 3] LANGUAGE OF ALGEBRA 3 10. Why is it not good form to write 25 for 2x 5? 11. Write the following without using a sign of multiplica- mion: S8x2z,bxX6,7xaxb,xxyxz,5xrxt,Ixmxnxv. 12. Indicate the subtraction of + from a By using the sign of subtraction. 13. Write the product of a, b, c, and din tres different ways. 14. If a is an integer, what is the next integer? What is the preceding integer? 15. A barrel contains 313 gallons. If cis the number of cubic inches in a gallon, what is the number of cubic inches in a barrel? 16. If x represents a certain number, what represents a number 10 greater than twice x? 17. If p is the cost of one article, what is the cost per dozen? | 18. If d is the cost per dozen, what is the cost of one? 19. What is the simple interest on $100 for three years at 6 % per annum? On z dollars for y years at m per cent? | 20. If x and y are the lengths of two lines, what is their combined length? | 21. The age of a father is 30 years more than twice his son’s age. If x is the son’s age, what is the father’s age? 22. A rectangle is twice as long as it is wide. Let x be its width. What is its length? Its perimeter? Its area? 93. A rectangle is 8 feet wide. If x is the length, what is the area; the perimeter? 24. Write in algebraic language the statement that the sum of two numbers, a and ), is divided by the number c. 25. Five times 6 plus three times 6 plus two times 6 equals how many 6’s? 26. 5a+ 3x+ 2x = how many 2’s? 27. 64+ 2a+a=? i> 98. 8¢—474+ 27+ 5c =? | 99. y+ by—3yt+hy=? 30. Oz)}+ $2 — 2+ g2)+ 62 — 22 = ? { 4 INTRODUCTION [ Cuap. Fill out the blanks in the following : 31. 1b+ 5b-—4b—-—b=() 0b. 32. 30+ ( ) a bt=— 07, 33. ( )a+3a— 2a— $a = 4a. 4. Use of letters in solving problems. The following examples show the further use of letters to represent numbers and to simplify the solution of certain problems. Example 1. The sum of two numbers is 128. The larger is 3 times the smaller. Find the numbers. Let n = the smaller number. Then 3n = the larger number, | and n+ 3n = 128, the sum. b Adding, 4n = 128. Dividing by 4, = = 32, the smaller number. be Then 3n = 96, the larger number. @ Example 2. A lot is sold for $720, which is 20% more than it cost What was the cost? 5 The following is a common form of the arithmetical solution. Let 100% of the cost = the cost. Then 120% of the cost = HEY. . 1% of the cost = 745 of $720 = $6, and 100% of the cost = 100 x $6 = $600. Hence, the lot cost $600. b The solution may be shortened by the use of letters. Let c = the number of dollars the lot cost. Then 1.20c = 720, and c = 720 + 1.20 = 600. Hence, the lot cost $600. Note that c is a number and is not the cost. EXERCISES AND PROBLEMS 1. A house and lot are worth $6000. The house is worth four times as much as the lot. What is the value of the lot? 2. During a January cold wave the price of soft coal in- creased 10%. At the end of the cold period the price was | $3.85 per ton. What was the price at the beginning? i i Arr, 4] USE OF LETTERS IN PROBLEMS 5 3. A book sells for $2.40. The dealer makes a profit of ‘25% of the cost price. What was the cost of the book to the dealer? 4. The sum of the edges of a cube is 48 inches. What is the length of one edge? 5. A merchant sells a hat for $3.50, making 25% profit. What did the hat cost the merchant? 6. For what number does xstand in the equality 3 +2 = 13? Hint: Here z is the number which added to 3 gives 13. In the following exercises, the letters stand for unknown numbers. Find the numbers. ih Cae Aas 15. 2 — 20 = 90. 8724 10= 12. 16. 12n — 5n+ n= 64. 9. 22=24. | 17. «+30 = 75. 10. 407 = 28. LB oyetG. 11. 2—5=0. 19. w—3.= ll. 12. 52 = 10. 20. 2+5=842. 13. 7y + 5y = 120. 21. 2+ 22+ 32 = 12. 14. 3w = 21. 22. Think of a number; double it; add 15. If the result is 52, what was the first number?. 23. If a certain number is multiplied by 4, the result is 5 more than 33. What is the number? 24. A boy bought 3 books. One. book cost twice as much > as the other two put togethers If these two books were the same price, and if he received 4 cents change out of a dollar, what was the price of each book? 25. Two men, Smith and Jones, do a certain piece of work for $300 at $2.00 per day each. Smith works 5 times as many days as Jones. How many days did each work? 26. In a fire, Smith lost 3 times as much as Jones, and Brown lost 4 times as much as Jones. If the combined loss was $5120, what did each lose? | 6 INTRODUCTION | [Cuar. I. 27. The greater of two numbers is 4 times the less. Their sum is 240. What are the numbers? Hint: Let » equal the smaller number. 28. The sum of two numbers is 264. The greater is 11 times the less. What are the numbers? 29. The sum of three numbers is 54. The second is twice the first, and the third three times the second. Find each number. 30. The perimeter of a rectangle is 216 feet. The rectangle is twice as long as it is wide. What are its dimensions? 31. A man buys 4 books costing x cents each, and 2 balls at x cents each. The whole cost one dollar and a half. What is 2? do. Use of letters to abbreviate statements. As pointed out in Art. 3, the use of letters to represent numbers enables us to write many statements in very brief form. A noticeable example is the use of letters to abbreviate the rules of arith- metic. In Art. 3, the rule:— The area of a rectangle is equal to the product of the base and the altitude —is stated in the algebraic form . Ae ab, in which A represents the area, a the altitude, and b the base of the rectangle. | Similarly, if we were given the length /, the width w, and the height h, of a rectangular solid, the rule for finding the volume is expressed in the form V = lwh. . The rule in arithmetic for the product of two fractions is: | The product of two fractions is a fraction having for its numera- tor the product of the two numerators, and for its denominator © a the product of the two denominators. If we let F and : repre- | sent the two fractions this rule becomes Ants. 6,7,8] AREA OF A TRIANGLE 7 6. Area of a triangle. In arithmetic it was learned that the area of a triangle is equal to half the product of the base and altitude. In symbols bh A= where A is the area, b the base, and h the altitude. 7. Circumference ofa circle. In arithmetic it was learned that the circumference of a circle equals nearly 3.1416 times the diameter. The number 3.1416 is approximately the ratio of the circumfer- ence to the diameter and is denoted by the Greek letter 7 (pronounced pi). In symbols b Fia. 1 C= 71d, where d is the diameter and c the circum- poet ference. % 8. Area of a circle. The area of a circle is one half the product of the circumference Fig. 2 and radius. Thus, cr A r= 9” where c is the circumference, 7 the radius, and A the area. EXERCISES AND PROBLEMS 1. If A is the area, x the base, and y the altitude of a triangle, write in algebraic symbols the rule for finding the | area. | 2. The base of a triangle is 40 inches and the altitude is 60 inches. Find the area. ) 3. A room is 18 feet long, 14 feet wide, and 8 feet high. ~ How many cubic feet of air are there in the room? 8 INTRODUCTION [Cuap. I. 4. State in algebraic symbols the rule for the length ¢ of the circumference of a circle when the radius r is given. ; 5 5. State the rule for the area of a circle when the radius r and the circumference c are given/’ 6. What is the interest on $300 for one ea at the rate of 6 % per annum. 7. Write in algebraic symbols the rule for finding the interest on a sum of money for one year. Let J represent the interest, P the sum of money in dollars, and r the ae on a dollar for one year. 8: What is the simple interest on $300 for 4 years at 6% per annum? 9. If P is the principal, r the rate, and n the number of years, write in letters the rule for finding the simple in- terest. } 10. State in words the rule of arithmetic for finding the quotient of two fractions. Bigs this rule into algebraic ; as the dividend and © 74 the divisor. 11. If we know the sum of two numbers and one of the numbers, how may we find the other? State the rule in alge- braic symbols. Let s be the sum, a the known number, and a the unknown number. 12. State in algebraic symbols the rule for finding one of two numbers when their product and one of the numbers are given. | 13. Write the rule for finding the altitude a of a rectangle ~ when its area A and its base 6 are given. : 14. The volume of a brick is 64 cubic inches. Its length is 8 inches, its width 4 inches. What is its thickness? 15. Write the rule for finding the thickness ¢ of a brick, when its volume JV, its length /, and its width w, are given. i 16. Write the rule for finding the dividend D, when the » divisor d, and quotient g, are known. | symbols using ers, 610/911 | FACTORS 9 9. Factors. Each of two or more numbers whose product ‘Is a given number is called.a factor of the given number. Thus,;in 2-3 = 6, 2 and 3 are factors of 6; in 2°3° 5 = 80, 2, 3, 9, 6, 10, and 15 are factors of 30. Similarly, 5, 2, y, 52, 5y, and «zy are factors of 5xy. 10. Coefficient. If a number is a product of two factors, then either of these two is called the coefficient of the other in the product. Thus, in 2°3, 2 is the coefficient of 3, and 3 is the coefficient of 2. In 3az, 3 is the coefficient of az, 3a of x, x of 3a, and a of 32. The numerical coefficient 1 is usually omitted. Thus, Lh Os In such expressions as 3az, the factor consisting of Arabic figures is often called the coefficient. | EXERCISES Give factors of 12; of 14; of 21; of 386; of 120. Write six factors of 2ab. Write ten factors of 3abc. Write twelve factors of 62xyz. What is the coefficient of x in 6ay ; of a in fa. . Write down factors of the following and give the coeffi- cient of each factor. 72; 6a; Sry; 8ab; ary; 9xz; abc; 12abc. Dorm wp 11. Exponents and powers. An exponent is a number written at the right of and slightly above another number called the base. When the exponent is an integer, it shows how often the base is taken as a factor. Thus, ae 424° 6 = 5-5-5, Pode Oe y= Bes yy. Y. A power of a number is the product obtained by using the num- ber as a factor one or more times. 10 INTRODUCTION [Cuap. I. ce We may read a™ as “‘a exponent m” or the “mth power of a.” Since 16 = 47, 16 is ‘4 exponent 2,” or the “second power of 4.’’ The number 2° is read “‘2 exponent 5” or the “fifth power of 2.” When the exponent is 1, it is usually omitted. Thus, a! =a. When the exponent is 2, ans power is called the square of the base, and when the exponent is 3, the DO is called the cube of the base. ee a? is usually read ¢ ‘a square,” and a* is usually read ‘‘a cube.” | The meaning of an exponent that is not one of the integers, 1, 2, 3, . . . will be explained later in the course in algebra. "EXERCISES Write the following products using exponents. Read the answers aloud. Lid 2 de 2: 130 G+ oe PERS Fes eB 3 8. 8-a-a@- 2. Of) carey 9. 2 2 - eee 4. ‘27. 10. 2-2-2 Gee Bis eat 11. 81-a-a-272epem 6. 7-3:-3-3:3. 12. 125-2-2-2-o aoe a. Write the following without exponents. 13. 6°. 16. 72. 19. (4). 22. a’. 14. 3°. ii 20. (.02)4. 23. "274". 15. 43. 18/325 21. (2)3. 24. abc. ~ If a= 2, b= 3, and n= 5, find the numerical value of the > following. 25. a’. 28. 6". 310% 34. an. 2G ome: 297 orb. 32. S7o: 35. 37b?n?. 27. 2". 30. ‘BF. 33. 2'n". 86. Gab? + a3. 37. What exponent is understood when none is expressed, . as in a, or 2, or ax? “Arr. 11]. PROBLEMS 11 MISCELLANEOUS PROBLEMS 1. The second of three consecutive even integers is n. What are the other two numbers? 2. How many square inches are there in a rectangle x feet wide and y feet long? 3. A brick whose height is x inches, is twice as wide as it ds high, and twice as long as it is wide. What is its volume? | 4. I take twice as long to walk from my house to the station uphill as to return downhill. If it takes 9 minutes to walk there and back, how long did I take each way? In the following the letters stand for unknown numbers. i Find the numbers and check the results. bare o> = 11. 10. 6. 14=27+ 10. 1 7. 84+2=5+0. — 12. 82 27-= 21: 13; 9. 2n-+- n= / — 3.6. 14. A certain number is multiplied by 3. is 42. What was the original number? Se AN A 4a 2y= 5+ 3. Adz = 5. 7y + 3y = 1. Twice the result 15. Find a number such that if twice the number be added to three times the number, the sum is 100. 16. The sum of two numbers is 60. The greater is 3 times the smaller. What are the numbers? 17. Four times A’s money is $3000 more than B’s. B has $8400. How much has A? 18. A man made a will leaving $10,000 to be divided among 3 daughters and 4 sons. Each daughter was to receive twice as much as each son. What did each son and daughter receive? * Some computers prefer to write 0.15 in place of .15. The form 0.15 places emphasis on the fact that the integral part of the number is zero and removes the decimal point from a place in front of the number to a ‘position where it is not so likely to be carelessly omitted. In writing decimals we shall sometimes use the one form, sometimes the other. , 12 INTRODUCTION [ Crap. I. 19. A man bought a number of baseballs, some at 75 cents each, and twice as many at $1.25 each. He paid $19.50 for the lot. How many of each did he get? 20. A boy has $2.75 in dimes and quarters. He has 3 times as many dimes as quarters. How many of each has he? 21. Which would you rather have, 3x+ 8y dollars, or 5a + 6y dollars (a) if « = 1000, and y = 800? (6) If x = 500, y= 500? (c) If x = 600, y = 700? 22. Write in symbols: The sum of the squares of a and b divided by the cube of c. 23. Write the powers of 10 from the first to the tenth power. 24. The number 343 is ati power of 7? 64is what power of 8? Of 4? Of 2? 25. Write as a power of 12 the number of cubic inches in a cubic foot. 26. Write ten factors of 8a?2’. 27. Write in symbols the weight of a rectangular tank of water. Let a, 6, c be the dimensions of the tank in feet, and w the weight of a cubic foot of water. 28. Using the answer to Problem 27, find the weight of a tank of water, for which a = 2, b = 3, c= 10, and w= 62.5. 29. A man saves 2 dollars the first year. During the second year he saves 100 dollars more than in the first year. In the third year he lost half of his savings for the first two years. What were his net savings for the three years? 30. If V is the volume of a cone, b the area of its base, and h the height, it is known that Give this result in words. 12. Historical note on symbols. The history of the early use of mathematical symbols is very interesting, and shows how mathematical progress was retarded on account of defects in symbolism. To appre-» ciate in a slight way some of these defects, we may well think of doing a ‘An. 12] HISTORICAL NOTE ON SYMBOLS 13 fairly long calculation with Roman numerals I, V, X, L, C, D, M. The so-called Arabic notation that uses the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 is of ‘Hindu origin, the Arabs having obtained it from the Hindus. The great ‘achievement of inventing a satisfactory method of writing numbers was not accomplished in a short time. It required centuries to perfect this elegant system. While the dates of the advances are in doubt, it is prov- able that the system was complete as early as the sixth century of the ‘Christian era. The basic idea in the system is that of assigning a place value to a ‘digit. A symbol for zero is necessary to the application of this notion; but the importance of a symbol for zero was not recognized until long ‘after symbols were being used for other numbers. After the principle of a place value was established, several sets of characters were used. In one of the systems, the circle O was used to denote one and the dot - was used to denote zero. It is held by some that the symbols used for digits fwere perhaps the first letters of early numerals, and that letters were thus used to denote numbers in the earliest attempt at a notation. The symbols +, -, X, -, +, = came into common use in the first half of the seventeenth century, and their origins are accounted for in various ways. For example, it is thought by some that the sign + came from the rapid writing of the letter p in the word plus, and by others that ‘it originated in warehouses in the marking of boxes that showed over- ‘weight. Thus, if a box, supposed to weigh 50 pounds, weighed 55 pounds, it was marked 50+ 5. Whatever may be correct about the origin of these symbols, it is doubtless true that previous to their use, the words ‘for which they stand were generally written out, and it is clear that the symbols add greatly to the brevity and elegance of our statements involv- ‘Ing numbers. CHAPTER, if - ALGEBRAIC EXPRESSIONS 13. Algebraic expressions. In algebra, an expression is a symbol or combination of symbols that represents a number. Thus, gt and x — 2y+ 8z are expressions. If g = 32.2 and | t= 10, the first has the value 322. If x=7, y= 2, and z=@ the second has a value 15. For different values of the letters, the expressions usually represent different numbers. EXERCISES If a=1, b=2;c=5, x = 4, and y = 2, find the value a each of the following algebraic expressions: 1. a+ 2b+4 3¢. 5. a+ 2bx + dy. 3b : | meh a 2. dab + be. 6. abay + Da ’ 3. Sabc + 6a. 7. ab— xy. 4, 2ab + 3ac + 4be. 8. Zab + Say + be — 3az. 9. Find the value of 3a+ 2ab— bc, when a=1, b = 2, endic = 2: 1 10. Find the value of 3a+ 2ab— bc, when a=, b= a anece =o. a iG © ae the value of 3a+ 2ab— bc, when a=7, b= i, and c= 4 14. Order of operations. If, to find the value of the expres- sion, 4+ 5-6, we perform the operations from left to right as | we come to each symbol, we obtain a result 54. If we perform the multiplication first, we obtain the result 34. This simple example shows that results depend upon the order in whi operations are performed. 14 i i 4 J ‘Ants. 14,15] | ORDER OF OPERATIONS 15 __ When nothing is said to the contrary, it is understood that in a series of operations involving additions, subtractions, multiplications and divisions, the multiplications and divisions are to be done in the order from left to right before any additions or subtractions. Then additions and subtractions are to be performed in any order. Thus, | 2056-4450 = 34, and 25 —-2-6-—442417 = 25 —- 12 —-2417 = 28. EXERCISES Carry out the indicated operations. 1. 84-2-34+4+2. 6. 16—4-2-3. 2. 56+ 2-—2-4+6-+ 2. Gre 2. 2 ee 3. 18+ 9 + 2. Te AO 42 4. 18+4-3+ 2. 8. 16-4+2-—4.-2. 9. xyz — 42+ 2y — 32 for z= 8, y= 2,2=1. 15. Use of parentheses. ‘To group the parts of an alge- braic expression together, parentheses ( ) are ordinarily used. Thus, | 3x + 5y + (w+ y) means ox + Sy + E+ Y. When one pair of parentheses occurs within another, it is con- venient to use different forms as follows: [ | called brackets ; {}called braces; and _—_— called a vinculum. These signs are often called signs of aggregation, but all have the same meaning and may be called parentheses. In the simplification of expressions involving parentheses, it is best to simplify first the expression within the parentheses. To illustrate, the removal of the parentheses from 8 + (4 — 2) gives 8+ 2 or 10. The expression 2(5+ 3) means 2 times 8 ; a(b +c) means the product of the number a and the number 16 ALGEBRAIC EXPRESSIONS [Cuap. Il. obtained by adding together 6 and c. The expression (a+ b)(e+d) means the product of the sum of a and b and the sum of ¢ and d. If in (a+b)(c+d), a=2, b=3,,c= 5, d=6, we have (24°3)(5 46) 295 lle bon EXERCISES Simplify the following : 1. 10+ (44 2). 2. 7+ (5+ 2). 3. c+ (d+ 2c). SoLUTION: c+ (d+ 2c) =c+d+2c=d+ 3c. 4. (a+b)+(c+b). 5. x +3(5— 2). 6. «+ 5(7 — 4). 7. (18 — 2) + (6— 2). 8. (8 — 4)(16 — 6). 9. (7+ 3)(7 — 8) + (5-38). 10. (7+ 3)(7— 3) + 5-3. 11. (138 —*6)(18 — 8+ 8 + 4). 12. 88+ 4-10. 13. (84 2)a+ 5(44 2)a. 14, Qa + 3y + 3a + 4y. 15. 2a+604+c+a-+ 204 Ze. 16. 2 [7(2+ 6 — 4) + 244 32]. 17. 2{(8+ 4) (6+1)4+ (24+83)}. 18. 5 [(8+7)4+3 (14+2+8)]. 16. Evaluation of expressions. It is frequently necessary to find the numerical value of an expression for certain values of the letters involved. This process is called the evaluation of the expression. Such evaluation will be used later to test the accuracy of the results of algebraic operations and to check the answers to certain problems. | ' | 4 ‘Anrs. 16,17] EXPRESSIONS IN ONE LETTER Ay t EXERCISES In each of the following, find the value of the expression for a = 3, b = 4, c= 5, d = 2, and simplify the results: Peo. 9. ab+ be — ad. 2. (a+ b)(c+ da). 10. 4ab — cd — d. 3. (a+ b)(c — d). 11. BF — c+ a. 4. 4a°b. 12. 3c? + 2a? — d. 5. 2ab(c — d). 13. 2b+ (5c —d) + (a+b). 6. 6ab— b—c. 14. 6b+ 10bc + 8a — d. 7. (a + 6)? — 2(c — d)?. 15. 4ab? — cd?. 8. 5c + (b— d) + 2(a+ Dd). 16. 5bc? — 16(5a — 30). | _-_—_ 17. If Q represents the number of gallons of water flowing from a pipe per minute, v the rate of flow of the water in feet per minute, and d the diameter of the pipe in inches, it is known that Q = .04vd?. Find the number of gallons of water flowing per minute through a pipe 2 inches in diameter when the rate of flow of the water is 100 feet per minute ; through a three-inch pipe when the rate is 75. 1%. Expressions containing one letter. Algebraic expres- | sions which contain only one letter form a class of great im- portance. It is desirable to note the change in the numerical value of the expression as a succession of numbers is substituted for the letter. Example. Show that the expression, 2? + 12—-— 7x, decreases as x Jakes on the succession of values, 0, 4, 1, 3, 2, 3, and 3. / Souvurton: If z = 0, then x? + 12 — 7x equals 12. If x = 3, then x? + 12 — 7x equals 8}, and so on. Ve can show the change in the expression by a table as follows: 18 ALGEBRAIC EXPRESSIONS [Cuap. II. x a+ 12 =e 0 12 } 8} 1 6 i 34 2 2 5 3 2 q 3 0 eg this table we see that as x increases from 0 to 3, the expression, + 12 — 7x, decreases from 12 to 0. EXERCISES 1. Find the value of 2x +3forz=0; forz= 1;forg= ae fOr 5, 2. Make a table showing the values of 5a — 3 when zx takes on the values, 1, 2, 3, 4, 5. 3. Make a table showing the values of 7 — 3y for y = 0, 2.4; .6,..8, anda: 4, Show that 5z— 1 increases as z takes on the succession of values, 1, 3, 2, 8, 3, 3, and 4. : 5. Show that 16 — 2z decreases as z takes on the same values as in Exercise 4. 6. Make a table showing the eR of 22+2+1, for Ter ae, 4: 7. Make a table showing the values of 2?+ 2 — 2z, for x = 1, 2, 3, 4, 5, 6, 7. Does the expression increase or Rerrenas . 8. Make a table showing the values of 2?+ 2 — 2z, for x= 0, 1, 4, 3, 1. Does the expression increase or decrease for these values of x? 9. If a is the length of one edge of a cube, and V is the volume of the cube, write in symbols the rule for finding the volume. From the result make a table showing the volumes of cubes with edges 4, 1, 13, 2, 25 and so on to 5. | 10. If $100 is it at 5% simple interest for a number of years it amounts to 100 + 5n dollars if n is the number of years. ; “Arts. 17,18] GRAPHICAL REPRESENTATION 19 From this make a table showing how $100 increases every year up to 10 years. 11. Show in symbols how $100 increases if cae at 6% simple interest. Tabulate the results as in Problem 10. 12. If v represents the velocity of the wind, and P is the pressure of the wind on a pane of glass one foot square, then it is known that P= 107 + 225, ‘Find the pressure when the wind is blowing at the rate of 15 miles per hour ; 20, 30, 40, 50 miles per hour. 18. Graphical representation. The way in which an ex- pression involving one letter changes “as the value assigned to ‘the letter is changed, can be represented to the eye by a dia- gram. This is shown as follows for the expression 2x + 3. On the horizontal line of Fig. 3 mark the numbers 1, 2,3,4... Make a table of values for 2x +3. Thus we have: | Pij2i3] 4] 5] Sa K 17 | 9 | 11} 18] 15 “At the point marked 1 draw a vertical line of length 27+ 3 when x = 1; that is, of length 5. At the point marked 2 erect a 0 I 2 3 4 5 6 Fiaes ‘vertical line of length 2x + 3 when xz = 2; that is, of length 7. Proceed in the same way for each value of x given in the table. a 4 20 ALGEBRAIC EXPRESSIONS [Cuap. IL. The diagram will then present to the eye the way in which 27 + 3 increases as X increases. Any table of numbers, whether representing an algebraic expression or not may be represented to the eye in a like manner. For example: During the opening day of a new shoe store a record was kept of the number of customers for each hour of the ten hours the store was open. | | Hour | Number of customers | HH) bo 11 2 This table is represented in Fig. 10-7 LD 8 E 7 =. 6 iS) 3 5 2 g4 a 3 2 af i we 3 4 5 6 7 8 a 10 Hours during which the Store is open. Fia. 4 In these diagrams any convenient unit can be used for the lengths of the vertical lines. EXERCISES 1. Show by a diagram the values of 5% — 3 when z takes on the integral values 1 to 6. Represent by a diagram the following expressions when x takes on the integral values 1 to 6. AE i Neb 4. 2P+a2+1, 3. 3x — 1. 5. v4 2— 2a. 6. The morning temperature record of a fever patient is given by the table: (Ber. 18] GRAPHICAL REPRESENTATION 21 _ [Morning [1 415 16171819 temperature 1.8 | 1 | 9 | 6 | 0 3 | Ts 2 Represent this table by a diagram. 7. The weights of a baby weighing eight pounds at birth for each month of the year are given by the table: Month |1 |2 |3 Weight | 93 tbo | Paes | eric s. ot 10 pli 142 | 153] 16] 18] 19] 192 | 20] 21 | 22 Represent by a diagram. 8. The heights of the same baby for the same months are 19 inches at birth, then 203, 21, 22, 23, 235, 24, 245, 25, 253, | 26, 263, 27 inches. ae By a ae EXERCISES AND PROBLEMS 1. The horse power of a certain style of gasoline engine is given by the expression | H=<.DN} where Z is horse power, D the diameter of the cylinder in inches, and N the number of cylinders. Find the horse power of a two- cylinder engine with 5-inch cylinders; with 6-inch cylinders. 2. Write in algebraic symbols: The volume of a sphere is four thirds the cube of the radius times 7. 825. 27—- 3° + 3=? 4.273) — 18-5 Sia 5. In the formula s = 16.12, ¢ represents the number of seconds a body has been falling, and s.represents the distance it has fallen. How far will a stone fall in 4 seconds? 6. Give the formula of Problem 5 in words. 7. Make a table showing the’ distance through which a stone will\fall in 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 seconds. Find the values of the letters in the following: Se oy=.25; - 10. 32 = &4. 9. 4+ a= 19. Hea ot 27 =120: * It is not expected in this and similar problems that the teacher will take the time to explain the physical principles involved. 22 ALGEBRAIC EXPRESSIONS [Cuap. IL. 12. A has $550 and B $150. How much must A give to B in order to have just twice as much as B? 13. If a is equal to 2 times b, express 6a + 36 in terms of 3 14. Write 6 factors of 2a(a + b). 15. 42? — 32°+ 5a?- w=? 16. If x is 3 greater than y, express 7x + 2y first in terms of x and then in terms of y. 17. Find the value of z{[(#+ 1)? y—1)]+ a when e=i2and ya 4. 18. 7 (a— b) — 2(a— b) — 38(a — BD) = (2) - (a — DB). 19. Write in symbols: The square of the sum of a and 6 divided by their product. 20. Find an expression for the surface of a cube if one edge is given. From the result make a table showing the surfaces of cubes with edges 1, 2, .. . to 6. 21. A bag contains x white balls, three times as many black balls as white balls, and twice as many red balls as black balls. How many are there of each color if there are 40 balls altogether? 22. Given that 254 centimeters equals 100 inches very nearly. Using c for number of centimeters and 7 for number of inches, show in symbols the relation between inches and centimeters. Tabulate the result up to 12 inches. 23. Show by a diagram the temperature record of a fever patient. The degrees of fever recorded were 2, 5.4, 4, 6.1, 4.5, 6.5, 5.3, 6.7, 4.5,.6.5, 5.9, 6.2, 7, 5, 6.7, 5.8, 6.107 24. The following table gives the length af a child’s. bare foot when the size of the shoe is known. ae of shoe Show by a diagram the relation between size of shoe and length of foot for children. CHAPTER III POSITIVE AND NEGATIVE NUMBERS 19. The use of a scale to represent the numbers of arith- ‘metic. The numbers of arithmetic may be represented by points on a straight line. Let OX be such a line (Fig. 5). Adopt a unit of measure, AB. Begin at O and divide OX into intervals of length AB. ‘We may thus obtain a scale of indefinite length. If the ends of the intervals are marked 0, 1, 2, 3, 4, . . . as in the figure, sei. oss 45> 6. y.8 8 Fiq. 5 we have a point on the number scale corresponding to each of these numbers. Fractions may also be represented by points on this line. Thus, there corresponds to. the number 4 a point midway between the points marked 0 and 1; to 33a point 4 of the dis- tance from 3 to 4; and, in the same way, for every fraction there corresponds a point on the scale.’ 20. Addition and subtraction on the scale. Addition and subtraction may be performed on the number scale. Thus, to add 3 to 4, begin at 4 and count 3 spaces to the right. To subtract 3 from 4, begin at 4 and count 3 spaces to the left. ; In general, to add a to b, begin at b and count a spaces to the right ; and to subtract a from b, begin at 6 and count a spaces to the left. 23 24 POSITIVE AND NEGATIVE NUMBERS [Caap. IIL EXERCISES 1. What number is represented by each of the following points? (a) The point 3 spaces to the right of 5. (b) The point 4 spaces to the left of 11. (c) The point midway between 4 and 5. (d) The point midway between 33 and 4. (e) The point one-third of the distance from 2 to 3. 2. State in words the positions of the points that represent the numbers 3, 4, %, %, 451, 0. 3.-On an ordinary ruler perform the following additions and subtractions: (a) 542. (e) 24+3-4. (h) 23+ 1. (b) 5-2. (f).7-2-3. (4) 8—@= 4; (c) 5—5. (9) $+2 (7) Ga ae} (dq) 24+344. 21. Positive and negative numbers. If, on the scale in Fig. 5, we attempt to subtract a number from a smaller number, say 6 from 4, we get off the scale. Let the scale be extended to the left of 0, Fig. 6. If we now attempt to subtract 6 from 4 by counting 6 units to the left of 4, we arrive at a point 2 units to the left of 0. Awe 3 -11-10-9 -8 -7 -6 -5 -4-3-2-1 012 3 45 67 8910 Fia. 6 The numbers that correspond to the points to the left of 0 are called negative numbers. The numbers of arithmetic, which are represented by points to the right of 0, are called positive numbers. The minus sign is used to distinguish the negative from the positive numbers. On this scale then, 4—6= -2. Ants. 21, 22, 23,24] POSITIVE AND NEGATIVE NUMBERS 25 Similarly, : 7-—10= —8, and 2—- 8= -6. The essential thing in the above representation of positive and negative numbers is that they are marked off in opposite directions. In Art. 1, we observe that the fraction is introduced to make division possible when the quotient is not an integer. In a similar manner, the negative number is here introduced to make _it possible to subtract from a number a greater number. 22. Illustrations. The student has had experience in arith- ‘metic with quantities measured in opposite directions. ' The thermometer gives a simple example. Temperatures ' above and below zero are measured in opposite directions. It is simple and convenient to regard temperatures above zero as positive and those below zero as negative. If the temperature is 10° above zero, or +10°, and falls 30°, we say it is then 20° below zero, or —20°. This result is obtained algebraically by saying 10° — 30° = —20°. North latitude is commonly called positive and south latitude negative.. If a ship is in latitude +20° and sails 30° south, it is then in latitude —10°. Algebraically, 20° — 30° = —10°. Positive and negative numbers are often called \signed numbers. \ ; +238. Numerical or absolute value. The numerical or ab- solute value of a number is its value without regard to sign. The absolute values of —10, +380, +6, —9, are 10, 30, 6, and 9, respectively. 24. Greater and less. The expressions greater than and less than which are common to every day life, when used in the precise sense of algebra, are easily misunderstood. For this reason, we point out their meanings on the number scale. A 26 POSITIVE AND NEGATIVE NUMBERS [Caap. III number a is greater than a number b when a is represented by a point on the scale to the right of that representing b. Thus, 1 is greater than —4; a temperature of —10° is greater (or higher) than one of —20°. Fia. 7 Exercise. Explain by the use of points on the number scale what is meant by the expression ‘‘a is less than b.” EXERCISES AND PROBLEMS 1. Locate on the number scale: —5, —2, 0, —8, —%, +5. 2. Find the point that corresponds to the following dif- ferences: 5—11; 7-7; 4-45; 6-5; 0-4. 3. Using temperature above zero, north latitude, west longitude, and assets as positive quantities, represent the following as signed numbers: 12° above zero; $10 debts ; 30° west longitude ; 40° south latitude ; $100 assets ; 1° below zero. 4. A goes 6 miles west, then 18 miles east, then 5 miles west, then 2 miles east. Using distarice east as positive, write the above distances as signed numbers. How far and in what direction is he from the starting point? Show how to find the answer by using the number scale. 5. If the temperature is +10°, falls 13°, and fen rises 5°, what is the resulting temperature? 6. The temperature is now 60°. What will it be after it falls: .(a) 20°; (6) 60°; (c) 70°? 7. If the temperature is —10°, what will it be after it rises: (2) 529,(b) 102 ae) 20g 8. Give the absolute values of +8, -—10, -—%, +443, —8.5. 9. What number is one greater than —8? What number is one less? | “Ant. 24] EXERCISES AND PROBLEMS 27 - 10. Show on the number scale the distance between (a) “+7 and +4; (b) —7 and —4; (c) —7and +4; (d) +7 and —4. 11: Ina game, A loses 10 points, gains 7, gains 31, loses 14, loses 36, gains 5. What is his final score? 12. A has $375 assets and no debts ; B has $200 debts and no assets. Consider debts as negative assets. What is the sum of their assets? How much more assets has A than B? 13. What should be taken negative if the following are con- ‘sidered positive? (a) West longitude. (f) Cubic inches of expan- (b) Dollars gain. sion of a toy ballon. (c) Points by which a game (g) Excess of water pumped is won. from a well over that (d) Miles northeast. flowing. into it in the (e) Excess of persons enter- same time. ing a room over those leaving the room. 14. What would be the meaning, if any, of a negative result in finding — (a) How much money you had gained in a trade? (b) The score by which your football team won the last - game of the season? (c) How much more a cubic foot of a certain liquid weighs than a cubic foot of water? (d) The rate at which a lion ran away from a hunter? (e) The increase of the population of a town in 10 years? (f) How far Philadelphia is west of New York? Historical note on negative numbers. The extension of the concep- tion of number to include the negative was an exceedingly slow process in the development of algebra. The Hindus recognized the exist- ence of the negative in working at the quadratic equation perhaps as early as 800 a.p., but they had little to say about such numbers except that the people did not approve of them. It was not until the work of Des- cartes (1596-1650) (see p. 182) that the rules for operation with negative numbers were at all well understood. 28 POSITIVE AND NEGATIVE NUMBERS [Cuap. IL (g) The height of the bottom of a well above sea-level? (h) The age of Washington at the time of the surrender of Yorktown? | 25. Addition of signed numbers. ‘Thus far it has not been indicated what is meant by the sum of two numbers when one or both are negative. Some illustrations will suggest how such sums should be defined. | Example. How many steps and in what direction from the starting point is a person who takes — (a) 8 steps east and then 5 steps east? (b) 8 steps west and then 5 steps west? (c) 8 steps east and then 5 steps west? (d) 8 steps west and then 5 steps east? So.tuTion: Let steps east be considered positive and steps west nega- tive. The answers may be obtained by counting on a scale. (a) To add +5 to +8 we begin at +8 and count 5 units to the right, arriving at +13. (b) To add —5 to —8 we begin at —8 and count 5 units to the left, arriving at —13. (c) To add —5 to +8 we begin at +8 and count 5 units to the left, arriving at +3. (d) To add +5 to —8 we begin at —8 and count 5 units to the right, arriving at —3. Hence to add a positive number we count to the right, and to add a nega- tive number we count to the left. The results just obtained may be stated as follows: +8 + (+5) = +18, Sid 8 eee sie +8 +(-5) = 43, —-8 + (+45) = -3. - The foregoing discussion suggests the following definitions of sum when signed numbers are added : The sum of numbers with like signs is the sum of their absolute values preceded by their common sign. The sum of numbers with unlike signs is the difference of their absolute values preceded by the sign of the one having the. greater absolute value. (ARTs. 25, 26 | il OF SIGNED NUMBERS 29 EXERCISES Perform the following additions by counting on a scale: * ‘me ) 10 ee G27 13) as ie oS t o 12 4 ~9 on 2 4 2;..—10 Ge.eh —6 1 2 ee! ek s i wae 3. —10 7. —2 —§ 14. 9 16 10 7 ly. = ~6 —12 —3 4 4, 10 8. 4 12. -—9 4 6 = =5 4 a a Answer the following questions : ast f= 13. 22. —4=-13+7 18.-?+8= 6. 23. ?+6=0. Ieee) = 7. 24. —124?=0. omen? — 12. $5. 3=-7+? Civetew = —10. Find the value of x in the following : 26. (7+ 8 = 12. 31. -—-l-—2z=-3. fe i= 3. 32. 20+2z=0. 28. x-—11=-8. 33. 13-—2=0. 29. 4-—2z=-16. 84. 2+(-—11) =15. 30, 1b= xz — 9. 35. 2-2 = —2, 26. Subtraction of signed numbers. As in arithmetic, subtraction is the process of finding one of two numbers when their sum and the other number are given. 6 because 6+4 = -14 because -14+4 = —10. 14 because 14+ (-4) = — 6 because -6+(-—4) = -10. Thus, +10 —- (+ 4) Pigeit+ 4) +10 — ( -4) 10= (4) 30 POSITIVE AND NEGATIVE NUMBERS [Cauap. III. EXERCISES Fill in the blanks in the following : 1. 47 = (+44) 2) eines (0) aaa 2. —7—(-—4) =(), since (—4)+ ( ) = -7. 3. +4-— (+7) = ( ), since (+7) + ( ) = +4. 4. +7— (-4) = (), since (—4)+( ) = +47. In a similar way perform the following subtractions : & 3-— (42). 13. 8 — (+48). 6. —3— (42). 14. 7-— (+49). 7 —3— (—2). 15. O-—(-—?7). 8 §—(—5d): 16. O- (47). 9, —4—(—4), 17. 27-—(-9). 10. -—9— (10). 18. —8—(+18). 11.-.—] — (+2): 19. 30-— (-8). 12, —1—(-lI). 20. —8— (+30). Answer the following questions : 21. ABs 25. 8—- ?=<-5. 22. ?—6=0. 26. —7= fas O35 7 LP = tae 27. —-5-—14=? 24, -—8—2=4. 28. —10— ?=5. 2%. Subtraction on a scale. We have seen, Art. 20, that to subtract a positive number on the scale we count to the left. Thus, in Art. 26, the result in Exercise 1 can be found by beginning at +7 and counting 4 spaces to the left, arriving at +3; and the result in Exercise 3 can be found by beginning at +4 and counting 7 spaces to the left, arriving at —3. We can also subtract a negative number by counting on the scale. The answer to Exercise 2, Art. 26, can be found by beginning at —7 and counting 4 spaces to the right, arriving at —3,; and the result in Exercise 4 can be found by beginning — at +7 and counting 4 spaces to the right, arriving at +11. Arc. 28 | _ RULE FOR SUBTRACTION ol _ We have then the principle: To subtract a negative number, count to the right the number of spaces indicated by the number. Exercise. Find the answers to Exercises 5-20 in Art. 26 by counting on the scale. 28. Rule for subtraction. Any problem in subtraction may be changed to a problem in addition. Examples. —© +10 — (+5) = +10+(-5) = +. 