University of California • Berkeley The Theodore P. Hill Collection of Early American Mathematics Books Digitized by the Internet Archive in 2008 with funding from Microsoft Corporation http://www.archive.org/details/easyalgebraforbeOOvenarich AN EASY ALGEBRA FOE BEGINNERS; BEING A SIMPLE, PLAIN PRESENTATION OF THE ESSENTIALS OF ELEMENTARY ALGEBRA, AND ALSO ADAPTED TO THE USE OF THOSE WHO CAN TAKE ONLY A BRIEF COURSE IN THIS STUDY. BY CHARLES S. VENABLE, LL.D., Professor op Mathematics in the University op Virginia, and Author op A Sekies of Mathematical, Text-Books. NEW YORK: UNIVERSITY PUBLISHING COMPANY. 1886 Copyright, 1880, Bt university publishing company. b *878 PREFACE This book is designed for the use of those for whom the High School (Elementary) Algebra may be too difficult, and is adapted also to those who can take only a brief course. It has been carefully prepared with a view to render an acquaintance with the essentials of elementary algebra easy of acquisition by the young beginner. The explanations are brief and simple. The examples are not difficult. Such definitions and rules as are common to arithmetic and algebra have been given without labored illustration, or have been assumed as already familiar to the pupil. As a preparation for the solution of problems by means of equations, well graded steps are given in the section preceding these problems for practice in the translation of quantitative statements from ordinary language into algebraic expressions. Only the leading and more easily understood principles of Radicals and Progressions have been introduced, and I trust that these more advanced topics are presented in a simple and attractive form. iy PREFACE. The Miscellaneous Examples for independent exercise on the subjects of the different sections will give the pupil greater familiarity with the methods learned in the text ; while the Review Questions will serve as a test of the accuracy of his knowledge of the principles underlying these methods and operations. 0. S. V. University of Virginia, November 10, 1880. CONTENTS. PAGE I. Definitions 7 II. Addition 10 III. Subtraction 12 IV. Multiplication 14 V. Brackets or Parentheses 18 VI.. Multiplication by Inspection 20 VII. Division 23 VIII. Factoring 28 IX. Greatest Common Divisor 30 X. Least Common Multiple 32 XI. Reduction of Fractions to Lowest Terms 33 XII. Reduction of Fractions to a Common Denominator. 35 XIII. Addition of Fractions 37 XIV. Subtraction of Fractions , 39 XV. Multiplication of Fractions , 41 XVI. Division of Fractions 43 XVII. Finding Numerical Values by Substitution 44 XVIII. Simple Equations 46 XIX. Translation of Ordinary Language into Algebraic Expressions 53 v vi CONTENTS. PAGE XX. Problems in Simple Equations 56 XXI. Problems — Continued GO XXII. Simple Equations with Two Unknown Quantities.— Elimination G3 XXIII. Problems producing Simultaneous Equations 67 XXIV. Simultaneous Simple Equations of Three or More Unknown Quantities 70 XXV. Involution or Raising to Powers 73 XXVI. Evolution or Extraction of Roots. — Square Hoot. 76 XXVII. Quadratic Equations 84 XXVIII. Solution of Affected Equations 86 XXIX. Problems giving rise to Quadratic Equations 89 XXX. Easy Simultaneous Equations solved by Quadrat- ics 91 XXXI. Radicals of the Second Degree 94 XXXII. Ratio and Proportion 101 XXXIII. Arithmetical Progressions 106 XXXIV. Geometrical Progressions Ill XXXV. Miscellaneous Examples 117 XXXVI. General Review Questions 134 Answers 143 EASY ALGEBRA. SECTION I. DEFINITIONS. 1. Algebra. — In algebra we use letters to denote num- bers, and signs to indicate the arithmetical operations to be performed on them. 2. Signs. — The usual signs, some of which we have used in arithmetic, are : (1.) The sign of addition + (plus), as a -f b. (2.) The sign of subtraction — (minus), as a — b. (3.) The signs of multiplication x, • , and simply writ- ing letters one after another, as a x b } a • b, and ab, all mean a multiplied by b. So, also, 5 x a, 5 • a, and 5a mean 5 times a. Note.— In writing figures we must remember that 57 means 5 and yL, and 57 means 50 + 7, and not 5 x 7. (4.) The signs of division -r-, and a line between the letters. Thus, a -7- b, and -r mean a divided by b. (5.) The sign of equality =, read "is equal to." Thus, -r = c is read " a divided by b is equal to c." 8 DEFINITIONS. 3. Coefficient.— As 6 + 6 + 6 + 6 + 6 is 5x6. So # + # + # + « + « is 5 x a, which we write 5a, and 5 is called the coefficient of a. Similarly, ab + ab + ab + ab + ab -h ab = Gab, and 6 is the coefficient of the product ab. Definition. — The coefficient is the number ivritten be- fore a letter or quantity to show the number of times it is taken. When no coefficient is written, the coefficient 1 is understood. 4. Algebraic Quantity, or Algebraic Expres- sion. — Any collection of letters with algebraic signs is called an algebraic quantity, or an algebraic expression. Thus, a, a + b + c — d, ab, hab + 2cd — 3ef are algebraic expressions. 5. Terms. — The terms of an algebraic expression are the different parts separated by the sign + or — . Thus, in the expressions a + b + c — d, hab + 2cd — 3ef, a, b, c, d, hab, 2cd, and 3ef are the terms. 6. Monomial, Polynomial, etc. — An algebraic ex- pression of one term only is called a monomial ; an expres- sion of two terms is a binomial ; one of three terms, a trinomial. In general, an expression of more than one term is called a polynomial. 7. Factors.— Just as 5 and 7 are factors of 35, so 7 and a are factors of 7a ; so a, b, and c are factors of the expression abc. 8. Power and Index, or Exponent. — When the same factor occurs several times, as a x a x a x a x a, or DEFINITIONS. 9 aaaaa, we write it, for the sake of shortness, a\ This a b , thus written, is called the fifth power of the number a, and is read "fl- tcLtlie fifth power." So, also, a x a x « is written a 3 , and is read "a tefcfa© third power." a x a is written a 2 , and read " a squared," or " a to the second." The 5, 3, and 2 thus written are called exponents or in* dices. Definition. — An index or exponent of a letter is a small number placed over it to the right to show its power, or how many times it is taken as a factor. When no ex- ponent is written, the exponent 1 is understood. Examples. — Bead a 7 , b 9 , 2% 3 4 , a 2 b\ b 3 c\ and write them with all their factors. 9. Positive and Negative Quantities. — All terms or quantities which have the plus sign, or no sign, before them are additive, and are called positive quantities. All quantities with the minus sign before them are sub- tractive, and are called negative quantities. 10. Like Terms. — Like terms are those which differ only in their numerical coefficients. All others are un- like. Thus, 7a, — 8a, and + 5a are like terms ; as, also, 8a 3 and — 6a 3 ; 16a~b and a~b. 11. Like Signs. — When two quantities are both plus, or both minus, they are said to have like signs. When one is plus and the other minus, they are said to have un- like signs. Note.