5. =10\— (+5) = -104+ (45) = — 5. Sere nyt 10 (5) =. ~ 15. +10 — (—5) = 410+ (+5) = +15. ry 4+ 1 Od We then have the working rule: To subtract any number change its sign and add the resulting number. EXERCISES Perform the following additions : al an 1g Toten Ba hacanl » eit ~9 2 3 EF 3 sly ai "eal Rad 62> .,-9 ) os ~8 pals pe 101 = ree 2: 7 3. —11 Goyal ¢ 6 4 ot eld -3 —21 Perform the following subtractions : ao —. 1. 14. 9 — 17. 17. —3 —(-14). seat): 16,9 — (—17).. 18. TT —/18. ots 7-8. TGA Dita hdd ata 19 pers Bh 3 (224), o2 POSITIVE AND NEGATIVE NUMBERS [Cuap. IIL. Answer the following questions : Sos See a7. -7— (2) 2 or R183) aoe foe yee ) pie Be 29, —17.— (1 GSO yess ae 1) 4. 1-24+34+?7?=4 $81. 104 2s 25. —21+11-—?=14. 32. ?+(-7) —(-7)4+7 =—7. 26. —-3 —? = (-7) =0. 83. —.14 .5—- (-.6) —-Q@ =l1. EXERCISES AND PROBLEMS 1. From the sum of —2$, —381, and 123, subtract the sum of 3, 5, and 84. 2. What is the difference between the temperatures of +85° and +40°? Of +36° and +14°? Of +438° and - 209 Of —43° and —51°? Of —24° and —6°? 3. If a man is worth $20,000, how much must ie lose to be in debt $8400? 4. B has assets of $300, $100, $800, and $1000, and debts of $200, $400, $100, and $600. Let assets be represented by positive numbers and debts by negative numbers. What is the net value of B’s property? What would be the net value of his property if (a) the courts should cancel the debts of $300 and $100; (6) he should lose $400 worth of property by fire; (c) he should gain $400 in a trade? 5. Suppose that weights to the amount of 300 pounds are attached to a balloon which pulls upward with a force of 400 pounds. Let pull upward be regarded as positive and pull downward (weight) as negative. What is the net upward pull? What would be the net upward or downward pull if (a) 200 pounds of weights were removed; (6) gas with a lifting force of 200 pounds were added; (c) 200 pounds of weights were added; (d) gas with a lifting force of 200 pounds were removed? 6. In the difference a — 6 = 6, the value of a runs through, the series of whole numbers from 10 to 0. Find the correspond- Anr. 28] EXERCISES AND PROBLEMS 33 ing values of 6. Make a table showing these corresponding values of a and b. 7. In the difference x — y = —4, the value of y runs through the series of whole numbers from —5 to +5. Find the corresponding values of x, and make a table as in Exercise 6. In the following problems + denotes A.D., and — denotes B.C. 8. The oldest mathematical manuscript known was writ- ten by an Egyptian named Ahmes. The date of the manu- script is thought to be about —1700. How old is it? 9. The Greek geometer, Euclid, lived about —300. How Jong was that before the birth of the French geometer, Des- eartes, who was born in +1596? ~ 10. Some characters from which our present numerals are thought to have developed, are found in inscriptions made in India as early as —250. The first undoubted use of the zero in India is said to have been in the year +876. How many years between these dates? 11. Archimedes was born in the year —287, and died in the year —212. How old was he when he died? 12. Through how many degrees of latitude does a ship sail in going from latitude —21° to latitude —56°? 13. Through how many degrees of longitude does a ship sail in going from longitude —17° to longitude +35°? State what single change will produce the same result as the changes mentioned in each of the following exercises. 14. The temperature rises 20°, falls 13°, then rises 6°. 15. An elevator goes up 60 feet, goes down 24 feet, goes up © 48 feet, and then goes down 84 feet. 16. A traveler goes 350 miles east, and then 420 miles west. 17. In a race a runner gains 12 feet on his opponent, loses 7 feet, gains 2 feet, and loses 9 feet. 18. A political party gains 724 votes, loses 328, loses 35, and gains 120. 34 POSITIVE AND NEGATIVE NUMBERS [Cuar. IL 29. Addition and subtraction of several numbers. In add- ing and subtracting several numbers we may proceed from left to right performing each addition and subtraction as we come to it. For example, ¥ +3 + (—2)—3-— (-1) = 7+ (—2) 3 Sia 5. Sean -2—-(€9) at It is usually shorter, however, to remove the parentheses, unite the positive terms and the negative terms, and then com- bine these results. Thus, 4+3 + (-2) —-3 —(-1) =443 —-2-34] =8-5 sro. The second method is usually more convenient when the numbers are written in columns. . EXERCISES Perform the following additions and subtractions: 1.62—-3+4+546-7-48. 2. —2 — (-3) + (-4) — (—5) + (-6) - (-7). 3. 1 + (—2) — 34 (—4) - (5). 4. 2a — 3a +a — 4a + 6a. 5. 54 — 64 + 2 — Tx + 2x — (42). 6. —1-— 2- 3- (-4) — (—5) — (-6). 7 —8 9. —59 11. —5m 13. ory a +7 +21 +9m —4.6xy —3 +22 —4m —.lry —11 Bak: +7m 5.22y 8. —17 10. +33a? 12. —45y 14. ee —25 —18a? —I13y 2a +40 —21a? —20y — —2a +43 +6a? +50y —a a ; "Ants. 29, 30,31] MULTIPLICATION IN ARITHMETIC 30 aie 15. 8.5 18. 3(2)? 21.. $2 24. —Txry —.75 5(2)? —2Qx gry _ 05 _(2/ _ 4a —rsty 16. —3.00 19. 4a 22. Ray? 2028 42 —8a — ery? P: —.01 _ 2a SR ex a | $3 a G 7.3 20. Sab 23, 4m 26. «+57 —4-3 —4Aab am —2.7 2:3 —ab —im —9.7 30. Multiplication in arithmetic. So long as the multiplier is a positive integer, multiplication may be defined as the pro- cess of finding the sum of a number of equal numbers by the ‘use of the multiplication table. The result in a multiplication is called the product. This definition of multiplication has no meaning when the multiplier is a fraction. Thus, to say that the product, # x 2, means finding the sum of ? equal numbers has no meaning. Hence, an extended definition of multiplication must be made when the multiplier is a fraction, and we say that the product of two fractions is the product of their numerators divided by the product of their denominators. 31. Multiplication of signed numbers. It is necessary again to extend the definition of multiplication in order that the products of positive and negative numbers shall have a meaning. For example, we are to define what is meant by such, pro- ducts as 3- —4, -—3-4, and -—3- —4. We have said that 3-4 =4+4+4=12. Using the same meaning of times, we say that 3 - —4 = (—4) + (-4) + (-4) = -12, 36 POSITIVE AND NEGATIVE NUMBERS [Cauap. IIL which extends multiplication to the case of multiplying a negative number by a positive number. Since to multiply by +3 we add, it seems reasonable to say that to multiply by —3 we subtract ; and we say that —3°4 means that 4 is to be subtracted 3 times. That is, —3:4= -4-4-4 = -12. Similarly, —3-—4 means that —4 is to be subtracted 3 times. That is, =3,-4=-(-) — Cj = Ach du Ape In general symbols this discussion may be summed up as follows : OME hopcefs))° a- —b = —ab, —a-b = —ab, —a-—b = ab. We may then state in words the Rule of signs for multiplication. The product of two num- bers with like signs is positive, and the product of two numbers with unlike signs is negative. It follows from the law of signs that a product is positive if it contains an even number of negative factors, and is-negative if it contains an odd number of negative factors. Thus, —2:3.-1-—5 ene and -1.4:-2+ 8.6. 22. EXERCISES AND PROBLEMS Perform the following indicated operations : 1. 4- —5. 5. 4-—2. 2. —4-5. 6. —2-—-a. 3.. —1 - —2. 7. 8-0. 4. —7-—-6. 8. —2--3-4. Arr. 31] EXERCISES AND PROBLEMS 37 9. 4--5:-6 17. 8--9--5-2. 10. 3-0: —2. 18.03 '°457 52 Oa et - 7: —3-2 19. (—2)?. 12. —2--a:7z. 20. (—1)°. mo 2-345: —4 21. (—3)*. 14. 3-104 (-3- -8) 22. (—2)?- —d. 15. —a-b-x-m x 93.0 (1) - (2)? (23)F. 16. —-2--3-4-2 24. (—a)*. m 25. -15 = —5-3 =5- —3. Find two pairs of factors of —21; 14; 95; -33; —ab; xy. 26. If y = 2x — 4, find the values of y when «x takes on the ‘integral values from —2 to 6. Represent the values of y by a diagram. SoLUTION: Pee oe iO Ly 2 | 8:)-4 1.5 | 6 -8 | -6 |-4 |-2|0[2/4/6|8 The corresponding values of x and y are given in the table. The diagram (Fig. 8) is made as in Art. 18. Notice that when the values of y are negative the lines representing those values extend downward. Fic. 8 27. If s = @, find the values of s when ¢ takes on the integral values from —4 to 4. Make a table showing the corresponding values of s and ¢, and represent these values in a diagram as in Exercise 26. ' 98 The formula C = 3(F — 82), gives the temperature, C, on a centigrade thermometer in terms of the temperature, 38 POSITIVE AND NEGATIVE NUMBERS [Cuap. III, F, on a Fahrenheit thermometer. Find C when F has the val- ues —20°, —10°, —5°, 0°, 5°, 10°, 28°. Make a diagram repre- senting these values of C and F. 29. If y = 2 — 62, make a table of the corresponding values of « and y, when x takes on the integral values from —2 to 8. For which of these values of x does y have the least value? The greatest? For what values of x is y equal to zero? Between what values of « does y increase? Decrease? 32. Division of signed‘numbers. Division is the process of finding one of two factors when their product and the other factor are given. Thus, 20 + 4 = 5, since 5° 4 = 20. The given product is called the dividend, the given factor the divisor, and the factor to be found the quotient. The application of this definition gives results as follows: +14 * F ra +7, because +7-+2 = 414; —14 =n +7, because +7--2 = —14; is = —7, because —7--2 = +14; —14 aes —7, because —7-+2 = —14. We have then the Rule of signs for division. The quotient of two numbers with like signs is positive, and the quotient of two numbers with unlike signs is negative. | In algebraic symbols, this rule is, +a a Fees; Bn =} aeagne ‘Arr 33] . FRACTIONS 39 sete ees aj Ba Me de Sse Y, 33. Fractions. A fraction is an indicated division. Thus, #means3 +5. The terms of this fraction are 3 and 5, 3 being the numerator, and 5 the denominator. In arithmetic we had the principle: The numerator and the denominator of a fraction may be multiplied or divided by the same number without changing the value of the fraction. This principle holds for the numbers of algebra, and is of great service in simplifying fractions. EXERCISES Perform the indicated divisions and check the results by multiplication. ee Eee 0d Rosie ad tk | 19° nee ty, A. eee ay 2 Ra 8 2 eee 1) SO. Sesnh = G. a Bes ot a 14. wals 18. AY meine 2 10.<—ab ~ a. —3 —2--3 oy ae ley ae be ioe Be Sg —¢ —* 2 . >i ere ‘Fe ahaa bs OY, ata —2X xy 2 Simplify the fractions : | a 4a = 102 ahs ates ys 12° 25. os ; =8-5 122 ae 22. Sans. 24. re 26. ab? 40 POSITIVE AND NEGATIVE NUMBERS [Cuap. IIL E 1 4 cy peasnasie 30. 4: Pegged vo = 4a Freee te ai ee sae 2 v DI 8 7x 3a+7 29. 32. or ae 36. What number divided by 2 equals 8? 37. Find 2 if 5 = 8. 38. What number divided by —3 equals 4.5? 39. Find 2 if —- 4.5. In the following exercises, the letters stand for unknown numbers. Find the numbers. x 6 48 40. 6 = 2. 45. 3a = 7 50. ye = 6. x 42 41. 7, - 2: 46. eS S68! Bi. = = -3. Lo HET as 7 ee 42. 5 = 5 47. = -5 52 3 = —0.08 Age ee iGo ee 5 8 44, = 5. 49. : eS MISCELLANEOUS QUESTIONS AND EXERCISES 1. How much greater is 9than 4? 9than —4? 52 than -2x? athan 7? 2. Write in the form of a fraction the quotient of 17 di- vided by 20. Also the quotient of a + b divided by m; of the sum of the cubes of a and 6 divided by the sum of a and 6; of. n divided by a number 6 less than 5 times n. J ‘Art. 33] MISCELLANEOUS QUESTIONS 4] 3. What is the meaning of a®? Of 6a? Find the differ- ence between a® and 6a when a has the values —2, 0, 2, and 3. 4. Write the following using exponents: 2-7-7-7-7; S-m-n-n-mM-mM; L-¥- yz LY 2-2-2 8-4-5 -4-5-3-3, 5. How is the sum of numbers with like signs found? Of numbers with unlike signs? 6. When is the remainder in subtraction greater than the minuend? | 7. Perform the following subtractions on a number scale : 8-9; -4-7; 8-—(-8); -5-(-6); -—7—-(-12). 8. What is the definition of multiplication when the mul- tiplier is a positive integer? To what other cases have we ex- tended the meaning of multiplication? 9. Give the rule of signs for multiplication. 10. Find the following products by adding or subtracting : Mee 4- 4-9; —5- —8. - 11. What is the absolute value of .—45; 34; —34; .07; —3? 12. Define division. State the rule of signs for division. 13. In 6abz, state the coefficient of abx; of x; of ab; of 6a; of b. 14. Give two illustrations showing how algebraic symbols may be used to shorten the statement of certain rules of arithmetic. 3 ee A af = an a find the corresponding values of y when x takes on the values —3, —2, —1, 0, 1, 2, 3. . 16. If xy =1, find the corresponding values of y when x takes on the values —2, —1, —.1, —.01, —.001. Are x and y increasing or decreasing? CHAPTER IV EQUALITIES 34. Members ofan equality. A statement that one expres- sion is equal to another expression is called an equality. The two expressions are called the members of the equality. Thus, in 5+ 38% =42, 5+ 32 is said to be the left-hand member and 42 the right-hand member of the equality. 39. Identities. There are many different ways of writing the same number. Thus, 5 may be expressed as 7 — 2, 8 — 3, or 2+ 2 +1, and in many other ways. Likewise, an algebraic expression may be changed in form and still represent the same number. Thus, 5% — 4 and 2% + 3x — 4 represent the same number no matter what num-— ber is represented by x. An equality in which the members merely represent dif- ferent ways of writing the same number is called an identity. Thus, the equalities, : 5a — 4 = 274 324 -— 4 (1) and | x+ 2% = 32 maa are identities. Since the members of an identity are simply different forms of writing a number, the members are equal for all values of the letters involved. For example, the members in (1) and (2) are equal no matter what number we substitute for x. Try moet 253. andl, EXERCISES 13 Show the 28 ia ae when a = 1, a = 5, a=0, and a = 10. 42 ‘Ant. 36] . EQUATIONS 43 2. Show that 3a + 2b =a+a+a+b-— 2b + 3b, if @= le -and 6 = 2: @=5 andb = 4; a = —1 and b = —2. =2 +2, whens =0,2=1,% =3,7=05, 8, Show that ~—< x— 2 and « = 10. 36. Equations. An equality in which the members are equal only for particular values of the letters involved is called an equation. Thus, the equality x — 1 = 2 is an equation; for, the mem- bers are equal only when z = 3. EXERCISES 1. Show that 7x —4=10 when x = 2, but not when @=1,orz = 3, orz = 4. | 2. Show that 27 + 4 = 3% + 1 when 2 = 8, but not when ae 2. 3. Show that 4% + 6 = x + 3 when x = —1, but not when — 1. Which of the following equalities are identities and which are equations : 4.2% =%2. 5B. oa + 42% = 7x42 -1. Hint: Try x = 2, and z = 1. 2 6. ao 7 7 22 =A. 8. 3a4+1=27+7+1. 9. 2? + 27 = -1. Hint: Try x=0. | 10. 242% =27?+ 34-72. a. 2° = 4. fee — or + 2 = 0. 44 EQUALITIES [Cuap. IV. 13. 4 SS, 14. 3a 4+ 2y = 3y —y+ 24442. 3%. Solution of equations. In an equation, a letter whose value is to be found is called the unknown letter or simply the unknown. To solve an equation is to find values of the unknown which when substituted will reduce the equation to an identity. Such a value of the unknown is said to be a solution or root of the equation. When an equation is thus reduced to an identity, it is said to be satisfied. Thus, 1 is the solution of x +1 = 2; for, if 1 is substituted for x, x + 1 = 2 is satisfied. EXERCISES Show that 2 is a solution of 2x7 — 3 = 1. Show that 3 is a solution of 4% + 1 = 13. Show that 0 is a solution of 4x + 2a = 7z. Show that — : is a solution of 7z + 9 = 5. foe Show that 2 and 3 are solutions of z? — 5a + 7 = 1. hehe Lact Cen Solve the following equations : 6. Pe 8. +—4=2. 7. 8+2=4. 9. ++1=5. 38. Principles used in solving equations. The value of the unknown in an equation is unchanged by the following : (1) Adding the same number to both members. Thus, if 2 be added to each member of 2 + 3 = 5, we have -~@+5=7. Both equations have one and the same solution 2. (2) Subtracting the same number from both members. Thus, if 2 be subtracted from each member of x + 3 = 5, we have x + 1 = 38, and both equations have 2 for the value of : the unknown. ED “Ane. 38] VERIFICATION BY SUBSTITUTION 45 (3) Multiplying both members by the same number other than ero. Thus, if we multiply both members of « +3 = 5 by 2, we | have 2x + 6 = 10, and 2 is the value of the unknown in both equations. The necessity of excluding zero as a multiplier may ‘be seen by multiplying by x each member of +3 =5. (1) ‘This gives 2 + 8x = 5a. (2) When zx takes on the value 0, equation (2) is satisfied, but 0 is ‘not a solution of (1). (4) Dividing both members by the same number other than zero. | Thus, if we divide both members of 2x + 6 = 10 by 2, we have x + 3 = 5, and 2 is the value of the unknown in both equations. Division by zero is excluded from algebra. ° Thus ‘ has no ‘meaning, and we should avoid attempting to divide both mem- ‘bers of an equation by an expression that is zero. To illustrate briefly the use of the above operations, solve the equa- tion 5z — 9 = 6. SoLUTION: 5r —9 = 6. Add 9, 5r -9+9 =6+9. Collect, be = 15, Divide by 5, f=3. 39. Verification of solutions by substitution. The above operations (1), (2), (3), (4), Art. 38, are useful in finding values of unknowns, but the solution of an equation is not complete until the value of the unknown found has been substituted in the equation to test the result. This is called checking by substitution or verifying the result. ¥ 46 EQUALITIES [Cuap. IV. Example. Solve the equation 32+ 10 = 28. (1) SOLUTION: az +10 = 28. \ (2) Subtract 10, 3x + 10 — 10 = 28 — 10. MG (3) Collect terms, 32 = 18. 8 Divide by 3, x = 6. CHECK: 3°6+10 = 28, 28 = 28. EXERCISES Solve the following equations and check the results: sR ae Ps ees 11. 9y = 45 + 4y. Te ets en 12. 1l3y = —dy + 36. 3.°2.4+3 = —Dd. 13. 54 +4 = —22+4 10. 4. x—3 = —5. 14. 5n —4 = 3n +18. 5. 7+4=9. 15. 8 — 62 + 12 + 82 = 24. 6-2 —'3:=' 12. 16. 372 +5422 —1 = 24. 7% 227+5=2-4. 17. +27 +10+2+422+10 = 140. 8 31+7=27-11. 18 32-3442 —- 16 = 68. 9. 524 +6 = 2x — 5. 19. 27 + 7 — 32x = 10. 10, 32 +.3 = 22 —b: 20. 3yi2=y+8. 40. Transposition. By the use of the principles (1) and (2), Art. 38, a number may be transposed from one member of an equality to the other by changing its sign. Thus, in the example, Art. 39, in deriving equation (3) from (1), 10 may be subtracted from both members by omitting the 10 in the left member and entering a —10 in the right member. Likewise, to solve z-—5=7, we transpose —5, and obtain s=1(+52 12; After a little practice, this process of transposition of num- bers will sometimes be used to advantage instead of principles (1) and (2), Art. 38. However, the term “transposing” is not very essential as the process is simply that of subtracting the number from each member. , Arr. 41] TRANSLATION OF ENGLISH INTO ALGEBRA 47 * Example. Solve the equation 5r+4 = 14. ~ SoLurion: 5a +4 = 14. abs ‘Transposing 4, we have 5a = 10. (2) . Dividing by 5, x =2. (3) CHECK: §°2+4=14, 14 = 14. 41. Translation of English expressions into algebraic ex- pressions. In order to give algebraic solutions of problems ‘stated in words, it is necessary to develop skill in the transla- ‘tion of English expressions into algebraic expressions. To illustrate, we give in parallel columns a few equivalent English ‘and algebraic expressions, together with some statements of equality. Let s denote the length of a side, p the perimeter and a the area of a ‘square. _ English expressions or statements Algebraic expressions or statements 1. Four times the length of a side 1. 10 + 4s. ) added to ten. (2. The square of a side plus 2. 2. s?42. 8. The perimeter of a square equals 3. p=A4s. four times a side. 4, The area of a square is equal to 4, a=s, | the square of a side. 5. The perimeter of a certain square 5. 4s = s?. (See 3.) | is equal to its area. ) 6. The perimeter of a certain square 6. 4s = 4s?. is equal to four times its area. | EXERCISES 1. A rectangle is x feet wide and / feet long. It is twice as long as it is wide. State this last sentence in algebraic sym- bols. What represents the perimeter? | 2. If the sum of two numbers is 8, and one of them is 2, ' what is the other number? 3. If x stands for the total number of men in a regiment, ‘and one-tenth of the regiment increased by 5 are sick, what | expression denotes the number of sick? Mg %. 48 EQUALITIES [Cuap. IV. 4. Write the algebraic expressions for x increased by m, decreased by m, multiplied by m, and divided by m. 5. The numbers 1, 2, 3, 4, . . . are consecutive integers, How much greater is each than the preceding one? 6. If n represents any integer, what will represent the next consecutive integer? 7. If n represents an integer, what will represent the next preceding integer? 8. If n is the middle one of five consecutive integers, what will represent the other four? | 9. The numbers 2, 4, 6, 8, ... are consecutive even integers. How much greater is each than the preceding? 10. If n represents an even integer, what will represent the next consecutive even integer? 11. If n is the middle one of five consecutive even integers, what will represent the other four? 12. If two numbers differ by 10, and the greater is x, what is the other? 13. If A has $100 more than B, represent the money of each in terms of x. Do this in two ways. Hint: First, let c = B’s money. Second, let x = A’s money. 14. If A is 5 years older than B, and B is x years old, represent their ages in terms of x (a) at present; (b) in 10 years from the present date. 15. Two men divide $1000 so that one shall have four times as much as the other. If ¢ is the smaller share, represent the larger share in two ways. PROBLEMS 1. A rectangle is twice as long as it is wide. The perim- eter is 180 feet. What is its width? Hint: Let the rectangle be x feet wide. 2. The sum of two numbers is 18, their difference is 4., What are the numbers? eer. 41) PROBLEMS 49 / 3. Two men are to divide $1000 so that one shall obtain ‘4 times as much as the other. What should each receive? 4. Divide the number 90 into two parts which are to each ‘other as 2 is to 3. V5. The sum of two numbers is 9, their difference is 15. What are the numbers? 6. The sum of two numbers is a, their difference is 6. What are the numbers? /7. The sum of two numbers is 32 and their difference Is —4,. What are the numbers? 8. If n represents an integer, what will conveniently ‘represent the sum of this integer and the next consecutive ‘integer? 9, Find three consecutive integers whose sum is 48. 10. Find two consecutive integers whose sum is 193. 11. A rectangle is 10 yards longer than it is wide. “Its perim- -eter is 80 yards. Find the dimensions. 12. A rectangle is three times as long as it is wide, and the perimeter is 128 feet. Find the length and width. \J8. The United States has 56,000. more miles of railway than Europe. The two together have 409,000 miles. Find the mileage of each. \%4. Three boys together have 120 marbles. If the second has twice as many as the first, and the third five times as many _as the first, how many has each? ~j5. A farmer has three times as many hogs as horses, and twice as many sheep as horses and hogs together. If there ‘are 120 animals in all, how many are there of each kind? vis. A plumber and two helpers earn together $7.50 per day. How much does each earn if the plumber earns four times as - much as each helper? 7, The sum of three consecutive integers is 78. What ~are.these three numbers? g. A merchant owes A three times as much as he owes B, he owes C twice as much as he owes A, and he owes D as much ‘ ‘ 50 | EQUALITIES [Cuar. IV. | as he owes A and B together. If the sum of his indebtedness to A, B, C, and D is $14,000, how much does he owe each? 19. The length of a field is 3 times its width, and the dis- tance around the field is 200 rods. If the field is rectangular, _ what are the dimensions? \20. If three times a certain number is added to twice the number, the sum is 35. What is the number? 21. A real estate agent purchased 3 houses, paying twice as ouch for the second as for the first, and four times as much for the third as for the first ; if the difference of the cost of the second and third is $3000, what is the cost of each? 22. A room is 15 feet long, 14 feet wide and the walls contain 464 square feet. Find the height of the room. 23. How much must be invested at 6 per cent simple interest to amount to $650 in 5 years? 24. Two motor cars can run one at 20 miles an hour and the other at 25 miles an hour. If the faster car sets out to catch the slower when the latter has 15 miles start, in how many hours will it catch up? 25. A golfer knows that it is 380 yards from the tee from which he starts to the hole. After playing two strokes with the brassie and one with the mashie, he finds the ball 5 yards short of the hole. Assuming that he played in a straight line, and that each brassie stroke is twice as long as a mashie stroke, what is the length of each stroke? ‘Arr. 41] REVIEW EXERCISES AND PROBLEMS 51 REVIEW EXERCISES AND PROBLEMS 1. Give a factor of each of the following and state its coefficient : 3mn ; ax; 49xyz; bez. | 2. Write the following expressions using exponents: ©@-@-@-Yy- 2-2; a 3-3-s-s; 125;10000; m-n-x-m-n-n-e. 3. Write the following expressions without using exponents: 2a’; Gy ab’; 3 xy 4, Add on the number scale: 5+ (-—2)+3+4+(-9) +24 (-2); —-§+64+24+(-5)+7+(-6). 5. Subtract on the number scale: -—4 — (-4) -6-1-(-9) We( —2); 8 —(-8) —7 —(-2) -10-(-1). 6. Show on the number scale that —2 — (+4) = —-2+4+(-4); and B- ( —4) =8 + (+4). | 7. Write a formula giving the cost c in dollars, of 40 chickens, aver- ‘aging p pounds, at r cents a pound. i 8. Write a formula which gives the cost c in dollars of n miles of wire ‘fencing at k cents a rod. 9. Write a formula which gives the cost c in dollars, of m miles of wire at d cents a pound, if one rod of wire weighs p pounds. . Perform the following multiplications and divisions : —18 100 4 10. -2-3- —5. 14, <——, | 2 ees Tecan ae certls ° 5 0 ‘ 6 4 3 11. -1- -17- -2. 156.. =—_—- : ( 5 19. =5 | peer ee it. 1. go a tlc 99, 2+4. | —2--2:-3 4 13. 10-0-—10- 2. 17. 22:0°3 91, 228 +8: 12, 2--3 ce ak ah 22. Define and illustrate the terms power, exponent, coefficient, abso- lute value. 23. If a is greater than b, is the absolute value of a greater than the absolute value of b? Give examples. 24. Give the results in the following: a+0;a-0; a-:1; a+1; a-0; O+a. 25. How many values of x satisfy 2x = 3x —2x? 26. How many values of x satisfy 2x = 2x — x? 27. Distinguish between an equation and an identity. 52 EQUALITIES LCuap. IV. 28. The equation x — 5 = 1 is satisfied by x =6. If 4 is added to the left member and 5 added to the right member, is the resulting equa- tion satisfied by x = 6? \ 29. State four principles used in solving equations. 30. Supply the missing term which makes the equality 5a +6-—2 =144+?-1244422 an identity. 31. Find the corresponding values of .5% + 2 when z has the values 0, 1, 2, 3, 4, 5. Represent these values graphically as in Art. 18. See 32. Find the corresponding values of : when z has the values +2 La hit hy Saeee 33. Which of the principles referred to in Exercise 29 are used in obtaining the second and third of the following equations : If 2 of A’s money = $400, (1) then 4% of A’s money = $200, (2) and of A’s money = $600. (3) 384. Upon what principles does transposition depend? Illustrate. 35. The beam AB supported at its : center (see figure), just balances when one end is weighted with x + 5 pounds and the other with 12 pounds. If 5 pounds be taken from the left side, how much must be taken from the right side in order that the balance be maintained? Hence, x pounds balances how many pounds? 36. Suppose the left pan of a balance contains x — 7 pounds and the right 24 pounds. Let 7 pounds more be put into the left pan. Then what must be done to maintain a balance? Hence, x equals how many pounds? 37. Illustrate with the balances the processes performed in solving (a) x+9=12; (0) 2x-4=2+46; (c) 12-2 =17 — 22. CHAPTER V ADDITION _ 42. Terms of an expression. An expression may contain one or more + or — signs which separate it into parts. Such a part with the sign preceding it is called a term. Thus, in the expression, ab + 2xy — 7c, the terms are ab, 2xy and —7c. 43. Monomials and polynomials. An algebraic expression ‘of one term only is called a monomial. Thus, 27, 5ab and abcd are monomials. _ If an expression contains more than one term, it is called ‘a polynomial. For example, ab + 2ry —7c, 3x — 3y, and a+b+care polynomials. In particular, a polynomial of two terms is called a binomial, and one of three terms, a trinomial. 44. Similar terms. Terms that have a common factor are called like or similar terms with respect to that factor. Thus, 2a and 7a are similar with respect to a; —52*y and 8x*y are similar with respect to xy ; 6abx and 5mnz are similar with respect to 2. EXERCISES With respect to what factors are the following pairs of terms similar? aeenr 12x. abe, xyz. hema, Cams". abc, ax*y. —ta’mn, 5mnv'. 36a2m5t*, —at*m?. 2 3. —4abx, Zamy. ew ae a cas 4 - or, 13 - 8. oO OM AD 53 54 ADDITION [CuHap. V. 45. Addition of monomials. We have already seen that 2a + 3a = 5a. Similarly, 7x + 4a + 6x = 172, and ax + bau +.cx = (a+b+4+c)z. The method is the same if some of the terms are negative. Thus, 8a +a — 5a — 2a = 2a, and am — bm +em —km = (a—b+e-—k)m. These examples illustrate the principle that the sum of like terms is the product of their common factor by the sum of its coefficients. EXERCISES Name the common factor and find the sums of the following : 1. 12a and 4a. 13. 2b, 9b, and —7b. 2. dy and y. 14. 8d, 0.7d, and d. 3. 8x and 2172. 15. 1.6m, 3m, and —3.5m. 4, —92x and 142. 16. az, —bx, and —cz. 5. —5r and 5r. 17. am, —2m, and m. 6. 207? and —7r’. 18. abc, xryc, and —2zyc. 7. 0.5m and 2.3m. 19. 22°, 1023, and —12z’. 8. 3-7 and 9-7. 20. 1r?, r2, and 27": 9. 4-25 and—4- 25. 21. 27r, 9rr, and —127r. 10. 6. —3 and -10--3. 22. >, i and - 11. —2n? and —7ni. 23. :, a and al 12. 8- 3? and —3- 37. 24. sa ze and =) eee: n 25. 3(a+y), —7(~@+y), and 13(x +y). Arr. 45] EXERCISES 55 26. 30(5a — 4), —9(5a — 4), —27(5a — 4), and 11(5a — 4). 27. a(x +y) and bia + y). 28. a(c — 4), 7(c — 4) and —3(c — 4). — 29. Let xy be represented by one unit on the scale. Add on the scale day, —10xy, 8ry, and —2xy. 30. Find the sums in the first five exercises by adding on the scale. 31. How much richer am I at the end of the day than at the beginning, if (a) I earn 3a dollars and spend 2a dollars? (6) I earn 2a dollars and spend 3a dollars? 32. One city lot contains 16x? square feet, a second lot is 82 feet long and 9x feet wide, and a third lot is 100 feet long and AO feet wide. How many square feet in the three lots? How many square feet if « = 12? | $3. If ¢ = 10, what are the values of #, #,-é*, é, ¢§, and ¢7? _ 84. Find the value of #+8+é+t+1,if¢ = 10. Also the value of 3¢4 + 6@ + 5 + 2+ 7, and of 9¢° + 90? + 9. 35. We may write 427 = 400 + 20+ 7 = 44 2¢+7, if t=10. Write in terms of ¢ the numbers 8699, 2941, 501008, and 345678. 242-431-110 -9 -8 7-6 -5 -4 -3 -2-1 012 3 4 5 6 7 8 9 10 11 12 Fic. 9 36. Show on the number scale that 5+3=3+45. Show that a + b = 6 +a when a and b have the sets of values 4, 7 ; ~3,8; —5, —6; 10, -13; —12, 12. 37. Show on the number scale that (8 4+5)+2=3+(5 +2). 38. Show that (a+b) + c=a + (b+ c) when a, 6 and c have the sets of values 1, 4,7; 6, -38,8; -—3, -—7, -2; —5, 8, —3. , 56 ADDITION [Cuap. V. 46. Simplifying polynomials. The terms of a polynomial may be arranged and grouped in any manner without changing the value of the polynomial.* By the use of this principle we are able to reduce many polynomials to simpler form. The result in the following ex- ample is found by first rearranging the terms and then adding together like terms. 8x + 2Qy — 3x + Ge — Ty + 82 = 8a — 3a + 2y — Ty + 2 + 32 = 5a — 5y + Ye. EXERCISES | Simplify : 1. + 4y — 2x + 5a + Oy — y. 9°. 364 BF 6G ads 3. 2.503 — 7a + 8a? — 3.50? — a? + 4a. 4.A—-B4C0 —2A43B —C: 5. 3m? — bn? + Tm2n? + 8n? — m’. 6. 4-5-8:-843- 5 — 0. 73-8446 — O- one: 8. 1q 4+ 3m — 4k + 3k — 4a + gm. 9. 247? + 35r + .6r? — Ir. 10. m— Ta n+5 11. Gry + 3ay? — vy — 40y? + Cte | Sarr? + Sarr? — Sar? + Tr’. -3.11-5-546-5-9:-11l —=5 4 * This principle includes two fundamental laws of addition : (1) The commutative law, or law of order, which says that a+b=b+4a. (2) The associative law, or law of grouping, which says that at+tb+c=a+(b+0c). Thus, 8+5-3+11=5+(11+8-38) =(-3+8) + (11 +5) =21. Similarly, 2a + 6a — 15a = 2a + (6a — 15a) = —15a + (2a + 6a) = —Ta. | Anrs. 47, 48] ARRANGEMENT OF TERMS 57 14. 4a — 4b + 4c — fa + 4b — $e. 15. 5ab + 8ac — 7bc — 2ab + 9ac + 2bce — 2ab. 16. ab" — 3a2"b" + ab” — 5a"b™ + Tab" — ab". 17. m* — 2n? + 5n® + 38m? — 4m? + n? — nn’. 18. 2(a+b) +5(a +). 19. 3(~ — y) + 9(¢ — d) — (wx — y) — 4(c - d). 20. (m+n)? —7(m +n) + (m+n) + 6(m 4+ n)?. 4%. Arrangement of terms in a polynomial. The polynomial 2 + 3x72 +. 3” + 1 is said to be arranged according to the de- scending powers of x; 7 — 38y + 7y? + 6y? is arranged accord- ing to the ascending Sirians of y; and 523 4+ 4a’y — 82y? + y' is arranged according to the descending powers of # and the ascending powers of y. In the addition, subtraction, multiplication, and division of polynomials which contain different powers of the same letter, it is usually advisable to arrange them according to the ascend- ing or descending powers of one letter. 48. Addition of polynomials. In the addition of polynomi- als it is convenient to write the terms in columns, each column containing only like terms. Example. Add 2x +y + 5z, 7x —y — 2z, and —4x + 3y —7z. SOLUTION: Cueck: Let + = 1, y=1,2=1. 2e+ y+ 52 8 7x — y—2z 4 3 —4x + 3y — 72 -8 5a + 38y — 42 + A check on an operation is another operation that tests the correctness of the first. The above solution may be checked by substituting any set of numbers for the letters. Let x=1, y=1,andz=1. The values of the given polynomials are then 8, 4, and —8, and the ‘sum of these values is 4. Since the sum thus found is the same as the numerical value of the answer for x = 1, y = 1, 2 =1, | 3% 58 ADDITION [Cuar. Ve! the answer is said to check. Such a check, though not strictly a proof that no mistake has been made, shows that the result is probably correct. Failure to check shows that a mistake has been made. EXERCISES Add: 1. 304+ Ty + 92 6. 2003 — 4a’%y + 12ay?+ y° 8a +4y+ 2 —323 + Bary — 84ay? — 5y? Or — 5y — 82 go-— sy+t+ aye 2. 8a — b+ 2c 7 1.6m?— 3mn + 3.05n? 7a+4b- ¢ —.9m? + 4mn — .45n? a- b- ¢ L.im+ mn — 2.25n? 3. 1872 -— 7x+3 8. 4(a —b)? — 8(a —b) +15 —477+ 2-9 21(a — b)? — 11(a — b) — 27 102? — 14% 47 —14(a—b)?+ (a-b)+ 4 ?(a—b)?— ?(a—b)— F 4. 44— 3B +200 9. 5(2 —6) + 12G7 79) ae “9A 13 B19 (x — 6) — 30(y +9) -— 2 5A a1 Bee — 2(x — 6) + 5(y +9) — 82 6. st — | ee wterty—zZ —w—-x—y-z2 10. 6(m — n)? — 9(m — n) + 11, 8(m — n) — 27(m — n)?, 43 + (m — n)? — (m — n), 9 — 9(m — n)?. . 11, 2¢— 30° +22 — 5e+8, 62! — 6a? +11, a + 142-7, lL —a2+ar+ 2 -— %, © + J, 12. 3ab — 4bc + cd, -—ab+6cd +12be, ab — be, ab + be + cd. 13. 62? — 42%y + bay? — y', —2a2 + y®, Gay — 12ay? + 3y’, 1423 — la’y + ry? — dy’. 14. ir —45 +t, 2t —4s — 1, 23s — it, 8r, —és, t. ‘Arr. 48] EXERCISES 59 m0. 270° — 40? + a— 10, 2a — 12+ 9a? — a’, 18 + 5a’, 6 — 3a° + 2a? — 9a, a — 2. We 16. 5(k + 1)? — 2(k + 2) — 7, 20144 +12? —-(k+) +5, —6(k +124 3(k 41) - 9. 17. 3" — 371 + 6, 5-3" + 4.387"! — 1, —2.3" — 4.371 + 8, 18. 4a —b + to, 2a + $b — $c, a—b—c, —3c + $b — 4a. pear 32 41, tt et + We —¢+32-8%, 41,17 -t-P-#-t-#, , 20. a + .1b + Olc + 001d, d°— 1c + .01b — .001a, la—b+c-—.ld. _ 21. a+b, c—d, d—a, a+b-—céd, b+d, -a-e, im d +5, a, }, ab, c. ime 22. (a+b) + (c — d) — (¢ +f), Bia b) —tliced).. Bets), (¢-d —2a+b) - Ue +f). 23. .lm+ .2n+ .3p + .4q, Olg — .02m + .03p — .04n, (001n — .002p — .003q — .004m, .085m — .167p — .225¢. 94. Find the sum of P and 3 times Q, when P=2 + dry +327 + 9’, and Q = 22° + Sa’y — 4ry? — y’. 25. Use the same values of P and Q as in the last exercise, and find the value of 2P — Oavhen 2 = 1, and y= 92: 96. Write 236, 327, and 413 in terms of ¢ = 10, and add. (See Exercise 35, Art. 45.) | 27. Write 8964, 50231, 100000,. 847, and 2140 in terms of ‘t= 10, and add. mas. Add: a®* + 3a%b" +'3a"b + b*", 8a*" — 8b", mr” — 1207"b" + 6a" b — b%, — a + 9a2"b” — 27a"b?” + 27b%". 29. How much is the sum 2 + y changed (a) by increasing x by 1, 2, 3, 4, and so on to 10? (6) By also increasing y in the same Taney: (c) By increasing x and decreasing y? ' 30. Answer the same questions as in the last exercise for the . difference 2x — y. 7 CHAPTER VI SUBTRACTION 49. Subtraction of monomials. As shown in Art. 28, a number a may be subtracted from a number b by changing the sion of a and adding the result to b. Example 1. From 10z take —6z. SoLurion: 10x —( —6x) = 10x + ( +6x) = 16x. Example 2. From —8a take —4a. So,uTION: —S8a — (—4a) = —8a + 4a = —4a. Example 3. Perform the subtraction —4zxy SoLution: —4zy — (+32ry) = —4cy — $ry = —'y' xy. EXERCISES Subtract : 1. 132 7. 14s 13. 7(x — y) 4x —8s —(% — y) 2. 20ab 8. abe 14: Tignes —5ab —4Aabc PET ahs 8. —d5xy 9. —-2x 15. 11:10 5a*y 9x —6° 10 4, —122° 10. —wv 16. 7(x +a) —42%3 16u7v 26(a + a) 5. —8mn 117 ame 17. —13(p? — q?’) —4Amn ym 7 —20(p* — ¢) 6. (=) 20D" 1 PZ 18; 22540 P| 200" —0.dxryz 3° 5 \ f Ars. 49, 50] SUBTRACTION OF POLYNOMIALS 61 19. 12°5m 26. «+ 4y S3eeeo —12:5m —sy —b 20. a+b Ze 1k 34, 57" EB & oe 21. 3x -—y 28. 7x 35. 8ab = 7T y Sa 22. 5a —™m 29. Q9v SGU —m 9 ag 23. a—2 30, 12 37. m—n : ix kk m 84. br +5 31. 28 86. ees 10 oe s 25: 3-2 SPP lie bf es Bars —5 oe pases 90. Subtraction of polynomials. In the subtraction of yolynomials like terms may well be written in the same vertical ‘column. If several powers of one or more letters occur, it is ionvenient to arrange the minuend and subtrahend according o the powers of a particular letter. Example. Subtract 3x3 — 7 — 52? — x from 1223 + 82? -— 4x 4+ 2. SOLUTION: CuHeck. Let z= 1. 127? + 82? —47+2 18 oa? — $2? — 2-7 —10 Oz? + 132? — 324+ 9 28 Beitract: EXERCISES 1, 1027 — 37 + 12 3. 2 — 9y— 2 827+ 24 —-— 9 x — Illy + 52 | 9, 5a + 4y — 3u 4, 3c — 42% — 8y3 2x — 3y + 9u 32° + 4a°y — 8y > |. } 62 SUBTRACTION (Cuar. VI. t Bi. gb eee ae 7. @-sbee —a + 8b — 5c — 8d —~at+bv+2 6. 12ab — 3cd4+ def 8. 6 (x + y) — 2(m? + n’) +a —ab + 17cd — 50ef Q(x + y) — 4(m? + n*) — U 2a +y) + 2(m? +n) +o) 9, 27(a +b)? + 5(a +b) - 9 (a+b)? —5(a+6) +1 10. (m +n)? + 3(m +n)? + 3(m +n) +1 — (m +n) + 2(m +n)? — 2(m +n) — 1 11. From 623 — y3 — 3a’y — 4ay” take bay? — 3y° + 2x3 — 92*y. 12. From 7a™ — b” + 5c? + 11d? take d? — 5cp + 5a™ — 2b". 13. From 8(a + b) + 12(@@ + y) -2 take 5(a + b) - A(x + y) + 102. 14. From 6(r +s)? — 7(r +8) +25 take 5(r + s)? — (r +8). 15. From 9(x —1) + 5(y4+ 2) - 4(z — 3) take O(a — 1) — 3(y + 2) — 9 - 3). 16. From 10(a — b) + 16(¢c — d) +e take 6e — 4(a — b). 17. Subtract 723 — 3y? — 2a°y — 7xy? from 7y? — 2 + zy” — xy. 48. Subtract 3ab + 2a%b? from — 7ab. 19. Subtract 2(2 + y) + 5(@ + y)? + 14 from 10 + “st. 20. Write 8639 and 27941 in terms of ¢ = 10, and subtract the first from the second. : 21. From 5:-72—4:-7+410-7 —7 take 4-73 + 7 — 6: 7. Simplify the result. 99. From 2-21 +3-31+4-41 take 5-41 —6-21+431 Simplify the result. . 23. How much greater is 7° + 2° + x than v3 + 2? + 2? 24. How much greater is 2* than 2? Answer this ques tion when x has the values 4, —3, 0, and 4. What is the mean. ing of a negative answer here? er. 50] EXERCISES 63 25. Find a value of x that will make 62° equal to, one that “ll make it less than, and one that will make it greater than 2. 26. How much greater is 0 than —11? Than 3x2 — 40? 27. From the sum of 5a? — 22ab — 38b? and 42a? + 19ab take 5a + 27ab — 50b?. 28. From 5m’ — .05m? + .7m — .001 take tom? — Am? — 3.2m — .31. 