— The pupil should now be practiced in reading algebraic expressions, and in writing them down from dictation. 10 ADDITION. SECTION II. ADDITION. 12. To add like algebraic quantities. Rule. — Add separately the plus and minus coefficients, take the difference of the ttvo sums, prefix to this the sign of the greater, and attach the common letter or letters. Examples — 1. (1.) (2.) (3.) (4.) 2a hax - 6b - lab 6a 4:UX - 3b - 3ab 4a ax - b - 2ab a 2ax - 10b — ab Ida 12ax - 20b - lOab (5.) (6.) (7.) (8.) (9.) (10.) To 3a 3a 3a -la 8a a add — a ±a ~7a~ 6a ha -10a -b 2a + 3a - 2a — 2a a — b (11.) (12.) (13.) (14.) hx 10a 2 lhab 12cx 2 — 4x 3a* - lOab — lex* + 2x - 4rr — ab - dcx 2 + ox + 9« 2 + Aab — ex 2 (15.) (10.) (17.) 5a — 3c ha - 3c ha + 3c 6a - 8c lc 6a + 8c 6a - 8c 11a - 1 11a + he 11a - he ADDITION. 11 (18.) 6a — 25 + 5c 4a + 85 - 2c 10fl 4- 6b 4- 3c OWL) 3a — 45 — c 6a 4- 9* - 7c - 5a + 25 — 4c (19.) a — b + c a + b — c 2a (20.) CC — y -f 2 a; 4- ?/ 4- ^ 2a; 4- 2z (22.) - 21a 2 - Uab + 20«c 2 4- 30ac 45a 2 - 20ab - 12ac* - \6ac — da* 4- 2ab + 25ac 2 - Mac Remark. — Unlike Terms. — When unlike terms occur, unite them in the sum with their proper signs. Thus, to add 8c — 5c? to 3a- — b, we simply write it 3a- — b + 8o — 5cZ ; again, the sum of 3a'—Aab and — 2a? is a- — 4ab. (23.) Add a + 25 - c and a — 6wi 4- 2c. Thus : a 4 2b — c a -f 2c — 6m. 2a + 25 -f c — 6w (24) Add 6a? s — 8a; + a and 3a; - # + 6. Thus : 6a; 2 - Sx + a 3a; — y -f 6 6ar — 5a; 4 a — ?/ 4- 6 (250 Add 3a 2 + 45a; - c 2 4- 10, - 5a 2 + 6ac + 2c 2 - 15a, 4a 2 - 3ac - c 2 4- 21. 3a 2 4- 45a; - c 2 + 10 — 5a 2 4- 2c 2 4- 6ac — 15a - 4a 2 - c 2 4- 21 - 3ac 6a' 4- 45a- 4- 31 4- 3ac — 15a 12 SUBTRACTION. Examples — 2. Add together 1. — Ga, &a, — loa, 3a, — 4ft. 2. x + y + z, — x + y + z, x — y + z, x + y — z. 3. 3 — a, — 8 — a, la -I, — a — 1, 9 + a. 4. 6 3 - 2«& 2 + a 2 b, ¥ + 3«6 2 , 2« 3 - ab 2 - a 2 b. 5. 2x* — W + 3, — 4z 3 + Gx 2 - 2a; + 7, a* 4 - 2z 3 - 4a;, 62; 3 - 9a; - 12. In like manner, by grouping and adding like terms, 6. Eeduce the polynomial a x — Ga/b + Gab 2 — 2b a + 5<2 4 — 3a# 2 + 6# 2 £ + b 3 — 4« 4 + 2« 2 # to its simplest form. 7. Eeduce 4«?/ 2 — 3^ + Sab + 7a# — Gab + c + 8## 2 + 4#2 + 7## — c + 7#y a — 822 — 9ab to its simplest form. SECTION III. SUBTRACTION. 13. To subtract + b from a, we have a — b. To sub- tract — & from a, we can write for a, a + b — b, as it is the same. Now — J taken from a + b — b leaves « + #. Hence, — b taken from a gives a + £. In like manner, b subtracted from — a gives — a — b, and — b subtracted from — a gives — a + b, — 2a sub- tracted from da gives + 5#. Hence, Rule. — Change the sign of every term in the subtra- hend, and proceed as in addition. SUBTRACTION. 13 Examples — 3. 1. From 4a 2. From 5x 3. From 4a take a take 4:X take — a 3a X ha 4. From b 5. From a G. From 5a take b take — a take — 4a 2a 9a 7. From -4a 8. From — 5b 9. From — a take a take ±b take a — 5a - 9b — 2a 10. From -4a 11. From — 5c 12. From — a take — a - da take — 4c take — a — c 13. From % +y 14. From b — c 15. From « -f bo take x -y take £ + c take a — ex 2y - 2c Ic + c# (16.) From 4a — 5b + 7c take a — ob + 10c — 4# ^i«s. 3a— 2b— 3c + 4z Ans. x + oy — 1 (18.) (19.) From a 2 + dab - 4c 2 From take 2a 2 - 6a£ - 8c" 2 take Ans. —a- + 9a Z> + 4c 2 Ans. — x*+x 3 + x" — 2x — 2 (17.) From Sx — 2y + 4z — 5 take 7a; - 5// -t- 4z — 4 z 3 -4.r+ &u - 11 z 4 -5z 2 + 10z - 9 14 MUL TI PLICA TION. Examples — 4. 1. From a take a — 6 — x. 2. From 2a + 3b — c - d take 2a -3b + c - d. 3. From 8a — I? — c take a — b + £c. 4. From 3« + 2a; — 56 take 2a + 3x + 46. 