29. What must be subtracted from 4b — 4b + 2c so that the smainder is 2a + 2b — 4c? 30. What subtracted from 22 will leave — x4 + 4a3 — 12x? — x? 31. What subtracted from x + y +z will leave 10x + 100y - 10002? ~ 32. From 2" + 6x" y —182"y? + 7y® take y® + 4a"—ly — 52x i: 4". 33. From 3-2” —6"+7-4"4+ 9.5" take 2” — 5.4” — 57, _ 34. Take x* — 3y? + 7xy — 4 from the sum of 32? — y? + 13 Od ry — x? — 4? — 4. 85. Take 2k? + 5h? + 11hk —-3h+4k from the sum of + 4k —h? —k? +1and k? + 5h? — hk + 38. 36. Take the sum of m? — 3m?n + 3mn? — n3 and 2 en + > — n>+ 3 from 3m? — 3n’. 37. Take the sum of — 2p‘ + 5g4+ r4 — per +1, ogr — 3p* + q—r', and p*+q*+ pqr—4 from the sum of 0* + 3q* — r4 and g‘ + par + pt + 3. CHAPTER VII PARENTHESES 51. Removal of parentheses. The expressions 6 + (7 +4) and 6 +7 + 4 are equal. Similarly, 9 + (6\— 2) = 9 eee In the same way, a+ (b-—c) =a+b-c. These examples illustrate the principle that the value of an expression is not changed by removing parentheses preceded by « plus sign. In subtracting the sum of 3 and 5 from 12 we may write 12 — (3 + 5) or 12 — Sia since we get the same remainder, 4,in each case. In subtract ing the difference, 7 — 2, from 19 we have 19 (7 = 2) =19 — since the value of each expression is 14. In general, such an expression as @ — (b — c), means tha b — cis to be subtracted from a. Hence, Art. 27, we chang the signs of the terms of b — ¢ and add. This gives, a—(b—c) =a+(-b+e)=a—-b+e. These examples illustrate the principle that parentheses pr ceded by a minus sign may be removed by changing the sign of eac term within the parentheses. Expressions often occur with more than one pair of pare: theses. When one pair occurs within another pair, bracke and braces may be used to avoid confusion (Art. 15). Allp 64 Arr. 51] REMOVAL OF PARENTHESES 65 ventheses may be removed by first removing the innermost pair vecording to the principles stated above; next the innermost jair of all that remains; and so on. Example. Remove the parentheses in 2a — {6b — 2c + [a — (b -c)] + 2b}. SOLUTION: ta — {6b - 2c + [a — (b-c)] + 2b} = 2a - {6b — 2c + [a—b +c] + 2} = 2a — {6b -2c+a-—b+c+42b} = 2a — 6b + 2c -a+b—c—2b = a—Tb+e. CHECK: Let a =1,b =2,c=3. Then, 2— {12-6+ [1 - (2-8)] +4} =1-1443. -10 = ~—10. EXERCISES Remove the parentheses and simplify the results by collect- ng like terms: 3a + (2a — 4). 5x + (7 — 2). 2x +1 —- (4 +7). b — (2a — 3b) +a. n—-1-—(1—7n). n—-1—(n+1). a+b—c—(c—b-a). 2x + (3x — 5z) + (52 — 2x + By). 1- {1+ [f1-(1+1) -1) 41} -1. (3a — 4b) — (a-—6 +4 2). . £— (2+2-—- (82 — 7)]. 7 . m— {3n + [2m —n — (5m —n+6)]}. . . 63 — [80 — (15 — 4) + 2] - 1. . 4x — [5a + (v — y) — y] + 4y. a ee ee eee ee e ee PrP oD BF © 66 15. 16. 17. 18. 19. 20. PARENTHESES [Cuar. VIL , 12ey + 3y? + [6ax — (2b + 7az)]. (22 —y +7) - [x- (y - 2)]. 4a + 3b — [ex ta+b—2y-(e+y)]. atm — ([6 + 4a — (6m + 2)] — (7m — 4a = 8)}. a—b— {(a—b) - [a—b — (@—b) - b-a)] —b +4}. p+q—r—([(p-—at+r) -(-pt+a-*l. Solve the following equations : 91 Or mela) pee 22. a + (2a — 1) = 59. 93. (32-—4) —2 =2+4 6. 94. + — 24+ (382 — 9) — 15 = 0. 25. (6m — 3) =9 — (4m + 12). 26. 22 —14 — (a — 12) = 6. 97. 4t--2=5-— (8t+7). 28. .5a2 — (.0la + 9) = 40. 29. Show that x — (m+n) =x —m-—n when z, m, and n have the values: 5, 1,2; 2,9,4; 0,3,3; —2, 2, 9. 30. Using the same sets of values show that a—-(m—n) =x2—-mM+N. 31. Using again the same sets of values show that x — (m +n) is not equal to x — m + n. For what particular value of 7 is a—(m+n)=x-—m+n? 52. Insertion of parentheses. Terms may be inclosed in parentheses with or without changing their signs according as the sign before the parenthesis is minus or plus. . Examples. e-2+b-y=xr2-(2-b+y). e+3—-b+a=2+(3-b+¢C). That these results are correct may be shown by removing the pa: rentheses, Arr. 52] EXERCISES 67 EXERCISES _ In each of the following expressions inclose the last three erms in parentheses: lL at+b—c+d. 7 —xX—y—Z2-W. » ab+pq —rs — xy. 8. a+b-—c-—d. 3 4—-2a+b0-y. 9. 1 — 4m? + 4mn — n?. ma? — & + 2bc — c’. 10. a+b+c-—ux-y-z. he ee = ty — 4. 11. 36 — 9m* — 6m?n — 1. —2Qey+y?—a@-—2ab-—b% 12. 4a%4+274+1. we . | 13. A rectangle is x + 20 units long and x + 7 units wide. Nhat is its perimeter? 14. Three points A, B, and C are in a straight line, and B s between A and C. If the distance from A to Bism +n+6 nehes, and from B to C is x — y +7 inches, how far is it ‘rom A to C? How much farther is it from A to B than from Beto C? 15. Write the remainder when the square of x plus the wroduct of « and y is subtracted from the cube of z ninus 7. 16. By what amount is m? greater than 2m — 1? 17. By what amount is a — 3 greater than 2a — b? 18. What is the smaller part of x* if x? + x — 1 is the larger dart? 19. By what amount does one million exceed + 544+ 8842+ 71+ 5 if ¢ = 10? 20. What number is 4¢ +8 greater than — 47? Find the ‘umber when t = 10. | 91, What number is 3d? +¢-+ 1 less than 300? Find the tumber when ¢ = 10. 22. Write in the form of a fraction the quotient of Ww + 12t + 4 divided by 3¢ + 2. Find this quotient when = 10. 68 PARENTHESES _ ([CHar. VIB 23. What is the error in the statement 7-y+2—-W= g—-(y+z—w)? | 24. What is the error in ax + by — ay — ba = (ax — bx) — (ay + by)? 53. Collecting literal coefficients. Add : ih ax 7. —6z 12. av? — 3mv bx — max —3v° + (a + b)x 2. cy 8. 3x? + bx 13. gt — ar — dy ka? + 9x —3t — br (c — d)y 3. —Ax 9. ap? =p 14. 42? — 7z vr —4p? + Pp ba? — az —(a + b)x 4. am 10. ax — by + cz 15. 3c? + 2d bm —2x%-—dy+ 2 Ic? + nd 5. 3m am 11. be—ay+dy 16. 390? — igt : —2be + xy — dy 29? + gt av <3; | ne Ti ae CHAPTER VIII MULTIPLICATION 54. Products of powers. It follows from the definition of an exponent, Art. 11, that a=a-a, G=4-a*a*a, and hence that a?-at =a-a-a-a-a:a=a' = @*4, ‘Similarly, Cee ea ee ee EL eH ee ele: These examples illustrate the following rule for combining exponents in multiplication : The exponent of a letter in a product equals the sum of the exponents of that letter in the factors. ; _ This rule may be stated in algebraic symbols in the form, anm-an=anrtn, (1) EXERCISES 1. Use the definition of an exponent and show the sree Mes c*; a7b?; Garty; 51?m?n; a”. Complete re following tidieated multiplications : 2. a?-at-a®, 8. (x)? 14. 102-108, 20. a?- a”. am b*-b- bd. 9. (m?)3. 16504). Ort be 4m-m-m'. 10. 3-3-3°. 16. (3). 22 mye. 7?-y. 11. (2°). 17. -($)?. Osa i Wee 2°, 12. —24.25.28 18. (3)-(3)% 24. (-a). | o —r°- 2°, 13. r8.72-79, 919. 20-208. 925. —a?- (—a)?. 26. Verify that a-b = b-a, when a and b have the values : N@7, 54; 18, 1.8; 229, 11; 341, 2.56. 69 70 MULTIPLICATION [Cuap. VIL | 27. Verify that a-b-c =a-c-b =c-a-b, when a, b, and c have the values: 3, 4,5; 25, 35, 41; .1, —%, .2. ! 28. Verify that (a-b)-c =a-(b: BY ener a, b, and c have - the values given in the last exercise. | 29. What must be the value of m so that (a) m-5-6 = 15; : (b) (m-6)- (3-5) = 180; (¢c) .2-6-m- 4 = 96? 30. Show that 5a*y’- 204 y = (5-2): (a3- a*)- (yy), when a =1 and y = 2. 31. Show that —722yz - 3ay%z- —x4y® = 212” y®2?, when x = 1, y = 2, and z = 3. | 32. Show that (a”)" = a™, when a = 5, m = 3, and n = 2. 33. Show that (a™)"=a™ when m and n are any) positive integers. | Hint: (am)n = am-qm-aqm.,.. ton factors. 55. Products of monomials. The factors of a monomial may be arranged and grouped in any manner without changing the value 6f the product.* Thus the order of the factors may be changed, asin 5-6 =6-5. Also, the factors may be grouped in different ways, as in (8-5)-8 =3- (5-8). We make use of this principle in finding the product of monomials. EXERCISES 1. Find the product of 72? and 82%. SoLuTION: 722 + Sat = (7-8) - (a? - x4) (Since the factors may be re- arranged and grouped in any manner.) = 567°. (Using the law for combin- ing exponents in multipli- cation.) * This principle combines two fundamental laws of multiplication : (1) The commutative law, or law of order, which says that a-b=b-a. (2) The associative law, or law of grouping, which says that ‘ (a-b)-c=a- (0-0). | Art. 55] EXERCISES 2. Find the product of —3am?n? and 6a3min. 71 SOLUTION: —3am?n? - 6a*min® = (-3 - 6) + (a+ a3) + (m2 +» m3) + (n? +n) =-—18a‘m?n’. Complete the following indicated multiplications : 1. a’®- ae. ate AL: - Opg: —g. . om-4m. . 2a- 5b. . 8u2v- —Tur?. 2 3 4 Dour say. 6 7 8. 12a%b4c® - abe. eae yee fiz - 2°22". toea—o0d.- —40.- —c. 14. lla’a’yz- —6y%2?- — fee + oe —2- 34r?, 16. 4°. 7?m?n?.4. mag. oF 8°p°- 2+ tp. 7mn. er eo 4 1. 1. 5 Bu". | P| 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. (x?yz)?. (3a°b*c)?. boar)": (—11m5n8p°)?. (—3aa%y®)*. (—5p?q’)*. 4. fa’. —5 - 2m'. 4mn —— 72 MULTIPLICATION [Cuar. VIL. 56. Multiplication of a product by any number. A product is multiplied by any number by multiplying any factor of the product by that number. Thus, 3.(2-3-5) =6-3-5=2-9-5 =2-3- 15. 5. (Qry) = 10xy = 2-5x-y = 2a- Sy. EXERCISES Perform the following multiplications in at least two ways. 1. 3(4-5). 6(5- 7). 10(7 - 8- 2). 12(4- 4-8). A(3ay). 3(12m?n3). HEHE Fia. 10 ST eee 7. 10(8- 25). 8. 123(4-5-7). 9. 334(62y). 10. —2(—5- 0). 11. —2(-2-6- —3). 12. —a(—b- —c-d) 13. Show that m(x# + y + 2) = mx + my + mz, when m, x, y, and 2 have the values: 2, 3, 4, 5; 3, 9 —4, 5; 6, 0, —6, —11; 0, 7, 2, I. 14. Show that m(a — y -2) = mx — my — mz, when m, a, y, and @ have the values given in the last exercise. 57. Product of a polynomial by a monomial. We know that A(5 +7 +3) = 20 + 28 ees | This illustrates the rule that to multiply a polynomial by a monomial, we multiply each term of the polynomial by the mono- mial and add the resulting products. f i! i i ‘Arr. 57] | diagram. POLYNOMIAL BY MONOMIAL This principle is stated in algebraic symbols in the form a(b +c+d) =ab+ac+ad.* The product of two numbers may be represented by a Example 1. The product, 3:4 = 12, is illustrated by Fig. 10. a| ab ac d h Example 2. The product, , b - Cc 4(5+7+3) = 20+ 28 + 12, is illus- ed by Fig. 11. z eave tt: is Example 3. The product, a(b+c+d) = ab+ac+ad, is illus- b+e+d ‘trated by Fig. 12. Fig. 12 EXERCISES AND PROBLEMS Find the following products and simplify if possible : i: Be oo bo — on pt ec) pill ah gl a —_ ae — > 8(5 +4 —7). 5(7 +9 — 6). 3(200 + 70 + 9). a(b+c+d). 6(8 + b — m). a’x?(a? + 2ax + 2x”). 5m2n(8m? — 4mn + n’). —3ac(a? — 9Yac — c?). 2y4(a? + 2ry — 6a%). —2ab(—a’x + 4ax — 122%). (—1): (a — b) + 4(8a — 26 — 4). . 8(—-x-—y) —3(24+y — 2). . (-3):( —s) + 2t4 8s - 4. 6(a? + 4a + 4) — 9(2a? — a — 2). . 6(7000 + 900 + 10 + 8). . 3a"(a2" — 2a"b" + 6"). We * This principle is known as the distributive law of multiplication. ant2h2n (inh? = 6a2"—-4h3-” ef 1la”—"b). 74 MULTIPLICATION (Cuap. VIII. 18. 2°(1 +.2" + 3"). 19. (x — y)*[(x — y)® — (@ — 9)’. 20. (a + b)*[(a + b)> — 6(a + B)?]. 21. 65 i i). 22. (5 +5- ’). SoLurIon: s(F+5-4) - at St SB = 4m + In — p. 23. 12(75 z ° mn 5 25. 30( FF e s 4 un). oy 10 a 9a? fi #), 26. 16(4a + 4b — $c — d). Se aadele ham 27. 6(§0 + fy — 32 + 2). 98. (52-2 BE) 29 16(7 4 oe . a), 30. 20( = 4 ue wate 6 1 Sa Ba ) 31. Find the area of a rectangle whose dimensions in inches are 3b and 6b — 7. What is the difference between the num- ber of square inches in the area and the number of linear inches in the perimeter? Check when 6 = 2. 32. Find the volume of a rectangular solid whose dimensions in inches are h, 2h, and 3h +4. Find the sum of the number of square inches in the surface, and the number of linear inches in the edges of this solid. Check when h = 10. 33. Illustrate by a figure that 4(5 — 2) = 20 — 8. 34. Illustrate by a figure that a(b — c) = ab — ace. 35. Show that (a + b) (x + y) = ax + ay + bx + by, when a, b,x, and y have the values : 1, 2,3, 4 ; 4,7,2,5; —2,3,4, —6. 36. Show that (m — n) (p — q) = mp — mq — np + nq, when m,n, p, and q have the values: 4, 2, 7,6; 1,1, 6,8; 0, —3, 2, —2. | 1) ee i | | i | ‘Arr. 58] TWO POLYNOMIALS 75 58. Product of two polynomials. The product, (8 + 4) (@ + y), may be found in two ways. (8 +4) (@+y) Also, (8+ 4) (w+ y) = 7(" + y) = 7x + Ty. =3(@+y)+4(r+y) = da + dy + 4a 4+ 4y = 12 Aye @ By using the second method, we find that (a+6)@+y) =a@+y)+ba@+y) =ax+ay+bxr+by. (See Fig. 13). | ‘These examples illustrate the following Rule. The product of two polynomials ws found by , multiplying one polynomial ‘by each term of the other and then adding these products. | a Example 1. Multiply x +3 by x +7. ax ay Fig. 13 SOLUTION: CHEcK: Let z =1. zx +3 4 Be ae te. x? + 3x 7x +21 x’ +10xr + 21 32 See Fig. 14 for a diagram to illustrate this exercise. Example 2. Multiply 27? + 3a — 5 by 4a — 7. SOLUTION: 207 + 32 — 5 Ar — 7 8x3 + 122? — 20x ae oly +55. 8a3 — 277 —417 + 35 CHeck: Let x = 2. 1 9 76 MULTIPLICATION LCuap. VIII. EXERCISES Perform the following multiplications and check the results by substituting numbers for the letters: a ee COND TS i) c—) oN bs wo nw Fe bo a OoONNNDN WN ono ont oa a ae QPP oo Oe ao m wr (x+y) (w+ y). (x — 4) (24 + 3). (a +b) (a — b). (80 + 7) (60 + 4). (2x + y) (3% — 2y). (6m? + 5mn) (m — n). (2? +2+1) (e - 1). (a2? — xy + y*) (w+ zy + ¥’). (a + b)?. (a + b)?. (30 + 1)?. (a+b+c)*. (a® + a%b? + b®) (a? — 6). (493 — a? + — 1) (2a? — 4x — 7). . (800 + 40 + 2) (70 + 6). . (xt — Qasy + 2x°y? — Qay + y*) (2? + zy +’). (g— 2) (e—3) ae . (e +a) (x +b) (x + 0¢). (a3 — 30% + 3ab? — 6°) (@ — 2ab + 02); . (m — §) (2m — 4). 1y — ty) (a + 4Y)- . (3x — ZY)? . (5a — 6b + 3c) (2a + b — 4c). (5 6 5) (§+3-5): 55 9 10) \ ou owe . (2m? — mn +n?) (2m — .5n). . (1.4402 — .72ab + .09b?) (1.2a — 3b). . (3.5m? — 4mn + .15n?) (4m — 6n). , (a — Barty + ay?) (w — y). (a + ab" + arb + 8") (a — bn), (x + yy, | i i } ‘Arr. 58] EXERCISES -- 31. What is the area of a rectangle that is 32 + 5 units long and 4x — 4 units wide? Check when z = 6. 32. Show that (8 + 3) (7? + 2t-+ 6) = 83: 726 if ¢t = 10. 33. What is the difference between the squares of two suc- cessive integers, the smaller of which is x? Is this difference an odd or an even number? 84. Illustrate the meaning of (a+ 6b+c) (x+y) by con- structing a rectangle a+b+c units long, and x+y units wide. _ 35. Show by a figure how much the area of a square of side a is increased by increasing the length of the side one unit. ' Solve the following equations : 36. (37 — 2) — 7x — (12 — 3z) = 18x. 87. 2(4 — y) = 5y — 18. 38. 3(a + 2) — (a -— 9) = 1. 39. n+2(n+1)4+3(n+4+ 2) = 91. 40. 7(4¢ — 3) + (7 — 82) = 1. 41. How much is the area of a rectangle of base x and alti- tude y changed (a) by multiplying its base by 2? (6) By divid- ing both the base and the altitude by 2? (c) By multiplying the base and dividing the altitude by 2? 42. How much is the area of a rectangle whose sides are v and y changed (a) by increasing both the base and the alti- jude by 2? (6) By decreasing both the base and the altitude oy 2? Draw figures showing the atte products : © 4s. (c+y) (e+ y). 47. (a + 2). 44. (x — 4) (22 + 3). 48. (30 + 1)? 45. (m+7) (m+ 5). 49. (a+b+4+c)?. 46. (2x + y) (3x — 2y). CHAPTER IX EQUATIONS AND PROBLEMS 59. Equations involving parentheses. Example 1. Solve the equation 3(4¢ — 1) — 5(2¢ — 3) = 18. SOLUTION: 3(4x — 1) — 5(2x — 3) = 18. Multiplying, (12% — 3) — (10x - 15): = 18. Removing parentheses, 122-3 — 10¢4+15 =18. Transposing and collecting like terms, 22 = 6, Dividing by 2, je ao CHECK: 3(4°3 —1) —5(2:°3 — 3) = 18, 33 — 15 = 18, 18 = 18. - In certain equations higher powers of the unknown than the first occur, but vanish as in the following : Example 2. Solve the equation (x — 5)(# + 3) — (32 —4) = (x — 1)”. SOLUTION: (x — 5)(a + 8) — (84 — 4) = @ - 1). Multiplying, g? —~27 —15 —384+4=2? -274+1, Transposing and collecting like terms, —3r = 12. Dividing by —3, z= —4, CHECK: (-4 — 5)( -4 +3) — [8: (-4) - |] = (-4 -1)’%, 9 + 16 = 25, 25 = 25. EXERCISES AND PROBLEMS Solve the following equations : 12D Dee 4(10 — 2x)°=3(2— 5). 3(9 — 2x) — 5(2% — 9) = 0. 7(4y — 3) + 3(7 — 8y) = 1. 8(3a — 2) — 7x — 5(12 — 3x) = 132. 6a — 7(11 — a) + 11 = 4a — 3(20 — a). 78 EL AH oth eA [ er. 59] EXERCISES AND PROBLEMS 79 5(m — 3) + 4(17 — m) = 11 — 7(8m - 6). . k — 2(4 — 7k) = 4k — 9(2 — 8k). 9. 80 — 6(4¢ + 8) = 7x — 3(6r + 1). 10. 3(y — 10) + 11(2y + 1) = 17(y — 12). 11. 82(82 + 2) — 27 = 4x(62 — 1) — 147. 12. (5 — 3a) (38 + 4a) = (1 — 4a) (7 +-3a) - 1. 13. 21 — 3p(10p + 3) = 45 — 5p(6p — 1). 14. (2n — 1) (n+ 5) —1 = (n — 6)? 4+ (n 47)? 15. n?+ (n+ 1)? = (n+ 2)?4+ (n+ 3)2, dale, — Eee Se Se f Peet 5 5-3)- 16. 4(2x — 3) 6(5 5) = 2. piv. 82x — %) = 5 + 5-4) -4(7 - 1). 4 18. 4(a — 3)? + 6(a — 2) = (2a — 5)? + 302. 19. (x-1)?+ 57 + 2) = (© + 1)8 — (a — 4) (22 +7) + 16. r 6 20. 27(x — 4) — 114( rn 1) S597 ee 6(2p—~ 2). 21. A line is divided into two parts, one of which is 20 | Aches longer than the other. Twelve times the shorter piece quals 8 times the longer. How long is the line? _ 22. A man paid $12.50 for 2 wooden golf clubs and 6 iron ‘mes. Each wooden club cost 25 cents more than each iron lub. Find the cost of each. . 23. The value of 31 coins consisting of dimes and nickels is 32.25. How many are there of each? 24. A tourist climbs from a certain point up the slope to he top of Pike’s Peak at the rate of 2 miles per hour, and ilescends by the same path at the rate of 4 miles per hour. If he round trip takes 6 hours, how long is the path? » 25. A man made two investments amounting to $4330. Mn the first he lost 5%, and on the second he gained: 12 %. Vhat was each investment if the net gain was $251? 80 EQUATIONS AND PROBLEMS [Cuap. IX. In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The hypotenuse is the side opposite the right angle. In Fig. 15, c? = a+b. Ifa=3,andb =4, then c? = 3?+ 42 = 25, and c = 5. By transposing, we get a? = c? —B? and ? = ¢c? - a’. 1 26. One side of a right triangle is 5 inches, and the hypotenuse is 2 inches longer than the other side. How long is the hypotenuse? SuaacestTion: Let z = thenum-— ber of inches in the unknown side. | Then x +2 =the number of inches in the hypotenuse, and (x% +2)? — a? = 25. : 27. A rope that is 8 feet longer than a flag-pole reaches: from the top of the pole to a point on the ground 32 feet from. the foot of the pole. Find the length of the flag-pole. | Historical note on the equation. The very earliest mathematical writer of whom we know, an Egyptian priest named Ahmes, solved equa- tions in one unknown. He lived at least as early as 1700 B.C. The un- known he called “hau” or heap. One of his problems reads: ‘“‘ Heap, its half, its whole, it makes 16.” When zx denotes the unknown, the Fig. 15 equation to be solved is x +5 =16. In Egyptian hieroglyphics this tae lame Oo Om Do Heap, its half, its whole, it makes 16 equation is written as shown. The Hindus used the word color to denote the unknown, the Europeans early used the word res (thing), and the Arabs used the word root in this sense. It seems that next the initial syllable of each word was used to denote an unknown. Thus ka (from kélaka = black) meant the unknown. It was not until the time of Vieta (1540-1603) that a single letter was used for the unknown. The con- ventional use of x for the unknown is due to Descartes (see p. 181). 7 i i ‘Arts. 59, 60] EXERCISES AND PROBLEMS 81 _ 28. One side of a rectangular field is 40 rods. The diagonal of the field is 20 rods less than half the perimeter. Find the area of the field. 29. The difference between the areas of two squares is 192 square inches. The side of one is 6 inches longer than the side of the other. Find the area of the larger. _ 30. One side of a rectangle-is 15 inches longer than the yther. The area of the rectangle is 450 square inches . greater than the area of a square whose side is equal to he shorter side of the rectangle. Find the area of the ectangle. _ 31. The sum of two numbers is 2. The niffercniom of their quares is 80. Find the numbers. 32. The difference of the squares of two consecutive integers s41. Find them. 33. The difference of the squares of two consecutive even ‘otegers is 436. Find them. _ 34. The radius of a circular flower bed is increased 2 feet, hus increasing its area 88 square feet. Find the radius of the wriginal bed. : : : 22 The area of a circle is rr?, where r is the radius. Use 7 = 7 : if 35. The difference of the areas of two circles is 423.5 square 2et. The difference of their radii is 3.5 feet. What is the ra- ‘ius of the larger? 36. Show that (2n)? + (n? — 1)? = (n? + 1)2, and hence that n, n* — 1, and n? + 1 may be used as the sides of a right tri- ngle. Give n the values 1, 2, 3, . 10, and find the corre- oonding values of 2n, aa nh i The formula is known as lato’s formula. 60. Equations involving fractions. We are able to solve rtain equations involving fractions by applying the principles ‘fractions learned in arithmetic. 82 EQUATIONS AND PROBLEMS LCuar. IX, Example 1. Three-fourths of a number plus 5 equals 47, What is the number? SoLutTion: Let xz = the number. Then, se +5 = 47. Transposing, a = 42, Multiplying both members of the equation by 4, we have, 3x = 168. Dividing by 3, x = 56. CHECK: am +5 = 47. 47 = 47. Example 2. One third of a number plus one fourth of the number equals 21. What is the number? SoLution: Let x = the number. el Then, 3 nr = 21. Multiplying each member by 12, 4% + 3x = 252, or 7x = 252. Dividing by 7, x = 36. : CHECK: 12 +9 =21, 21 = 21. EXERCISES AND PROBLEMS Solve: 7. 1 per a! 1. = =4. pies: aig RE I - 6 3 + 6 2 x Vi rg Bc 2. —=3. ~na-5 = WZ. y 9 3 oh iy n—3 Ss; 9 + 2 =F ise 8. 4 <3 5. 4 —6 1+ 4 4. = 2s 9 re ee ae 8 5 a 12 bites Som Lies 10. — =4. 5 5 +8=13 0 - 4 | : i i i i) ] ‘Arr. 60] EXERCISES AND PROBLEMS 83 Bee OF) g 16. ee @ 1 Fac lieee — i — —_—_ — —= ese) * a3. om aaa 5 = 9) ae 5m. 18. 32 = =. 2808 ee 14. 100 + 3000 = 3028. 19. igs 0.004 | Lote! Susu 15. ens 20. 5 + 3 = 2s. _ 21. An article was sold for $5.40, thus losing one-sixth of the cost. Find the cost. _ 22. The increase in the value of certain farm land is 14 times the value 15 years ago. What was the land worth 15 years ago, if it is now worth $220 an acre? | 23. After a decline of 5% in the price of an article, it was worth $7.98. What was the value before the decline in price? 24. The difference between yy of a number and .05 of the aumber is 20. What is the number? 25. A bushel of corn and a bushel of wheat cost together 61.50. The corn. costs ¢ as much as the wheat. Find the cost of each. 26. A’s age is # of B’s age. The sum of their ages is 48 years. Find the age of each. 27. One-half of the time past midnight equals the time till 100n. What time is it? Hint: Let x = the number of hours past midnight. Then, 5 = the number of hours till noon, and +5 = 12. { | 28. One-third of the time till midnight equals the time vast noon. What time is it? 84 EQUATIONS AND PROBLEMS LCuap. IX. 29. Three-fifths of the time past midnight equals the time till noon. What time is it? | | 30. Twice a certain number is 7 more than { of it. Find the number. 31. If a certain number is diminished by 99 the result is the same as if it is divided by 10. What is the number? 1 32. The sum of the three angles of a triangle is 180°. In a triangle the second angle is 3 times the first, and the third is 1 the first. How many degrees in each angle? 33. In aright triangle one acute angle is 3 as large as the other. How many degrees in each? 34. In a right triangle one acute angle is 12° greater than the other. How many degrees in each? 35. In a triangle the second angle is 3 the first, and the third is twice the second. How many degrees in each? 36. At what time between 3 and 4 o’clock are the hands of a clock together? SuacEstion: Let x = the number of minutes past 3 o’clock when the hands are together. Since the hour hand travels 7; as fast as the minute hand, 6 is the number of minute spaces over which the hour hand has passed since 3 o’clock. But at 3 o’clock the hour hand was 15 minute spaces ahead of the minute hand. Hence, : 37. At what time between 5 and 6 o’clock are the hands of a clock together? | 38. At what time between 2 and 3 o’clock are the hands of a clock pointing in opposite directions? 39. At what time between 7 and 8 o’clock are the hands of a clock at right angles to each other? Two answers. 40. If it takes the author 3 times as long to make up a problem as it does the student to solve it, and the making and solving together take 12 minutes, how much time would be saved each by omitting the problem? ‘Arr. 60] PROBLEMS 85 41. In a recent election 46 less than two-fifths of the votes were cast by women. Three-sevenths of the women voted a certain ticket, thus casting 540 votes. How many men voted? 42. A man travels for 25 hours in his automobile. At the end of that time something happens to the engine, and he has to travel at half his former speed for 13 hours to reach a garage. His speedometer shows that he has traveled 65 miles. What was his speed at first? _ 48. The total operating revenue of the Pennsylvania Rail- road Company for a certain year was $6,000,000 more than 4 of the total operating expenses. The total operating revenue was $157,000,000. Find the total operating expenses. _ 44, A man has a certain sum of money invested at 7%, and = as much at 6%. The income from the two investments is $248. Find the amount of each. | CHAPTER X DIVISION 61. Division of monomials. Since a?- a? =a', it follows from the definition of division, Art. 32, that a’ +a? = a7, Similarly, since q™"-.qr = a", then, ms pn — pm-n (Ge Fa Ss fe ‘ We have then the following rule of exponents for division: The exponent of a letter in the quotient equals the exponent of that letter in the dividend minus its exponent in the divisor. In all cases the quotient must be given the proper sign according to the rule given in Art. 32. To multiply 3ry? by 42%y2, we form the products of the ‘numerical coefficients and then the products of the letters. Hence, to divide 12z%y* by 3xy?, we divide 12 by 3, then x* by a, then y’ by y’, and obtain 12z°y' oxy’ =47°/", If there are factors in the divisor that are not in the dividend, the result is left in the form of a fraction. Thus, 18-23), 2mink Sie 12 2’ 6mnz 3mz 86 ‘Ars. 61, 62] DIVISION OF A POLYNOMIAL Complete the following indicated divisions, and check the EXERCISES results by multiplication : 3h2 a3 is Fe QD 14: 120%? 21. (=2) (-y), 3ab x = 87)? | aby _p)2 AD. 6 12. —lbmin* 22. (~a) (—6)? —5m3 —b ’ Sxy (a + 1)% 3. 18a + 3a. 13. ce 23. (a +12 0.52%y (x — y)? rteapel Saati Sa 4. 30a? + —2. 14. WES 24. (Ga) = pet eR) eh 23 + 32-5 5. TYe + @. 15. Bis ore 25. er a ae ase (—m) (—n) (—p)? 2 6 7 ° 6. 3a? + 3b. 16. yp 8B. Sr yi2" (a — 3) (b + 4)? 37, 2 m2 ; ° 7. xty + 2°, 1. ee 8 G3 O ED 4a*b*c? 122°y724 ate. 2 - ° 8. 3a? + 2a. 18. he 28. a oe 12p°q —(loxy2 2; a 6 +5. 19. zy ae 29. 3022” — of2 Myn Pei 13 211. 20, —. a6 — 3st art 62. Division of a polynomial by a monomial. Since a(b+c+d) =ab+ac + ad, t follows from the definition of division that (ab+ac+ad)+a=b+c+d ' This illustrates the rule : To dwide a polynomial by a monomial, divide each term of he polynomial by the monomial and add the quotients. 88 Divide and check the results by multiplication: rs ee & on & Oo i ie 16. 17. 18. 20. 7.908 22. 23. OE OC Oe A ae ee DIVISION EXERCISES — 3a? + 4a by =z. 10y — 15a by 5. 6x? + 8x by 2z. ay + 4ay by wy. 14abx — 49bcex + 7bx by 7b. 2a — 6b + 10c by 2. ge — x? + a4 by &. m3 — 2m@x + ma* by —m. 48x3y2z — 18a°y?2 + 120xy?z* by Oxyz. 66min? — 1.2m?n by .2mn. _ 25a7b%e> — .5a>dbtc? + 2.5a°b?c? by .25a7b*c?. . 6a+2z2+ (a+2) bya+e. . 12(m — n)3 — 9(m — n)? + 6(m — n) by 3(m — n). 4a? — 6a + 2 gc aU ? 15. ie a —b —ce ad —A —1 3p? — 6p’g + 38pq" _ » Spy Alot. di AC a elias dae ee Siqcus =? 19 3 36(a — 1)? + 12(a — 1) aes 12(a — 1) ; 18a3y? — 302x7y bry 18(a — 1)? — 30(a — 1)? sis 6(a — 1) 98(a — 5)? + 35(a — 5)? — 7(a — 5) es 7(a — 5) =f, ae [Cuap. X. et 7? — 9 9a + 664+ 3 ee IN it My ‘Arts. 62,63] DIVISION BY A POLYNOMIAL 89 24. seen | 2°" ay, a?b3 4 oa ab” 26. Select monomial divisors for the following and divide: (zy — 1447+ 7x, 3a + 6a? — 9a’, 8xy + 42°y — 16zy’. 63. Division by a polynomial. Example 1. Divide x? + 2z?y + 2xy? + y’ by x? + ry + y?. SOLUTION: B+ 2ey+2ey+y wt+ayt+y t w+ xyt+ xy? a+y | ey+ ry +y vy + pty? me CHECK: Let z=1, and y=1. Then 2° + 2a’%y +22y? +73 = 6, e’+azy+y =3,andzx+y=2. Since 6 + 3 = 2, the solution checks. Explanation. (1) It is convenient to arrange the dividend and divisor according to the descending powers of x. (2) The highest power of x in the dividend divided by the highest power of x in the divisor gives the highest power of x in the quotient. Dividing x’ by x”, we get x for the first term of the quotient. (3) Since the dividend is the product of the divisor and quotient, it contains the product of the divisor and each term of the quotient. Hence, we multiply 2? + x2y + y? by x and subtract the product, 2 + xy + xy’, from the dividend. The ‘remainder, xy + xy’ + y’, contains the product of the divisor and the remaining terms of the quotient. (4) The first term of the remainder, «?y, divided by the first term of the divisor, x?, gives the second term of the quo- tient, y. Multiplying the divisor by y and subtracting, the ‘remainder is zero. The division is therefore completed, and the quotient is x + y. 90 DIVISION [Cuap. X; Example 2. Divide 1225 — 262‘ — 152’ + 8a? - 4x + 9 by 423 — 277 + | x-l. SOLUTION: 1225 — 262! — 1523 + 82? —4274+9 Mo? — 22? +27-1 124 — 67° = Se 52" Ba? —5r — 7 — 20x4 — 182? + 11x? — 42x — 2027! + 102° — 5a* + 52 —2873 + 162? —- 97 + 9 —2823 + 142? —724+7 2a? — 2a + 2=Remainder EXERCISES Divide and check the results by substituting numbers for the letters: . a+ 2ab+ 0? by a+ b. —2ry +y? by x — y. e+ 54+6byr+3. m? — 7m +12 by m — 4. y? — y — 20 by y — 5. a + 2a — 35 by a+ 7. v’+9byx+4+3. . 2+ bax + 8a? by x + 2a. a? — Jab + 146? by a-— 2b. 6a? + Tam + 2m? by 2a + m. . 202? — Try — 3y? by 5x — dy. a —bya— b. . 4a? — 25y* by 2x + dy. . & + 3b + 3ab? + B® by a + b. . m+ 5m?n — 24n? by m — 3n. . 4a? + 4a? — 29a — 21 by 2a — 3. . 8a? + 12a%b + Gab? + b3 by 2a + b. . 12u3 — 23u%v + bur? + 5v3 by 4u — 5v. a — bby a — b. } om? + 8n3 by 3m + 2n. SO ON Sd ee ae aoe gies So ee ee SODNARTAPEWNHH OS lar. 63] EXERCISES 91 21. x*—y* by x — y. 22. v+ytbyxz+y. 23. p? + g? by pt + qt. 24. 6x* — 1l3az* + 13a’2? — 13a*x — 5a‘ by 2x? — 3ax — a?. 25. 4y° — 26y* — 9y*? + 41y? + 2y — 12 by 4y? + 2y — 3. 26. 21a* — 16a*b — 5ab? + 16a7b? + 2b4 by 3a? — ab + B®. 27. 4m°n — 4m'n? + 4m?n*t — mn® by 2m’ — 2mn + n’. 28. a? + 6% — meg ee ore 0: 29. <2? — tee —iby 3 aE +4, 30. 8x3 + 27 by 32 4+ 2. 31. gya* — ayo'y + rery? —gy y® by 32 — ZY. 32. 4m4 — 2m? + 23m? + 3m + qe by Fm? — 3m — 33. .16m‘4n? — .01p*q* by .4m?n + .1p%¢’. 34. .0403 — .12y° + .17xy? — .12%*y by .2% — By. 35. -.002482° + 1 by .32 + 1. 36. 1 by 1 + 2 to five terms of the quotient. 37. 1 by 1 — x to five terms of the quotient. 38. 22" + Qarry” + y?" by x” + y". 39. a3" + 3a2"b" + 3a"b?” + 6" by a” + b*. 40, a3rt3 _ yont9 Hy grtt — y2nts, 41. Show that ‘pata baat 5 AE os | 42. Show that. 2m* — 5m'n + 6m?n? — 4mn? + n* divided by m? — mn + n? equals 2m? — 3mn + n? when m = 1, and n=1. 43. Show that a® — b® divided by a — b is equal to at + a*b +b? + ab? + bt when a = 1, and b = 2. 44. The area of a rectangle remains 1 while the base and | change. Find the corresponding values of the base ‘when the altitude has the values 3, 2, 1, .1, .01, .001, .0001. Which of these rectangles has the fangs verter? Which ‘has the greatest peuetr! : 45. In the fraction + — let 2 take on the values 1, 2, 4, 8, 16, P| =2x+ywhenz = 1, and y =2. 32, 64, and soon. So ae as possible, represent the correspond- 92 DIVISION [Cuap. X. ing values of the fraction on the number scale. What value is approached by the fraction as © becomes larger ? 46. Find the corresponding values of the fraction ca when x takes on the values 1, 2, 3, 4, and so on. So far as | possible represent the values of the fraction on the number scale. Find a value of x that will make the value of the frac- tion greater than .999. What value is approached by the fraction as x becomes larger? 64. Literal coefficients. Subtract : vk; ax 6. av Lion ba Sie. —Nx (a — b)x 2. cy 7. —62 12. ap + bq —dy —M2z 5p — mq (c+ d)y 3. ax Ser 13. 32? + bx : —bx Ss ka? — 9a (a + b)x 4. am 9. 8xy 14. nr? — mr bm axy 1 at 5. 3m 10. ar 15. ar —by +c am Mie —2x — dy —2 In each of the following expressions the terms have a com- mon factor. Find this common factor and write each ex- pression as a product. 16. ax + ay — az. SoLturion: The common factor is a. Then, ax + ay —az=a(r+y—2). it i | i ker. 64] EXERCISES 93 17. bx + by. 24. a*y + 2aby + b’y.. 18. 6x — by. 25. x — 3x + an. 19. a? — 3a. 26. 15a — 20ab + 5a. 20. p+ prt. 27. aba? + aby — ab. 21. y — 3y. : 28. amx + abm — 4amy. 22. 7n — 14m + 35k. 29. 2x — by +2. 23. nr — dr? + Sr, 30. ax — bx + cx — &. The following exercises illustrate the use of the division of yolynomials in the solution of equations. Solve and check results: 31. ax — a’ — 3ab = 2b? — bre. SOLUTION: ax — a? — 3ab = 2b? — br. ax + bx = a? + 3ab + 2b?. (a + b)x = a? + 3ab + 207. _ @ + 3ab + 20? a+b = a-+ 2b. Cueck: Substituting a + 2b for x in the given equation, we have, a(a + 2b) — a? — 3ab = 2b? — b(a + 2b). Multiplying, . a? + 2ab — a? — 3ab = 2b? — ab — 20’. Jollecting terms, —ab = —ab. 32. 2x + 3ax = 10a + 15a?. 88. ax + bx = c(a + bd). : | 34. ax — bx = a? — 3ab + 26°. $5. ax + bx = c(a + b)(c + a). (36. ax + 2ab — & = a? + ba. 87. ax +4=a8+ 30+ 3a+1. 38. ax — bx = a’ — B. 39. mz +3m=2+m? + 2. 40. x +1 = 267+ 5b — be. a a DIVISION . [Cuar. 38] REVIEW EXERCISES AND PROBLEMS 14. Find the sums of (a) 62, —2x, -32; (b) 5(m+n), —(m + ai —8(m.+n) ; (6)/ 73,3, lle, (d) an, 38n, —bn; (e) zy, y, —cy- 2. Arrange 622 —« + 9x5 — x! +5 according to ascending powers of x. : 3. Arrange a> + 4ab? + ab — a‘b4 + 6a*b? according to descending - powers of a; of b. 4. In each of the following pairs of numbers tell which number is_ the greater, and which has the greater absolute value: (a) —4, 7; -(b) —9 —1; (c) -6,0; (d) —5,5 5. State a rule for adding like terms. 6. State a rule for removing parentheses; (a) when preceded by the’ sign +; (b) when preceded by the sign —. 7. State the rule for combining exponents in finding the produ am-aqn-ar. Complete the following indicated multiplications: 8. m3-m™-m. 12. (3)?-4. 16. 6(27—4y + 3). 9. 2a3xy - Saxy2?. 13..(— 2)" (=a. 17. sx( 20 aed 5): 405 10. (a3)? - (ab)*. 14. 3ab- 4a. 18. 3(2-3 +4-3? — 3%)+ : ae 11. (—a4)8- (2ax)*. 15. mn. ‘19. State the rule of exponents in dividing am by an. Complete the following indicated divisions: 20. 12a%ty'+4atmy’. 24. (2a+2d) +2 28. aot Q1. (28-4) +2. 25. 3m + 3mn 29 bee jive 3m ap ae | 6 +14 (a — b)? 92. (23 +4) +2. 26. : eter Lat ee) 6 349 ent i 23. (2a-+2b) +2. Teese 91, Sa 5 2a 30. asb - ab? ab Zs Simplify: « — 3y — (@ +3y) - {1+ [e@+y -@+2y —2) +4] — 72x}. Arr. 64] REVIEW EXERCISES AND PROBLEMS 95 34. Simplify: 2a + (3b -a +4) — [2 + (3b —4) —(1 +a). 35. Multiply 500+70+5 by 200 +30+4, using the form for multiplying one polynominal by another. Compare this result with that obtained by multiplying 575 by 234 in the usual way. 36. Compare in a similar way the products (42 + 3)(18 +3) and 42? X 18%. Solve the following equations for z: Bier 44 3 — 1) = —2(x — 2). 38. 27 — a= 3x +b. 39. az + 0? = br + a’. 40. ™ 41. mz — 3n = 3m — nz. | ‘42. In the equation 2x + b = 3, give b numerical values such that she value of x shall be (1) a positive integer; (2) a negative integer ; 3) a positive fraction ; (4) a negative fraction ; (5) zero. Which of these values of x have no meaning if x is the number of points scored in a foot- all game? 43. Find six terms in the quotient 1 + (1 —). What is the difference »etween the sum of these six terms and i : = when « = 4? Whenz=.1? When x = .01? 44. Show that the numbers of the series .3, .33, .833, .3333, and so on proach nearer and nearer the value }. 45. If A is the area of a square whose side is s, then A = s?._ By what 3A multiplied if s is multiplied by 2? If s is multiplied by 3? By 4? 3y 10? By a? 46. 3 t= 10, what number is represented by 3¢4 + 52 + 8¢ +5? ope 0, Wat) ess ay! _ 47. Write 86423 and 23090 in terms of powers of ¢ = 10, and sub- ‘ract 5 times the second from the first. Check for ¢ = 10. 48. Write 67731 and 211 in terms of powers of ¢t = 10, and divide the : rst by the second. Check for ¢ = 10. CHAPTER XI LINEAR EQUATIONS 65. Linear equations. A great many of the equations we have solved can be reduced by the use of the four fundamental operations to the form : ax +6 =0, where x represents the unknown, and a and 6 are numbers that may have any value except that a cannot be zero. Thus, 42 +3 =a —7 can be reduced to 3x + 10 = 0, which is of the above form, where a = 3,b=10. Again, iy +2 =3(1 + dy) can be put in the form ay +b = 0, where a = 3,6 = -1 Equations which can be reduced to the form ax + b = 0 by use of the principles given in Art. 38, are called linear equations in one unknown. Such equations are often called simple equa- tions. EXERCISES AND PROBLEMS Write the following equations in the form az + b = 0 and point out the values of a and 6: | 1.. 8¢ + 2'= 62-4 6. 2. 62 — 5 = 92 + 2. 8. 5¢4 —3 + 21r7 = 18 + 42. 4. 3(2 —7) =“ +415. 5. 81 =—4(e 4+) =—2-7 6. 25 — 6(x — 6) = 20 — (2% — 13). 7. 13(2% — 1) = 5(52 + 4). 8 a, aes secre 3 7 —3(@-—5) | 9. m = |], 96 i | i if | Arr. 65] EXERCISES 97 y 10. x - 2-34 = WGr +1). E Hint: Multiply each member by 36. Which of the following equations can be reduced to the type form ax + 6 = 0? 11. x(x —1) = 404 + 3. SotuTIon: Collect terms and this equation reduces to 2? — 5a — 3 =0. ‘Since it contains a term in 2? it is not of the form ax +b = 0. m 12. (x + 3)(a — 2) = 4(a@ — 2). ‘— 13. v—44+5=54 2", my 14. (27 — 1)(82 + 1) = (62 — 12)(@ + 3). 1b. (x — 5)(27 — 9) = (& — 4) (22 — 6). fe 16. x(x? — 1) = 242 + 3. m 17. (22 — 1)(144e + 5) — 267 = 367(8% + 1) +11. ; x AP x x d5xe+4+1 19. (x — 2)@ +2) +3 = iG + 30. 20. (« + 1)? + ( — 2)? = («@ —1)(@#+ 5) + 2. Solve the following equations : 21. 9x + 5x + 21 — 68 — 6 = —22 — 5B. 92. 3° 5(x + 6) + 5° 7(1 + 2x) — 7: O(x — 8) = 827. 23. { — 5a = @ — 28a. 24. i -etz- G7 18. 25. 8e —5 = +5 + 153 26. a(x - §) - (5 +5) 7. S+5t1-5-5 98. ett ts 98 LINEAR EQUATIONS [Cuar. XL 29. +32 =3+4 92. 30. 4a + 5a = 5(a@ +). 31, (2 eo) eee $2. (x — 2)? — (4 —3)2 =@. $3. 2+a2+1=(¢+4+1). 34. 244242 = (& + 2). oe x 35. f +20 = (5 +1): 36. (e+ a)(@ — a) = 2 + 2x + 2a’. 87. (~@+ nS oa 2) = (*% + 2(5 + 1); 38. 922 — 1 = 3(82? + 52). 39. (2x — 5)(27 + 5) = (4r — 11)(a@ + 1). 40. When 4 is subtracted from twice a number the result is 30. What is the number? 41. When 4 is subtracted from a number and the result doubled we get 30. What is the number? 42. The sum of two consecutive integers is 27. What are the numbers? 43. The sum of three consecutive integers is 27. What are the numbers? : 44. Is it possible to find 4 consecutive integers whose sum is 27? 45. Find 4 consecutive integers whose sum is 46. 46. Find three consecutive odd integers whose sum is 39. 47. Find three consecutive even integers whose sum is 42. — 48. A teamster contracts to haul 2100 bags of flour. He makes 22 trips with his dray carrying the same number of bags each trip except the last one when he carries 42 bags, How many bags does he haul each of the first 22 trips? ; 49. The perimeter of a rectangle whose length is 4 feet longer than its width is 28. Find the dimensions. 50. The length of a rectangle is 3 feet more than twice the width. The perimeter is 42. Find the dimensions. ~ (Arr. 65] PROBLEMS 99 51. A cubic foot of pure water weighs 62.5 pounds. Twenty cubic feet of water weigh 15 pounds less than 22 cubic feet of ice. What is the weight of a cubic foot of ice? __ 62. One pound of ice occupies 30 cubic inches of space. Five pounds of ice when melted decrease 12 cubic inches in volume. What volume does 1 pound of water occupy? 53. EKighty gallons of water from the Dead Sea weigh 1 pound less than 100 gallons of pure water. The weight of a gallon of pure water is 8.33 pounds. What is the weight of a gallon of water from the Dead Sea? I 54. A father 54 years old has a son 21 years old. How any years ago was the father 4 times as old as the son? ' 65. What number is to be subtracted from both the nu- ‘merator and denominator of 3+ in order that the new fraction may be equal to 4? 56. The denominator of a fraction is 12. When 3 is sub- tracted from both numerator and denominator the value of Mie fraction is decreased by 34. What is the numerator of the first fraction? 57. A number is the sum of two parts. The first is 3 ‘greater than half the number, while the second is 2 greater than one-fourth of the number. What is the number and how was it divided? _ 68. What number has the property that when multiplied by & the result is greater by one than when multiplied by #? CHAPTER XII IMPORTANT TYPE PRODUCTS — 66. Certain algebraic products occur so frequently and | are so useful as models for other multiplications that they | should be memorized. | 67%. Square of a binominal. Multiplying a +b by a + ‘a | we find | (a+b)? = a@ + 2ab re b?, ~ which may be translated into words as follows : | The square of the sum of two numbers is the square of the | first, plus twice their product, plus the square of the second. | In a similar way, we find (a — b)? = @ — 2ab + OB, or in words : The square of the difference of two numbers is the square a | the first, minus twice their product, plus the square of ie | second. | In these formulas, a eit b represent any two numbers, or any two expressions. Example 1. (10 +5)? = 10? +2:-10°5 + 5 = 225. Here a = 10, and b = 5. Example 2. [ay? + (x + y)]? = (xy)? + 2(@zy)@ +y) + (@ +9). Here a= ary’, andb=2x+y. An algebraic expression which is the product of two equal factors is called a perfect square. 100 . ‘Arr. 67] EXERCISES 101 ; EXERCISES Square the following binomials by the above rules: 1 ¢v7+/y. 4. “2+¢. 7 1+2. 5. 24.3. 8. 7+ 5. 6. 2 — 3. 9. 646. Fia. 17 10. Let ABCD (Fig. 16) be a square whose side is of length \a+6. Its area is then (a+b). Let AEFG be a square of side a. From the figure show that (a + 6b)? = a? + 2ab + b?. ) 11. Let TUVW (Fig. 17) be a square whose side isa. Let |) XU =b. Then TXYZ is a square whose side is (a — b). (From the figure show (a — b)? = a? — 2ab + B*. Square the following numbers by expressing each number _as the sum or difference of two other numbers and then applying ‘the above rules. Work each exercise in two different ways. 12. 39. SOLUTION: 39? = (30 +9)? = 30? + 2: 30-9 + 9? = 1521. 39? = (40 — 1)? = 40? —-2- 40-14 12 = 1521. Sees Oy. 15. 1, 16. 0. 17.51. 18. 99. | Square the following according to the above rules and ‘verify by actual multiplication : H 19. a? — y. 21. ry — 2. 23. cy +2. 20. x + 3y. 22. ab + xy. : 24. 2a + 32. 102 IMPORTANT TYPE PRODUCTS [Cuav. XIL. | 25. xy — y’. 28. (a+b) +6. 31. x+y - 2. 26. 5k — 3h. 29. 999. | 32. m—n—?. 27. a+ 110. 30. r+y+2. 33. 2h +k — 3l. The following are squares of binomials; find the two equal | binomial factors: | 34. 2 4+2cd+a@. 38. 4a? + 12ab + 9b. 35. 2? — Qry + y’. 39. 922+ 6241. 36. 7? — 427 + 4. 40. 252? + 20x + 4. 37. y? + by + 9. 41. 49r? — 42r + 9. - 42. Give a rule for finding whether or not’a trinomial is a perfect square. : | Hint: Two terms of the trinomial must be perfect squares. What is | the other term? Some of the following are perfect squares; find them and | give the factors: 43. c+ 2cd — d’. 48. 9 + 42 + 49. 44, x? — 102 + 25. 49. 4-54 25. 45. 16a%y? — 8ry + 1. 50. 49a?y* + 1l4ay? + 1. 2 46. 9a2 + 7a +1. 61. Gtue+4. 47. vy?+ay +1. 52. + B48 68. Product of the sum and difference of two numbers. By carrying out the multiplication, we find (a+b) (a — b).= @ — BW, In words, this formula reads : The product of the sum and difference of two ee: 1s the difference of the squares of the two numbers. i ' “Arr. 68] EXERCISES 103 i EXERCISES Form the following products by the above rule and verify by actual multiplication : 1. (vx —y) (@+ y). 7. (3a + 2b) (8a — 2b). 2. (c +d) (c —‘d). 8. (x — y") (4 +»). 3. (x +2) (x — 2). GF (le ey (1 42) 4. (3x + y) (32 — y). 10.8010 1) (10 4-1), 5. («@ — y’) (@+ y’). 11. (a? +3) (@ — 8). 6. (2 +2y)(-r+2y), 12 (+2) (-2). | The following binomials are the products of the sum and Wiifierence of the same two numbers. Find the numbers. 13. a? — 4b. 15. 16 — 2". 17. o**. — 7! 14. 42? — 9y’. 16. 9 — 25. 18. xt — y'. Perform the indicated divisions : 19. (2? — y*) + (@+ y). / 20. (at — b*) + (a? — bd). 51. (s? — Or?) + (s — 3r). 22. (16n? — 36m?) + (4n — 6m). 23. (a? + 2ab+ 0? —c’?) + (a+b-4+0). 24. (x? — Qry +.y? — 92?) + (x — y + 82). 25. Give arule for telling whether or not a binomial is the product of the sum and difference of the same two numbers. Which of the following are the products of a sum and dif- ference? Find the factors of those which are such products. 26. (a — b)? — ce’. 29. a +0? — c?. a7. at +B. 30. a2 +a +1)- Wes. 84 2241-42 81. a + 2041-2 104 IMPORTANT TYPE PRODUCTS ([(Caap. XID 69. Product of two binomials having acommonterm. ‘l'wo binomials having a common term can. be written in the form z+aand«#+b. By actual multiplication, we find re+a a+b x + ax ba + ab x? + (a+ b)x + ab or (a +a) (wv +b) =a? + (a+ b)x + ab. In words this reads : | The product of two binomials having a common term equals the square of the common term plus the product of the common term by the sum of the other terms, plus the product of the other terms. EXERCISES Expand by the above formula : 1, (2 +2) (« +3). 4. (x —1) (&@ — 2). 2. (a+ 2) (a +3). 5. (a — 4) (a + 5). 3. (x + 5) (@ +2). 6. (n — 3) (1 — 39). T.. (22 + 3) (2x +1). SoLuTIon: (22 + 3)(2a 4+ 1) = (2x)? + (8 +1)2a4 +3°1 = 4x? + 8x +3. 8. (3y.+ 1) (8y + 2). 10. (4 + 3a) (4 — 2a). 9. (2a +b) (2a +c). 11. (av + 13) (ax — 1). The following trinomials are products of two binomials having a common term; find the binomials: 12. 2° + 37 +2 = (x +2) ( ). 18. y+y-—6. 13. 2? + 42 + 3. 19. a? — 2ab — pos 14. 2+ 67+ 5. 20. 25+ 45+ 14. 15. n?+ 7n + 10. 21. w? — 12u + 32: 16. a? —6a4+ 9. 22. (x +1)? + 8(@+4+1) -4. 17. «x? — 3x + 2. i | i ‘Arrs. 69, 70] | MISCELLANEOUS EXERCISES 105 MISCELLANEOUS EXERCISES Form the following products according to the foregoing rules: 1. a _ v) G + v), 9. (2c +1) (x + 4). 10. (x +3) (x + 3). it: . (2P + 4) (2P + 6). edie bite Ls: (xy + 2) (ay — 2). io: (z+ 6) (+7). 14. (x — 1) (x + 2). 15. (a — 4) (a +4). 16. (ab — xy) (ab + xy). (x +3) (x — 16). Ge 9) a — 47°): (1 +c) (1-2). (9r242 + 1) (9r?? — 1). (a+b) +c] [@ +b) —c]. (x?y? + 7) (xy? — 2). The following expressions are either the products of the sum nd difference of the same two numbers, or products of two binomials having a common term; ss the factors of each : LT. 18. 19. 20. 21. 70. x? + 5a 4+ 4. a? + 7a + 12. 100 — 9a?. a? + 4a +4 4. (x? — Qay + y”) — 2. 22. — 9x + 20. 23; + 2ab + b? — s? + 4s — 4. 24. p> + 7pq + 104°. 25. 16 + 20x + 627. Cube of a binomial. By actual multiplication, we find (a+b)? = (a+b) (a+b) (a+b) =a+6 a+b a? + ab ab + 0? a? + 2ab + 6? a+b a? + 2a2b + ab? a’b + 2ab? + 6 a? + 3a7b + 3ab? + 63. 106 IMPORTANT TYPE PRODUCTS — [Cuap. XII. — That is, the cube of the sum of two numbers a and 6 consists — of four terms as follows : a The first term = a the cube of the first number. The second term = 3a*b = three times the square of the first multiplied by the second. _ The third term = 3ab’ = three times the first multiplied by the square of the second. The fourth term = b? = the cube of the second. As a formula, we have (a+b) = @ + 3a°b + 3ab* + BD. In a similar way, we find | (a — b)? = & — 3a’b + Sab * — B. EXERCISES Expand the following by the foregoing rule or formula : 1. (c + d)°. 4. (« + 1). 7. (3a — b)3, 2: (27 —y)*. 5. (2 + 3)3. 8. (52 —2)*. 3. (a + 2)%. 6. (2x + y)’. 9. (xy + 1)%. 10. (xy + 22). 71. Square of a trinomial. By actual multiplication, we find (at+b+eP=040? 4c? + 2ab + Zac + 2be. That is, the square of a trinomial equals the sum of the squares of its terms plus twice the product of each term by each succeeding term. ; EXERCISES 1. By actual multiplication prove: (a+b—cP=0+h+4c + 2ab — 2ac — 2be. 2. Prove (a—b—c)? =@+0?+¢ — 2ab — 2ac + 2be. 8. Prove (a—b+c)=@+4+0?+c? — 2ab + 2ac — 2bc. PQ Ta ar See ‘Arr. 71] | EXERCISES 107 By use of the foregoing formulas expand : 4. (x+y+z)?. 8 (142+ 3)?. mo. (c— y + 1)?. 9. (1+1+1)?. 6. (a + 2b + 3c)’. 10. [(a +b) +2+y]?. q. (ar — y + 32)". AT eo 0 ae), MISCELLANEOUS EXERCISES Perform the following multiplications and divisions, using the type products of this chapter whenever possible. Check the results by substituting special numerical values. 1. (14+ a)? 14.. (1 +a + 2a)’. 2. (1 +n) (1 + 2n). 15. (402+ 44 + 1) + (Q4 41). les. (1 —7n) (1+ 2n). 16. (2? — 32 + 2) + (x — 2). 24. (1 —n) (1 — 2n). 17. (a-— 4) (a+ 3). 5. (1 +n) (1 — 2n). 18. (a + 4) (a — 3). 6. (1 — 3a) (1+ 3a). ~ 19. (a+ 4) (a+ 3). m7. (1 + 2)2. 20. (a — 3) (a — 3). eo. (1 + a)*. ate AL 2 4 — 16¢ 9. (2x — 3)?. 22. Ti pSaae ett — 2 — y) (1—2z +4). 11. (xy — 8) (xy + 1). 23. (101) (99). 2. (ct+y—8) (xt+ytl). 24. (a? + 262 +169) + (r+ 18). 13. (1 + 2)%. 25. (a? + ax — 2a”) + (a + 22). | b | | | Fig. 18 K Fia. 19 26. What are the dimensions of the rectangles in Fig. 18? Show that the two rectangles can be joined together so as to 108 IMPORTANT TYPE PRODUCTS [Cuar. XI. make a rectangle whose length is a + 6, and whose width is a—b. From the figure show that (a + b) (a — b) = a — B’. 27. Fig. 19 is a rectangle of width x + a and height x + 0b. From the figure show that (v + a) (v + b) = a 4+ (a+ b)x + ab. Fill out the blanks in the following : 28. ab? — 64c2d? = (ab + 8cd)( ). 29. a3 + 3a2x + 3az? + 23 = (a? + Zax + x*)( ). 30. 44:27 4 2y + 2y = 2 Paya 31. (xy? — Qey +1) = (zy — 1)( ). 32. (2x +a) (24 +b) = 427 +( ) +8. Remove parentheses and unite terms where possible : 33. (2% — 3)? + 3(82 — 2). 34. (xy — 2)? + (ay + 2). 35. 6a(a — 1) + 2(8a — a)?. 36. 62(a — 1) + 2(a — 3z)?. 37. a(b — a) — (2 — a?): 38. (a? — b?) — (a — ). 39. 6(2? — 37) — 12(2 — 3) 40. (5% — 3%) — (5 — 3)’. 41. (a? = b*) — (a — Bb)’. 42. Show by multiplying that a(a — x)(y + 2) + x(a — y)(a — 2) — 2ayz =a(a — y)(@ +2) — y@ —- a)(a — 2) — 2axe. CHAPTER XIII FACTORING 72. Prime factors in arithmetic. One of the problems of arithmetic is that of finding the factors of a given number. A factor is one of two or more numbers whose product is the given number. Thus, 30 may be written 2-15, 5-6, 3-10, or 2-3-5. The numbers, 2, 3, 5, 6, 10, and 15 are all integral factors of 30. If we consider fractions we may go on indefinitely, for we may write 30 = 7-24, 30 = 4-$-36, 30=4-32-195 and sO on. The important problem in this connection, however, is find- ing the prime factors. A prime factor is an integral factor which is the product of no two integers except itself and unity. There are many sets of numbers whose product is 30, but there is only one set of prime factors, that is, 2,3, and 5. Any integer has only one set of prime factors. _ By the factors of a number we shall often mean its prime factors although the word prime is omitted. EXERCISES Separate the following numbers into products of prime factors; find for each number two.sets of factors which are not prime: i 12. Baro. 5. 32. (ER MENE 9.570; i 2. 15. 4, 120. 62-31. 8. 100. ) ) | _ @3. Prime factors in algebra. The expressions which we propose to treat in this chapter involve a definite number of Additions, subtractions, multiplications, and divisions, but no other operations. Such expressions are said to be rational 109 110 FACTORING [Cuap. XIII, expressions. A factor of a rational expression is one of two or more rational expressions whose product is the given ex- pression. An expression that, is rational with respect to. a given letter contains no indicated root of that letter. 3 : Thus, a, : +=, a? + a are all rational with respect to a and x. The expression @ + ax + ./zx is rational with respect to a, but not with respect to x. The expression involves, besides the operations of addition and multiplication, the operation of extracting the square root. An expression is integral with respect to a letter if this letter does not occur in any denominator. 3 Thus, a? + = is integral with respect to x, but fractional with respect to a. A factor is rational and integral if it is rational and integral with respect to all the letters contained in it. Thus, a? + ay + 5a? is rational and integral. In this chapter the word factor is to mean rational and integral factor. : Thus, the factors of 22 — y? are x —y and x+y. Although 2? —¥ = V2 — y- Vx? — 2, we shall not consider V2? — y’ as a factor of 2? — y?, Again, the factors of } — 2? are — x and 3 +2; the factors of 2x3 + 22? + 2x are 2, x, and a7 +2 +41. A factor is said to be prime if it contains no factor except itself and one. The different prime factors of 15a%b’c are 3, 5, : a,b,andc. 15, a’, b?, ab are also factors, but not prime factors. In ARGS as in arithmetic, the most important sees of factoring is that of finding the prime factors. 44. Factors of monomials. The factors of a monomial are evident by inspection. Thus, the different prime factors of 21a?z* are 3, 7, a, 7 % | \Anrs. 74, 75] MONOMIAL FACTORS 114 EXERCISES Name the different prime factors of the following : mi. Ga*b*. - 4. Qlab?cid*. Dh abecsd?, a 2. 12a?y%z. 5. 32p7q°r. 8. 169272". (8. 42meniaty. . a. 33 + D2sH5. 9. 1728(abc)?. | ! 10. —108(xy?z*)3, _ 11. If ab?’ is considered as one factor of 6a2b2, what is the other? | 12. If 42m?n*xy is considered as the product of two factors and one factor is 6mz, what is the other? | 13. 32p’@r is the product of three factors. Two of them re 2p and pqr. What is the other factor? | 14. If a®b’c'd’ is obtained by multiplying together three oxpressions, two of which are a’c? and bd, what is the third? 15. The expression 9lab?c’d! is apraned by multiplying ‘together three expressions. One of these is 13bcd. Write down i; three possible pairs of the other two factors. | 75. Monomial factors in polynomials. If each term of a dolynomial contains the same monomial factor, then this enomial factor is a factor of the polynomial, and the problem | s to obtain the other factor. (See Arts. 45, 53, 62.) A type ‘orm of expression coming under this class is | ax+ay—az=a(r+y-— 2), 17 the factors of ax + ay — az area andz+y —z. : Thus, in factoring 6a*b -- L2a*%* — da*b? we find that each 6a‘b — 124%! — 3a°b? = lon — 4b — b), EXERCISES Factor the following : 1. mx — my. 3.00? 4, 2. 4x — 8. te ad) ee) 112 FACTORING [Cuar. XI) 5. 6a? + 16a. 8. 2x + 4y — 8z. 6. 32 + Ory + 1827. 9. 6a? — 9a? + 3a. . | 7, a + a°b + ae. 10. 5a + 202? + 152°. 1 11. 2x +2y +ax +ay =2e+y)+ae+y) = (2+a)( ), 12. 4ax + 4ba — 3a — 3b + Say + Sby ~ 4r(a +b) — 3(a +b) + 5y(at 6) =( )(). 13.- 3a + 3b + 3c — 2ax — 2bx — 2cx -3(a +b +e) —2e(at+b+e) =()(). 14. Write down a rule for factoring polynomials havi a common monomiat factor. 6. Factors found by grouping terms. As shown in Exer- cises 11, 12 and 13 of the last article, a polynomial may have. Pe enomial factors which may be found by grouping the terms properly. A typical example of this class is aw + ay agl + by. Collecting the terms in parentheses, we get ay - (ax + ay) + (bx + by) =a(e+y)+b@t+y). TF Each group of terms contains x + Y, which then may be factored out. This gives a(a+y) +b(@ +y) = (x + y)(a + 9B). 2. -- + eke gteeyyepatlinn age oman. Hence, a] ax + ay + bx + by = (a +y)(a + 0). 1 ) i EXERCISES Factor the following : | 1. 2(a +b) +2(a +d). 3. 2(22 + y) + x(2u + y). | 2. ax +y) + (e+ y). 4. a(b—c)+b(b—c). “@l 3a(2y — 32) — 4y(2y — 32). 2(4a — 3c) + 3(4a — 3c). Watb+c)+alat+b+c). . a(x +y) + y(a ty) +2 +). — ee CO nae aeecealal ft f ‘Arts. 76, 77] DIFFERENCE OF SQUARES 113 Hint: 9. a(a +b) + 2a(a + b) + b(a +). 10. n(x — y) + my — 2). Write the expression n(x — y) — m(x — y). MISCELLANEOUS EXERCISES Factor the following : Le — 3a(b + c) — 4(b +c) + 2(6 + €). 2. aa+b+c)+b(a+b+c). pial Pica et geld sen Aa 10. 4a*b — 10bc + 6ab. 8a? + 403 — 2x4 4+ 62°. a(a — b) + 2(b — a). a*b — ab? + a*b® — ab‘. afa+b+c)+b(a+b+c)+c(a+bd-+0c). Avty — 12x4y? — 1l6xty® + 8arty?. Write the prime factors of —62527y’. Write the prime factors of 27xy(a + y)’. _ %%. Difference of two squares. The typical form for this ease is 2 2 a? — 6, which we have seen to be the product of the sum and difference of aand b. Hence, aw — b= (a+ b)(a— D). EXERCISES Factor the following : i ee 2 8. x* — 16. 2a°7 — 1. 9. x*— 1. 3. a — 4. 10. 16a* — 81. 4, n?— 9. 11. (2a — b)? — 6. By te — a: | 12. 25 — 4(x — y)?. oe 9 — 4. 18. (a +6)? —-@+y)*: 7. «4 — y'. 14: (2a — b)? — 9(@ — 1). Sotution: (#4 - y’) = (+ yY)@ -y?) =@+y)@+y)@ —y). 114 FACTORING [Cuap. XIII. MISCELLANEOUS EXERCISES Factor : | 1. 3abc + Ya*be. 2. atz* —1. 3. p(p — g) —a(q — DP) + 29 — P). 4, atx — dy’. 10. wv — y%. 5. vt — y’. 11. ab + b& + ac 4+ cz. 6. 362° — 1. 12. 2° — 1. 7. 2xryz — 8a*yz — 16zry’z. 13. 17ab’xy? — 34a*b*27y?. 8. mx + my + nx + ny. 14. 16924 — 7°. 9. a® — D®. 15. a*x? — a. 48. Trinomial squares. We have seen that | a? + 2ab + b? = (a + b) and a2 — 2ab + b? = (a — D) from which we see that if in a trinomial the middle term 1s twice the product of the square roots of the other two terms, then the trinomial is a perfect square; that is, it is a product of two equal factors. Thus, in xy? — 2xy +1, we have 2ry = 2° /xy?* /1 and the trinomial is a perfect square, (ry — 1)’. EXERCISES Test the following trinomials to see if they are perfect squares. If they satisfy the tests, find the two equal factors. 1. 22+ 2ry 4+ y’. 5. a + 2a+1. 2. x? — Qry + y’. 6. 2? + 3ry + y’. 8. v2 —ayt+ y’. 7. 40? + 4ary + 2y?. 4. v2 + Qry + y’. 8. 44+ 16+ 16. 9. 70? +2:°70°646. “Arr. 78] MISCELLANEOUS EXERCISES 115 In each of the following expressions replace the parentheses “by a term which will make the trinomial a perfect square: 10. 2?+()+y%. 13. 400+ ( ) + 32. a. + ( ) +I. 14. 2? 4+ 42y +(). 12. 407+ () +a’. 15. 25y?+ ( ) +4. 16. 169n? + ( ) + 169m?. | 17. The middle term of a trinomial is 2xyz. Find three possible pairs of terms which will make the trinomial a perfect “square. Each of the following terms may be considered as the middle term of many trinomial squares. Find three such trinomial squares for each exercise. 18. 2abc. 19. 6ryz. 20. 4a%b. Separate each of the following into two factors : 21. 225x?y* + 120x*y* + 162xy’. 22. 36a°b? + 60a*b® + 25a*b4. 23. 9a8 + 42a’b + 49a°b?. MISCELLANEOUS EXERCISES Factor if possible : = oy — 2. —ab?c'd! + 3a4b?c2d + 6a°b’c?d?. Find two factors only. ax — bx + ay — by. . 4a? — 12a + 9. a’b? + 4ab + 4. a’b? + 2abx + 2. an — 1. ab? + 10abry + 2527y?. x? + xy + Qu 4+ 2y. vt — 627 + 9. 3a(a ++b+c) —(a+b+4+0). x§ — 27° + 1. areata tere See Bee war SS 116 FACTORING [Cuap. XIII. | | ' In each of the following replace the parentheses by a term | which will make the trinomial a perfect square: 13. vy? +()+ 2. 14. 4a? + 4a+( ). 15. ( ) + 42ab + 9. 16. Find three trinomials which are perfect squares and which have 22?y for a middle term. 17. Factor a? + 2ax’y + x+y’. 18. Factor x?" — y’. 19. Factor 2ax + 2ay — 3bx — 3by. 20. Factor 169x* — 182a?y + 49y?. 79. Trinomials of the form a? + (a+ b)x + ab. We have found that the product of two binomials having a common term is given by the formula (c+ a)(a+b) = 224+ (a+ b)x + ab. Here it is to be noted that the coefficient of x is the algebraic sum of a and b and that the last term is their product. For example, (2 +4)(¢4 +3) = 22 + 7x + 12, (x —4)(2 —3) =2? — 7x + 12, (cx —4)(2 +3) =2 —-z- 12, (c¢ +4)(@ —3) =2? +2 —- 12. If a trinomial comes under this type, it is possible to find two numbers whose sum is the coefficient of « and whose product is the last term. These trinomials can usually be factored - by inspection. EXERCISES From the following, pick out those trinomials which are of | the form 2? + (a + b)x + ab and find the factors. BF ersten pit i), Sotution: We are to find two numbers whose sum is —5 and whose product is 6. Such numbers are —2 and -3. Hence, a? —5¢+6= © (x — 2)(a — 3). { fe | “Arr. 79] EXERCISES 117. 2; 2? + 52 + 6. Bee a G, : Sotution: We are to find two numbers whose sum is —1 and whose product is —6. Such numbers are 2 and —3. Hence, x? — x —6 = (x + 2)(x — 3). 4, 2? +2 -6. 5. a + 8x + 2. SoLuTIon: It is impossible to find two integers whose product is 2 and whose sum is 8. 6. 227+ 7x + 2. 9. 2? — 2x — 2. 12. a* + da-+ 1; 7. 224+ 32 4+ 2. 10. +2 —-2. 133 a ae: 8. 2? —3r44+2. 11. a@+4a+1. 14. y?> + 6y + 8. MISCELLANEOUS EXERCISES : Factor : im 1. 4x(a — b) — 3(b — a) + 5y(a — b). 2. 22 — 62+ 8. 6. ax? — 8ax + 1da. a. zc — 22 — 8. 7 n+n— 12. 4, 6? + 2b —- 8. Sip at 14": 5. bry + 4by — cry — 4cy. 9. p?+7p+3 10. 2? +47 +3. 11. Fill out the parentheses to make ( )+4ay+y? a perfect square. 12. Fill out the parentheses to make 8la’x? + ( ) + 4a‘ a perfect square. Factor : 3. 5 + 62+ 27. 16. az? + Yar + 20a. 14, a? + 122. — 28. 17. 9a? — 54. 15. 274 — 2° + 40 — 2. 18. y? — 4y — 21. 118 FACTORING [Cuap. XIII. | 80. General quadratic trinomial, ax? +ba+e. If we mul- tiply together the two binomials 3a + 5 and 2a + 3, we find: 3x +5 3D eo 24 +3 x 6x? + 10x 2x 3 9x + 15 622 + 19x + 15 The two products 27-5 = 10% and 3xz-3 = 9z are called cross products. The product 6x? + 19x + 15 is a trinomial of the form O07 be: While the product of two binomials like 32 +5 and 2% +3 is a trinomial of this form, yet all trinomials of the type ax? + bx +c cannot be factored into such binomial factors. If the trinomial can be factored it is often easily done by inspection. Example. Factor 67? + 192 + 15. The product of the first terms of the binomial factors is 6x2. The first terms are then 2z and 32, or 6x and 2, if the coefficients are integers. The product of the second terms of the binomial factors is 15. The second terms are then +3 and +5 or +1 and +15, if the second terms are integers. Since the middle term of the trinomial is positive we keep only the positive terms 3 and 5, and 1 and 15. We have now to pick out two binomials having 2x and 32, or 6x and x for first terms and with 3 and 5 or 15 and 1 for second terms in such a way that the middle term of the product is 19z. That is, the binomials are chosen so that the algebraic sum of the cross products is 19%. By | trial, we find the factors to be 32 + 5 and 2z + 3. EXERCISES Find the following products : 127 +3) Tote 4) 5. (2a + 1) (8a + 1). 2. (22 — 3) (52+ 4). 6. (2c — 4) (2c + 8). 3. (2% — 3) (5% — 4). 7. Qe +5) (22a 4. (2x + 8) (5a — 4). 8. (Sy + 4) (2y + 8). 113 6x? + 7x + 2. 6a? — 8x + 2. 62? — 7x + 2. 10x? + lla + 3. 10a? — 132 + 3. 1527 4 132 Ps: Show that the trinomial x? + (a + b)x + ab is a special z*.— (2 — 18: zy? — 15a — 34. PP — 27 — 7 + I. 4. 8x7? + 227 + 15. 8a? — 23x + 15. ?+(x+y)t + ry. Arts. 80, 81] EXERCISES Factor the following : 9. 277 + 7x 4+ 3. 15. 10. 322 + 5x + 2. 16. 11. 527? + 7x + 2. 17. 12. 5a? — 7x + 2. 18. 13. 227 — 5¢ + 3. 19. 14. 2x? — 7x + 5. 20. ® 21. ease under the type az? + bx + ¢. Factor: | 22. 22+ 92 + 14. 24. ) 23. 2? — 4x — 21. 25. | MISCELLANEOUS EXERCISES 1. 2(xe+y)+a(x+y). 3: 2. 9(x + y)? — 4. 5. 8x? + 23x + 15. 6. 4x(a + b) — 8(a + b) + Sy(a +). 7. p?>+4p — 21. 14. 8. a? + 9ab + 80. 15. 9. a®b? — a®b? — ab+ 1. 16. cation, and ax + by — ay — bx. - (a+b)? — ee. . (24 + 3y)? — 1. . 8a? + 1212 4+ 15. 81. iy & 18. 19. 20. 3x? — bax? + x — 2a. re tgs WPA Bae a Ua 82? + 43x + 15. (a+b+c)?— a. 4ax® + 8ax — 8a — 4az’? Sum and difference of twocubes. By actual multipli- we find a+b) = (a+ b)(a@ — ab + db’), a — B3 = (a — b)(a@ + ab + Db’), Any expression which may be written as the sum or difference of two cubes can be considered as the product of a binomial factor and a trinomial factor. 120 FACTORING [Cuap. XIII. EXERCISES Factor the following : 1. x? + 27. SoLution: We may write the expression in the form z? + 33 from which we get x? + 33 = (x + 38) (2? — 32 4+ 32), v427 = (x4 + 3)(2? — 32 49). The factor 2? — 3x +9 cannot be factored since there are no two integers whose product is 9 and whose sum is —3. 2, v= 24. 6. 8 —1. 3, a — 1. 7. Sa° = ize: 4. a +1. 8. y® — 1252%. 5. § +1. 9. a’ + b°. Find two factors only. Hunt: b%= (b?)3, 10. x? — y®. Find two factors only. 11. 2? — y’. Find two factors only. L2e or Hint: This expression can be written as the difference of two cubes or the difference of two squares. Factor by both methods. Which is_ the easier? 13. 64275 — y, 14. a® + 6°. Find two factors only. 15. 1252%y? + 82%. 16. x’ + y". Find two factors only. 17. a” — x§. Find two factors only. 18. x" — 1. Find two factors only. 19. x" + y’". Find two factors only. 20. (a + b)? + (a — b)3. 21. (a — x)* — 2. 22. (a — x)? + 23. 23. (x? — 1)? + (a? + 1)%. Find four factors. 24. 125(a + y)? + 82%. 25. 1000 — 1. Anrs. 81,82] | SUMMARY OF FACTORING 121 26. Give in words a rule for factoring the sum of two cubes. 27. Give a rule for factoring the difference of two cubes. 82. Summary of factoring. No simple general rules for factoring can be given, but a few suggestions will be helpful. (1) First take out all monomial factors, not forgetting factors expressed in Arabic numerals. (2) After the monomial factors, if any, have been re- moved, the number of terms will usually be the best guide in factoring further. (a) Binomials are factored as — The difference of two squares, a? — b?. The difference of two cubes, a? — 6°. The sum of two cubes, a’ + 6°. (b) Trinomials are factored — As trinomial squares, a? + 2ab + b?. By inspection, x? + (a + b)x + ab, and ax? + bx +. (c) Polynomials of four or more terms are usually factored — By grouping. As the difference of two squares. (3) Sometimes an expression needs to be rewritten in order to show the type of factoring. Before concluding that an ex- pression cannot be factored, see if an arrangement of terms will bring it under any known type forms. (4) Test each factor to see if it can be factored furthoe (5) It is convenient to remember that x? + y’, x? — ry + y’, w+ ay + y?, 24+ cy — yy’, 2 -— zy — y’ are prime. MISCELLANEOUS EXERCISES Factor: : 1. ax + ay + 2abz. 5. xv? — 216y°. 2. 2a? — 2m’. 6. 348 4+ 1. 3. 2? + y? — Qay. Tape: br. S. 4, x? + 11x — 42. 8. 27a%b? — 18ab?. 122 21. FACTORING (4 — 92) P(e 15. Nets es ted be “16. . a +192 + 18. 17 . 9-6a+a’. 18 . 62? — 18x — 28. 19 tei 0. 20 Sat ot [Cuap. XIII. dx — Sry? — Bry + 10zry?. a® + 2a* + 1. . x — 8x'. . 16 + 8ab + a’b?. . 62? + 3x — 3. . 12 — 10a — 2a’. Hint: Collect the terms thus, (1 — x*) + (82? — 3x) and factor expres- sions in parentheses. 22. 23. 24. 28. 29. 30. 31. 32. 35. 36. 37. 38. 39. 40. 41. 42. 43. 51. 52. 53. 54. a’ + 2a? + 4a + 8. x? — 132? + 36. e—x+ uy — y. a®b®ct — 125c?. 26. 27. 25. 4a? — 20ab + 46. 26x? — 63x — 5. 4x? — 28xry + 49y?. Find four factors. lla? + 9x2 — 2. xty — xry'*. 24a4y? + 26 + 14427y*. Find four factors. 5abx® — 5a?x?. oo: 16x + 8abe + ax. 34. Sq} =. q'2, 622 — x(a + 2) — (a 4 2)?. 15a? — 142° — 82°. v—a+u-—a. 25 — (2? + 2ry + y?). 36 — 24 + 2a?y? — y®. 16ab — 24abx + Yabz?. 343p3g? — 729p°. 5la? — 45a — 6. 50x? + 60xy + 18y?. (24 +3) (w@+y) +4(a +9). 4a3 — 3ry — 8a?y + By’. Aa’b? + 36a7ba + 8laz’. 55. 56. 3a° + 375a?. . 2xry® — 102?y? — 282°. . a+ (b— 2b2?)ay — 2b?2y8 . 5a* — 5a*b — 5ab — 5a. . L— ab? — xy? + 2absy. . «2 — (a — b)x — ab. . 22a? + 59x + 39. . (Qa +3)3 — (Qe — 3). a (32 + y)® — (2% — y)*. 7 — 6% —Z4ee “Arr. 82] EXERCISES 123 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. ais 12. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. = 83. 7x? — 21cx — 280c’. 9a? — 6a? + a’. (a? + 2ab + 0?) — (a? — 2Qay + y?). xv? — Qay + y? — 49. xy—-l+a-y. sr + st —? — rt. 320° — 48x?y + 182y”. 2 — a? — b? + y? — 2ry — 2ab. e+24e. n? + 4mn + 4m? — 16. 2ry —2?—-y +1. Qy? — a(x + y). 8 + (x — 2). x§ — 1324 — 902’. 2abx? + 10a*bx — 28a%*b. Sa2x? — 56a?xy + 98a7y’. 16924 — 1562%y? + 36az%y°. 27a*" — 125. Find two factors only. a> — ab? — a’b? + 6°. 3 — 27 — 9a? + 272. 4+ y> + 4a? 4+ dry — y’. ep+y+ta— yy. 2524 + 102° + 2”. 28abn? + 384abnm — 12abm?. Sy’? — 80y?x + 200y”. 3 — 2 +323 — 2°. 6x? + ( ) — 14. Find three different expressions replac- ing the parentheses which will permit the trinomial to be factored. : 84. Write down at random three trinomials of the form ax? + bx + c and try to factor them. 124 FACTORING [Cuap. XIII. | Fill the following parentheses so that the trinomials will be perfect squares : 85. 49a2x? + 1l4ax + (\). 88. ( ) + Garry + y’. 86. ( ) + 12pqrx + 2’. 89. 22+ Qryz+(). 87. 1607+ ( ) + 49y’. Factor : 90. 5a? + 10x — 5a” — 10. 93. (1 + a)? — (1 + 2a)’. 91. 2? + 38y — 3x — zy. 94, (x + 2)? — 25(x + 3). 92. 144v?y?2? + 24ryz 4+ 1. 95. Write down three perfect square trinomials whose middle term is 4abe. Factor : 96. 100ax? + 90a?x — 90a’. 97. 10x? — 2524 — 2”. 98. ax? + ba? + aty? — bty?. 99. — 2cy — 3bz + 4ay + cz + Oby — 22. 100. atx? — b’x? + aty? — b*y’ 101. 102. 103. 104. 105. 2° 106. 2” 107. 108. 109. eo+y+aeyt+aoay+rt+y. B4ty+te—syt+y’. a® + 2ab + b? + 3a + 3b. a‘(a + x)* — (ax)*. —~ay —xvyt+y’. — 2(a — b)x — 4ab. ax? + ba? + ax + bx. Show that x? — 16 is a special case under az? + bx +. What values have a, b, and c when 2? — 32 is consid- ered as a special case of ax* + bx +c? Factor : 110. 2ry + 3yz2 + by + xz. Bik (e+ y +2)? —1. 112. 49p%q? + 42pq + ? = a perfect square? EXERCISES 125 . y — 8y — 65. 117. (a + b)§ — (w@ + y)3, toe or — 12. 118. 82? + 297 + 15. meee re ts 2. 119. 8a? -- 62a 15: - 1728c3 + 643. 120. 82? — 29% 415, - (a+b)? —-(@+y +2)? - (2+4+2)? -(24+274y)2. . 4abed — 24cdx — 3abmn + 18mnz. . ov +1 — 22 — Be. - 2ax? + 8ary — 2bary — 3by?. -9-(8-—2-y)?. 128. 82? + 267) 4-15: . 8x7 + 34x + 15. C29 a | : (a + 6)§ — (a — 6), 131. In arithmetic we find that the sum of the odd powers of two numbers is divisible by the sum of the numbers. What case of factoring comes under this rule? 132. What case of factoring comes under the rule that the difference of the odd powers of two numbers is divisible by the difference of the numbers? Factor : 133. a + bx + ay? + bay’. 134. 10% + 10y — 5ax — 5ay — 15a2x — 15a?y. 135. x** — 1. Find two factors only. 136. 3a* — bab — 4a — 8b. 137. a° +1. Find two factors only. 138. 20m? — 9nm? — 20n2m?. 139. 140. 141. 142. ae “Ere F..4e x? + 2a*x + 384 + 6a?. 492? — (1 + 2? 4+ 2)? a? y? 4+ ax? +4 ay? + aa? + ay? ax + ay + az + bx + by + bz + cx + cy + cz, CHAPTER XIV EQUATIONS SOLVED BY FACTORING 83. Quadratic equations. If, in 2? — 4x + 3, we substitute different numbers for x, we find different values for the expres- sion. Thus, when x = 0, the value of 2? — 4a +3 is 3; when z = 1, its value is 0; when z = 3, its value is ?. We give a table showing the value of the expression for all integral values of a from —4 to +4. Suppose we ask what values of « make 2? — 4x + 3 equal to zero. By referring to the table we see that the expression is zero for at least two values of a; that is, when x is 1 and when az is 3. We have really asked a question equivalent to the following: What values of 2 satisfy the equation w—4r+3 =0? This equation is different from the’ equations previously con- sidered. In the left-hand side we find one term containing 2, one containing z, and a term containing no unknown. In other words the equation a? — 4x + 3 = 0 is a special case of # class of equations of the form az? + bx + ¢ = 0, where a =], b = —4, and c = 3. Equations which can be reduced to the form az? + bx +¢ = 0, in which a, b, and ¢ are any numbers whatever, except that a cannot etn zero, are called quadratic equations. 126 PO > fy Arts. 83, 84] | EXERCISES 127 EXERCISES _ Make tables showing the values of the following expressions for the integral values of x from —4 to +4. Indicate if possi- ble those values of x which make the expressions equal to zero. 1. x? — 34+ 2. 6. 277 + 8x — 10. oe — 2. ,. Y ev pies Si geod © aa r — 6. Fee rare AB a. x? — 7x + 12. 9. (v + 2)(2x — 8). B..2? + x — 12. L005 ( = 6) tz — 3): _ Reduce the following quadratic equations to the form - w+ br+c = 0: a1. x? — 3x = 2: 16. 27 — 52? = 32 — 722. 12. 277 —- 444+ 7 = 9. 17. 13+ — 4a? = 27? +4741. is. 2+27 = 2, 18. 2 — 77 + 2? = 27? + 2. a4. OF = (2x? — 4. 19. 2 -—3r+4=4 —- 3z. 16. 77 + 3827-1 =2. 20. pv? +qzut+r=retqt+re. «684. Factoring applied to the solution of quadratic equations. Jo solve a quadratic equation, it is not necessary to proceed y trials as in Art. 83. If the quadratic expression can be asily factored, the method illustrated by the following example » an be used. _Example. Solve 2? —- 4% +3 =0. (1) SoLuTioN: Factoring the left-hand side of this equation we find (x —3)(x — 1) =0. (2) The products of these two factors is zero if - 2-3 =0, orifz —1 =0. (3) From (3), . #& =3or 1. The equation has then two roots, 1 and 3. Checking by substitution, (1)? -4-14+3=0,. id (3)? -4°3 +3 =0. = 128 EQUATIONS SOLVED BY FACTORING [Cuap. XIV. If in any product, any factor is zero, the whole product is zero. Conversely, if any product is equal to zero, some factor of that product must be zero, and any factor which contains an unknown may be equal to zero. Therefore, in solving any quadratic equation in which one member is zero and the other’ member can be factored, we find values of « which make each of the factors zero. That is, we may equate each factor to zero) and solve for the unknown. Thus, to solve 2a? = 7x —5, we write the equation in the form or — 72 +5 =0. Factoring, we have (2% — 5)(x —1) =0. Equating each factor to zero, 2c —5 =0, x —1 =0, and we find x = 5 and x =f. EXERCISES Find two solutions for each of the following quadratic equations : | : 1. (@@ —2)(¢— 7) = 90. 10. 22? + 22 = lige oe 2. (x +3)(a —1) = 0. 11. 42 +2? = Alger 3. (2x + 5)(a + 6) = O. 12. 6 — (i374 4. (5a +6)(22 —4) =0. 18. 32? — 14% = 2° + 60. 5. (x +1)(82 +7) = 0. 14° 2° =a 6. 2? —324+2 = 0. 15. 2? 40; 7, a + 7a — 30 = 0. 16. 42? = 81. 8. 2? + 132 = 30. 17. 22? + 5¢ =3 = 9. g? 4102 +24 =—-—427. 18. 6 = 62 + 352. Solve first for x and then for m in Exercises 19 and 20. 19. 22? = 3mx — m’. 20. x2 — 9max + 14m? = 0. 21. a? — Zab + 10b? = 0. Solve for a and for b. @99. 77? + 13% = 8 + 32. 23. 202? + lla —3 =0. 24. 2a +a —3 =0. Arr. 84] PROBLEMS 129 25. 2? + mx+3x+3m=0. Solve for zx. \) 26. 2? — mz —nzr=0. Solve for x. ‘) 27. 2? — 4a? =0. Solve for x and then for a. | 28. 2? +ax+bx+ab=0. Solve for x. ~ im 29. x? + px =0. Solve for x. , 30. av?+axr+br+b6=0. Solve forx. 31. ax?+abr+2+b6=0. Solve for z. ly 32. az? + acr + bx + bc =0. Solve forc. PROBLEMS The following problems involve the solution of quadratic quations; find all solutions possible for ‘he equations and determine whether or ‘ot each solution is a reasonable result: y b 1. Inaright-angled triangle the longer +g is two feet shorter than the hypotenuse = ut two feet longer than the shorter leg. c= a7 40° Vhat is the length of the longer leg? Fra. 20 SoLuTion: Let x = the length of the longer leg. hen x — 2 = length of the shorter leg, ad x + 2 = length of the hypotenuse. rom page 80 ( +2)? = 2? + (x — 2)2, : (See Fig. 20, also Fig. 15). ence m@+4e4+4 =a 422? —-47 44, x’ — 8x = 0. hence 2 =, or =-8; The answer x = 0 must be cast aside, for it has no interpretation when iplied to the problem in question. 2. In a right-angled triangle, the short leg is two feet orter than the hypotenuse and one foot shorter than the oger leg. ‘Find the length of the short leg. * 3. A positive number when multiplied by a number 5 nes as large becomes 405. What is the number? 130 EQUATIONS SOLVED BY FACTORING [Cuap. XIV. 4. The square of a certain number plus twice the number itself is equal to eight times the number. Find the number. 6. The product of two consecutive integers is 306. What are the numbers? 6. The sum of the squares of two consecutive integers is 41. What are the numbers? 7. The sum of the squares of three consecutive integers is 50. What is the smallest of the numbers? 8. The square of a number is 20 more than the number itself. What is the number? Is there more than one answer? 9. The area of a rectangle.is 18 square feet. The length is one foot longer than 4 times the width. What are the di- mensions of the rectangle? 10. A rectangular floor is 4 feet longer than it is wide, and its area is 320 square feet. What are its dimensions? 11. The perimeter of a rectangular field is 60 rods, and its area is 200 square rods. What are its length and width? 12. Make up a rectangle problem whose solution will in- volve a quadratic equation. - | 13. A photograph is one inch longer than it is wide. It is mounted on a card so that there is a 1-inch margin on all sides. The total area of the margin is 2 square inches greater than the area of the photograph. What are the dimensions of the photograph? 14. It takes 96 square inches of paper to cover a cube. What is the length of one edge of the cube? 15. The dimensions of a closed rectangular box are con- secutive integers. The entire outside surface of the box is 52 square inches. What are the dimensions of the box? 16. A paving brick is 4 inches longer than it is wide. The thickness is 4 inches. The volume of the brick is 128 cubic inches. What are the length and width of the brick? 17. A rectangular box is 5 times as wide as it is deep and twice as long as it is wide. The total surface of the box is 18C square inches. What are the dimensions? | | | ‘Arr. 84] PROBLEMS 131 18. A rectangular solid is twice as long as it is wide, and the width is 3 inches more than the depth. The total surface is 160 square inches. Find the dimensions of the solid. 19. Make up a problem about a rectangular solid or brick whose solution will involve a quadratic equation. 20. A club had.a dinner that cost $60. If there had been 4 persons more, the share of each would have been 50 cents less. How many persons were there in the club? CHAPTER XV HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE | 85. Greatest common divisor in arithmetic. The largest integer contained as a factor in two or more integers is called | in arithmetic the greatest common divisor. This number is easily found by separating the numbers into their prime factors and multiplying together those found in each number. ‘Thus, in) 12 =2°2°3, 18 =2°3:°3, 3d0= 2a the greatest common divisor is 2:3 = 6. 86. Highest common factor. In algebra, a number or expression which is a factor of each of two or more expressions, is called a common factor. Thus, in 102?y, 4xy, 8xy?, the com- mon factors are +2, +27, +y, =ay, ==Z2ay pus the positive factors alone are considered. The factors 2, x and v are the common prime factors. ; The product of all the common prime factors of two or. more expressions is called their highest common factor. (H.C.F.). For the above expressions 102*y, 4xy, 82y’, the H.C.F. is 2zy. Ww Two expressions that have no common factor except 1, are” said to be prime to each other. In algebra, the word “highest” is preferred to the wort ' ‘“‘oreatest’’ in connection with common factors. The highes' i" common factor is the common factor which contains the highe number of prime factors. In the above example, 2zy is the highest common factor, but if « =1 and y = 3 it is not th g greatest common factor. 132 Arr. 86] EXERCISES 133 EXERCISES Find the highest common factor of the following sets of expressions : 1. 8a‘b3c?, 4a7b?c?, 16a2bic!, The common prime factors are 2, 2, a, a, b, b, c, c. Hence the H.C.F. is 4a2b2c?. UL ye, Ory" 2°. 5. dmn*p*q*, 10m n’, 2Omnpq. pada 210*b’c*. 6. 12a°b?, 6a7b®, 18a%b?, 9a‘b4. 4. 3a’b, bab, 3b’. G Say, 16a", 92747 7": 8. 14abe, 8a*b?, 7ab?c?, 2a5b?. Pot yor, y*, 1 2r5y°- 249%,)4, 10. 2a? — 2ab?, 4a? + 8a2b + 4ab?. SOLUTION: 2a’ — 2ab? = 2a(a? — b?) = 2a(a +b) (a — b). 4a5 + 8a*b + 4ab? = 2:2-a(a +b)(a +b). The common prime factors are 2, a, (a +b). Hence, the H. C. F. = 2a(a + b) or 2a? + 2ab. 11. 2? — y’, x — 2ry + y?. iz. 2 — y*, x? — 7’. 13. 2+ y', vy + ry’. 14. 2 — y*, w+ xy + ry’. 