5. From xij + 2a;* 2 + 3if take £«/ — 2a;' 2 + 3y\ G. From 4?»w + 5m — Gn take 2mn + m + w. 7. From 3# 2 6 + 4« 2 c — Gc 2 take ft* 2 6 — a/c — 8c' 2 . 8. From \ab — \bc -f -J take \ab + foe — §. 9. From a 3 — 30« 2 a; + 51 x # 4 is aaaaa x «««« = a 9 . Hence, to multiply powers of the same letter, we add the exponents. 15. 1. -h 3a x + 2b means + 3a added 2b times, or + Gab. 2. — 3a x + 26 means — 3a added 2b times, or — Gab. 3. + 3a x — 2b is the same as — 2b x 3#, or — Gab, as above. 4. — 3a x — 26 means — 3a subtracted 26 times, that is, — Gab subtracted, which by the rule of subtraction (Art. 13) gives + Gab. M UL T I PLICA TION. 1 5 Hence, summing up, we have + oa x + 2b -- + Gab, - 3a x -f- 2b = — G«#, + 3« x — 2# = — 6«5, - 3« x - 26 = + Gab. Hence, + by + gives + ; — by - gives + ; + by - gives — ; and — by + gives — . Therefore, 16. To multiply a monomial by a monomial, Rule.— I. For Coefficients : Common multiplication. II. For Signs : Like signs make + ; Unlike signs, — . III. For Exponents : Add the exponents of the same Let- ters. Examples : 1. 3a 2 x 4« 3 = 12a 8 . 2. hob x ±abc = 20a*b*c. 3. 4«' x — axy — — Aa'xy. 4. — 5x*y*z* x — 2x 1 yz 1 = lOx* y 3 z*. 5. 2ab x — 3cy x — a'Vy — 6a 4 J"cy\ 6. axy x b ; 3ab x — x ; — 3mm x awi ; — xxf x — xif . 17. To multiply a polynomial by a monomial or single term. Rule. — Multiply every term in the multiplicand by the multiplier. Examples — 5. 1. 5«V - Gabx 1 + 3b*y* 10« 2 foV + 12ad a cV - G£ s cy 1 G MUL TIPLICA TION. 2. 12a - 7b. 3. 5b - 8a. 4. 10x 2 - - 5ax - -3a\ 9a - 12a 4-x 2 Multiply 5. 2ab — Aac + Gbd by — 2x. 6. ftc + 2Z>6' by oa. 7. 2az + 5% - dcz by - 2^ 2 . 8. ox H — 2.Z + Ax 2 by — 7# 2 . 18. To multiply a polynomial by a polynomial. Rule. — Multiply all the terms of the multiplicand by each term of the multiplier. Then add these products. Ex. 1. Ex. 2. Ex. 3. 3a + 2b x + 3 o 1 + 2z 2 ha - U x — 2 3a 2 + a 2 15a a +10a£ rr + 3rc 3a 4 + 6aV -12a5-85 a -3a? — 6 a 9 x* +2x A Prod. 15« 2 -2tfZ>-86 2 Prod. ^ 2 +^-6 Prod. 3« 4 + 7aV + 2a; 4 Ex. 4. 2a + 36 - 5c ' a -V b — c 2a 2 + oab — 5ac + 2ab + 36 2 -5bc — 2«c - Sbc + 5c 3 2a' 2 + 5afl - 7«c + 3b 2 - Sbc + 5c\ MULTIPLICATION. 17 Examples — 6. Multiply 1. a + x by c + y. 2. 5a + 4 by a — 2. 3. a; — 5 by x + 4. 4. 3a - 4 by 2a - 3, 5. 1 — 2a by x — a 2 . 6. ac — b 2 by c 2 — ab. 7. - 11a - 3a by - 10a - 8a. 8. 1 + 3a + 2y by a — y. 9. «# — be -\- ac by 2« — 5. 10. a 3 + a 2 + x + 1 by a — 1. Multiply 11. 5 + 2a + a 2 by 5 — 2a + x\ 12. a + 4 — £/ by a + 4 + y. 13. 3«V + Wy by 3aV - 2%. 14. 2a 3 + 4.r + 8a + 16 by x - 2. 15. «V — a*x*y + « 2 a 2 ?/ 2 — aay 3 + y* by #a -f y* 1G. a 2 - 2a6 + 2¥ by a: + 2ab + 26 2 . 17- « 2 + # 2 + c 2 — «# — ac — 6c by a + b + c. 18. 1 - 2a + 3a a - 4a a 4- 5« 4 by 1 + 2a + a\ 19. a 2 - 5a - 9 by a 2 - 5a + 9. 20. Find the continued product of a — 2 by a — 2 by x - 2. 21. Multiply a 9 + 2a - 1 by a* — w + 1 and by a' - 3« — 1, and subtract the second product from the first. 22. Multiply 1 - 2a + 3a 2 - 4a 3 by 1 - a ~x\ 18 BRACKETS OR PARENTHESIS. SECTION V. BRACKETS OR PARENTHESIS. 19. Two or more terms are sometimes put in brackets or a parenthesis, and considered as a single term. The sign before the brackets indicates the operation to be performed on all the terms in them. If we remove the brackets, the operation must be performed. Brackets in Addition and Subtraction, 20. When the + sign is before the brackets, the terms are to be added. Thus, a + (b + c — d)=a + b + c — d. When the — sign is before the brackets, the terms are to be subtracted. Thus, a — (b + c — d) = a — b — c + d. Therefore we have the Rule. — When we remove brackets tvith the + sign be- fore them, the signs of the terms within remain unchanged. When we remove brackets with the — sign before them, we must change the signs of the terms within. Ex. 1. 