15. a? + b?, 2a? + 4ab + 2b?, a? — b?. 16. av —y + xy — a, ax? + ay —-a—y. 17. vy —y — 22+ 2, ry —2z2—y —z@. 18. 1584, 1728. 19. 861, 615, 984. 20. 156, 130, 182. 21. 9 — 2x’, x? — x — 6. ieee. 0 — 27, 7? — x — 2, 2x? — x — 6. im 23. 2* — xy", 32° — xy — Qry?, x? — Qa%y + ay? 24. ax + a’y + abx + aby, 2ax? — 2ay?, 3ax* +6axy + 3ay?. 25. abx? — abxry + aby’, abx? + aby®, aba? + abry + b?y?. 134 H.C. F. AND L. C. M. [Cuar. XV. 87. Lowest common multiple. An expression which con- tains each of two or more given expressions as a factor is called a common multiple of the expressions. Thus, if xz, xy, 102, 2y, dxy” are given, then 10zy? is one common multiple. There are many more, for example, 20x?y’, 100zy*, 402%y%z?. | Among all the common multiples of a given set of ae | sions, there is one which is most important. It is called the lowest common multiple (L.C.M.). The lowest common mul- tiple of two or more expressions is the product of all their’ different prime factors, each factor being used the greatest number of times it occurs in any of the expressions. For example, consider the expressions x, xy, 10x, 2y, 5ay?. In por the different prime factors are xz, y, 2, 5. The factors 2, 2, and 5 occur only once in any expression. The factor y occurs . hates in 5xy”, since we may write it Sry: The L.C.M. is then 2-5-xa-y-y = 10zy’. ) | EXERCISES i 1. What is the least common multiple of two or nal numbers in arithmetic? 2. In connection with common multiples in algebra wh y should the word “lowest” be preferred to the word “ least’? Find the L.C.M. of the following sets of expressions : : So LY Bae aes 8. ay? a eee j 4. 3xryz, 5ax, 10a%2°. 9 («+y),24+y*, 2 — y’. q 6. 12a%bc*, 6ab*z*, 8abc’. 10. x? — 77, a 4 Qay ey, 6. (n’m'*, 13an’m, 2amn. 11. 2° 4+ 32 +2) ae ee | (Peni Gn lponnl an 12. 6x? + 13a + 6, 42? — 9. 13. 32? — 10% 4+ 8, a? — 44 4 4. 14. ab — Bb, a + ab, ab + DB. 15. 67? — 5% + 1, 8a? — 62 + 1, 102? — 7x +1. 16. anes 3x? + 8x + 4, oo ae 17. — y*, 2 — y*. 18. uk ee Arr. 87] EXERCISES 135 19. 244+ zy, — xy + ay,2+ y. 20. a — b, (a? — b?)?, (a + Bb). 21. ax? + ax, ax* + 3ax 4+ 2a, 2? + 4¢ 4+ 4. 22. 3(ab? — b°)?, 2b(ab — x?)3, 6(a2b? — a). 23: x2? — xy — Sax + By, x? — 10x + 25, 2? — 25. 24. av + a+ 2x +2, 5a? + 7a — 6, 5ax + 5a — 3x — 3. 25. Gar + 9a + 4x + 6, 2ax — 4x 4+ 3a — 6, 3a? — 4a — 4. 26. a*b — ab*, 7a? + Tab + 7b?, 14a3b + 14a2b? + 14ab3. 27. 1—-m+n—mn,1—m—2x4+mz7z,1+n—2— ne. 28.2? +2? 4+274+1,2724+ 7, 7? + 27 +1. | 29. ax+y — by + ay — br +2, ax + ay — bx — by, ¥—ab+a-ab+0b? -b. 30. ax +a — bx + ay — b — by, (x+yP+ut+y, w + ay — by — bx. a a® + 63, ac — be + ad — bd, a’c + abc + b’c + a’d + abd | ae CHAPTER XVI FRACTIONS 88. Fractions in arithmetic. If a unit be divided into 7 equal parts and 3 of these are taken, we denote the amount taken by #. This fraction may also be understood to mean 3+ 7. The fraction # is thus the answer to the questions : 1. What part of 7 is 3? 2. What is 3 + 7? But a fraction such as 5 cannot be regarded in the first of the 4 F two ways mentioned, since a unit cannot be divided into 22. equal parts. The fraction a is thus taken to mean. 3 + 2% that is, the fraction x is an indicated division. 4 89. Fractions in algebra. Similarly, in algebra a fraction is an indicated division in which the dividend and divisor are algebraic expressions. Thus, 4 60-3) C(t iT bc t4on ee are fractions. The dividend is called the numerator, and the divisor the denominator of the fraction. The two are often called the terms of the fraction. The fraction i is read “x over y,” “x divided by y.” The rules for fractions used in arithmetic apply 1 in alechil In particular, much use is made of the 136 i. _ Arr. 89] | EXERCISES 137° . Principle. The numerator and the denominator of a fraction may be multiplied or divided by the same number without changing the value of the fraction. Zita 4 a, 373375 a, 141014 7D Similarly, a oa7 73 : i y pkey © yesh gre a and e-8 (@-b)+(a-b) a+b (a-b? (@ —b)? +(a—b) ~a_-b 2 EXERCISES 1. State two meanings of : 7 0G - of a and of © : g ‘if a, b, and c denote integers. 2. In his four-year course, a student spends for books a dollars the first year, b dollars the second, ¢ dollars the third and d dollars the fourth. What fraction denotes the average expense per year? 3. What part of a piece of work can a man do in one day, if he can do the entire work in 6 days? In 53 days? In a days? 4. The grades of a student in two subjects differ by 5 points. The lowest grade is x. What is the average grade? By multiplying or dividing both numerator and denom- inator by certain numbers, replace the following fractions by equal fractions. Give at least three solutions for each exercise, | 7 2 r+y —14 | | 5. 3" 8. 5 iB Ee zy 14. ee | 4 x a ab | 6 9 9. y 12. a BS b 15. ab ‘ m2 2 | ae TAS tre ies Eee fonts 2° " BF * ab? cd? 138 FRACTIONS “=| Crapo x V ies | 90. Division by zero. The numerator of a fraction may be any number whatever, including zero. In the case of the denominator, however, we have one exception. The denom- inator cannot be zero, for division by zero is excluded. That is, an expression. with denominator zero is not considered as a number. Care must be taken not to give such values to the x+4. 73s a number for any value of x except x = 3. For x = 3, we have letters as will make the denominator zero. ‘Thus, 5 which has no meaning. EXERCISES — Give the values of x for which the following expressions have no meaning : +2. 3a + 4: 1 a pe e 32 — 4 4 a(x — 1) x 1 3°. 52a a} 4 c+1 a; rt yi 3x + 5a? x+1 x+4 2 Se ct : 3 f (2x — 1)(x + 2) 91. Signs in fractions. In any fraction, there are three | signs to consider: the sign before the fraction, the sign of the numerator, and the sign of the denominator. Thus, i 7. Historical note on fractions. . Fractions offered very great difficulty to the ancient nations. In their operations with fractions, the Babylo- nians reduced all fractions to the denominator 60. Similarly, the Romans reduced fractions to the denominator 12. The Egyptians and the Greeks reduced fractions to the same numerator. In fact, Ahmes (see p. 80) seems to have restricted the term fraction to those with numerator equill to 1. Other fractions such as 3 were expressed as the sum of fractions with.numerators equal to 1. Thus, “‘} and }”’ took the place of 3. This probably seems rather awkward to us, but it shows that eadinngs offered real difficulties to the early students of mathematics. | Arr. 91] ' SIGNS IN FRACTIONS 13973 ‘have the three signs attached. Since the names numerator, denominator, and fraction mean dividend, divisor, and quotient respectively, the laws of signs for wes must hold for fractions. , Hence, we have 2 SCRE b pies hin = Ach eh See V2) ee (2a) ow ; -($) = +55 - (5) = +2; and eae b In words, we have: (1) Any two of the three signs affecting a fraction may be changed without changing the value of the fraction. (2) The sign of a fraction is changed by changing the sign of either numerator or denominator alone. | Thus, * and = are both positive ; = and * are both negative. In changing the sign of the numerator (or denominator) ‘when it is a polynomial, be careful to change the sign of cay term of the polynomial including the first. Thus, changing the sign of the fraction ty 3x +y by changing the sign of the numerator gives -o+y 3x +y EXERCISES Reduce to equal fractions with positive signs in both numera- tor and denominator : 4 ae -2 =a). aye 2. ris , 3. ae { | | 140 FRACTIONS [Cuar. XVI. ! —4c 6 y+9d hear ih ae fa ahi SO: 3 —x%-—2 —m — 2 —a—b Shea Bipir te eg eae a eee in ee 10. Show that —. =—~—- Give reasons for each step. a—b b-a : 11. Show that 2 sy as l-y y—-1 2a i 2a bbs a) - Lee 12. Show that 92. Reduction to lowest terms. A fraction is in its lowest terms when no factor except 1 is contained in both numerator and denominator. Thus, : and = octet are in their lowest terms, while 2 and fee +] 8 e+: not. Pee a epee to its lowest terms we use the principle of Art. | 89 and divide both numerator and denominator by the factors common | i to both. For example, v2—ax x(x —-1) oat 24+ (ce +1) eee are EXERCISES Reduce the following fractions to lowest terms : 21 Pape nk co ls xv — @ 28 e 203 + 222 cc 2(x — ay. 2 4ab?ct 5 30 — 8 P 21 + 10x + 2? " 2abe " a+1 ; v9 | 3 250 6 aan 9, ae (v +4) Lgsio " 327 + 62% + Ox — 16 . ax? + ary ve ax’ — axy? ! i ' Art. 93] CANCELLATION 141 93. Cancellation. The process of dividing the numerator ‘and denominator by the factors common to both is called “cancellation by division. It is merely an application of the “principle of Art. 89. The Baoeed ure is illustrated by the following examples. Example 1. Reduce aan ; to lowest terms. wx De 4 ae 14a4b%c6 V AB a’bc? i SOLUTION 2 abc “3 Tepe 3 Example 2. x3 -2y? 2x(e—y)(x + y) z+y a — xy) ey) (x2 +2y +y*) x(a? + ay +9) x - , ‘ ‘Cancellation can be used only with factors of the numerator ‘and denominator. 4 ae GS Y) 2+2 _ Thus, in the fraction a (ray We may cancel the 2’s, but in | “(i +y) 2+ we cannot cancel the 2’s ; a here 2 is not a factor of the numerator or denominator. To illustrate, H = at To strike out the 2’s gives 5 ig, 5 3 But gis not equal to 6 EXERCISES Reduce to lowest terms: 4,5 4 2 he Be 6. r xry 44 4% 6 + 3x ae rogue ab 9+ 3x 3, Tab%ctds Rian, " 25be?d3 tay te be 4 28xry?2° 9 4+ 20 "14226 ' 24+2 14 — 7x Seca 10 * Qe + Qry eT — Te 142 s/ FRACTIONS [Cuar. XVI.4 11, Bet Sy, 18, 2 " 3x — by ' az + 3x + 2a+6 2+ 14y oe 12. 7 Dae 4203 + Way a® — ab? i Bs 14y 20. a® — ab? 14 Mateo a1, Aaah + Sate + Bot 2 Yh ° 2 Quy + y’ , 2ax + 3a (2X4 dau 8a2 + 8a + 2 ax + 3a — bz — a0 15. 16a2a a a? .+-.bo 40 16, mmm gg BOax + 45ay — 4a — 5y { * mn? — 2m?n + m? c 8x + 10y j ab — 36b 12ax — Gay — 50ca + 25ey Agha ee 24. ——— ee a? + 6a 227 + xy — y? | 94. Reduction to common denominator. Two fractions are said to be equivalent when one is the result of multiplying or dividing both terms of the other by the same number. 2 Thus, : and < . an = = and — =? are pairs of equivalent fraction | Any set of fractions may be changed into a set of equivalent vegans all having a common denominator. 5b ee 2 5 Ge Soy Gy(a? +B) yO aed Thus, by’ Bby’ ‘Gby may be written This may be done in many different ways, but when th common denominator is the lowest common multiple of all the~ denominators the process is called reduction to the lowest com | mon denominator (L. C. D.).. The lowest common denominator of any set of fractions is the L. C. M. of the denominators. | Arts. 94, 95] EXERCISES 143 EXERCISES Reduce the following sets of fractions to the lowest common ‘ denominator: hes a | y 1 Dey sy ye ) . 25 8 * y-T yd) aay a’ 3a. SO a ne Le leer a+ba—b a. a+l1,a+2 3. Zab? ) ab 10. 2, db b2 ) b3 . a+ba—b r+] 2-1 | 2 aa 2 EG 2x 3 a b 1 ab Pastimes : r+2 274+ 524+6 ee as a ab 2 ee ag ca ga Bat aval ty y ty -—y ry-x eee 1 Ui, «2a ge Le aa x-1l2x+124-1 L-Y y-xuX ~-~u-y ‘a? 3 _@+5a4+6 2(a? +30 + 2) | ai rn ary a — yp CY x+y 17. a +2, m + 4i. 18. 54 +a, a+b — 2; ale 3 ike os 95. Addition and subtraction of fractions. In adding or ubtracting algebraic fractions we proceed as in arithmetic. lust as 3 feet + 4 feet. = 7 feet, 3 4 7 | eae ite 11 | a a a 144 FRACTIONS [Cuap. XVI. | EXERCISES Perform the following additions and subtractions: 1 2 ~ page isco = ee pe : acts a 0. . 05 ee 6 ay 2x Slee eres = 22 127 2 1. 1 ee Tin De eS at be see St seaee 3. Geri § 1 re 8 cee amaaey a 8 8 eee SOS aaa 0. (2 m mM nN n 1 To add expressions involving different units, as 3 feet and 24 inches, we reduce them to the same unit. Thus, 3 feet + 24 inches= 3 feet + 2 feet = 5 feet. Similarly, to add } and % they must be reduced to the same denominator. Thus, 3:1 it 2am Sie Le 479 12> eee In like manner, aC std Oe ad + be | btd ba bd mabe Similarly, for subtraction. Thus, | mae Bt) Aa Sey — 3a 4y? — (Bry — 32) | y-1° 4y 4yy—1) 4yy-1) © See i _ 4y? — 3ry + 32 qi ~ ayy = | When this is put in the form of a rule, we have : | To add fractions, reduce the given fractions to a common de- nominator, add together the new numerators and place this sum over the common denominator. rs 95, 96] EXERCISES 145 A similar rule holds for subtraction except that we subtract one numerator from the other instead of adding. EXERCISES Perform the following additions and subtractions : wes i. 5 -. | ica ea ae | 3. 24% ; 13. fas =o ; _ 4 rete s “Strato | 5 Stet. 15. Stat et iti. i P or+6 3x+a 16. at 2a 14 21 elie a — | Batsty (tet tae ¢ aes “ Ga = Ge ee aot eta (1. T+ et 20, 24 2-—*. 96. Multiplication of fractions. As in arithmetic the pro- ect of any number of fractions is the fraction whose numerator the product of the numerators and whose denominator is the ‘oduct of the denominators. 146 FRACTIONS | [Cuar. XVI. ; ‘ ; lee 6610 es Thus, in arithmetic pe tea) is In algebra, 2 £+3. £2242 Qte+3) - 2+T) 4 z 2£+1 2 +4e4+3 ceFHDeraaet+l ze+1) Expressions not in fractional form can always be considered as fractions with 1 for denominators. EXERCISES Perform the following multiplications and reduce the an- swers to the lowest terms: 7 4 3 co r+y BY = 29 52 prema 2G 10, 2%. OYE WY. 25 go tO by aa? ba —3 2 27 me ee” 2 2 3. ie LOR t At. ye 3ab 4ax 22\* 9 f24\ eae Are A ae —. |e ad 2ry 9b = S e (3) 5b? 8ab —a\? (ab)? oot 25 - & at a bed x\? (x+y) 6. aoe ee aS 14. x (2) 3 72.4.2 15, 29 oe ; y & x y Y a 22-2 x+y 8. P| a 16. 7-y oo 1 e+ 2Qry+y? a®+ 2ab + 0B 1. @ias6t: pee 18 (a+1)(@+2) @+2)@+3) 1 , a+3 a+1 a+2 ART. 96 | 19. 20. at, ye 23: (m5) (m+5) MISCELLANEOUS EXERCISES 2 e EXERCISES 147 ax + 2? x? — a? ax — a? 2? + 83ax + 2a? e+e ry—x yt yp 0 ea 2_ 1] a+ 4a+a? es oe BE Ys my: 2 ar (x? — Qry + y?). (ate b \. a+b b a+b/ In the following Exercises, 1—6, the letters are assumed to ‘be positive integers. By Hich is larger, = or =? Why? * or 2p Why? : Le im 8. or earn Which is larger, or? Why? zor a Why? Which is larger a Which is larger, = or » Which is larger, : or OL or 1 —1 ee pla ee Explain. + Ls a a! Explain. 3 9 ane Explain. Which is larger, y or 9, zoey a Does x of x and a? gq + r+3 . Does = = = ae all values of x and a? : for all values of x and a; for any values 148 FRACTIONS [Cuap. XVI. x 9. Is it true that ae = ? eg! bine ae re) 8a _ ; ee 2 : 10. Does [oy Dae Explain. a i A)» ala Does erga TR Explain. 12. Which will give the greater product, if x and y are posi- tive integers, to multiply a positive number by 3 or to multiply the same number by mt Explain. 13. Arrange in order of size, if a and x are positive, (a) if x is greater than 1; (b) if x is less than 1. a a a a | e etl 243 2 14. Fill the blanks in the following : oh ins (gO ea , b hab 9) ae (b) le 2s 1p 2 2 Saas c-y 2£-y 1-2 2-2 eee Loe | x ? ? ) @-D(b-) ~ ©=a) C-8) @—-H)-H) 15. Express 2) 8) 7 as fractions with denominator 24. 16. Express = m : as fractions with denominator abe. 17. Express — as fractions with denominator 2Aabe ia’ 6B Be 18. ‘lixpress:'= 3 an Oa a as fractions with denominator 9a°t?, x+y x-—dsy 2a — 2b 3a + 3b as fractions with denominator 19. Express 6(a? — 6). “Arts. 96, 97] | DIVISION OF FRACTIONS 149 3a 4b é (a — b)? (a + b) (a—b) (a+b)? ' Reduce to a single fraction and simplify (leaving the de- nominator factored) : 20. Find the L.C. D. of and | 1. 5+ oa. LO eer ea a2, Wo. Lupeeree yaar 23. a ae ieee 24. ay se sa eS 25. “4S. a | 26. ee 32. i ee 33. : @-2)@-9)' @-HW-4* W-) &-® 97. Division of fractions. The method of dividing one fraction by another is the same in algebra as in arithmetic. By the definition of division, the divisor times the quotient gives the dividend. Hence, to divide 3 by ? is to find a number ¢ such that 3 2 ees (1) Multiply both members of (1) by 4. This gives DAL eS. In general, to divide : by means to find a number gq such ‘that 2) Cc a 2 at 150 FRACTIONS [Cuar. XVI. . To find g, multiply both members of (2) by : - This gives ad) ode (3) Sha on hee From (38) we note that the quotient is obtained by multiplying the dividend by the divisor inverted. EXERCISES Perform the indicated operations and simplify: ) | 3 2 a a | 12a—-6 2a-1 Brake Pe ye ese Yh i. “ao ee 3 BS: aOR 12 _4 40 4Ge eta 2 DGS ipoaeee ’ a(a?-—a—12) 30° tom 4, oath , 4c 43 21 GeryO owe cOGs ° gt — OP 2+ 3y 5, Oe. fey. i 2 tee 62° Aa * oS 1 44 ieee 6 (Sy + (Bab)? 1s, Ue. ee SANE (26) ° g— yay? 33 ey re Gar eevee ab = fa+b b ) 7 (F+4) ; 16, + ; ot or a A + 8. (5 ‘) yh (m— =) + (m+=) Caec e e—x—-6 2-2-3 Ged ue ve—x—-2 w#+2-2 Arts. 97, 98] COMPLEX - FRACTIONS 151 Tere it si Be fang) *(*-2 53) Cra Bb + 4ab(a + ti 20) a—b G+ 9)-G-1-4)] [0-9] 1 ten Vee 2 ao 25. }—+—4+-—]+]/-4+-4+°-}]. eee Tee” 42 ae wifin 98. Complex fractions. A complex fraction is an indicated division in which either one or both terms are fractions or con- tain fractions. A complex’ fraction can be reduced to a simple fraction. 2 23. (a 2 prs 2a? + 2b(b + Sin Sa 2a*7-—3ab+P Example 1. Reduce = to a simple fraction. oma polco SoLtuTIoNn: Multiply both terms by 10, the L. C. D. of the numerator and denominator. Then eS ee 15, SORT eae 243 Example 2. Reduce =~ to a Sauls fraction. ne Soxvution: Multiply both terms by 3zy, the L. C. D. of the numerator and denominator. Then 24 szy(2 ae 3a FRACTIONS FCuar, XVM EXERCISES Reduce the following complex fractions to simple fractions: 23 ae Lp | 1. 32 at ul 10. 5 = 9 3,+4 ke "244 Z a? — 6? 4 +27 a 3. ah 11 ae 2a 1 m—n toe. +1 4. <. 12) ae Fie! i = (ys m+n 1 man, mtn 5 0 13 mtn m—nN ; b 1 he a m+n m—n 1 aaewit 14, —— ge Sh sea al ar x—2+- ae 15. i 1? sata Lee x Cee eb 8. areas xy ees 16. 4: a z+e rhode i 9 ee 17 1 an a b * tere hae baa 1m m - Art. 98] MISCELLANEOUS EXERCISES 153 1 2 @+1) +25, ate ), eae oes ie ae tia, 242 meat i 2 3 : cee ae eee og rp peas ) 19. : r+5 b eee b ered ta 3 5 1 1 1 rf eae ; aang oer a+b b2 1 + ; peat 1 +a G0 24° 2u.-- Z or -b Peale AG MISCELLANEOUS EXERCISES 1. Which is the greater —) when xand y are positive numbers? 2. Which will give the greater quotient, if x ia y are posi- j 55 or 3 tive numbers, to divide a positive number by i or to divide the same number by mi Explain. : : 22 3. Fill the blanks in the following: ang pena | sy Seas Cay wy(2 + 2?) 4. The reciprocal of a number « is That is, the recipro- eal of a number is 1 divided by that number. What is the product of a number and its reciprocal? 154 FRACTIONS [Cuar. XVI « 5. From the definition of ep in re ; give the reciprocals of the following: (a) 2; (b) ® 43 (c Ne i (d) = ae ; (0) 5: 6. Show that to divide by a rane we Eee simply multiply by its reciprocal. | Perform indicated operations and simplify : 3a a geet 1 (Serre eae 13. 4x + 2 _ 2a +1 . 3 3 mits = ST 15. ne 2 pe va x 4+ ax 16 aa e ti Pi a _- <4) es (x — a) x—a 1 erty? x+y «u-y 1 eee 11 . re pppoe “ath _2+y ey 3 ———_——_ r-y «x+y Ye ee 3 eee | Ox 19. (eo ~ 1) zs (sey — z = . a2 ete *) ae a “Art. 98] EXERCISES — R 23. 24. 25. 26. 27. 28. 29. & m2 m—n? 1 wn n 05 ax — bx + ay — by axy — bay 6xyz + 3x72 + By" 2x72 + 4aryz + 2y?z2 ax — bu + cx + ay — by + cy axy — bry + cry Sayz — l5yz + 10xz — 302 5a’z — daz — 302 4ax* + l4ax — 30a 62? + 21a? — 45x ve 4s 32° — 6x2? + 12x Gaz? — ax? — 12ax__ 6ax* — 17ax*? + 12ax (mate iat ee Tes mat), m n m+n (e+) e-)- Gra) 32. 33. 34. oe or paul | res Ss oe a GR \m-n m+n w@—xr-—6 2-2r-3 2 -2r4+1 ar ey 7 tr 2 e@+2e+4 +8 wv 4 y? — 9 +2 y= 8 442 (a +6)? a a Aa’ bi ac—a*—ab (a—c)?—8 be —ab +b? 155 47. 48. Ge 4129) (aly eo Pe a ae dan Aelia eee Cine COON (L — q)* ; iGsaed (y — a) [op Orn 1 ee ee x 1 3 '@-0@_- @-24-2) 12 aseee 1 lo 1 Baa eae I pat Loe ne aa 3 ; ey at ay — 2y? 9 4 Baye a 2(1 — a) l—a FRACTIONS [Cuar. XVI. x+1 = x+4 (w — 1) (2-2) a+y a-% (4-2) (a—-y) — 1 1 1 Gob) G=0 wehbe 1 1 1 1 1 1 2 3 f +3442 708+44+3 atasbomee iy tt i) “Ant. 98] REVIEW EXERCISES 157 REVIEW EXERCISES 1. Give an example of a linear equation in an unknown x ;1n an un- known y; in an unknown t. 2. State whether, according to the definition of a factor, (a) } is a factor of 4; (b) * is a factor of x7; (c) mn is a factor of mn. Name all the factors of az?; of 12. Find the following products. Bea — 3)(a + 7). 7.. (% +0). 11. (60 + 7)3, tr 1)(27— 4). 8. (0 — x)? 125 bere 5 Qr+3)Qr-3). 9 (4045) 13. (4h + 6¢ + 8)2. , 6. (4¢ + w)?. 10. (70 +1)(70 -1). 14. (82 +7)(4e — 5), 15. Tell which of the following are rational and integral with respect tor: 3a2x3; a + mr +53 5V/z +17; 14Vazr + 2°. 16. Tell which of the following are not trinomial squares and give rea- ‘sons: (a) x* + 2ax +a’; (b) m? — 2mn — n?; (c) a? —ab +b?; (d) 4 +8b + 16b2. | 17. Describe the way in which the terms of a trinomial square are made up. Combine each of the following into a single fraction: Es aS ry Weg te a -a ab a+b rss qr 1 won, 24 o : 19. +2 22, 5 45 25. 5 -- +2 Aj ae age 26, ~ _¥. yr a+b inet Simplify the following: b 1 4a 3b 1 27. a-7- 30. (a+1)-—. apie $6.5 ot) a a V4 ORAS Ge. oa al ear b x a 2 ee x al - 20. =-10c, 92. °F = 12py 35, 2 +3 Los 5 52 Zi 38 a Or. 9 = | $ 158 FRACTIONS [Cuap. XVI-%} 39. Find the remainder when the sum of the squares of a and 6 is_ taken from the square of the sum of a and b. 40. Find the remainder when the difference of the cubes of a and b- is subtracted from the cube of the difference between a and 6. Perform the following additions and subtractions: Oia ak a=—o 43 a b oS gh Ro 0 eS ‘ @+2ab a +4ab + 4b x7 2x a-2x b+e os 6a? 452-4 10x*4+ 7x —6 a az +bon +exr (a+b? —-2 Find two solutions for each of the following equations: 45. x? — 62 = 27. AY. 32 —4z =a. 46. 427 —- 16 =0. 48. 32727 + 7x = 6. 1 —=2 1-2 qg= 17 =f 49. Add Tre ees Fete: Wek yp 3 1 vito 3 50. Add G— end)’ C-Had--) @-)e=e) CHAPTER XVII FRACTIONAL AND LITERAL EQUATIONS | 99. Clearing equations of fractions. We have solved ‘(Chapter IX) some equations in which fractions are involved. -It is often convenient to clear such an equation of fractions by ‘multiplying each member by the L. C. M. of the denominators. = To illustrate, solve the equation 32 a 3(@-1) 99 2 baie #110) @) SOLUTION : Multiply each member by 10, the L. C. M. of 2, 5, and 10. This gives 15z + 6(2 — 1) = 99. (2) IPransposing and collecting, 21%°= 105, (3) —_ 3 3(5 —1 L5- 2127. 99 CHECK: Bo a 224? _w EXERCISES AND PROBLEMS Solve the following equations and verify the results by substitution: 1. §@—2-t§ =44(8e4+1). 6. 2(2 +1) 4+ 52 = 46. 47 4 By = 42 a: B 2. $2 + gr = 4g. 6. 2p 2 a8 _5 3 r+3 2r+1_. ee 9 7, 4 Wwt+6 4x+1 1 epee 7 | 4, = feeeee = 4 Direei--3 1 7 5 Ga ts(*-3) +a ~ 9 | ae oe DY = Se Cas bear’ 10 Sire eae Ce 160 FRACTIONAL AND LITERAL EQUATIONS [Cuapr. XVII. | 11. («© + 3)(2x + 1) = (@ + $)(2e — 3). 12. (ce -—3? -@+9)@—® =O. 13. (x — 2)(a — 3) — x(a — 4) = 9. e-1 a¢=—2- Visi 7.3) (2 15. (x — 3)(a+ 4) — 2° 4 Or = 3. 14. 100. Unknowns in the denominator. In some equations the unknown appears in a denominator or in both numerator and denominator. In these equations as in the equations of Art. 99, the first step in the solution is clearing the equation of fractions by multiplying each member by the L. C. M. of the denominators. Why the L. C. M. is used rather than any common multiple of the denominators, will be taken up in the second course. Example. Solve 22-1 3-2) 22 — 3) Sa SouuTion: The L. C. M. of the denominators is 2(x — 3) (8x — 1). Multiplying both members of the equation by this expression we obtain - (2x —1)(82 —1) = 6(% — 3)(@@ — 2), or 62? — 5a +1 = 62? — 302 + 36. Then 252 = 35, and | x= f. Substituting x = “ in the original equation, we find the equation satisfied. a EXERCISES 10 4 aera, 2 at ae TEN EVP Gy 1. 73 eee Pion yo ee gee) 3 ei epee Mie a erage 7 PRE par SYST 2” F(a +3) Arr. 100] EXERCISES AND PROBLEMS 161 2 ae 227 = 1 4 ar a z—1 ens | 7, 3@+4)_ 3Qe-1)_, Seen 5 el ee eae , ea ae en ie een Gee Mee tT 1 eo Gh oe 4t2 2+ | 4 3 3 4 ee ss ia 4 9 6 1 are ny I ae fe mee 6 9 4 1 Se ets pee ae fs 13. 2 3 4 11 eter 2.748) 7a +4) PROBLEMS 1. One-half of a certain number plus one-third of the numberis10. Find the number. 2. The sum of two numbers is 66. One-half of the smaller plus one-seventh of the greater is 18. Find the numbers. 3. The difference between one-third of a number and one- fifth of the number is 6. Find the number. 4. The sum of a certain number, its half, its third, its ‘ourth, and its fifth is 274. What is the number? 5. What number must be added to the numerator of #7 in order that the resulting fraction shall be equal to 4? 6. What number must be added to the denominator of *y in order that the resulting fraction shall be equal to 4? 162 FRACTIONAL AND LITERAL EQUATIONS [Cuap. XVII. 7. What number must be added to both numerator and denominator of 38; in order that the resulting fraction shall be equal to #? 8. A yardstick is cut into two pieces. One piece is ¥ the length of the other. What is the length of each piece? 9. What number added to both numerator and denom- inator of the fraction 2 will double the value of the fraction? What number will halve the value of the fraction? 10. Find two numbers which differ by 4, and such that one-half the greater exceeds one-sixth of the lesser by 8. 11. $1000 is divided between A and B in the proportion 3 to 8. How much more is B’s share than A’s? 12. A man leaves one-third of his property to his wife, one- fifth to each of his three children, and the remainder, which was $1200, to other relatives. What was the value of his estate? 13. An estate of $5300 was left to two heirs. The first received one-third more than the second and $400 additional. How was the estate divided? 14. A man has two sums of money at interest, which ton gether amount to $19,000. One sum brings 5% interest, the other 3%. From the latter he receives $250 more income than from the former. What is the amount of each of the two sums? 7 15. The width of a room is two-thirds of its length. If “the width were five feet more and the length 8 feet less the room would be square. What are the dimensions of the floor? 16. Each year a merchant increases his capital one-third, but takes away $4000 for expenses. At the end of the second year, after deducting the second $4000 he finds that his capital has increased one-half. What was his capital when he began ae | 7 . The denominator of a fraction exceeds its numerator by 3 If 1 is added to both numerator and denominator the resulting fraction will be equal to 3. What is the fraction? 18. A lazy boy is asked to ide one-half of a certain. number by 8, and the other half by 10, and add the quotients. F Anv. 100] PROBLEMS 163 _ To save work, he divided the number itself by 9, and obtained ~aresult too small by one. Determine the number. 19. At the time of their marriage a man’s age was to that of his wife as 3 is to 2. Nine years later, it was as 4 is to 3. What was the age of each at the time of their marriage? Hint: Let 3x = age of the man at marriage. 20. A tank is emptied by two pipes. One can empty it in 30 minutes, the other in 25 minutes. In what time can the _two together empty it? : | 21. A can do a piece of work in 3 days, and B can do it in Od days. In what time can they do it together? Hint: Let x = the number of days it will take A and B together, then . x = the part A and B can do in one day. | 22. If A can do a piece of work in 8 days and B in 10 days, in what time can they do it together? 23. Smith and Jones have forgotten the scores in the foot- ball game between Chicago and Illinois in the fall of 1914. Jones remembers that the difference of the scores was 14, and Smith remembers that the difference of the scores was half as ‘much as the sum of the scores. What’ were the scores of the two teams? : 24. Smith and Jones have forgotten the scores of the foot- ball game between Chicago and Illinois in the fall of 1913. Smith remembers that Chicago won by 16 points, and Jones remembers that Illinois’s score was 2 less than half that of Chicago. What were the scores? 25. A man is now 45 years ‘old and his son is 15. How many years must elapse before their ages will be as 13 is to 7? 26. Find a fraction whose numerator is greater by 3 than half of its denominator, and which when reduced to its lowest terms is 2. 27. Find the age at which the Greek mathematician Dio- phantus died. His epitaph reads as follows: Diophantus 164 FRACTIONAL AND LITERAL EQUATIONS [Caap. XVII. « passed + of his life in childhood ; +; in youth and + more as a bachelor ; five years after his marriage was born a son who died four years before his father at half his father’s age.* 101. Literal equations. A literal equation is one in which some or all of the known numbers are represented by letters. It has become customary to represent the unknowns by the | last letters of the alphabet, while the known numbers are usually represented by the first letters. Thus, the general linear equation (Art. 65) ax +b =0 is a literal equation. Other examples of literal equations are —+b=6, (a—b)z=b+c, (8 —a) +2 +a EXERCISES Solve the following equations, the unknowns being repre- sented by the last letters of the alphabet: 1. -4b=c. fe Sotution: Multiply both members by z, and obtain a+bx =cx. (1) Transposing and collecting terms, we have (6 —c)x = —a, and | dn a (2) CuHEcK: Substitute in (1), a+ c = xt < a ac —ab + ab = ac. 020: 2. (a—b)t=b+. 4, ee x x 3. (8 —a) + (24+ ))r =. 5. ay +cy =2(a+b). || * Father’s age here means his age at his death, and not at time of his son’s death. \ 4 | A J \ t i “Arts. 101,102] | SUBSCRIPT NOTATION 165 6. az + bz = 3(a +b). 10. as —a® — 4 = 3a — sg. b i dx = 2 — d?. a= ———: eed 1 = C a sarees , 2 ae ip eee U U 2 9. a(2a+x—1+d)=2b. 13 en ae Needs 1 he be 14. (a+ 2)(b—2x) —- (a—2)(b+2) = 2. c+z2z.c-2 1 0 ce aa a naa r+a r iL a 16. - 2a Ea a+b at—b by-a 17. a+ ee ii ; Solve for y and then for ¢. 18. = @ 19. = . 2 2 3 8 i ay} 1 Ne 8 Mee cor rar +b 2—a 102. Subscript notation. It is often convenient to repre- sent related numbers by the same letter with small numbers (called subscripts) written at the right and below the letter. Thus, if 7 represents temperature, the temperature of a body pt two different times may be represented by 7), 72. The weights of three bodies may be represented by Wi, Wo, Ws. These letters are read “7 sub 1,” “T sub 2,” “W sub 1,” and Bo on. Another notation less often used is the prime notation. Two or more letters are distinguished from one another by 166 FRACTIONAL AND LITERAL EQUATIONS [Cuar. XVIL marks (called primes) placed at the upper right hand. ‘Thus, temperatures at two times may be represented by 7’, T", and weights of different bodies by W’, W”, and W’”. ‘These are read “7 prime,” ‘7 second,” “ W prime,” ‘‘W second,” and “W third.” Both the subscript and prime notation are used in physical and engineering formulas. | TRANSFORMATION OF PHYSICAL FORMULAS AND PROBLEMS In the following physical formulas, solve for the letter indicated. The custom of using the first and last letters of the alphabet for the knowns and unknowns respectively is not followed here. 1+2 2. H = 1082 + .3805t. Solve for é. | 8. C = 8(F — 32). Solve for F. | 1. h= Solve for 2. én 4 = - Solve for n. R+o 5. C= eae uae Solve for r. R19 W+R 6. Q = W+tRtw Solve for R. W'(H — t+ 32) 1... Wi 966 Solve for ft. + 7d ; 8s. s7hitiene Br Solve for d and for @. W —- W'’ 9. A= H Wiw+w” Solve for W. 1 10. F = ( : , (",) Solve for 7’. j Vy ae 5 / | Arr. 102] EXERCISES AND PROBLEMS 167 Read aloud the following formulas, then solve for the letter indicated. a Fu fe C(ts a t1) i W(t, — t1) = W a(t — ts). Solve for ts. 12. Vi; = Vo(1 + .0087¢,). Solve for ,. (i) a Ba Ct See . 3 Big St, Solve for 7; 0, ~0;) = Ee Baise fan re 1 7.65 765 : Laide Bee T pr Solve for 7”. 16. A = $bh” + 4a(h” +h’) + 4ch’. Solve for h’. V1 / | oes ” P v2 17. PP” = — Solve for ve. pier 2 202 18. If a and b are the altitude and base of a triangle and A the area, then A = 2. Express a in terms of 6 and A. 19. If r per cent is gained on an article that costs ¢ dollars, the selling price S is given by the formula S = c + in Find c if r = 10 and the selling price is $181.50. 20. Write a formula for the selling price when r per cent is lost. Find r when the selling price of the article in Problem 19 is $138.60. 21. The proceeds P of an amount of a dollars discounted for ¢ days at r per cent a day is P=a — sa00" Solve the equation for a; for r; for ¢. 22. If a cents be divided between two boys so that one boy has 6 cents more than the other, how many cents has each? 168 FRACTIONAL AND LITERAL EQUATIONS [Cuap. XVII. 23. The pressure of water per square inch at a depth d feet cae 2. nS is given by the formula P = ad a d. At what depth is the pressure ten times as great as the pressure of the air at the surface? (Pressure of the air to be taken as 15 pounds per square inch.) 24. If two quantities of water mm and m, at temperatures t; and tf. are mixed, the temperature, ¢, of the mixture is 7 mt + Mato M1 +- Meg How much water of temperature 48° must be mixed with 6 gallons at 150° to make a mixture at 110°? CHAPTER XVIII RATIO, PROPORTION, AND VARIATION 103. Ratio. The ratio of a number a to a second number 6 is the quotient obtained by dividing a by b. The ratio of a - to 6 is usually written in the fractional form “, but the form > b ) a:6 is often found. All ratios of numbers may be considered _as fractions. The two numbers a and b in a ratio are called the terms of the ratio. The numerator is called the antecedent and the denominator the consequent. EXERCISES In the following exercises name the antecedent and the consequent for each ratio; write the ratio in fractional form and simplify by reducing the fraction to its lowest terms: Ratio of 4 to 12. Ratio of 12 to 4. Ratio of 33; to 6. Ratio of 74 to 7%. Ratio of 3 minutes to 2 minutes. Ratio of 3 minutes to 2 seconds. Ratio of x? to xy. Ratio of a2 —b toa+ 0b. Ratio of 16 pounds to 26 pounds. Ratio of 16 pounds to 26 ounces. Sager) Bahai tapeiles pte oo SEs = a Kalco ch — iS) 8 coo xR 169 170 RATIO, PROPORTION, VARIATION [Cuap. XVIII.’ 13. (1 _ “) : (1 a “|. y y y" 14. (u+2 U) (4° -¥). 104. Proportion. A proportion is an we equality of. two ratios. If the two equal ratios are and - then the proportion is and a, b, c, d are said to be in proportion. The proportion is often written | C50 00 An older notation sometimes found is a:b::e:d. In any notation the proportion may be read “‘a is to 6 as ¢ is to d.” | The first and fourth terms of a proportion are called the extremes, and the second and third terms the means. ‘Thus, in the proportion, a and d are the extremes and 6 and c the means. Since a proportion is an equality, operations that may be performed upon the two members of an equality may be per- formed upon the+two ratios of a proportion. Thus, in the proportion, both members may be multiplied by bd, giving ad = be, which proves the theorem: In any proportion the product eo the means equals the product of the extremes. Ht ' Arrs. 104, 105,106] MEAN PROPORTIONAL 171 EXERCISES In the following proportions test the above theorem: eis Pics 6. 35 ' 137 408 ee a 85a Bri 1 " 17y = 289xy a 5 ax+ay _b ' bn —~ by a-—y r+y Find the value of x in the following proportions: _ ee 11. a:b=Z:¢. 5 ee ees 1 ety Cz: x o 8) 73:07 = 3:8. 1385 377-2 = (ix +3. g, =:3) = 3:8. 14. —-2:7-4=3:2742. 104).0 = C > 2. 15. +:40 = 100 7:4.7.- 1065. Mean proportional. If the means of a proportion are the same number, this number is called a mean propor- tional between the two extremes. Thus, in the proportion, 2:6 = 6:18, 6 is a mean proportional between 2 and 18. In the proportion a:z = «:b, x is a mean proportional between a and b, and from this we obtain x= + V/ qb, 106. Third and fourth proportional. A third propor- tional to two numbers, a and 6), is the number z, such that b x be b 172 RATIO, PROPORTION, VARIATION [Cuar. XVIII. | Thus in 3°; = 33, 48 is a third proportional to 3 and 12. A fourth proportional to three numbers a, 6, and c taken in the order given is the number z if Thus in 3 = 45°, 9 is a fourth proportional to 5, 3 and 15. EXERCISES Find the mean proportionals between the following pairs of numbers: 1 1 1. 2 and 8. 5. F and 50° 2. 4 and 9. 6. 2 and y. 3.. 2 and 32. if 3 and = xe ci 1 4. 1 and 64. 8. = and 2. Find third proportionals to the following pairs of numbers : 9. 2 and 4. 12. (a + b) and (a — BD). | by at gic 1 1 10. 3 and : 13. e and ii 11. Bog —1. hy = a 3 7 u—y y Find fourth proportionals to the following sets of numbers in the order given: ee SOAS 18. %¢, Daa: 16s), 2) 19. a+b, ab, a — b. 17. a, b,c. 20.\2;° Sie Cee | Ars. 107, 108, 109] PROPORTION 173 10%. Proportion by alternation. If, in the proportion eo fs eo _ both members of the equation are multiplied by °, the equation becomes after reduction That is, in a proportion the means may be interchanged with- out destroying the equality. In this case the second propor- tion is said to be obtained from the first by alternation. | j | 108. Proportion by inversion. From the proportion, we have from Art. 104, (1) be = ad. | Dividing both members of this equality by ac, we have i b d ae: 2) _ It may be noted that the members of (2) are obtained from (1) by inverting the fractions. For this reason the second propor- tion is said to be obtained from the first by inversion. 109. Proportion by composition. If, in the proportion, a c¢ | ees (1) 1 be added to both members, we find | a tr a+b cec+d fee ot 1 or =) eae ee (2) | In this case, the second proportion is said to be obtained from the first by composition. 174 RATIO, PROPORTION, VARIATION [Cuar. XVIII. | 110. Proportion by division. If 1 be subtracted from both members of the proportion, there results ah ts and the second proportion is said to be obtained from the first by division. 111. Proportion by composition and division. From the proportions by composition and by division we obtain, after dividing members, a+b ct+d a—b c=—d This proportion is said to be obtained from the proportion : ae by composition and division. Composition, division, and composition and division are often very appropriately called addition, substraction, and addition and substraction respectively. EXERCISES Write proportions obtained from the following by (a) alternation, (b) inversion, (c) composition, (d) division, (é) composition and division: . 5. 3:4=(e+y):(@-y). From each of the following equated products obtain five proportions: ; 24 2 4 6. 2-10 = 4-5. 8 5 me 7 ¢-3= 21-1; 9. 2-(4+y) =3:(z—-y) Pieter cs. “Ants. 111, 112, 113] VARIABLES AND CONSTANTS 175 10. 3:4=24-1. Pas *a1- do = b1 « be. 1h A 3 = 6. Vos fe PT» = 1° XV. 14. met prove ~~ 4 _ oe. Ma — a | by — by eae iy no iss 16. is = 4S prove X1 + 322 M Yi + BY2_ Y2 v2 Y2 ay by Te it ay by’ Prove ! Find the value of x in order that the following proportions : i eW+1):@—1)=2:3. 19. @+2): (8244) =5:6. / fete 1)s (4 +2) =2:3. 20. | 112. Variables and constants. If ¢ represents the time “since a Chicago train left New York and s its distance from |New York, we note that each of these symbols s and ¢ takes ‘different values during the progress of the train. On_ this ‘account, ¢ and s are said to be variables. If the train should Tun at a uniform speed of v miles per hour, we know that Sk | In general, a symbol is said to be a variable if it may repre- sent different numbers in a discussion or problem. It is con- stant if it represents only one number. The idea of a variable ‘plays an important réle in algebra (Chap. XIX). 113. Direct variation. When two variables are so related ‘that their ratio is a constant, either one is said to vary as or to lat directly as the other. In symbols, i Sakon ker. when & is a constant, may be read ‘‘y varies as 2.” | * See Art. 102. 176 RATIO, PROPORTION, VARIATION [Cuap. XVIII. . ¥ In describing the progress of the train, (Art. 112) we write ! s = ut, and say that the distance from New York varies as the time since leaving New York. Suppose v = 40 miles per hour, then s = 400. EXERCISES Write in the form of an equation each of the following statements : | 1. The weight W of the water in a tank varies as the volume V of the water. 2. The simple interest earned on a principal P varies as the time ¢. 3. The speed v of a falling body, started from rest, varies as the time ¢ since it began to fall. 4. If y varies as x, and « = 2 when y = 6, find y when en), SoLution: The statement y varies as 2 means y =kx (k constant). (1) To determine k, make x=2,y =6. This gives 6 = 2k, (2) or k =3. (3) From (1) and (8), y= Sf. (4) Substituting, y = 30. 5. If y varies as x, and x = 4 when y = 12, find y when A 0; 6. The area of a circle varies as the square of its radi If a circle whose radius is 10 feet contains 314.16 square feet, find the area of a circle whose radius is 8 feet. a 7. In using a spring balance, the principle applied is that the stretch s of the spring varies as the weight W to be de- termined. A weight of 10 pounds stretches a certain spring 1 inch, how great a weight is required to stretch it 2% inches? Arr, 113] EXERCISES AND PROBLEMS 177 8. The distance through which a body falls from rest “varies as the square of the time in seconds. If a body falls 16 feet the first second, how far does it fall in 10 seconds? 9. The weight of a wire varies as its length. It is found that 100 feet of a certain wire weighs 3 pounds. Find the weight of one mile (5280 feet) of this wire. EXERCISES AND PROBLEMS Find a mean proportional between 3 and 48. Find a fourth proportional to 5, 6, and 35. Find z in the proportion 50:75 = 90: x. In a certain business transaction, A gains $200 and B _ loses $50, and then A’s capital is to B’s capital as 4 tol. If _A’s original capital was $1200, what was B’s? 5. Find two numbers, one being twice the other, such that their sum is to their difference as 5:1. 6. How may $10 be divided among three boys so that for every dollar the first receives, the second shall receive 15 cents and the third 10 cents? 7. The sides of a rectangle are in the ratio 2 to 3. Find the ratio of the area of a square of the same perimeter to the area of this rectangle. 8. In the state of Illinois in the 1910 census, the ratio of foreign-born white inhabitants to native-born white was ap- proximately 27.8 per cent. What number of each if the total white population is 5,527,000? . 9. A business worth $37,000 is owned by three men. The ‘share of the one man is a mean proportional between the shares of the other two. If his share is $12,000, what is the share of each of the other two? 10. Prove that no four consecutive integers such as n, n+1,n+2,n +3 can form a proportion. 11. The Ist, 3rd, and 4th terms of a proportion are x + y, x — y, and (x + y)?; required the 2nd term. ye oo pO be q 178 RATIO, PROPORTION, VARIATION [Cuapr. XVIII.” 12. If four numbers 2, y, 2, and w, not all equal, are in proportion, show that no number different from zero can be added to each which will leave the resulting four numbers in proportion. 7 13. If ax — by = cy — dz, find the ratio of x to y in terms: of a, b, c, and d. 14. Divide 44 into two parts such that the less, increased by one, shall be to the greater, decreased by one, as 5 is to 6. APPLICATIONS TO PROBLEMS FROM MENSURATION Two triangles that have the same shape are said to be similar. In two similar triangles the sides of the one taken in any order are proportional to the sides of the other taken in the same order. 15. The sides of a triangle are 12, 15, and 20. In a similar triangle, the side corresponding to 12 is 18. Find the other sides. oN Hint: Let 15z = one required side — and 20x the other. . His 16. The sides of a triangle are 12, 15, and 20. The perimeter of a similar triangle is 188 feet. Find its sides. The areas of similar triangles are in the same ratio as the squares of the lengths of corresponding sides. 17. A triangular field has sides 30, 40, 50 rods. Find the sides of a similar field of four times the area. 18. A triangular field has an area of 20 acres. What is the area of a similar field with twice the perimeter? 4 19. A triangular lot has one side 10 rods and has an area of 40 square rods. What is the area of a similar lot whose corresponding side is 25 rods? ; 20. The areas of two similar triangles are 121 and 144, respectively. One side of the one triangle is 22. Find the corresponding side of the other. >, 4 & Arr. 113] EXERCISES AND PROBLEMS 179 La ee ie wae Bie A horse tied with a rope 40 feet long in the center of a pasture eats all the grass within reach in 4 days. If the rope were 20 feet longer, how many days would it take him to eat all the grass within reach? Hint: The area of a circle varies as the square of the radius. 22. The volume V of a sphere varies as the cube of the radius r, and the volume of a sphere of radius 10 inches is 4189 cubic inches. Find the volume of a sphere of radius 8 inches. EXERCISES IN THE NOTATION OF PHYSICS * 23. The ratio of the force F to mass m is a number k. Express each of. the letters /, m, and k in terms of the other letters. | 24. The product of the pressure p by the volume v of a gas divided by its temperature 7 is a number k. Express each of the letters p, v, and 7’ in terms of other letters. 25. Given horn uae solve for R, h, and r in turn. eo: Sar 2 ay ats 26. Given = = es solve for G. 27. Given toes ok = solve for x. 28. Suppose v varies as ¢ and v = 128 when ¢ = 4, find v when ¢ = 25. 29. Given that s varies directly as the square of ¢ and that s = 64 when t = 2. Write this in the form of an equation, and find s when ¢ = 10. 30. The force F acting on a body varies as the acceleration a produced in the motion. Write the relation between force and acceleration when the constant ratio is the mass m. * The teacher is not expected to take the time to explain the physical meaning of the relations given. The exercises are given to familiarize the student with the use of other letters than x and y for unknowns. CHAPTER XIX GRAPHICAL REPRESENTATION OF THE RELATION BETWEEN TWO VARIABLES 114. Introduction. When the corresponding changes in two related quantities, as for example the temperature at successive hours of the day, are to be described, it is useful to represent the quantities by lines and points (See Art. 18). Such a representation is said to be graphical. By this method, the corresponding changes in the two quantities are presented to the eye in a very vivid way. For example, a daily paper gives the following temperatures for Chi- cago at successive hours of a certain November day : BRA Mi Soke 8 aM. 23° 2 P.M. ‘dog 8 P.M. 37° DA MeL 9 aM. 28° 3 P.M. 41° 9PM. 37° 4 aM. 20° 10 am. 31° 4pm. 40° 10 p.m. 38° Db AM.. 20" 1 pW) es 5 p.m. 39° 11 p.m. 38° 6 am. 21° Noon. 35° 6 p.m. 39° Midnight. 37° 7AM. 22° 1pm. 38° 7 PM 38" 1am. 36° The changes in temperature with respect to time are readily grasped by the representation of these numbers on cross ruled paper (See Fig. 22). . 115. Axes, Coérdinates. In graphical work, much use is made of two fixed perpendicular lines of reference. The lines of reference X’X and Y’Y (See Fig. 23) are called coérdinate axes and their intersection is called the origin. The horizontal line X’X is called the X-axis and Y’Y is called the Y-axis. The horizontal distance from the Y-axis to a point P is called the abscissa or x-value of the point. The vertical distance from the X-axis to P is called the ordinate or 180 ae ee / Art. 115] . PLOTTING OF POINTS 181 y-value of the point. The z-value and the y-value of a point are together called the coérdinates of the point. It is the custom to take distances measured to the right from Y’Y as ; “Rees teaum cB Rie Wie | | ‘ /< Sena mal \ » / Se ee Be Ra ee Re ; a SePeer ATE ; ot EERE : PERE tpt sreseiiatarts M. 1 ye Fia. 23 182 GRAPHICAL REPRESENTATION ([Cuap. XIX. “i positive and those to the left as negative; those measured upward from X’X as positive and those downward as negative. 116. Plotting of points. If we have given two numbers, — say 2 and —5, we can find one and only one point that has the first number for its abscissa and the second for its ordinate. To find the location of the point for the numbers x = 2, y = —5, we start at the origin O and measure two units to the right along the X-axis, and from this point, we measure downward > a distance 5. This point may be represented by the symbol _. (2, —5). The symbol (a, b) denotes a point whose abscissa , is a and whose ordinate is b. When a point is located in the manner described above, it is said to be plotted. 117%. Use of coordinate paper. In plotting points and obtaining the geometrical pictures we are about to make, it is convenient to use codrdinate paper. ‘This is paper ruled both horizontally and vertically as shown in Figs. 22 and 24. EXERCISES AND PROBLEMS 1. Plot the points (3, 4), (3, —4), (—8, 4), (—38, —4). 2. Draw the triangle whose vertices are (3, +1), (0, 5), (—4, —2). ) 8. Draw the quadrilateral whose vertices are (2, —2), (—8, 4), (-6, —8), (3, 4). Historical note on graphical representation. The discovery of the method of representing functions and equations graphically is due to René Descartes (1596-1650), the French philosopher and mathematician. The discovery of this graphical representation of equations marks one of the greatest. advances ever made in mathematics. He showed that dis- tances measured in opposite directions could be used to represent positive and negative numbers, and through such representation brought mathe- maticians to see that negative numbers are indeed very real and useful. ¥ q y : Z » Arrs. 117, 118, 119] VARIABLES 183 4. If a point moves parallel to the X-axis, which of its codrdinates remains constant? 5. If a point moves parallel to the Y-axis, which of its coordinates remains constant? 6. A line joining two points is bisected at the origin. If the codrdinates of one end are (4, 5), what are the codrdinates of the other end? 7. Draw the triangle whose vertices are (8,0), (0, 5), (—3, —2). 8. Given a north and south line, and an east and west line , for reference lines (Y and X-axes respectively), the following fy coordinates of points on a river indicate its general course : (0, —1), cp —2), ee —23), (2, —13), (3, We (4, 5), (5, 10), (-1, 0), (—2, iy (—3, 2), (—35, 1), (—4, —1), (—5, —3). “Map the river from x = —5 to x = +9. 118. Variables. A variable is defined in Art. 112. To illustrate again, if ¢ represents the time of day measured from 2 a.m. (Fig. 22), and 7 represents the temperatures, we note that each of these symbols changes in value throughout the 24hours. They are therefore variables. As a further illustration, we may think of a particle of matter whose position is given by (2, y) moving to various positions.in the plane of the codrdinate axes. As such a parti- cle moves along a curve x and y vary in value. 119. Definition of a function. If two variables xz and y are so related that when a value of one is given, a correspond- ing value of the other is determined, the second variable 1s called a function of the first. Thus, the area of a square is a _ function of its side. This function may be expressed algebra- ically as — 72 ete where z is the length of a side. 184 RELATION BETWEEN TWO VARIABLES [Cuap. XIX. The simple interest, denoted by J, earned on a principal of $100 at a rate of 6 per cent per annum is a function of the time tin years. This function may be expressed algebraically by Leal. We have had many examples of functions. In particular, the idea of a function of 2 taking different values when x changes has been well illustrated in Arts. 18, 31, and 83. 120. Graph of a function. By a method similar to that employed in Ex. 8, Art. 117, to map a river, a function may be represented with respect to codrdinate axes. ‘This repre- sentation of a function is called the graph of the function. Fig. 22 gives the graph of temperature 7’ as a function of the time ¢. Example. Obtain the graph of $v +4 for values of x between —5 and +5. Let y = 34 +4. The object is to present a picture which will exhibit the values of y that correspond to assigned values of x. Any assigned value of x with the corresponding value of y determines a point whose abscissa is 2 and whose ordinate is y. Assigning values for « and computing the corresponding values for y, we obtain the following table : e|O|1 |2125 13 | 4] 5 |-1 |-15 | -2|-3 | -4| -5 y14155|717.75|85| 10|115| 25 |-1.75] 1 |=0.5 ))—oq0—une and so on. The corresponding values of x and y are plotted as coérdi- nates of points in Fig. 24. It should be noted that there is no limit to the number of : corresponding values which we may compute and imagine plotted in a given interval along the X-axis, and further that to small changes in the values of x, there correspond small changes in the values of y. These facts suggest the idea of a continuous line or curve — to represent the function much as a continuous curve is used in mapping a river. The line in Fig. 24 is the graph of the function 3x + 4. 3 F, oR) ba , Arr. 120] EXERCISES | 185 Two spaces = 1 unit Fic. 24 EXERCISES Construct the graph of each of the following functions by plotting a number of points for each function, and drawing a continuous curve through these points: 1. 32 +4. 4. 27 —1. 7. 62-1. Beer 4 6. 5.5 +5. 8. 7x +2. 8. 4a — 3. 6. 4” + 2. 186 RELATION BETWEEN TWO VARIABLES ([Cuaap. XIX. 9. The temperature readings of a Fahrenheit thermometer that correspond to Centigrade readings are given by y = $x + 32°, where x refers to Centigrade readings and y to .Fahrenheit readings. Draw the graph to represent Fahren- heit readings to correspond to given Centigrade readings. 10. A certain kind of cloth costs $2 per yard. What function gives the cost of any number of yards if c is the cost and n the number of yards? Plot the graph of this function. 11. A man drives a car at the rate of 20 miles per hour. Write the function that gives the distance d he drives in ¢ hours, and construct its graph. 12. Butter costs 30¢ a pound. Draw a graph to show the cost c of n pounds. 121. Graph of an equation. If x and y are involved in an equation, say y—3¢-—4=0, (1) we often speak of the graph of the equation. If we express y in terms of x, we have y = 32 + 4, and may plot the graph of this function of z. The graph of this function is often spoken of as the graph of equation (1), since coordinates of points on the graph and of no other points satisfy — the equation. To construct the graph of an equation in xz and y, we have therefore merely to express y in terms of x and to construct the graph of the function thus obtained. The graph of an equation is perhaps better called the locus of the equation, since the codrdinates of points on the graph and of no other points satisfy the equation. bs \ Arts. 121,122, 128] GRAPHICAL SOLUTIONS 187 EXERCISES Construct the locus of each of the following equations : 1. y — da = 4. 4. 4x — 2y = 3. ~. ae2+y = 7. 2.04 = © = 6. 5. o2-+ 4y = 9. 8. y — 32 =.2: 3. 3x + 4y.= 5. 6. 2a Sy = 4. 9. 2y + da = 4. 10. 62 — 3y = 3. 122. Locus of a linear equation. An equation of the form az + by +c =0 is said to be linear in x and y because its locus is a straight line.* To find the locus of a linear equation, it is only necessary to find two points of the locus and to draw a straight line through them. Thus, to find the locus of «+ y = 6, choose x = 0 and find y = 6; choose y = 0, and find « = 6. We may then locate the points (0, 6) and (6, 0) and draw a straight line through them. EXERCISES Draw the locus of each of the following equations : for oy = 11. 3. 4r—sy=90.. 5. 84 —y =3: ane Aly = 1. 4°62 + Sy = 4. 6. 6y — 2 ='8: 7. Write an equation in which y increases as x increases, and plot its locus. 8. Write an equation in which y decreases as x increases, and plot its locus. 123. Graphical solution of equations. If we construct the loci of two equations, say of 3% — 2y = 6 and x + 2y = 10 as shown in Fig. 25, it is seen that the point of intersection of the loci is (4,3). As this point is on the locus of each equa- * No attempt is made here to prove this statement. The proof is given in analytic geometry. The fact is well illustrated in the above examples and may be taken for granted for the present. 188 RELATION BETWEEN TWO VARIABLES [Caap. XIX. Two spaces == 1 unit Fre; 25 tion, the values x = 4, and y = 3 satisfy both equations. A pair of values, such as (4, 3) that satisfies each of two equa- tions is said to be a solution of the pair of equations. Exercise. Show that x = 4, y = 3 satisfies the given equations. Thus, the graphical solution of two equations is the point of intersection of the loci of the equations. Since the graph of a linear equation is a straight line, and ~ since two straight lines intersect in only one point, there is in general only one pair of values of x and y that satisfies a pair of linear equations in x and y. EXERCISES Plot the graphs of the following equations, and find solu- tion of each pair of equations from the graph. Test the solution by substitution in the equations. ey, »» Apts. 123, 124] GRAPHICAL REPRESENTATION 189 1 xr+y =8, 7. 3x + 4y = 10, x—y =4. 5x — y = 9. 2. x2-—y =4, 8. 5a — 4y = 11, 2x + dy = 15. 4x + 2y =14. Baur —y =—.0, 9. 2x —y = 6, y+3z =0. 4x — 2y = 8. 4,.x+y+6 =0, 10. 2x + y = 3, z—y = 0. 6a — 2y = 14. 5. x — 2y =4, 11. 2x + y = 0, z+ 2y = 8. —y + 5x = 0. 6. x+4y = 9, 125 37 Ay a7. ot — y= 14. 6x4 +y = 32. 124. Graphical representation of scientific data. The method of this chapter for representing a mathematical function has been adopted by nearly all scientific men to show simul- taneous changes in related quantities. Thus, physicists, chemists, engineers, statisticians, economists, historians, and others, use graphs to present to the eye relations that could not be shown otherwise without considerable effort. PROBLEMS APPLICATIONS IN LIFE INSURANCE 1. The number of persons per hundred thousand living at age 10 that reach certain assigned ages, as given by the Ameri- can Experience Table of Mortality, is shown in the following table to the nearest thousand : Ages | 10} 15 | 20] 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 No. of thousands living| 100 | 96 | 93 | 89 | 85 | 82 | 78 | 74 | 70 | 65 | 58 | 49 | 39 | 26 | 14| 6: fSk Exhibit the relation between the number living and age. SuagEstion: Let 20 years be represented by 1 inch along the horizon- tal, and 20,000 persons living by 1 inch along the vertical. 190 RELATION BETWEEN TWO VARIABLES [Cuap. XIX. 2. A man aged 20 may have his life insured in a certain company by the payment of $18.50 per year for $1000 of in- surance. The following table gives the amounts that must be paid if the insurance is taken at other ages: Age. Payment. Age. Payment. - 20 $20.10 A5 $37.80 30 22.92 50 46.65 30 26.55 55 58.84 40 31.38 60 75.77 Construct the graph to show the relation between the payments and the age of the person when insurance is taken. APPLICATIONS TO HISTORICAL STATISTICS 3. The population of the United States as found from the various Censuses is given by the following table : Year |1790| 1800] 1810/1820] 1830|1840|1850|1860|1870|1880|1890|1900|1910 Population in millions | 3 | 4.3 | 7.2 | 9.6 [12.9 |17.1 |23.2 |31.4 [38.6 150.2 |62.6 |76.3 |92.0 Represent the population graphically. 4. The marriage rate in England per thousand population for years from 1872-1890 is given by the following : Year | 1872 | 1873 | 1874 | 1875 | 1876 | 1877 | 1878 | 1879 | 1880 | 1881 Rate | 17.4 | 17.6 [17.0 [16.7 | 16.5 | 15.7 [15.2 |144 [149 | 154 Year__| 1882 | 1883 | 1884 | 1885 | 1886 | 1887 | 1888 | 1889 | 1890 Rate [15.5 [15.5 [15.1 [14.5 [142 [14.4 [144 [15.0 | 155 Represent this table by a graph. 5. The following table gives the production of coal in the United States in millions of tons for various dates : Date | 1850 | 1855 | 1860 | 1865 | 1870 | 1875 | 1880 | 1885 | 1890 | 1895 | 1900 | 1905 | 1910 Tons of coal| 6.3 | 11.5] 13. | 21.2 | 29.5 | 46.7 | 63.8 | 99.3 [140.9 |172.4 |240,.8|350.6 [447.6 Construct a graph to show changes in coal production from 1850-1910. es ie Be Apr. 124] APPLICATIONS 191 6. In considering the different items that enter into the high cost of living, a man records the rents per month he has paid for a house during the past twenty years as follows : 1893 | 1896 | 1900 | 1903 | 1905 | 1909 | 1913 $15 | $20 | $24 | $28 | $30 | $40 | $55 Construct a graph to show increase in rent. APPLICATIONS TO WEATHER REPORTS 7. The following table gives the temperatures on a Fahren- heit thermometer at a certain station at various hours of the day : emer) | 3 |4|/5|6|7{8| 9 | 10) 11| 12 |lam] 2 [48 10° | 11° |10.5°|9.5°|8°|6°}4°|2°|0°| — 3°|— 6° — 9°|— 12.5°|— 15° |-18°|— 20° Meise) 6 | 7 | § | 9 | 10 |-11 [Noon ee 22 91 .5°[— 21°|— 20°|— 18°|— 15°|— 12°| — 89 _ Construct the graph to show the changes in temperature with respect tothe time. Sucha graph is called a temperature curve. By reference to the graph, give approximately the lowest and highest temperatures of the day. At about what time was the temperature lowest? 8. Observe the weather report in a daily paper, and draw the temperature curve for the 24 hours covered by the report. CHAPTER XX SYSTEMS OF LINEAR EQUATIONS 125. Solution of equations in two or more unknowns. By a solution of an equation in two or more unknowns, we mean a set of values of the unknowns that satisfies the equation. Thus, fs % i is a solution of x + y = 5. A single equation in two unknowns has many solutions. Thus, the equation x+y=5 : x =0 Z=2 x=3 x=A4 has solutions, E e 5) (* 3 ai & z al C i iy and so on. In fact, an equation such as x + y = 5 in two unknowns has an unlimited number of solutions, since if we assign any value to x, we can find a corresponding value of y that satisfies the equation. Graphically, we may say the coordinates of each of the unlimited number of points on the line in Fig. 26 satisfy the equation 2 + y = 5. EXERCISES Find four solutions of each of the following equations : 1, 2 + /2y = 9. 6. s+i = 12. 11. 27 + dy = 5. 2. y+ 3x = 8. 7 2r+s=7. 12. 8& —y = 6. 3. 4% + 3y = 5. 8. 27 — 3y = 10. 13. 54 +49 = 4. 5 + 2n = 11. 9708 ob ais. 14. 6% — y = 5. 5 v—we=i7. 10. /+4m=12. 15. 1+m=10, 192 « iJ fe Ants. 126, 127, 128, 129] SIMULTANEOUS EQUATIONS 193 Fig. 26 126. Simultaneous equations. In the last article, we have shown that an equation in two unknowns has an unlimited number of solutions. Two such equations, say x + y = 5 and 2y —x = 4, are said to be simultaneous equations when at least one pair of values of x and y satisfies both. The graphs of these two equations are shown in Fig. 27, and we have found graphically the solutions of some simultaneous linear equations in Art. 123. In the present chapter, such equations will be solved by algebraic methods. 127. Independent equations. Two equations are said to be independent if they have distinct graphs. Thus, the equa- tions x + y = 5 and 2y — x = 4 (Fig. 27) are independent. 128. Dependent or equivalent equations. Two equations are dependent.or equivalent if they have the same graph. Thus, the equations x + y = 5 and 2x + 2y = 10 are equivalent. 129. Inconsistent equations. Two equations such as x+y = 5and 2x + 2y = 18 are said to be inconsistent because 194. SYSTEMS OF LINEAR EQUATIONS [Caap. XX. ~ Y’ One space = one unit Hig-v2¢ there are no values of x and y that satisfy both equations. The graphs of « + y = 5 and 2a + 2y = 18 are parallel lines. (Fig. 28.) | By a system of linear equations, we mean a set of two or — more linear equations that are to be treated together. A set of values of the unknowns that satisfies each equation of a system is said to be a solution of the system. 130. Elimination. Combining equations of a system in y such a manner as to get rid of one of the unknowns is called elimination. , Example. Find two numbers such that 2 times the first plus 3 times the second equals 12, and 4 times the first minus 3 times the second equals 6. SoLuTion: Let x = the first number, and y = the second number. Then 2x + 3y = 12, (1) 4x — 3y = 6. (2) Adding, 6x = 18, 2 eo. | a : | Arts. 130, 131] ELIMINATION 195 BGbetitutine 3 for x in (1) gives . 6 + 3y = 12. Then 3y = 6, and OD ' CHECK: 2°35 +3°2'4 12, 4-3 —-3:2 =6. ae 3 and 2 are the numbers sought. 131. Elimination by addition and subtraction. The exam- ple of Art. 130 is there solved by a method of elimination known. as elimination by addition. To apply the method to a somewhat more general system, solve ox + 2y = 17; (1) 4x + 38y = 24. (2) These equations are marked (1) and (2). We shall find it convenient to write, for the sake of brevity, such statements 196 SYSTEMS OF LINEAR EQUATIONS [Cuapr. XX. > as (1) + 5 to mean the members of (1) divided by 5, and (1) - 5 to mean the members of (1) multiplied by 5. More generally, the symbol (n) is used to denote the members of equation marked (n). (1) - 3 gives 9x + 6y = 51, (See Art. 38) (3) (2) - 2 gives 8x + by = 48, (4) (3) — (4) gives GS (5) Substituting in (1), 3-3 + 2y = 17, (6) | 2y = 8, (7) ¥en (8) 4 CueEcxk: Substitute x = 3, y= 4 in (1) and (2). This gives 3°3+2:°4=17, 4°34+3°4 = 24, Hence x = 3, y = 4 is the solution sought. Explanation. In elimination by addition or subtraction we (1) multiply or divide the members of the equations by such numbers as will make the coefficients of the unknowns to be eliminated numerically equal. (2) We then eliminate by addition if the resulting coefficients have unlike signs and by subtraction if they have like signs. EXERCISES Solve the following systems of equations, making elimina- tions by addition or subtraction: A. 37 + 4y = 10, SK 42 + 5y = 12, t= 62 —y +16 =0. \Z. 5a + 3y = 8, 6. 3a 4+ 7b = 7, 4e + 5y = -1. 5a + 3b = 29. 3. 62 — 8y = 20, 7 5y—2=242, 5x + 2y = 8. 2y — 2x +4 =0. 4. x + 3y = dy — 7, 8. ua =a ; x + 2y = 13. 3u — 2v = 10. fe j i i I ea j |, Arts. 131, 132] ELIMINATION 197 Seer os. = 1, 12. 15k + 201 = 10, 4r — 12s = 4. 25k + 141 = 11. 10. 6p — 5q = 9, 13. 6c + 15d + 6 = 0, 7p + 2q = 34. 14d + c = 22. init —i on = 20, 14. 5s + 6t = 17, 18m + 6n = 18. 6s + St = 16. 15. If the coefficients of the letter to be eliminated from _ two simultaneous equations are prime to each other, what is + the simplest multiplier for each equation? Answer the same question if the coefficients are not prime to each other. 16. 3x + 4y = 10, 18. 18u + 10v = 60, ox + by = 16. 12u — 15v = 105. 17. lox + 14y = 18, 19. 4m — 15n = 12, 25% — 2ly = 163. 9m — 10n = 122. 20. 18u + 10v = 55, 12u — 15v = 15. 132. Elimination by substitution. This method of solving a system of linear equations is illustrated in the following. Example: Solve the system of equations, 82 —4y =14, © | (1) ox + 2y = 32. (2) SoLuTion: From (1), 3a = 4y + 14. (3) 4 14 From (38), L= te . (4) _ Substituting wae for x in (2) gives Bey +) + 2y = 32. (5) _.(5)* 3 gives 5(4y + 14) + 6y = 96. (6) Multiplying, 20y + 70 + 6y = 96. (7) _ Collecting, 26y = 26. (8) y =1. (9) Substituting y = 1 in (4) gives 2 =6 (10) CueEck: Substituting x = 6, y = 1 in (1) and (2), we find the equations are satisfied. 198 SYSTEMS OF LINEAR EQUATIONS [Cuar. XX. Solve the following systems by the method of substitution and check the results by substituting in the original system of equations : 7 1. + + 2y = 11, 8. $p + 3q = 11, or — dy = 3. q-—p= oR, 9. eo ny —4, ox + 4y = 7. j Sap ae 3. 5h + 2k = 0, 10. mi + 5m: = 19, 8h +k = 3. 6m, — 7m, = 3. 4. Tr — 8s = 5, 11. 27 + 4y = 14, r+ l1s= 25. 3a — $y = 24. t ae 5. 8&4 +y =7, fa oa 5 ae llz + 2y = 9. it. ooo 3 aed 6. 127 + dy = 14, 13. .037 + .70y = 10, 3x — 10y = 26. O72 + Lely Sie Lo aoe gh ea i. 2 a 15a — 5y = 20. Ll? ee —~—~+—-=2: t.4 Hint: Solve first fope and ©. Ciah amen) : tO es eee! 17, (1? ee PEATE at LP Y = ieee eee 19, 2 See ites 0 vt Y 2 ie Malt o-3 yaaa ct ae + Arts. 132, 133] GENERAL LINEAR EQUATION 139 19. 67 + 3y = 19, 21. 4x — Sy = 8, 8a — 2y = 15. 62 + 3y = 12. 20. 32 — 7y = 15, 22. 7x + 9y = 2, ba + 4y = 11. ox + OY = 2. = 49, =a 17; 133. Standard form ax + by +c¢ =0. While some of the equations given in the following list of exercises may appear complicated, they can all be made to depend on the solution of a standard form, ax + by +c = 0, ‘in which all the terms containing x, those containing y, and those that contain neither unknown, are collected. When the following equations are not given in this form, they should usually be reduced to that form as a first step in the solution. EXERCISES — After reducing the following equations to standard form, solve the systems by either process of elimination : _1-4y cy c+y 16 5 ple PS . ’ 5. e 3 apie 3x + 4y = 15. x— 2y = 1. 7a —15 | 7p+5 Tq-1 2. 3 = ¥, 6. Bur ats = —9, 15a — 9y = 27. 10p — g = 2. x-3 yt8 3, 2 - -2, (s m + 5 vy Ly eet 2y—4 \ | z+ 10y = 5. 9 7 =), me, — 8. 5a + Ty = 37, = a 6 —2y =8 2x — 2y = 10. ae oraet lena 200 SYSTEMS OF LINEAR EQUATIONS [Cuap. xx. 9 2x — 1 Cp aS 13. .3p — 2g = 23, Oh Se es ee : 2p Ai | ae 5a — Ay = 12: 3 Gate Doe eee 2 + 4 eee 10. 6 — iF = (), 14. B} 4 = 34, i ee ee ot + 45, 27d Dee eS is.) ee 11. 5x + 9y = 40, da +1 40 ae Deaths). 15. 5 ae 3 = 153, 3 ek 4a — 2b =2 12. 16x — 4y = 150, 16, he 2x —1 AG er 3 3 4 ab ah air Qs + 5r = 45. PROBLEMS 1. Find two numbers whose sum is 100 and whose differ- ence is 28. 2. In a meeting of 484 voters a motion was carried by a majority of 32. How many voted aye, and how many no? 3. A farmer paid 3 men and 2 boys $8 for a day’s work, and afterwards paid 5 boys and 2 men $9 for a day’s work. What were the wages of a man and what the wages of a boy? 4. ‘The admission to a circus is 50¢ for adults and 25¢ for children. If the proceeds from the sale of 5000 tickets is $2200, ; how many tickets of each kind were sold? | 5. The perimeter of a rectangle is 200 feet and the length is 28 feet more than the width. Find length and width of the rectangle. 6. A part of $2500 is invested at 6% and the remainder at 5%. The yearly income from both is $141. Find the amount in each investment. 7. A part of $5000 is invested at 5% and the remainder at 4%. 'The income from the part at 4% exceeds that from the |, Arr. 133] PROBLEMS 201 part at 5% by $52. Find the number of dollars in each invest- ment. 8. If A should give B $10, then B would have twice as muchas A. If Bshould give A $10, then A and B would have the same amounts. How much money has each? 9. In a certain family each son has as many brothers as sisters, but each daughter has twice as many brothers as sisters. How many children are in the family? 10. A bottle and its contents cost 60 cents, and the contents cost 44 cents more than the bottle ; what was the cost of the bottle? 11. A mechanic and an apprentice together receive $50 for a piece of work. The mechanic works 8 days and the apprentice. 12 days; and the mechanic earns in 5 days $2 more than the entire amount received by the apprentice. What wages per day does each receive? 12. A mule and a donkey were going to market laden with wheat. The mule said: “If you give me one measure, I should carry twice as much as you ; but if I give you one, we should have equal burdens.”’ Tell me what were their burdens. e Tradition says that Euclid gave this problem in his lectures at Alexan- _ dria 280 B.C. 13. A and B can do a piece of work in 2 days ; but an equal _ piece of work when A puts in only half his time and B only one-third his time requires 43 days. How long would the work take A and B each, working alone? 14. A farmer bought 100 acres of land for $4000. If part _ of it cost him $34 an acre and the remainder $49 an acre, find _ the number of acres bought at each price. ; 15. A plumber and his helper receive $4.80. The plumber i works 5 hours and the helper 6 hours. Working at the same - rate per hour at another time the plumber works 8 hours and the helper 95 hours, and they receive $7.65. What are the wages of each per hour? 202 SYSTEMS OF LINEAR EQUATIONS [Cuar. XX. . PROBLEMS INVOLVING THE LEVER Fia. 29 If two weights w and W balance when placed on a bar at distances d and D respectively from the point of support F (called the fulcrum), then - w:d=W-D 16. Two weights balance when one is 5 feet and the other 8 feet from the fulcrum. If the first weight, increased by 25 - pounds, be placed 4 feet from the fulcrum, the balance is maintained. Find the two weights. 17. Two children weighing 35 and 49 pounds just balance a seesaw board 12 feet long. Where is the support of the board placed? 18. T'wo children, playing on a seesaw board 15 feet long, just balance when the support is 9 feet from one end. If the child on the long end of the board weighs 50 pounds, what is the weight of the other child? 19. ‘Two unknown weights balance when placed 8 and 10 feet from the fulcrum of a lever. If their positions are reversed, 5 pounds 10 ounces must be added to the lesser weight to restore the balance. What are the weights? PROBLEMS ABOUT DIGITS 20. In an integer of two digits let ¢ represent the tens’ digit and wu the units’ digit. The number is then 10¢+ 4. (a) What are t and wu in 46? (6) In 85? (ce) Write each number in the — form 10¢+ wu. (d) What is 10¢ + wif t =6 and u = 3? (e) If t=landu=97 , Arr. 133] PROBLEMS 203 21. In an integer of three digits let h, t, and wu be the hun- dreds’, tens’, and units’ digits respectively. (a) Write the num- bers. (6) What are h,¢and win 865? (c) In 506? (d) In 189? (e) Write each of these numbers in the form 100h + 10¢ + wu. (f) What is 100h + 10¢+uifh=9,t=4,u=3? (g) Ifh=1, b= 8, u = 0? 22. A number contains two digits. The units’ digit is 3 greater than the tens’ digit. ‘The number equals 4 times the sum of the digits. Find the number. 23. The sum of the digits in a certain two-digit number is 8. If 18 be added to the number, the result is expressed by the digits in the reverse order. Find the number. 24. Two numbers are written with the same two digits; the difference of the two numbers is 45 and the sum of the digits is 9. What are the numbers? 25. The numerator and denominator of a certain proper fraction each consists of the same two figures whose sum is 9, written in different orders. If the value of the fraction is 4, find the numerator and denominator. PROBLEMS ABOUT COINS 26. A man has 22 coins amounting to $10, all dollars and quarters. How many of each denomination has he? 27. A collection of nickels and dimes, containing 121 coins, amounts to $7.90. How many coins of each kind are there? 28. A man met some tramps and wished to give them a quarter each, but found he had 23 cents too little for that. He _ therefore gave them two dimes each and had 42 cents over. How much money had he and how many tramps were there? PROBLEMS INVOLVING MOTION 29. Two boys run a race of 440 yards. In the first trial A gives B a start of 65 yards and wins by 20 seconds. In the second trial A gives B a start of 34 seconds and B wins by 8 yards. Find the rates of A and B in yards per second. 204 SYSTEMS OF LINEAR EQUATIONS [Cuap. Xx. 30. Inamile race A gives B a start of 44 yards and is beaten by 1 second. In a second trial A gives B a start of 6 seconds, and beats him by 9% yards. Find the number of yards each runs in a second. 31. A train traveling 30 miles an hour takes 21 minutes a longer to go from A to B than a train which travels 36 miles — an hour. Find the distance from A to B. 32. A steamer makes 50 miles downstream in two hours, and returns in 23 hours. Find the rate of the current and the rate of the steamer in still water. Hint: Let x = rate of steamer in still water in miles an hour, and y the rate of the current. Then the rate downstream is x + y and the rate upstream is x — y. 33. A boat goes downstream 72 miles in 3 hours, and up- stream 48 miles in 3 hours. Find the rate in still water and the rate of the current. PROBLEMS ON MIXTURES 34. A grocer has two kinds of sugar, one worth 5¢ and the other 6¢ a pound. How many pounds of each sort must be taken to make a mixture of 25 pounds worth $1.40? 35. What quantities of silver 72% pure and 84.8% pure must be mixed to give 8 ounces of silver 80% pure? 36. If 25 pounds of sugar and 10 pounds of coffee together cost $5.50, and at the same price 25 pounds of coffee and 10 pounds of sugar cost $10.60, what is the price of each per pound? 37. A farmer finds that one day with 200 pounds of milk and 40 pounds of cream, he gets 24 pounds of butter. On ee) another day with 150 pounds of milk and 25 pounds of cream, i he gets 16 pounds of butter. (a) What per cent of his milk is © butter? (6) What per cent of his cream is butter? 38. How many gallons each of cream 40% fat and milk | 5% fat shall be mixed to produce 30 gallons a the mixture — 162% fat? | Arts. 133, 134] LITERAL EQUATIONS 205 39. A pound of tea and 25 pounds of sugar cost $1.75. If sugar rises in price 20 per cent and tea 10 per cent, the same amounts cost $2.05. Find the price per pound of each. 134. Literal equations containing two unknowns. When the coefficients of the unknowns are letters, the equations with two unknowns can still be solved by means of elimination, but the coefficients of the unknowns appear in the results. EXERCISES AND PROBLEMS In the following exercises, consider the letters a, b, c, d from the first of the alphabet as known numbers ; solve for the x, y, 2, w and check: 1. © + 3y = 2a, (1) 10x — y = 3a. (2) SoLuTION: (2) - 3 gives 30x — 3y = 9a. (3) (1) + (8) gives slz = lla, (4) ~ ‘Lin aay be (5) re lla Substituting in (1), 31 + 3y = 2a, (6) ae an, _ la si egy , lla Iva _ 1ila+5la_ 62a_ CHECK: 31 +3 Thos 31 = The 2a. tig, 17a 110a—17a 93a Seviaoesl so 8i, kr at Ot 2. 4x — 2y = 5b, 4. 3.5” + 3y = 5a, 6x+3y 7b. x+y = 3a. 3. ax + 3y = 10a, 5. ox — 4y = lla, 4x — 2y = 6a. x + 3y = 6a. 6. 8x + Sy = 4a + 96, x a — 126 ey rnd) 2 a 206 SYSTEMS OF LINEAR EQUATIONS [Caap. XX. 7 se 0 eee 9. ax + by =<, a Cpe . dx + ey =f. bz fw 38. ee Cie om 10. —+-— = 2a, x» 8. ax + by = 0, 2 3 _ 3p xty+tc= bc 11. The base of a triangle is 10 and the altitude is 8. Find the area. 12. The base of a triangle is a and the altitude is h, what is , the area? 13. The base of a triangle is a and the altitude is 10. What is the altitude of a triangle of the same area but with a base a+ 2? 14. The altitude of a triangle is h and the base is a. If the base be increased by b, how much must the altitude be decreased so as to leave the area unchanged? 15. The sum of two numbers is s; their difference is d. Find the two numbers. 16. Two persons, A and B, can complete a certain amount of work in 8 days; they work together 4 days; B finishes it in 5 days. Find the time each would require to do it alone. 17. Two persons, A and B, can complete a certain amount of work in 1 days; they work together m days, when A stops ; B finishes it in n days. Find the time each would require to do it alone. — | 18. Determine 6 and c so that 2? + bx + ¢ is equal to 1 when x = 1 and is equal to 2 when z = 2. 19. Two books cost a dollars. The one cost b dollars more 7 than the other. Find the cost of each. 20. If A gives d dollars to B, they have equal sums. If | B gives e dollars to A, then A has 3 times as much as B. How — much has each? 135. Linear systems in three or more unknowns. ‘To solve a system of three equations, involving three unknowns, one of ae Ss |. Arr. 135] EQUATIONS IN THREE UNKNOWNS 207 ine the unknowns must be eliminated between two pairs of the equa- tions. The problem is then reduced to one of two unknowns. Likewise, to solve a system of four equations, involving four unknowns, one unknown must be selected for elimination, and it must be eliminated from three pairs of the equations. We have then three equations with three unknowns and proceed as above to reduce them to two equations with two unknowns. Example : | Solve 22 + 4y + 32 = 8, (1) x—-d5y+z2 = -4, (2) 3x2 — 10y + 5z = -3. (3) Soturion: Eliminate one unknown, say z, between (1) and (2), thus: 2) > 3, 3a — 1l5y + 32 = —12, (4) (1), 2x + 4y + 32 = 8, (5) (4) — (5), x —19y = —20. (6) Now eliminate z from (2) and (3) as follows: (2).- 5, 5a — 25y + 52 = —20, (7) (3), 3a — 10y + 52 = -3, (8) (7) — (8), 2a — ld5y = —-17. (9) The equations (6) and (9) contain only the unknowns 2 and y. (6) ~ 2, 27 — 38y = —40, (10) (9), 27 — 15y = —17, (11) (10) -— (11), — 23y = —23, (12) (12) + —23, y =1. (13) _ Substituting 1 for y in (9), 27 —15 = -17, - (14) Solving (14), z= -l. (15) Substituting z = -1, y =1 in (1), we get a+ 4 4 oe = 8, (16) or 9) Ba'= 6. (17) (17) +3, 2 = 2. ; (18) Cueck: -2+4+6=8, or 8 =8; -1-54+2=-4, or —4 = -4; Seeetio +10 = —3, or | -3.= -3. 208 SYSTEMS OF LINEAR EQUATIONS [Cuap. XX. By EXERCISES AND PROBLEMS 1. ty ee ee 5 1, 2= ee 3x + 3y — 2z = 60, Te 2 4 10x — 5y — 32 = 0. 5 8 See 2. 8x2 — Sy — 22 = 14, 2 ox — 8y — z = 12, 7 4 8 c— 3y — 32% 1. ier 3. ¢+y+2=6, ae 27 —-y—2= -8, Hint: Solve first for ie x + 2y + 32 = 14. 4. 4x —2y +2 = —6, 6 r+yt+z=4a, xt+y+z=0, 3x — l2y — z = 2a, 2x — 3y + 32 = 2. ox + 3y —zZ=4. 7 crc+y+e2e+uw=d0O0, 8 r+yt+2+w =, x+2y—z2+3w =0, x + 2y + 32 + 4w = 17, 2x +y —-22+w =4. 4x + 3y + 22+ w =8, 2x + 4y —2+ 3w = 5, x — 2y+ 32 — 2w =83. 9. Find three numbers such that the sums formed by tak- ing them in pairs are 30, 40 and 50. 10. The sum of three numbers is 76. The sum of the first and second is 4 greater than the third number, and the differ- ence of the first and second is one-third of the third number. } Find the numbers. 11. The perimeter of a triangle is 74. The sum of two — sides is greater by 10 than the third side, and the difference of — the same two sides is 10 less than the third side. Find the sides of the triangle. = 12. Divide 1000 into three parts, such that the sum of the first, 3 of the second, and /5 of the third shall be 400; and the sum of the second, § of the first, and 75 of the third shall be 450. — 13. Three cities, connected by straight roads, are at the — vertices of a triangle. From A to B by way of C is 112 miles; _ ~ Arr. 135] EXERCISES AND PROBLEMS 209 from B to C by way of A is 116 miles ; from C to A by way of B is 104 miles. How far apart are the cities? 14. Separate 400 into 4 parts such that if the first part be increased by 9, the second diminished by 9, the third multiplied by 9, and the fourth divided by 9, the results will all be equal. 15. In a race of 500 yards, A can beat B by 20 yards, and C by 30 yards. By how many yards can B beat C? 16. Between two towns the road is level one-half of the dis- _ tance, and the speeds of a motor car are 9, 20 and 18 miles per hour up hill, on the level, and down hill. It takes 54 hours to go and 6% to return. What are the lengths of the level and inclined parts of the road? MISCELLANEOUS EXERCISES AND PROBLEMS Solve the following equations for x and y when the solution is possible. Show by means of a graph why the solution is impossible in Exercises 4, 6, 8, and 15. 1. 6x — 3y = 15, 6. Se nN Mea 2x + 7y = 45. 6 5 r+d yt+6 x-1_ (y-3) Dpewiwgagiay oF ae aaa 5) = 4) 2— y= 2, mec tt > 2x — 2y = 10. B ul 8. y + 22 = 7, ame — 9 = 23, 2y + 4x = 4. az + 5y = —13. 9. ax+y=a+2, x+ay = 2a +1. ome Ya 10. ax + y = 2a, 3x + dy = 6 2ax —y =4.. De fe 11. 3x — 4y = 2a, Beata * 4e + 3y = 11a. foes -'5 12. ax — by = 0, Pes... bx — ay = b? — a’. 210 SYSTEMS OF LINEAR EQUATIONS [Cuap. XX. * 13. ax + y =, 15. 2 + 3y = 2, bx +y =a. 32 + 9y = 15.. dame pesd Pas ot 8 2 oo 1d. = ep ee ey 2 nly eieh si 0) db 4 ee ye 17. Find two numbers whose sum is 60, and such that one of them exceeds twice the other by 6. 18. A board 18 feet long is cut into two pieces whose lengths are in the ratio 1 to 3. How long are the pieces? 19. A banker changes $5.00 into dimes and nickels. ‘There ~ are 73 coins in all. How many dimes and how many nickels are there? 20. There are two numbers whose sum is 63. If the greater is divided by the smaller, the quotient is 2 and the remainder 9. What are the numbers? | 21. A board a feet long is cut into two pieces whose lengths are in the ratio b to c. How long are the pieces in terms of a, b, and c? 22. A grocer bought oranges, some at 20 cents a dozen and some at 18 centsa dozen. He paid for all $6.70. He sold them at 25 cents a dozen and cleared $2.05. How many oranges / did he buy at each price? | 23. After an examination a teacher decided to raise each grade from x to y by the formula y = mz + b, where m and 6 are to be determined by the facts that a boy who made 50 is to receive 65 and one who made 60 is to receive 77. Find m and b, also the new grade of a boy who received 75. . 24. A bird flying with the wind makes 60 miles an hour, but when flying against a wind half as strong it makes only _ 45 miles an hour. Find rates of the two winds. Also the rate _ of the bird in still air. | AssuMPTION: It is to be assumed that the rate of the wind should be added to the rate in still air when the bird goes with the wind and sub- % tracted when it goes against the wind. if Reisen ee ie ee _ Art. 135] PROBLEMS 211 25. The report from a pistol travels 1080 feet per second with the wind, and 1040 against the same wind. Find the rate of the wind and the rate of sound in still air. 26. An aéroplane flies with the wind at the rate of 80 miles per hour and against a wind twice as strong at the rate of 50 miles per hour. Find its rate in still air. 27. A certain sum of money is invested at 5 per cent and another at 6 per cent. The annual income from both invest- ments is $98. If the first sum had been invested at 6 per cent and the second at 5 per cent, the income would have been $2 greater. What are the two sums of money? 28. The grocer sold Mrs. Brown 3 quarts of strawberries and 2 quarts of cherries for $0.65. He sold Mrs. Jones 2 quarts of strawberries and 5 quarts of cherries for $0.80. Find the price of a quart of strawberries and of a quart of cherries. 29. The grocer sold Mrs. Brown 8 quarts of strawberries and 2 quarts of cherries for a certain sum of money. He sold Mrs. Jones 1 quart of strawberries and 5 quarts of cherries for the same sum of money. Compare the price of a quart of straw- berries and the price of a quart of cherries. 212 SYSTEMS OF LINEAR EQUATIONS [C#ap. XX. REVIEW EXERCISES AND PROBLEMS 1. Find amean proportional to each of the following pairs of numbers : —3, -12; 8,4; mr, wR; 12%, a? ; 52%, 20y?. 2. Solveforz: ax =b;—=b; a+x=b;2x-a=b. Tellin each case which of the principles 4 Art. 38 is used in solving the equation. 3. Give an example of a linear equation in two unknowns. What is meant by a system of linear equations? Give an example. Define a solution of an equation in two unknowns. How many solutions can be found for 2x — y = 10? When are two linear equations in two unknowns said to be simultaneous? Give an example. When inconsistent? Give an example. 4. What is the locus of an equation of the form az + by =c? How many points must be found to determine the locus? Are the a pairs of equations simultaneous? Do their loci intersect? x+y=4 4 b= 2 z=1 @ {e+ seks Oo ike @{% ag © {9 a 5. What changes of sign can be made in the erases ; Without chang- wr 3 A a ing its value? Answer the same question for = 6. Draw a pair of codrdinate axes, XX’ and YY’, which intersect at O and divide the paper into four parts. Show the part in which a point is located if (a) its codrdinates are both positive ; (b) both negative ; (c) the abscissa is positive and the ordinate negative ; (d) the abscissa is nega- tive and the ordinate positive; (e) both zero; (f) the abscissa is zero ; (g) the ordinate is zero. 7. State an equation giving the relation between # and y if (a) y is twice as great as x; (b) yisk times as great as x; (c)if y varies as 2 (d) What does the last relation become if y= 16 when x = 8? 8. If y varies as the les of xz, and y = 8 when zw = 2, find y when x has the values 0, .01, .1, 4, 2, 1, 2, 3, 10. 9. Make a statement concerning the proportionality of area and one concerning the proportionality of areas of similar triangles. Tlus- trate by figures the meaning of each statement. > #3 ~ 10. The hypotenuse of a right triangle is 5 inches and one side is | } inches. Find the other side. The hypotenuse of a similar triangle is 10 inches. Find the other sides. | : ee } a | ; } Art. 135] REVIEW EXERCISES AND PROBLEMS 213 11. A man 6 feet tall is standing 10 feet from a lamp post which is 15 _ feet high. Find the length of his shadow. 12. A tree casts a shadow 60 feet long when a post 10 feet high casts ° a shadow 8 feet long. How high is the tree? 13. Show that in the proportion a:b = c:d, the product of the means divided by either extreme equals the other extreme ; and that the product of the extremes divided by either mean equals the other mean. Solve the following equations : 14. 2x -— 3x = $4 — 3 -— 40 +2. meets — se 7 1 a'3 82 12 42 16. ax+br =m+2. F 17. (a —2x)(b — 2) = 2’. ox —-5 Se-1l ax-4 een ye 7! 1 19. 5% + 3y + 2 = 0, oz + 2y +1 =0. . 20. 242 = 33y + 4, 2ly = 33a — 47. 21. +1 =2(y +1), y+2=4(2+4+1), 2+3 =2(% +1). 22. Determine 6 and c in the equation y = x? + bx +c, if y=1 when x = —1,and y = 5 whenz = 1. 2, 23. Determine a, b, and c in the equation y = az? + bx +c, if y =1 when x = 0, y = 3 when x = 1, and y = 6 when gz = 2. 24. The volume of a cylinder varies as the square of the radius when its height is constant. When the radius is 1, the volume is 153. Find the volume when the radius is 6. CHAPTER XXI SQUARE ROOT AND APPLICATIONS 136. Definition of a square root. A square root of a num- ber is one of its two equal factors. Thus, 2 is a square root of 4, since 2-2 = 4, and 2a is a square root of 4a?, since 2a : 2a = 4a’. Since a? = (—a)? = —a- —a, it follows that every square has two square roots, differing only in sign. Thus, —2 and +2 are both square roots of 4. 137. Radical sign. The radical sign ~/ —_ is _ used to indicate the positive square root of the number under it. When a negative root is to be taken, the radical sign is preceded by the sign —. Thus, ++/4 or./4 means 2, and —/4 means —2. EXERCISES Find the following roots by inspection : ih, cy eli} 5. +/49. 9. 81. 2. —/64. 6. /a°b?. 10. —+/225; 3, —/al. 7. /100. 11. —+/169.m 4. 4/121. 8. —4+/144. 12. +4/2R6 138. Square root of monomials. The square root of a product may be found by finding the square root of each of its factors, and then taking the product of these roots. The type form may be written | Vab = Va‘ Vb. q 214 A Arts. 138, 139] SQUARE ROOTS 215 This principle is of much value in finding the square root of a monomial, if it consists of factors each of which is a square. In fact, we may find the square root by simply dividing the exponent of each factor by 2. Thus, \/225 =/9-25 = /#8 - & =3-5 =15, and / Marys = /9 /B/P = Ba3y'. EXERCISES Find expressions equal to each of the following and free of radicals : 1. /144. 8. —+/36rty%2®. 15. \/64r oy? 9. 4/256. 9. /8latt® 16. +/64a%®. gee 4/025. 10. 1/490 17. . v/xtysz!0, Wee s/ 1225. 11. —V/2y"_ 18. —+/25- 36zty®. 5. -V/Oaty’. 12, fa 19. /64° 8127! 6. —+/16a°d®. 13. ~/3%r8y4, 20. —/amb?c!. Tee oe at. 14. —V/4exef 21. 22527. 139. Equations solved by finding square roots. Since +2 and —2 have the same square, 4, the equation , y= 4 is satisfied by both +2 and —2. That is, ih £V4 = +2, are two solutions of the equation x? = 4. EXERCISES Solve the following equations : to =.20. bg Dike goer = 121. 6. a? = 81a‘d§. 3. 2? — 144 = 0. 6. x? = 42a‘b?. 216 SQUARE ROOT AND APPLICATIONS [Cuap. XXI. 1 Tate 12. x? — 64a7b® = 0. 5.8 a= 0r Us 13. x? = 16(a — b)?. 9. a? — 225a‘b8 = 0. 14. 2? = 49a%D°, 10) 7 = 3ba°O: 16.2" = Sige 11. x? = 100(a + D)?. 16, =a 140. Square roots of trinomials. If a trinomial is a perfect square, its root is easily extracted by comparison with the | familiar formula a? + 2ab + b? = (a + b)?, at b. from which a/a? + 2ab + 6b? Thus, to find 4/92? + 1l2zy + 4y’, we may make the comparison by putting /92? = 3x =a, and /4y? = 2y = b. Since 12ry = 2ab, we have /9x? + 12ry + 4y? = 8x + 2y. EXERCISES Extract the square root of the following by inspection : 1. 4a? + 12ab + 967. 4. 42° + 4zy + y?. 2. 16x? + 247 + 9. §. 477+ 47 + 1. 3. c@ — 4ac + 4a’. 6. 9m? — 6mx + 2. 141. Process of finding the square root. Given that a + 6 is a square root of a? + 2ab + b?, it is well to follow a certain process by which a + 6 may be obtained from a? + 2ab + 6?. PROCESS a +2ab +0? | a+b a? Trial divisor = 2a Zab + Bb? Complete divisor = 2a +b 2ab +b? . } i } | Arr. 141] SQUARE ROOT OF POLYNOMIALS 217 To follow the process indicated, let us note that the first term, a, of the root may be obtained by taking the square root of a certain term, a”, of the given expression. If a2 is subtracted from the given expression, the remainder is 2ab + b. The second term, b, of the root may be found by dividing a certain term of the remainder by 2a, which is twice the part of the root already found. | On this account, twice the root already found is called the trial divisor. iN Since the remainder 2ab + b? = b(2a +b), the complete divisor which multiplied by 6, produces 2ab + 6’, is 2a + 6. The complete divisor is thus found by adding the second term of the root to the trial divisor. Before trying to extract a square root of a polynomial, the terms should be arranged according to ascending or descending powers of some letter. Example 1. Extract the square root of 162? — 24xy + Sy? by following the process just explained. | PROCESS : 162? — 24ry + 9y* | 4a — dy 162? Trial divisor = 8x —24xy + Oy’ Complete divisor = 8x — 3y —24zry + 9y? Example 2. Extract the square root of 1622 — 24ry + Oy? + 16az — 12yz + 42’. (1) In squaring 4% — 3y + 2z, we may treat 4x — 3y as a single term, and write { (4% — 3y) + Qz}2 = (4x — By)? + 42(4e — 8y) +42’. (2) Hence, in extracting the square root of (1), we may find first the square root of the first three terms as in Example 1. Then to find the next term, 2z, it is seen from (2) that we should use 2(4x% — 3y) as a trial divisor. That is, twice the part of the root already found should be used as a trial divisor. 218 SQUARE ROOT AND APPLICATIONS [Cuap. XXI. PROCESS 16x? — 24xry + Oy? + 16xz — 12yz + 42? [4a — By 4 22 162? Trial divisor = Sr —24ary + 9y? Complete divisor = 8r — 3y —24ry + 97? Trial divisor = 8x — by 16xz — 12yz + 42? Complete divisor = 8x — by + 2z l6rz — 12yz + 42 Explanation. We first proceed as when the root is a binomial, and find, for the first two terms in the root, the expression 4a — 3y. \ y Take twice this root already found as a trial divisor. Hence, the trial divisor is 8a — 6y, and the next term of the root is 2z. Adding this to the trial divisor, we obtain 8x — 6y + 2z for a complete divisor. Multiplying this by 2z, and subtracting the result as shown, we have no remainder. In case the root contains more than three terms, the process indicated is continued. EXERCISES Extract the square roots of the following and verify : 25a? + 30a + 9. 4. Quy? + 12xyz + 42. 494 — 282? + 4. 5. 9a*t — 4223 + 4927. Ox? + 80ry + 25y?. 6. 25y? — 40y + 16. at — 2a?b + 2a®c? — 2bc? + b? + c*. 9a? + 256? + 9c? — 30ab + 18ac — 30be. 4y*y? + 12a*y + 92? — 30ry? — 20xy? + 25y4. 10. 1 + 2% 4+ 7a? + 623 + Oat. 11. 9a* — 12a3y + 3407y? — 20ry? + 25y4. 12. x* — 10a? + 212? + 20% + 4. ie ete 2 2216 14. 4x4 — 1223 — 72? + 242 + 16. SO eam Poa: c. 13; 24) 223.4 / Anrs. 141, 142] SQUARE ROOTS 219 15. 9x? — 302 + 4 ~ i desi 16. 252? + 30ry? + 9y* — 2.5x2 — 1.5y?z + 0.06252". dara? Qax® x4 asd 30 8 at 4x3 xy 2 Y ae om 3 gs oo 18. 9 g +3 +4 ry + 7 19.. x — 40° + 5a? — 27 + 5. | s 4 3 “ee 1 f 20. 92* — 122 +42? —-- + = + 6, zZ a at 2 lyf 4 + a®xe.+ 21. =+% z ue 22. a® — 6a* + 15a* — 20a? + 15a? — 6a + 1. He 98. 1 + Qr + 3a? + 423 + 5at + 405 + 3x8 + 227 + 2. 24, 1624 — 2473 + 25a? — 122 + 4. 25. x" + Qarry” + y". ih 142. Square roots of numbers expressed in Arabic figures. | The positive square root of 100 is 10; of 10,000 is 100; of 1,000,000 is 1000; and so on. Hence, the square root of a number between 1 and 100 is between 1 and 10 ; the square root of a number between 100 and 10,000 is between 10 and 100; the | square root of a number between 10,000 and 1,000,000 is between 100 and 1000; andsoon. That is to say, the integral part of the : square root of a number of one or two figures, contains one figure; of a number of three or four figures, contains two figures; of a number of five or six figures, contains three figures ; and so on. Hence, if an integral number is separated into periods of two figures each, beginning at the right, its square root has as many — digits as the number has periods. 220 SQUARE ROOT AND APPLICATIONS [Cuap. XXI. + 143. Explanation of process of finding square root in arith- metic. ‘The process of finding square roots of numbers of arithmetic may now be explained by the process just used for polynomials. _ Example. Find the square root of 576. SoLtutTion: Separating into periods of two figures each as indicated in Art. 142, we have 5’76. There are thus two figures in the integral part of the root. To explain the solution algebraically, we may use the formula : (¢+u)? = + 2tu+w = + (2 + u)u, ct in which ¢ is the number represented by tens’ digit, and wu is the number represented by units’ digit. The processes in the columns at the right and left below are alike if t = 20 and u = 4. O4+2iu+wvilti+u 5’76 | 20 +4 = 24 e 4 00 Trial divisor = 2¢ | oe 2t = 40 | 176 Complete divisor , =2i+u|l 2tu+u 2t+u =44| 176 144. Numbers with more than two periods. The method © just explained applies to numbers that separate into more than two periods of two figures each. In making the application, we consider ¢ in the typical form — t* + 2tu + wu? to represent at each step the part of the root already found. Example. Find the square root of 44944. 4’49’44 | 200 + 10 + 2 = 212 4 00 00 ; 2t = 400 4944 | 2 +4 = 410 4100 1 2t = 420 844 2t +u = 422 844 ‘? Arts. 144, 145] . SQUARE ROOTS OF DECIMALS 221 Omitting zeros, we may condense the process so as to appear as follows: ; 4'49/44 | 212 4 4 49 41 41 145. Square roots of decimals. Decimals are separated into periods of two figures each from the decimal point towards the right; for, if the square root of a number has decimal places, the number itself has twice as many decimal places. } Example. Find the square root of 5220.0625. 52’20.’06’25 | 72.25 49 When a number is not a perfect square, we may annex periods of zeros and continue the process of root extraction. EXERCISES Find the square root of each of the following : 1. 3025. 9. 599076. 2. 508369. 10. 87.4225. 3. 930.25. - 11. 0.015625. 4. 96.4324. 12. 0.00321489. 5. .000625. 13. 75570.01. 6. 2985984. 14. 20880.25. ; 7. 65536. 15. 1849. ! 8. 13107.9601. 16. 250500.25. 222 SQUARE ROOT AND APPLICATIONS [Cuap. XXI. 25 144 2809 17. 10.9561. 19. 36 21. 576. 23. 5184 225 289 361 18. 0.001225. 20. 956. Pe 394 24. 1600 ® 25. To illustrate square root by a diagram, let us think — of a square board that contains 529 square inches (Fig. 30), and try to find the number of inches in a side. What is multiple of 10 inches that can be cut from the board? When 7’ is found, what is the length of one of the rect- angles U? How many square inches of the 529 are left after T is removed? Neglecting U’ as small compared to the U’s, how can the width of U be found from the area and total length 2 x 20 = 40 of the 2U’s? What is 2 x 20 called in the process of finding the square root? If a side of U’ be added to the total length of the two U’s, what is the total length of the 2 U’s and U’? What is this number called in the process of finding square roots? 146. Approximate square root. Example. [ind the square root of 2, correct to two decimal places. 2.’00’00’00 | 1.414 011900 2.824] .011296 Norte. — To get the root correct to two decimal places, it is necessary the largest square 7’ whose side is a to find the figure in the third decimal place, and to note whether it is : larger or smaller than 5. In the present case, it is smaller than 5, and 1.41 is the approximate value sought. That is to say, we mean by “cor- é rect to two decimal places,” the same as ‘correct to the nearest hundredth.” — 4 4 Arr. 146] APPLICATIONS 223 | EXERCISES Find square roots of the following, correct to two decimal places : ‘ies 6. 24. tier: 16. 200.02. Be D. i 18. Loe, 0.02: L7eeo: aa .6. S27. 13. 0.002. 18, 2. 4, 8. 9. 2.92. 14. 0.005. 19. 2. 5. 10. 10. 0.305. 15. 0.307. SS. ee ee 20. Show that \/4 +9 = V4+4 V9. * APPLICATION TO PROBLEMS FROM MENSURATION Nore. Unless otherwise stated, find approximate values in the follow- ing problems, correct to two decimal places. - Using the letters as shown in Fig. 31, supply missing values - of letters in the following : gee 0 eS, c= 7 3. a = 20, b = 300, c=? Geo 1, c= 2,b=7 4. a a3). = 20), C=! 5. A boy having lodged his kite in the top of a tree, finds that by letting out the whole length of his line, which he knows is 225 feet, it will reach the ground 180 feet from the foot of the tree. What is the height of the tree? MK 6. A tree 80 feet high was broken off by a storm, the top striking the ground 40 feet from b the foot of the tree, and the broken tebe end resting on the stump. Assum- Fia. 31 ing the ground to bea horizontal plane, what is the height of the part standing? 7. Two vessels start from the same point, one sailing due northeast at the rate of 15 miles an hour, and the other due * The sign ~ is read “ is not equal to.” 224 SQUARE ROOT AND APPLICATIONS [Cuap. XXTI. southeast at the rate of 18 miles an hour. How far are they apart at the end of 24 hours, supposing the surface of <—_—___—. 60 —_———» _ the earth to be a plane? 8. Find to thenearest tenth of area 1200 square feet. 9. Find the diagonal (line hee BD of Fig. 32) of a rectangle B C of sides 60 feet and 30 feet. Mee 10. Find the diagonal of a square of sides 50 feet. 11. The diagonal of a square is 24 feet, what is the length of its sides? 12. The dotted line in Fig. 33 indicates a path across a field. How many rods aresaved by taking the path instead of following the road? 13. Find approximately (to the nearest tenth of a rod) the sides of a square having an area equal to that of a rectangle whose sides are 40 rods and 50 rods. 14. Find the sides of a square having an area equal to that of a triangle of base 70 feet and altitude 35 feet. Facts from Geometry: (1) The area of a circle is mr?, where 7 = 3.1416, and r is the radius of the circle. (2) The surface of a sphere is Amr’, where r is the radius. (3) The convex surface of a cylinder is 2mrh, where r is the to “a radius of base and h the altitude. Road 15. Kind the radius of a circle whose area is (a) 50.2656 — square feet; (b) 1000 square feet. 16. A hase is to be tied to a post in a pasture by means of a rope just long enough so that he can graze over 3 acre. ¢ of a foot the sides of a square © & | a. 7 » Fi » Arr. 146] PROBLEMS 225 ~ How many feet long should the rope be if an allowance of ' three feet is made for tying? (One acre = 48,560 square feet.) 17. A given circle has a radius of 4 feet. What is the radius of a circle whose area is twice that of the given circle? 18. A bowl 8 inches in diameter in the form of a hemi- sphere is made by pressing from a circular piece of brass. What is the diameter of the piece of brass required to make the bowl? AssumPTION: In Problems 18 and 19, it is assumed that the areas are the same before and after pressing. 19. A shoe-blacking box (without lid) 4 inches in diameter and 3 inch deep is pressed from a circular piece of tin. Find the diameter of the piece of tin required. CHAPTER XXII RADICALS 14%. Radical. An indicated root is called a radical expres- sion, or simply a radical. Thus, \/a is a radical expression or more briefly a radical. The radical sign is used to indicate other roots than square roots by means of a figure called the index of the root. Thus, in V27, the number 3 is the index and shows that the third or cube root is indicated. In general, the nth root of a number a, one of its n equal factors, is written ~/a. 148. Rational and irrational numbers. A rational number is an integer or the quotient of two integers. Thus, 2, 2, 3.333 are rational numbers. Exercise. Express 3.333 as the quotient of two integers. It is found useful to extend our number system to contain numbers that are not rational numbers. For example, if we attempt to find the side of a square whose area is 2, we may write the result as 1/2, but this is not a rational* number. Stated in another form, we have a solution for the equation x? = 2 only when we extend our number system to include more than what we have defined as rational numbers. Any number which is not a rational number is called an irrational number. Thus, V/2, V5, /10, ¥/7 are irrational numbers. 149. Surds. A surd is an irrational number that is a root of a rational number. Thus, v/2, /3, \/5 are surds, but »/4 * See Rietz and Crathorne’s College Algebra, p. 18. 226 } ‘ i t | iat ny, A ¥ Arts. 150, 151] SIMPLIFICATION OF RADICALS 227 is not a surd since 1/4 = 2, and 1/2 + +/3 is not a surd since 2+ +3 is not aa rational number. Surds that express square roots only are called quadratic surds. 150. Square root of a fraction e Since a fraction is squared by squaring the numerator and denominator separately, it follows that the square root of a fraction is given by extracting the square root of the numerator and denominator separately. In symbols, ai Vi-% 6b y/o 151. Simplification of radicals. The form of a radical expression may often be changed to advantage without chang- ing its value. For example, V8 = V/4-2 = 2/2; and if we know that /2 = 1.4144, we have /8 = 2(1.414+) = 2.8284. Further, CMe ee and the latter is easier to compute to a given number of decimal places than the former. To continue with illustrations, let it be required to express \ 3 2PPTOX- - imately as a decimal fraction. Three plans suggest themselves, for we _ may write 2 _ v2 V2 = v066064 are a a aor REN ea EO NG 1) N 353: ¢ 33 VJ/2_ 1.4144 | Li By the first plan, Oe TRAD S 0.8165 — . By the second plan, 1/0.6666+4 = 0.8165 —. By the third plan, ve = ae = 0.8165 —. The first plan clearly involves more labor in computation than either the second or the third plan. 225) RADICALS (Cuar, XXII EXERCISES Find, correct to two decimal places, the following square roots; do Exercises 1 and 5 by three plans: 1, 4/3: 4, /8. 2.

6. Vt. 8. /3y- 11. Vz: 3. Vt 6. /3- 9 12 a ° 5 ° 8 . 4/5 e /7 13. The area of a garden plot in the form of a square is 1° square rods. Find the length of a side to two decimal places. 14. A board in the form of a square contains 4,4 square feet. Find the length of a side correct to two decimal places. 152. Meaning of simplification of a radical. The expres- sion under a radical sign is often called a radicand. Expressions involving square roots are, in general, said to be in simplest, form when they are reduced so that the radicand (1) is in in- tegral form, (2) contains no factor which is a perfect square, and (3) contains no radical in the denominator. For example, | VJ/27 = /9°3 = 3, Vite = Velney = ey Vay and VE = VE3 = V3. The right-hand forms of expression are simplest for purposes of computation. This fact will be appreciated if we compute the values of the different expressions to a certain number of — decimal places. ? Arts. 152, 153] EXERCISES WITH RADICALS 229 EXERCISES Given 1/2 = 1.4144, V3 = 1.732+, compute the following, correct to two places of decimals: Pw Aris: Bar a2: 5. »/108 + V5. an 1 se a 2. /27. 4, et 6. /75 + V3. Simplify the following : 7 +/50. 11. ./1 — (4)? 15. +/200a°0?. ne ce 8. 1/20a2b. 12. ./2 + (#)?. 16. / ie (F) 9. 3/5. 13. 4/1227; Le vise 10. 5/2. 14. /63zy%2?. 18. 3,/2. Put each of the following in a form without a number written outside the radical sign: ALO ee Sotution: In this case, 2.\/3 = 4: /3 = V12. F oe ae 4 DS ak 20. 5/3. 24. 32~/2" — y’. 27. y V3 Zt x ee 3 21. 24/3. 20% >= * 28. 3/4. EV ea ivV% x /y ee ee - -/= 26. =~ 2. 29. —v/7°. 22 =e 6. SV 9. Vi 23. 3q./2p- 153. Addition and subtraction of radicals. When two radical expressions, such as 3./a and 5,/a, have the same expression for a radicand or can be given the same radicand by simplification, they are said to be similar radicals. Similar 230 RADICALS [Cuap. XXII. radicals can be combined into one term by additions and subtractions. Thus, V/8+ /32 = 2,/2 + 4,72 =(2 + 4)v2 = 6/2. Again, /4a + /9a + /l6a = 2V/a + 3a +4Va = 9V/a. EXERCISES Combine the following and simplify as far as possible with- out approximating roots : V18 + V50 + V2. n/ Loe wy ieee Toe /20 + »/45 — +/80. V/128 — V18 + V72. /108 — 147 + V/75. /700 — »/28 + +/63. /32a — /8a? + +/18a?. /32a — V/8a + V/18a. Va + 4a — /166 + 90. /d.— 0 +'./4a = 4b x/0a Ob . V(a — b)?m + V/9(a — b)*m — V/4(a — b)?m. . V3202b? + 1/12802b? + »/162026°. . V27 + V108 + V/45 + 80. V5 + -V125 + V9 4+ V 18. . W522 — 50x + 125 + ~/5a? — 10x + 5. OAV nk a VA PA EN Oy . 2/8 + 40/60 + V/15. EN AN BEN OG See Coa este sen ae ea Oa Ee en a Ss Se — = = ONDA FF WOW DY FH SO © Anns. 153, 154] QUADRATIC SURDS 231 19. ~/4ab? + bv/25a + Va(a — 3b)?. 20. 5x/300a3b?2 — 3+/243ab?. 21. Find the sum of the lengths of diagonals of three squares of sides 10 feet, 20 feet and 30 feet. 154. Multiplication of quadratic surds. To multiply one quadratic surd ~/a by another +~/b, we use the principle, Art. 138, : hs. Re bit iab: That is to say: The product of the square roots of two num- bers 1s the square root of their product. Example 1. Find the product of »/3 and ~/27. SoLuTION: +/3- 1/27 = +/3-27 = V/81 = 9. Example 2. Multiply 2\/3 + 3/5 by V5 - V3. SoLuTION: B2Ao 3/5 Roe Ne 2/15 +15 34/15 \— 6 ~V/15 + 9 Example 8. Multiply \/a +a —6 by Va - Va —b. SoLuTION: /a+Va—b N/a N/a. 0 a + +a? —ab — v/a? — ab — (a — bd) a-a+b Hence, the product is b. EXERCISES Perform indicated operations and simplify results : fas V7. Lig PUAN EA 2. /8-/12. 5. Va. V 2%. 3. v/a - Vda. BEV se’. 232 RADICALS [Cuar. XXL. TV 5-V 4. 9. Vrsit3 - ~/ rst. 8. 1/2. 738. 10.--V xyz - / EI a 11, 5 (34/2)= AV BBA Az 81 12. (5V3 + V7)(5V3 — V7). 13. (20/5 + 8/3) (4/5 = 5+/3). 14. (Va + V/b)(Va — Vb). 15. (3V/a + 2V/b)(5V/a — 6y/b). 16. (VW2+ V3 + V5)(VW2 - V3). 17. (R- SV2)(R + *y/2). 18. Find the value of 2? if = 1/5 + V/10. 19. Find the value of x? + 5z — 1 when x = —5 +4 Vv 29° 2 Slee 3 21. Find the value of 2? + 6a — 4 for x = a mS: 22. Find the value of 32? + 32 — 5 when x = ; + us 23. Does 1/3 + ~/2 satisfy the equation 2? — 4% + 1 = 0? 24. Does —3 + V14 satisfy the equation x? — 6x — 5 = 0? 155. Division of quadratic surds. In dividing one radical expression by another, the division is usually indicated by using the dividend as the numerator and the divisor as the denominator of a fraction. 20. Find the value of 32? + 27 — 2 when z = ye Thus, to divide ~/a by ~/, we write VE We may also make use of the following principle: The quotient of the square roots of two numbers is the square root of the quotrent of the numbers. _ | That is, Ba Wie V/b Arts. 155, 156] RATIONALIZATION 233 EXERCISES Express each of the following as a fraction under one radical sign and reduce to simplest form: v2. AGM A Neon * V8 ane 2 ey. v5 V10 "V5 " 4/20 " 4/30 156. Rationalization of denominators. While we may Li ' thus indicate (Art. 155) the quotient of two quadratic surds in two ways, it is frequently necessary to go further and present the result in a simpler form or find an approximate value of the quotient. Thus, it may be necessary to find the values yp se fea all to mae V5 - V3 three places of decimals. To find the approximate value of Ye three methods are shown in Art. 151. It is a saving of work to multiply the numerator and denom- inator by the factor 4/3 so as to make the denominator rational. Pay bs V5 - V3 square roots of 5 and 3, then subtracting the square root of 3 from that of 5 and performing the division, involves three rather long operations. The labor of two of these can be avoided by multiplying the numerator and denominator by V/5 + V/ 3. Thus, V5 V5 4 V8 5+ VIB _ 5+ V15 To find the approximate value of by first extracting the eens Vorye 6-38 2 Finding the value of sa ee wee involves but one long operation. The process just explained is called rationalizing the denomi- nator. 234 157%. Rationalizing factor. RADICALS [Cuap. XXII. The factor by which the terms of a fraction are multiplied to make a denominator rational is called a rationalizing factor. If the denominator is of the form +/a, then clearly ~V/a is a rationalizing factor. If the denominator is ~/a + VW b, then Va — Vb is a ra- tionalizing factor. EXERCISES Find equivalent expressions with rational denominators: 1. Vig BVB42v8 V5 Rew cyen«: Wee biel 4/20 AY Die Va 5 es cae Vb "4/5 — V3 Wala Richey a/3b2 4/3 — V2 V8+V2 4) 8+V5. WN tea vat 2/15 - 6 V5 2N/2 eee V5 EV/2 5 +2V2 4 — 24/2 re Vine See . BV/5 — 2/3 ik 12. 13. 14. 15. Find the value of the following to four places of decimals, avoiding as much lengthy computation as possible : 9 V8+V5. 1 i Af Da V7 a Re. 18. 8V3 + 2V2 V3- V2 6a 20. 5 +272 42070 1 ——] when « = V2. f > Arr. 158] EQUATIONS INVOLVING RADICALS 235 22. If a regular decagon is inscribed in a circle of radius 7, one side of the decagon is (V5 —1). Find the ratio of the ra- dius to one side (correct to three decimal places). 158. Solution of equations involving radicals. Equations involving radicals may sometimes be conveniently solved by squaring each member. This operation is justified on the ground that each member is multiplied by the same number. Example 1. Solve Vx —1 =5. SoLuTion: Square both members and we have, x —1 = 25. Hence, x = 26. CHECK: V/26 -—1 =5. Example 2. Solve Vz -1+ /x—4 =2. (1) SoLution: Transpose one of the radicals, say ~a — 1, then We aaa 2 — Vx —, (2) Square both members, eet A Aas = Pee = 1s iG) Simplify, Av/az — l= 7. (4) Square both members, 16(2 — 1) = 49. a 42/45) Hence, 16x = 65, a = 4y5. (6) CHECK: /4}, —-4 =2- A/a Ls or + —=2 -—1} ee paale ' Note that if radicals are involved, it is well to get a radical alone on one side of the equation before squaring. It is spe- cially important that all results be checked by substitution when the members are squared in the course of the solution. This is necessary for the reason that solutions may be intro- 236 RADICALS : [Cuap. XXII. duced by the operation of squaring both members which do not satisfy the original equation. Thus, given x = 5, (1) let us square both members. | This gives 8 25, (2) Extracting the square root of both members, aim But x = —5 does not satisfy equation (1). EXERCISES Solve and check results: 1\/ 2210 2. Wx+10 = VW 22 — 6. 38. 24-5 = V2 — 4¢ + 23. 4 — Ve— 74 4+4=V2+2— 90, 6 Ve t+1+ V2 —6 =7. 6 VWe+9=14+V2—5. 7 VePX+_7+2=9. 8 6 — Va = Vx +10. 9 Vy 420 —/7 a fo 10. V2+543 = 8-4/9 ct Vid A yen Bobs Vata=Va2 +a. 13. \/ 2e = 8a 44/22 = 8a7u 14. V/4a+¢ =2Vb+2e4+V-2. © Arr. 158] EXERCISES AND PROBLEMS 237 Vat4 x+4 v/x + 20 z+ 20 V2e+1 V2e+10 CE Wee J ie Sara. 17. V/0+-2-— 716 42 = 0. 18. VWr4+25=14+ v2. 19. The time ¢ in seconds for the complete vibration of a simple pendulum is given by 15. ae} ea ™ V 39.9" where | is the length (in feet) of the pendulum, and 7 = 3.1416. Solve the equation for / in terms of ¢ and find the length of a _ pendulum that makes a complete vibration in two seconds. 20. The velocity of a falling body starting from rest is given by v = ~/ 2gs, where s is the distance passed over and g = 32.2. Express sin terms of v and g, and find the distance s that corresponds to a velocity of 128.8 feet per second. CHAPTER XXIII QUADRATIC EQUATIONS 159. Quadratic equations solved by factoring. In Art. 84, we have solved by factoring some special equations of the form ax? + bu +c = 0, and have learned to call such equations quadratic equations. Take, for example, the equation 2a? + 5a —3 =0. The factors of the left hand member are easily found. They are 2x -1 and « + 3, and we may write our equation in the form \ (2x —1)(x@ +3) =0. Any value of x that makes either factor zero satisfies the equation. If x = 3, we have , (2-2 -1)@ +3) =0-( +3) =0. Again, if « = —3 we have [2(-3) -1](-3 +8) = [2(-8)-1]-0 =0. Hence, ¢ and ~3 are solutions or roots of the given quadratic equation. EXERCISES Solve the following equations: 1. 2? — 7a +6 =0. 7. 22 +4 = 3 = 2 2 BS 5 =U, 8. 3a? — 2x — 8 = 0. 3. 6a? — 1387 + 6 = 0. 9. 67? + 352 — 6 = 0. 4, 3x? — 5x = 0. 10. 377+ 74 +2 =0. 5. 2y? — y — 36 = 0. 11. 8? + 2-1 =0. 6. 7s? + 10s — 8 = 0. 12. 227 — 7x — 15 = 0, 238 “Ants. 159, 160, 161] COMPLETING THE SQUARE 239 13. 8? —10¢-—3 = 0. 17. 2y? —7y +3 =0. 14. 2? — 52 = 52 — 25. 18. 9 — 5s? — 12s = 0. 15. 2? — 227 — 48 = 0. 195 2572 — G0 307 16. 2? — 207 + 51 = 0. 20. 277 — x = 3. 160. Completing the square. From b\* b\? (s+5)-e+b2+(5), 2 we note that the last term (5) is the square of half the co- 2 efficient of x. Hence, the expression x? + bx 2 lacks only the term (5) of being the square of « + z Therefore, if the square of half the coefficient of x be added to an expression of the form x? + bx, the result is the square of a binomial. Such an addition is usually spoken of as completing the square. EXERCISES Complete the square in each of the following : 1. 2? + 42. 8. 2? 4 122. 5. x? + 9a. oe 2 82. 4, 2? + 32. 6. 24+. 7. «x? — 2az. Hint: Make b = — 2a in the above discussion. 8. x? — 4a. 11. (27)? + 4(2x). 14. 162? + 8z. 9. x? — 32. 12. 4a? + 82. 15. 92? — 122. 10. (2x)? + 2(2x). 138. 162? — 2(47). 16. 92? + 92. 161. Equations solved by completing the square. Equa- tions of the form az? + be +c¢ =0 240 QUADRATIC EQUATIONS [Cuap. XXIII. | may be solved by a process that involves completing the square on ax? + ba. Example 1. Solve the equation x? — 6x — 7 = 0. (1) SoLuTion: ‘Transpose —7, and we have v— tr =7. (2) Add 9 to each member to complete square in left member. This gives xv -—-6r4 +9 =16. (3) Taking the square root of each member, zr-3=+4. Hence, zx =7or —I: Check by substitution in (1). . Example 2. Solve the equation 2y? + 84 +1 = -2e +4. (1) SoLtuTion: Transpose and divide each member by 2, a + x =} (2) Add 73 to each member to complete square in left member. This gives m+ 30 +43 = $3: (3) Taking square roots, zr+%=+F: (4) Hence, x = ¥, (5) and also r= -—3 (6) Check by substitution in (1). Solve and check : Wage ea 1. 2? — 37 — 2 = 0; 7. x? + 27x + 140 = 0. 2. 2 —67%+4=0. 8. n(n — 1) = 210. S$... 3? — §—4= 0). 9. y® — 10y = 75. 4. x? — 6x = 40. 10. —8 = 2x? + 102. 5. 2? — (x 6 = 0; 11. ¢t(¢+ 4) =7. 6. 32? — 1074+ 3 = 0. 12. (n+ 1)? — 8(n +1) = 16. 162. Solution by Hindu method of completing the square. In case the coefficient of a? in the equation is not unity, as in 2a? — 42a — 7 = 0, both members may be divided by this coefficient to obtain an equation in which the coefficient of 2? is unity, and the equation may be solved as shown in Art. 161. | 7 » Art. 162] HINDU METHOD 241 However, the following method sometimes avoids the 7 introduction of fractions until the last step of the work. Example. Solve the equation 22? — 4x = 7. (1) Multipy each member by 8 (four times the coefficient of 2?). This gives 16x? — 32a = 56. (2) or (4x)? — 8(4x) = 56. _ Complete the square, 16x? — 32% + 16 = 56 + 16 = 72. (3) 4 Extracting square roots, 4x —-4 = +/72 = 4+6vV2. * Hence, z= 1+ anes (4) Check by substitution in (1). } It should be noted that the number 16 added to complete the square is the square of the coefficient of x in the original equation. This method of completing a square on aa? + bx _ after multiplying by 4a is known as the Hindu method. EXERCISES Solve and verify by substitution in each case : 1. 52? — 3a —2 = 0. 16. 3s? + 4s = 95. 2. s* + 2s = 120. 17.. Q2 — 3) = 62 +-1L. 8. x2? + 227 = —120. 18. m(m +4) = 7. 4, 2? —111+ 28 = 0. Lover Jer = 1: 5. 2n(n + 4) = 42. 20. 15a? — 1474 + 3 = 0. 6. 62? — 52 —6 = 0. 21. 2? +67+5=0. ot — OL = 8. - 22. 22° — (= 42. 8. 2m? + 3m = 27. 23. s? +12 = 8s. 8.0.22" + 0.92 = 3.5. 24. 37? +r = 200. | 10. 03x -O7e=1. 26. + if =2. 11. 42? — 192 = 5. 26. (x +1)? — 8(x +1) = 16. : 12. 2s? — 5s = 42. 27. 3(v? — v) = 2v? + 5v 4+ 4. (13. Se — 14t = -8. 28, s = 4542. tat + 27 = 32. 29. 8m — 10 = m?. 15. 18v? + 6v = 4. 80. $x — $07 +2 =0. 242 QUADRATIC EQUATIONS [Caap. XXIII.” 163. Type form of a quadratic equation. The typical form of a quadratic equation is ax? ++bx+c=0, (1) where a, 6, and c do not involve z, and may have any valuce with the one exception that a may not equal zero. Since the result of multiplying the members of an equation in this typical form by any given number is an equation in the typical form, the a, b, and ¢ can be selected in an indefinitely large number of ways. EXERCISES Arrange the following equations in the typical form ax? + bx +c = 0, and indicate the values of a, 6b, and c in the resulting equations: 3 5 4a? — 5 + 50 = Sn? +m. 3 Soutution: By transposing and collecting terms, ae +32 —(5+m) =0, from which a = 4°, b = 3, c = —(5 +m). 2.92 (me bt 8. (2x — 3)? = 67 4+ 4. = 2 2 4. 22+ (ax + 6)? = 0. 10. 1547+ 2 =1 0 5. 3a? + 52 — 7 = x? — 2a. 11. 1 -—2Qy+y* =2. 6. (y — 3)? + 4y = 16. 12. 30? — 7x = a(x + 1). 7. aa? + 2eu = —d. 13. (27 — 3)? — 64 +1) =38: 14. r(v7 +4) = 7. 15. 4m2x? + 3k? — 8mx + 82 —-m+k =0. 164. Quadratic solved by formula. The process of ‘com- pleting the square’’ when applied to the typical quadratic az? +ba+c=0 i i] } f ; | Arr. 164] SOLUTION OF QUADRATIC 243 leads to a very useful formula which may be used to solve all quadratics, and which affords the most convenient method of _ solving many quadratics. To obtain the formula that gives the solution of ax? + br +c = 0, (1) transpose c and divide by a. This gives A b Cc e+ -7 = —-. a a 4 Add (5) to both members to make the left-hand member a perfect square. Then b b2 Cc b? b? — 4ac 2 = ae wes ae ae a7 4q? a 4a? 4q2 ? ( 6b } _b- 4ac a a ATi antag ye . ie Extract the square root, x + a = + Vb = dao . 2a 2a —b+ 4/ b? — 4ac or 4 (eee ee 2a Therefore the roots of the general quadratic equation, : az? + br +c = 0, are — b+ a/b? — 4ac 2a and —b — v/ b? — 4ac 2a as may be verified by substitution in equation (1). —Bb+ 4/ b? — 4ac he expression ~ Bae EXD 2a | may therefore be used as a formula for the solution of any quadratic equation. [Cuap. XXUI._ 244 QUADRATIC EQUATIONS Thus, to solve 27? —4¢ —7 =0, we substitute in the formula, a = 2, b = —4, ¢ = —7, and obtain path Vib = 432m 4 are the roots of the equation. These values of x are identical with the solutions given in (4), Art. 162. EXERCISES Solve the following equations by the use of the formula, and verify by substitution: 1. 152? — 1474 +3 =0. 16. 227 = 9 — 3z. 2. 6x2 = 192 — 10. 17. x(2¢ +3) +1=0. 3. 72 — 24/382 +2=0. 18. a2 = —4(4 —3). 4. 1627 — 347 + 15 = 0. 19. 4(2x + 5) = 2. 5. (2x-+ 5)? = 452. 20. 4 = «(3x + 2). 6. 2a +7 = = 21. 422 — 32 —2 = 0. 7. 3° +7 = 200. 8. 92? + 32 = 2. 9. 62? + 5” = —1. . 2(22 +3) +1 =0. . 1 sae 2 3 2 = 10. 722 + 2a = 32. 5. ee a4 4 1 PO8E 27 — 2 ee LAE Tt thea eer 16. ie +5 12. 2y? — 5y — 150 = 0. 27. 2s? + 3s = 27. 13. 42? — 17x = —4. 28. (2x — 3)? —- 6(4 +1) +15 = 0. 14. 47 -27 -3 =0. 99. 22+ (5 — 2)? =e 15. 322 + 7x = 110. ~t+V24+6 = 14. Hint: See Art. 158. ‘é Arr. 165] QUADRATIC EQUATIONS 245 165. The special quadratic aw? +¢= 90. When b =0 in the typical quadratic equation, the solution is very simple. Thus, from az* +c = 0, (1) we have Ci t= 6, Y= ay a and X= Ves a Equations of the type of (1) are solved in Art. 139. MISCELLANEOUS EXERCISES AND PROBLEMS Solve the following equations : ia — Liz + 30 = 0. 16. 4s? + 12s+5=0. 2. (x? + 22 = 32. 17: a2? 4-22 4.='(), 3. 2? — 127 = 28. 18. 427 — 287 + 49 = 0. 4, x? — lor = 0. 19. 6— 3x2 — x? = 0. 5. 27 — 25 = 0. 20. 64 + 2? — 162 = 0. . + Deen et A. 6. a? — 4.37 + 3.52 = 0. oh Vp ly ea Lea oe — 0:25 =.0.15. B27 GF — 350 - G =.0. = 8. s* — 123s.+.27 = 0. 23. 72+ 10¢ —8 = 0. 2 pee? Log 24. 20s? + 11s — 3 = 0. 4 3 * 4 ee+2 «24-5 ae 5 tO 2b: Dees ea Se 41, 42? — 282 + 45 = 0. 26. ay eg) t-—3 x 1l+2 2-1 2 “Ss ee = ie pesieaa my eer ie AS 12, 7 +2-@-a=0. 61 -FSap Sar ee 13. 6y? + 35y — 6 = 0. 28. 0.42? — 0.27 — 0.2 = 0. 14. 1522+ 162+ 4 = 0. 20 sept Dba 1 5G al), 15. 67? + 5¢ = —-1. 30. (1 — e”)a? — 2mz + m = 0. 246 QUADRATIC EQUATIONS [Cuar. XXIII. & 31. The sum of two numbers is 30, and their product is. 176. Find the numbers. SOLUTION: Let x = one number. Then 30 — x = the other. Since their product is 176, and 2(30 — x) = 176. Solving the quadratic, x = 8, or 22, and 30 — x = 22, or 8. Hence, the numbers are 8 and 22. 32. Divide 50 into two parts whose product is 600. 33. Divide 8.4 into two parts whose product is 17. 34. Find two consecutive integers whose product is 72. 35. Find two consecutive integers the sum of whose squares is 145. 36. A square field contains 10 acres. What are its dimen- | sions? | 37. A rectangular field is two rods longer than it is wide and it contains 6 acres. What are its dimensions? 38. Some boys are given 3500 square feet of land in rec- tangular form for basket-ball grounds. If the grounds are 20 feet longer than they are wide, what are the dimensions? 39. The dimensions of a picture inside the frame are 14 by : 18 inches. What is the width of the frame if the area of picture with frame included is 320 square inches? 40. A piece of tin in the form of a square is taken to make an open box. The box is made by cutting out a 3-inch square from each corner of the piece of tin and folding up the sides. (Fig. 34). The box thus made contains 192 cubic inches. Find the length of the side of the original piece of tin. 41. A farmer has a 10-acre wheat field in the form of a square. In cutting the wheat, he cuts a strip of uniform Fig. 34 |. Anrs. 165, 166] | QUADRATIC FUNCTIONS 247 " ~ width around the field. Find the width of the strip when one- half of the wheat is cut. 42. A stream flows at the rate of 5 miles an hour; a crew rows 8 miles down the stream and back to the starting point in 4 hours and 40 minutes. What is the rate of the crew in still water? 43. The distances through which a body falls in different periods of time are to each other as the squares of those times. In how many seconds will a body fall 400 feet, the space it falls through in the first second being 16.1 feet? 44. A and B distribute $1200 each among some poor people; A gives to 40 persons more than B, but B gives $5 more to each person than A gives; find the number of persons helped by A and by B. | 45. A rectangle is 15 by 20 inches. How much must be added to the length to increase the diagonal 3 inches? 46. One leg of a right triangle is 6 feet, and the other leg is one-half the sum of the hypotenuse and the given side. Find the sides of the triangle. 47. A rectangular park 56 rods long and 16 rods wide is surrounded by a street of uniform width. This street contains 4 acres. What is the width of the street? — 48. The circumference of the fore wheel of a buggy is 3 feet more than that of the hind wheel. If the fore wheel makes 125 more revolutions than the hind wheel in going a mile, find the circumference of each wheel. 49. What is the area of a square whose diagonal is one foot longer than a side? 50. An automobile round trip of 250 miles was made in 11 hours. On the return portion of the trip, the speed was 4 miles an hour more than on the outgoing portion. Find the rate each way. 166. Graphs of quadratic functions. Any quadratic func- tiion, say 3a?-—5x%-—2, may be represented graphically | 248 QUADRATIC EQUATIONS [Cuap. XXIII. (Fig. 35). The graph may throw considerable light on the solu- tion of the equation | 3x? — Sa — 2 = 0 formed by equating the quadratic function to zero. In order to plot the graph of 32? — 5a — 2, we first fen a table of corresponding values as follows (compare Arts. 18, 120): | To represent the numbers in the table conveniently, it is best to use different scales for x and for the function 3a2 — 5a — 2, since for a range of values from —5 to +6 on 2, the function takes values from —4 to +98. Plotting points from our table of values and drawing a smooth curve through these points, we have the graph in Fig. 35. - oa — ba —2 | oe CoP WNW KH © ar = ae ech lee Leet arP WN O On bo S y’ Horizontal - 2 spaces = 1 unit f ; Vertical - lspace = 4 units S Arts 166, 167] IMAGINARY NUMBERS 249 It should be observed that the graph crosses the X-axis at two points. The values of x that correspond to these points make the function zero, and are therefore the roots of the equation formed by equating the function to zero. The function becomes zero both when x = — 3 and when @ = 2. EXERCISES 1. Solve the equation 32? — 5a — 2 =0, and explain the meaning of the roots by the use of the graph (Fig. 35). 2. To ask for the values of x that make 32? — 5x — 2 equal to 10 means what on the graph? 3. Form the equation whose solution answers the question in Exercise 2. Solve this equation. Plot the graphs of the following functions : 4. a? — 32 + 2. 7. 244743. 5. aw? — 7x + 12. 8. 32? + 7x + 4. 6. 277 + 32 +1. 9. 2? — 5a + 4. 167. Imaginary numbers. Certain quadratic equations, for example, z?+1=0, and a —-6%+15=0 demand for their solution an extension * of our number system to include the square roots of negative numbers. From the equation x? + 1 =0, we have 7 = |, and the equation may be said to ask for a number whose square is —1. It is useful to create such a number and it is customary to write it+/—1 or i. That is, 7 is to be thought of as a number whose square is — 1. The square roots of negative * For other extensions, see Arts. 1, 2, 21, 148. 250 QUADRATIC EQUATIONS [Cuar. XXIII. numbers are called imaginary numbers, but we shall see in the chapter devoted to these numbers in the advanced course that this designation is a misnomer. It is usually convenient to re-— duce any imaginary number to the form az, where a is a real number. EXERCISES Reduce the following numbers to the form az: 1. VY -4. Soutution: VW —~4=Y%—1-4= VW4-VY—1=2:-V -1= 2%. TH ee Bye 2. V/—-9. $247 216) 7 ey Ga 9. ~/—a?. 13. VW—(2 + @) rey ea 6..4/—5 10. ~/ —2c. Solve the following equations : 14. 22+ 1 = 0. 17.. 22° 4 eee 15. 2?°4+4= 0. 18. 3274+ 2 = 0. 16. 277 + 6 = 0. 168. Graphical meaning of imaginary roots. a). It will be instructive at this point to note an important property of the graphs of quadratic functions that give equations with imagi- nary roots when the functions are equated to zero. | Arr. 168] IMAGINARY ROOTS 251 To illustrate, plot the graphs of x? — 67 +C (1) when C = 0, (2) when C = 5, (3) when C = 9, (4) when C = 15. Wes Horizontal - 2 spaces =1 unit Vertical - Ilspace =1 unit Fia. 36 The four functions plotted differ only in the value of C. They are similarly shaped graphs; but, as C is increased from 0 to 15, the graph is simply moved upward on the paper. From the graph for C =0, the roots of the equation 2? — 6x = 0 are seen to be 0 and 6. From the graph for C = 5, the roots of x? — 6x +5 = 0 are seen to be 1 and 5. We note that the graph for C = 9 merely touches the X-axis at the point x = 3. | ) The graph for C = 15 does not at all intersect or touch the X-axis. _ This is the case with the graph of the function when the roots of the equa- _ tion formed by equating the function to zero are imaginary numbers. 252 QUADRATIC EQUATIONS [Cuar XXIII. To illustrate, solve the equation a —6¢ +15 =0. (1) Weatad oe =3+ V-6, or3 +i V6. (2) These solutions are imaginary. Example. Operating with 7 as with any other number, and remember- ing that 72 = — 1, verify by substitution in (1), that 3 +7 V6 and 3-1 V6 are solutions of the equation. Thus, (3 + 1/6)? — 6(3 +iv6) +15 =9 4 6/6 + 6? — 18 - 6ivV/6 +15 =9-6-18 415 = 0. EXERCISES Plot graphs of the following functions and solve the equa- tions formed by equating these functions to zero: Lotte 6. x — 4a. 2. x? — 34 + 2. 7. 32? + 5x — 2. 3. vw — 4443. 8. 3x7 + 5x. 4. x? — 474 4. 9. 3a? + 5a + 4. 5. 2? — 4x 4+ 10. 10. 42? + 32 — 1. PROBLEMS 1. The difference of two numbers is 5, and the sum of their squares is 325 ; what are the numbers? 2. The altitude of a triangle is 6 inches more than its base, and the area is 108 square inches. What is the altitude? 8. Divide a line 386 inches long into two parts such that the rectangle whose sides are equal to the two parts has an area of 315 square inches. Arr. 168] PROBLEMS 253 4. The perimeter of a rectangle is 24 inches, and its area 35 square inches. Find the length and breadth of the rectangle. 5. The perimeter of a rectangle is 14 inches and the area is 25 square inches. Find the length and breadth of the rec- tangle. Explain why this solution is imaginary while that in Problem 4 is real. 6. A square pond is surrounded by a gravel walk with a uniform width of 2 yards. The area of the walk is equal to that of the pond. Find the dimensions of the pond. 7. To get from one corner of a college quadrangle (rec- _tangular) to the opposite corner, I must go 140 yards, around the sides ; if I were allowed to cut diagonally across the grass I should save 40 yards. What are the dimensions of the - quadrangle? g. At what price per dozen are eggs selling when, if the price were raised 5 cents per dozen, one would receive twelve _ fewer eggs for a dollar? 9. The sum of the base and altitude of a triangle is 6 inches, and its area is 10 square inches. Find the base and altitude. Explain why the results are imaginary. 10. A certain man has an income of $5000 more than his exemption of $3000 from income tax. After deducting a per- centage for federal income tax on the $5000, and then an equal percentage from the remainder for a state income tax, the income is reduced to $7900.50. Find the rate per cent of the income tax. 141. A man bought a number of $100 shares of R.R. stock, when they were at a certain rate per cent premium, for $3500; and later, when they were at the same rate per cent discount, sold them all but 5 for $1200. How many shares did he buy, and how much did he pay per share? 12. The telegraph poles for a certain line are set at equal distances. If there were 4 more per mile, the distance between them would be decreased by 66 feet. Find the number of poles per mile. : 254 QUADRATIC EQUATIONS ([Cuar. XXUIL. . 13. A beginner wishing to simplify (x + 5)(a — 2) just leaves out the parentheses and writes x +5a2—2. Are there any values of x for which the two expressions are equal? 14. If a ball is thrown upward with an initial speed of v feet, per second, it is known that after t seconds its height will be ot — 16.1¢ feet. If such a ball is given an initial speed of 100 feet per second, after how many seconds will it be at a height of 100 feet? After how many seconds will it return to - the starting point? 15. A balloon is 1 mile from the ground and is descending at the rate of 5 feet per second when a sand bag is dropped. If the formula 5¢ + 16.1@ gives the distance that the bag will fall in ¢ seconds, find the number of seconds required for the bag to reach the ground. 16. A farmer is plowing around a field 160 rods long and 80 rods wide. How wide a strip must he plow around it to make 10 acres? Draw a.diagram of the field and verify your result. 17. Find the side of a square field whose area is equal to that of a rectangular field whose length exceeds a side of the square by 40 rods and whose width is 20 rods less than a side of the square. : 18. The base of a triangle is 6 longer than the altitude, and the area is 176. Find the base and altitude. 19. In a right triangle, the hypotenuse is 80 inches, and one leg is 16 inches longer than the other. Find the dimensions. 20. Divide the number 20 into two parts whose squares are in the ratio 4 to 9. 21. An open box is made from a square piece of card-— board by cutting square pieces out of the corners and then folding up the flaps. Find the size of cardboard that is used to make a box 4 inches high to contain 144 cubic inches. 22. From a cardboard a box twice as long as wide and containing 240 cubic inches is made by cutting 5-inch squares from the corners. Find the dimensions of the cardboard. Arts. 168, 169] PROBLEMS 255 23. A stream flows at the rate of 5 miles per hour. A crew ‘can row 6 miles with the stream and the same distance back ‘in & hours. What is the rate of the boat in still water? (See Problem 32, p. 204.) 24. A crew can row upstream against a current of 2 miles an hour, for a distance of 10 miles upstream and back again in 22 hours. What rate should we expect the crew to make in still water? 25. The numerator and denominator of a given fraction ‘together equal 80. If we increase the numerator by 8 and decrease the denominator by 8, the resulting fraction is $ as large as the given fraction. What is the fraction? f 169. Historical note on quadratics. Problems that involve quad- ‘ratic equations were solved by Diophantus, the Greek algebraist, who lived in Alexandria about 300 A.D. But he gave only one value of the unknown. He did not recognize the meaning of a negative result, although he solved some quadratics by a method not unlike that of completing the ‘square. When Diophantus came upon an equation both of whose roots are negative, he rejected the equation as absurd or impossible. When only one negative root occurred, he merely rejected that root. When both roots were positive, he took only the root that would be obtained by taking b+ V0 — 4ac +4. In 2a fact, Diophantus looked upon a quadratic equation as having either one or no root. About five or six centuries after Diophantus, the Hintlus solved quad- ratic equations, and observed that they have two roots. They did not regard an equation as absurd because its roots are not positive, but merely rejected the negative rcots on vague grounds illustrated by the following : In solving the equation the positive sign before the radical in the formula a — 452 = 250, Bhaskar gives x = 50 and x = —5 for roots, but he says, ‘‘ The second value is in this case not to be taken, for it is inadequate; people do not approve of negative roots.” The Hindus did, however, observe that negative numbers may be taken to relate to debts if positive numbers relate to assets. It was not until the work of Descartes (see p. 181) became known that the theory of the quadratic was well understood. CHAPTER XXIV SYSTEMS OF EQUATIONS INVOLVING QUADRATICS 170. Introduction. The typical form of a quadratic equa- tion in two unknowns is ax? + bry + cy? +dx+ey+f =), where at least one of the coefficients a, 6b, or cis not zero. Ex- amples of quadratic equations in two unknowns are x? — 2ry + dy? — 11 = 0, T+ ay +203 =0, sy —4=0. One such equation is satisfied by an indefinite number of pairs of values of « and y. For example, each of the pairs of values (0, 9), (1, 7), (2, 6), (4, 5) satisfies the equation, xy — 34 + 2y — 18 = 0, as can easily be shown by substitution. EXERCISES Show that each of the following equations in two variables is satisfied by the pairs of numbers given: 1. czy — 34 + 2y —18 =0, (10, 4), (-14, 2), (—6, 0), (-3, —9). 2. 2x oe 2xy a y? eo 1. 2 = 0, ae 1), (Is —4), (0, —2), (0, 1). 3. x +y’ ot 0, (0, BS (0, —1), (3, 5), (-3, —$), (7s, V3). 4. is x + 44 —3= 0, (0, 3), GL 0), (2, or 1), (3, 0), (2+ V3, 2), (2-V, 2). ) 256 _ Arts. 170, 171,172] SIMULTANEOUS QUADRATICS 257 Find at least four pairs of values of x and y which satisfy ~ each of the following equations : 5. at+xex—y—6=0. SoLution: Substituting x = 1 in the equation, there results 1 + 1-y —6 =0,ory = —4. Hence, x =1, y = —4, satisfies the equation. Again substituting x = 2, we find y =0. Hence (2, 0) satisfies the equation. In this way any number of pairs of values satisfying the equation can be found. yy +22e%-—-y-2=0. 2x? + dy’ = 4. etaeayty+2e4-—y—2=0. 9. 2 +2742 =(44+1) (y - 2). PID 171. Solution of simultaneous quadratics. Although there is an indefinite number of pairs of values of x and y which satisfy one quadratic equation in two unknowns, yet there are never more than four pairs which satisfy two different equations. _ For example, the four pairs of numbers, (3, 4), (—3, 4), ‘Gamer ay _ (—8, —4) satisfy both the equations, 162? + 27y? — 576 = 0, e+y? — 25 = 0. No other pairs of numbers can be found which will satisfy both of these equations. The general problem in systems of simul- taneous equations involving one or two quadratics is the finding of all the pairs of numbers which satisfy both equations. This problem is in general quite difficult, but there are some types of these equations which can easily be solved. A more extended discussion, together with the graphical interpretation, will be given in the second course in algebra. 172. One equation linear and one quadratic. A system of two equations in two unknowns in which one equation is linear and the other is quadratic, can be solved by the method of substitution. In this case there are in general two solutions. 258 EQUATIONS INVOLVING QUADRATICS [Cuap. XXIV. > EXERCISES Solve the following pairs of equations : 1227 — Ff = OY, ee ay te SoLtution: From the second equation, we find ei Te 2y, Substituting in the first and reducing, we have 10y? — 69y + 98 = 0. Solving this equation for y we find y =2 or 4.9. Substituting these values in the second equation we find x =3, or —2.8. The solutions of the two equations are then (3, 2), (—2.8, 4.9). CHECK: (coe iia 3 4+2.2 =7, 2. (—2.8)? — (—2.8)(4.9) = 6- (4.9), { BeBe bp re Wh) rc 2. 27+ 47% — 25 = 0, 6. cy = 15 ty ea: rt+y =2. Be ty la a ee 7. 237 “yy? ace se a 2x +y = 9. 4.0 Py = 2; 8. XY == eae A rn VP mmps x+ty=dsl. 5. Seale 9. LY —=.S08 2x +y = 3. x — 2y =7. 10.%2yi— 2 = 0, 3x — y = O. 11. zy + 7y + 6x — 38 = 0, x+y =. 12. 20? — 3rzy — y’? = 1, 64 + y = 3. | Arrs. 172, 173] SIMULTANEOUS QUADRATICS 259 ie yr? =. 0(x + y) + 2, ee Dike 14. 2(e + y)? — (w@ + y)(@ — 2y) = 70, 2(a + y) — 3(@ — 2y) = 2. 15. The sum of two numbers is }%. Their product is 3. What are the numbers? 16. If the figures in a number of two digits are reversed, the new number is 27 greater than the given number. The product of the two numbers is 1300. What is the given number? 17. The difference between the numerator and the denomi- - nator of an improper fraction is 4. If 2 be added to both - numerator and denominator the resulting fraction is less by 3's than the original fraction. What is the fraction? 18. The diagonal of a rectangle is $ yards. The perimeter is 2 yards. What are the dimensions of the rectangle? 173. Equations containing x? and y? only. The type form of this case is ax? +cy2+f =0. If, instead of x and y, we consider x? and 'y? as the unknowns, the method of solution is that for linear equations. In general there will be four solutions for a pair of equations of this type. Solve: EXERCISES or y= 2, xv? — 2y? = —4l. Soutution: Solving for x? and y? we find 2? = 9, y?= 25. Hence x = + 3, y = +5. The four pairs of numbers, (8, 5), (—3, 5), (3, —5), (—3, —5) will be found upon substitution to satisfy the two equations. 2 OE & = 20, fo ay = 11, : : x? 4+ Dy? = 22. 28 OE Ean 9 260 EQUATIONS INVOLVING QUADRATICS [Cuap. XXIV. |. 4°32" = by? = 3; 6. 22? — dy? = 6, xv? + y? = 25. 3x” — Qy? = 19. 5. a+y* = 16, 7. ax? — by =a + BD, 4x? + 25y? = 100. ba? + ay? = a + B? 2 8, ne 4% — 2( ~— +3), m m v f = 2 mae 9. The square of the diagonal of a rectangle is 34. _ If two such rectangles were put end to end, the square of the diagonal of the new rectangle thus formed is 109. What are the dimensions of the first rectangle? 10. Two rectangles of the same size and having diagonals 2% feet long, are placed end to end. The square on the diag- onal of the new rectangle thus formed is 5; square feet greater than the square on the diagonal of the rectangle formed by putting one of the rectangles above the other. What are the dimensions of the first rectangles? 174. Special methods. The solution of two quadratic equations in two unknowns is usually in the nature of a puzzle for which no special rules are given. Often no solution can be found without the use of mathematics beyond the courses given in high schools. Many systems may be solved by special devices in which the aim is to find values for any two of the expressions « + y, x — y, and xy, from which the values of x and y may be obtained. Various manipulations are performed in attaining this object, according to the form of the given equations. Whatever method is used in solving simultaneous equations, it must be kept in mind that the ultimate test of a solution is substitution in the given equations. \ Arr. 174] EXERCISES 261 Example 1. Solve the system ev +-xy = 12, (1) y + zy =4. (2) SOLUTION: Adding (1) and (2), v+2ry +y? = 16, (3) whence x+y = +4, or -4. (4) Subtracting (2) from (1), vey =8. (5) Dividing (5) by (4), x—y = +2, or -2. (6) From (4) and (6), oe 3, or —3; y =1, or -1. By substitution in (1) and (2) we find that the two pairs { ae and i % is satisfy (1) and (2). Example 2. Solve the system 22 +7 = 16, (1) y? = 62. (2) SotuTion: Substituting 6z for y? in (1), we obtain x? +62 = 16, or 4+ 62 — 16 =0. (3) Solving (3) by formula (Art. 164), jae 6 + V96 + 64 Bae = 2, or cao If x =2, we have y = +V/12 = +273. If x = —8, wefindy =+ 4/—48, = + 41v/3, which are imaginary numbers. r=2 z=2 fra 3 The possible solutions are then = 2 on Pe Set have ey a ., each of which should be tested by actual substitution. y= 4iv/3 EXERCISES Solve and check: 1 OX, eae = 20; v—y+8 =0. rin IA Wh 2. 2? + y? = 85, 4, 5x2? — 9y? + 121 = 0, oot a) ocecnad A 7y? — 32? — 105 = 0. 262 EQUATIONS INVOLVING QUADRATICS [Cuar. XXIV. - Dee ya), 9. x? + 4peaee, ry = —7, v—ayty =3. 6.27 Pay =36, 10. (x + y)? + (x — y)? — 50 = 0, xy +y*? = 45. (x —y)(a@+y) +7 =0. Ten det ry een Li Li) 2 yee sy+y’ =6. Ty sate 8 p—g=9, 12. 2° — aio 4p? = 25¢. : DY reas Hint: Divide the members of one equation by the corresponding mem- bers of the other. 13. 2? — 7? = 1, 16. x? + dry + 2y? = 15, x—y =3. r+y =3. 14. x7 — y? = 25, 17. 2m? + mn — n? = 2, x+y = 10. 2m —n=1. 15. 2? — xy — 6y? = 16, 18. a? +7 = 4ab + 5b?, x + 2y = 16. QO? a= jeans MISCELLANEOUS PROBLEMS 1. The sum of two numbers is 5 and the sum of their squares is 143. Find the numbers. 2. A rectangular field 68,200 square feet in area, is sur- rounded by a 4-wire fence ; 5840 feet of wire were required for the fence. Find the dimensions of the field. 3. The sum of the squares of two numbers is $2, and their product minus @ is equal to their difference. What are the numbers? 4. The hypotenuse of a right triangle is 25 feet. The sum of the other two sides is 35 feet. What are the lengths of the sides? 5. The area of a rectangle is 50 and the perimeter is 54. What are the dimensions of the rectangle? i Art. 174] PROBLEMS 263 6. A piece of wire 48 inches long is bent into the form of a right triangle in which the hypotenuse is 20 inches long. Find the other sides of the triangle. 7. A group of students club together to rent a suite of rooms for $400 per year. By adding 5 new members to the group the assessment was $4 less per member. How many students were there in the club at first? g. A certain number of two figures, when multiplied by the left digit, becomes 54; but when multiplied by the right digit, it becomes 189. What is the number? 9. The annual income from an investment is $72. If the principal were $240 more and the rate of interest 1% less the income would remain unchanged. What are the principal and the rate of interest? 10. A sum of money placed at simple interest for 6 years amounts to $5580. Had the interest been increased 1% it would have amounted to $45 more than this in 5 years. What are the principal and the rate of interest? 11. A rectangular field is 145 yards long and 50 yards wide. How much must the width be increased and the length decreased in order that the area remain unchanged while the perimeter is decreased 30 yards? 12. A certain kind of cloth loses 5% in width and 4% in length by shrinking. Find the length and width of a rec- tangular piece of the cloth whose shrinkage in area is 2.2 square yards and in perimeter 2.1 yards? 13. An 8 by 10 photograph is enlarged until it covers twice the original area, keeping the ratio of the length to the width unchanged. Find the sides of the enlarged photograph. 14. A man loaned $9000 in two unequal sums at such rates that both sums yielded the same annual interest. The larger sum at the higher rate of interest would yield $250 per year, and the smaller sum at the lower rate, $160 per year. How was the money divided and what were the rates of interest? 264 EQUATIONS INVOLVING QUADRATICS [Cuap. XXIV. - 15. A dealer sells a number of books for $1125, receiving — the same price for each book. If he had sold 150 books less but charged 25 cents more, he would have received the same sum. Find the price and the number of books. 16. A dealer purchased a certain number of sheep for $175 ; after losing two of them he sold the rest at $2.50 a head more than he gave for them, and by so doing gained $5 by the deal. Find the number of sheep purchased. 17. Both the numerator and the denominator of a frac- tion are increased by their squares. The new fraction reduces to 3. If both the numerator and the denominator are divided by their squares the result is 3. What is the fraction? 18. The differences between the hypotenuse of a right tri- angle and the other two sides are } and 1 respectively. Find the sides of the triangle. 19. The area and the perimeter of a rectangle are both 25. What are the dimensions of the rectangle? 20. A farmer is cutting wheat in a field which is twice as long as it is wide. After cutting a strip 10 rods wide around the outside of the field he estimates that 4 of the work has been done. What are the dimensions of the field? Art. 174] EXERCISES AND PROBLEMS 265 REVIEW EXERCISES AND PROBLEMS Extract the following square roots : : Ves, Dap, ~/9-16, V9 +16, Vat + 20% +B. = 20n? +25 | /16- 100 - 36, VJ (a? +y)? —4(2? +y) +4. or +3)? me : : — 1 a oY 3. Simplify: W/4a, ~/8a3, 3 Vi Ve V 45a?b’. 4. If sis the side of a triangle having equal sides, what is the altitude? What is the area? ; SOLUTION: The altitude, h, bisects the base AC. B st. Ss* OR ee 5 a “3: — SS Then. At = 8 (5) s aieg | 382s 2 i s ane h 4 5V3. We have then for the area, 213 3? J Area = 5-5V3 = 7 V3. A Ds C 5. The side of a triangle having equal sides, is 10 inches. Find the altitude and the area, correct to two decimal places. 6. Find to two decimal places the side of an equilateral triangle - whose area is 100 square inches. 7. Solve the equation A =r? forr. Find r if A = 50 square inches. 8. Add ee and y -, and express the result as a fraction 24375 4-V5 with a rational denominator. 9. Make up quadratic equations with the unknown x so that the values of x shall be (a) positive integers; (b) positive fractions; (c) negative integers ; (d) negative fractions ; (¢) both zero ; (f) one zero, and one a positive integer. 10. Solve for x the equation 22? + 5a -4 =0. Calculate the roots to two decimal places. 11. If the diameter of a circle is increased by 3 feet the area is doubled. Find the diameter, correct to two decimal places. 266 EQUATIONS INVOLVING QUADRATICS [Cuap. XXIV. 12. The perimeter of a rectangle is 82 inches. The diagonal is 29 inches. How long are the sides? 13. The hypotenuse of a right triangle is 20 inches. If the altitude is multiplied by ~/2 and the base by +/3, the hypotenuse is multiplied by $. | Find the base and the altitude. a+b : b4 a + ab op) (a eae 14. Simplify > . a : (So tg | 15. What number added to the denominators of ; and “ respectively will make the results equal? Under what condition is a solution impossi- ble? 16. Plot the loci of the following equations and determine the solution of each pair from the graphs : y = 6, ee, ps ee Oe eee (0) 2x +y = 10; ©) — dy =3. 17. Solve the equation s = ut + $f? for t. Find ¢ when wu = 10,f = 32.2, and s = 200. 18. Plot the locus of the equation y = x? for values of x from 1 to 10. Then solve the equation for x. This gives x = ~/y. If now we choose a value of y to be 4, and read off from the graph the corresponding value of x, which is 2, we have the square root of the chosen value of y. Thus determine from the graph, as accurately as possible, the square root of each of the following: 16, 25, 36, 7, 2, 3, 60, 42. 19. Which is the larger 1/3 or SE | V/11 aan 20. Find the values of ——~——— to three significant figures (1) b ite i gures (1) by using the square roots of 5, 7, and 11; (2) by first rationalizing the denom- inator. 21. Evaluate to three significant figures — V7 -2 Abscissa, 180 Absolute value, 25 Addition, associative law for, 56 commutative law for, 56 of fractions, 143 of monomials, 54 _ of polynomials, 57 of radicals, 229 of signed numbers, 28 on the number scale, 23 Ahmes, 80, 137 Algebraic expressions, 14 Antecedent, 169 Arabic notation, 13 Axes of coérdinates, 180 ’ Base of a power, 9 Binomial, 53 cube of, 105 square of, 100 Braces, 15 Brackets, 15 Cancellation, 141 Checking, an operation, 57 a solution, 45 Circle, area, 7 circumference, 7 Coefficient, 9 Completing the square, 239, 240 Complex fractions, 151 INDEX [The numbers refer to pages.] Consequent, 169 Constant, 175 Coérdinates, 181 Cube, of a binomial, 105 of a number, 10 Denominator, 39, 136 rationalization of, 233 Descartes, 27, 80, 181 Difference, of two cubes, 119 of two squares, 113 Distributive law for, 73 Dividend, 38 8 Division, 38, 86 by zero, 138 law of exponents-for, 86 of fractions, 149 of monomials, 86 of polynomials, 87, 89 of quadratic surds, 2382 of signed numbers, 38 rule of exponents for, 86 rule of signs for, 38 Divisor, 38 greatest common, 132 Elimination, 194 by addition and subtraction, 195 by substitution, 197 Equality, 42 members of an, 42 268 Equation, 43 clearing of fractions, 159 graph of an, 186 historical note on, 80 involving fractions, 81, 159 involving parentheses, 78 involving radicals, 235 linear in two unknowns, 187 linear or simple, 96 literal, 164, 205 locus of, 186, 187 principles used in solving, 44 quadratic, 126 solution or root of, 44 solved by factoring, 126, 127, 238 Equations, dependent or equivalent, 193 graphical solution of, 187 inconsistent, 193 independent, 193 simultaneous, 193 solution of simultaneous linear, 188, 193 system of linear, 194 system of quadratic, 256 Evaluation of expressions, 16 Exponent, 9 Exponents, law for division, 86 law for multiplication, 69 Expression, 14 terms of an, 53 Extremes, of a proportion, 170 \ Factor, 9, 109, 110 common, 132 found by grouping, 112 highest common, 132 integral, 110 ae INDEX Factor, monomial, 110, 111 of trinomials, 116, 118 prime, 109, 110 rational and integral, 110 rationalizing, 234 Factoring, 109 equations solved by, 126, 127, 238 summary of, 121 Fractions, 39, 136 addition and subtraction of, 143 clearing equations of, 159 complex, 151 division of, 149 equations involving, 81 historical note on, 137 lowest terms of, 140 multiplication of, 145 reduction to common denom- inator, 142 reduction to lowest terms, 140 signs in, 138 square root of, 227 terms of, 39, 136 Function, 183 graph of, 184, 247 Graph, of a function, 184, 247 of an equation, 186 Graphical meaning, of imaginary roots, 250 — Graphical representation, 19, 180 historical note on, 181 of positive and negative num- bers, 23, 24 of scientific data, 189 Graphical solution, of equations, 187 Greater than, 25 INDEX _ Highest common factor, 132 ’ Historical note, on fractions, 137 on graphical representation, 181 on negative numbers, 27 on symbols, 12 on the equation, 80 Identity, 42 — Imaginary numbers, 249 _ Index of a root, 226 _ Integral expression, 110 Irrational numbers, 226 _ Less than, 25 Linear equations, in one unknown, 96 in three or more unknowns, 206 in two unknowns, 187 locus of, 187 standard form in two unknowns, . 199 systems of, 194, 206 Locus, of a linear equation, 187 of an equation, 186 Lowest common multiple, 134 Mean proportional, 171 Means, of a proportion, 170 Members of an equality, 42 ~ Monomials, 53 addition of, 54 division of, 86 factors of, 110 product of, 70 square roots of, 214 subtraction of, 60 Multiple, common, 134 lowest common, 134 Multiplication, 69 269 associative and commutative laws of, 70 by a monomial, 72 distributive law of, 73 in arithmetic, 35 law of exponents for, 69 of a product, 72 of fractions, 145 of monomials, 70 of polynomials, 72, 75 of powers, 69 of quadratic surds, 231 of signed numbers, 35 rule of signs for, 36 signs of, 2 Negative numbers, 24 historical note on, 27 Numbers, graphical representations of, 23, 24 imaginary, 249 irrational, 226 positive and negative, 24 rational, 226 signed, 25 Numerals, Arabic, 13 Roman, 13 Numerator, 39, 136 Numerical value, 25 Order of operations, 14 Ordinate, 180 Origin of codrdinates, 180 Parentheses, equations involving, 78 ‘forms of, 15 insertion of, 66 removal of, 64 uses of, 15 270 INDEX Polynomials, 53 Proportional, addition of, 57 fourth, 172 arrangement according to as- mean, 171 cending and _ descending third, 171 powers, 57 division of, 87, 89 factors of, 111 multiplication of, 72, 75 simplifying, 56 subtraction of, 61 Positive numbers, 24 Power, 9 Powers, ascending and descending, 57 product of, 69 Prime, expressions prime to each other, 132 factor, 109, 110 Product, 35 cross, 118 of a polynomial by a mono- mial, 72 of monomials, 70 of polynomials, 75 of powers, 69 of the sum and difference, 102 of two binomials with a com- mon term, 104 Products, important type, 100 Proportion, 170 by addition, 173 by addition and _ subtraction, 174 by alternation, 173 by composition, 173 by composition and division 174 by division, 174 by inversion, 173 by subtraction, 174 Quadratic equations, 126, 127, 238 in two unknowns, 256 ; solved by completing the square, 239 solved by factoring, 127, 238 solved by formula, 242 special forms of, 245 type form of, 242 Quadratic surd, 227 division of, 232 multiplication of, 231 Quadratic trinomial, 118 Quotient, 38 of monomials, 86 of polynomials, 87, 89 Radicals, 226 addition and subtraction of, 229 division of, 232 equations involving, 235 multiplication of, 231 similar, 229 simplification of, 227 Radical sign, 214 Radicand, 228 Ratio, 169 Rational, expression, 109, 110 number, 226 Rationalization, of the denominator, 233 Rationalizing factor, 234 Reciprocal, 153 Reduction, of fractions to lowest terms, 140 | to common denominator, 142 INDEX * seght triangle, 80 ~ Roman numerals, 13 _ Scale, in representing numbers, 23 Signed numbers, 25 addition of, 28 division of, 38 multiplication of, 35 subtraction of, 29 | Signs, in fractions, 138 rule of for division, 38 : rule of for multiplication, 36 | imilar triangles, 178 | fmaneon equations, 193 of second degree, 256 . olution, ‘| ofan equation, 44 of a pair of linear equations, 188, 192 of simultaneous quadratics, 257 (quare, -\ completing the, 239 of a binomial, 100 of a number, 10 of a trinomial, 106 trinomial, 114 uare root, 214 of decimals, 221 of fractions, 227 of monomials, 214 of numbers expressed in Arabic figures, 219 of polynomials, 217 of trinomials, 216 process of finding a, pa ubscripts, 165 tbtraction, 29 of fractions, 143 271 of monomials, 60 of polynomials, 61 of radicals, 229 of signed numbers, 29 on the number scale, 23, 30 rules for, 31 Sum, of signed numbers, 28 of two cubes, 119 Surd, 226 division of, 232 multiplication of, 231 quadratic, 227 Symbols of operation, 1 historical note on, 12 System, of equations involving quad- ratics, 256 of linens equations, 194 Term, 53 Similar or like, 53 Transposition, 46 Trinomial, 53 general quadratic, 118 Square, 114 Square of a, 106 Square root of a, 216 Unknown, 44 Variable, 175, 183 Variation, 175 Vineulum, 15 Zero, division by, 138 origin of, 13 Ae. “ uly a i, ®: ire i Be i) 4 be : * ‘ ; . fs ~ ‘ , ~ * , ‘ ’ 4 ; J - a ae 4) a x iy i . 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