2a — b + (b — a) = 2a — b + b — a = a. Ex. 2. 2 - a - (2 - 2a) = 2 - a — 2 + 2a = a. Ex. 3. 3 + a + {- a — 3) = 3 + a - a - 3 = 0. Ex. 4. 1 - b + ¥ - (- 1 + 2b) - (- ¥ + 1) = 1 - J + ¥ + 1-2* 4- V -1 = 1- db+ 2b\ Examples — 7. Simplify 1. 6 - (5 + 3) + (2 - 4) - (3 - 10). 2. a — b — c — (a + b — c). BRACKETS OR PARENTHESIS. 19 3. 2x - dy - {2x + 3y). 4. 1 - b + b % - (1 - b + /r - 6 3 ). 5. 9a + 126 + c — (a + lb + c) — (Sa A- Ab). 6. a — [b — c — (d — c)] = a — b + c+ (//- c) — a — b + c i-d — c=a — b+d. 7. Qa - [U - (2a - b)]. Brackets in Multiplication, 21. When brackets are used in multiplication, they mean that the number or letter before or after the brackets is to be multiplied by all the terms in them. Thus, a (Jj — c ) means a multiplied by b — c. (a + b + c)x means a + b + c multiplied by x. Notk.— Instead, of using brackets to indicate multiplication, we often use a line over the polynomial as follows ; a . b +c. This line is called a vinculum. 22. When two pair of brackets are used in multiplica- tion, they mean that all the terms in one pair are to be multiplied by the terms in the other. Tims, (a + x) {a + 2x) means a + x multiplied by a + 2x. (a + b) 2 means (a + b) (a + b), i. e., a + b multiplied by a + b ; (5a)* == (5a,) (5a), i. e., 5a x 5a. 23. Hence, in multiplication we have the Rule. — Wlien we remove brackets, we must first perform the multiplication indicated. Thus, a (b — c) = ab — ac ; a x b + x — ab + ax ; (a + x) (a + 2x) = a{a + 2x) + x{a + 2x) = a 2 +■ 2ax + ax + 2x°- - a 2 + 3ax + 2x' 2 ; 5 (x _ 2) - 6 (x - 3) = 5x - 10 - 6z + 18 = - x + 8. 20 MULTIPLICA TION B Y INSPECTION, Examples — 8. Simplify 1. 6 (x - 5) + 3 (x - 4) - 5 {x - 2). 2. a (b — 6') — b (a — c) + c (a — b). 3. (x - 5) (x + 5) - G (x* - 25). 4. (a + b + c) x — {a + b — c) x. 5. (3x - 2) (3 - ab + 6 2 « 2 - 2«6 + 6 2 or (a - b) 2 = a 2 - 2ab + b 2 . . . . (B), which expresses the Rule : — TJie square of the difference of two quantities is the square of the first, minus twice the product of the first by the second, plus the square of the second. Ex. 1. (x - 5) 2 = x 2 - 10s + 25. Ex. 2. (da - 2b) 2 =, (3a) 2 - 2 x 3a x 2b + (2b) 2 =s 9a 2 - 12a5 4- 46 2 . T/*e Swm o/ Two Quantities Multiplied by their Difference. 27. (a + b) (a — b). Operation, a + b a — b a 2 + ab - ab - b- a 2 ^~b 2 or (a + b)(a-b) = a 2 - b 2 . . . .(C), which expresses the Rule : — Tlie sum of two quantities multiplied by their difference is the square of the first minus the square of the second. 22 MULTIPLICATION BY INSPECTION. Ex. 1. (as + 5) (x -5)= x* - 5 2 = .r - 25. Ex. 2. (2a + 3b) (2a - 3b) = (2a)"- (3b)' = 4« 2 - W Ex. 3. (46 + 3) (46 - 3) = (U) 2 - 3 3 = 1G6 2 - 9. The Product ofx + or — one Number by x + or another Number. 28. (x + a) (as + b). Operation, x + a x + b x + aas + bx + ab x" + (« + 6)a + a# or (as + a) (x 4- 6) = x 2 + (a + J)as + «6 Also, (as — a) (x — b) = x* — (a + b)x + ab (x + a) (x — b) — x 1 + (a — b)x — ab J (D). Hence, the Rule: — The product of xplus or minus a number, by x plus or minus another number, is x 2 plus or minus the (algebraic) sum of the numbers multiplied by x, plus or minus the product of the numbers, according to the sign. Ex. 1. (x + 2) (x + 3; x 1 + 5x 4- G. 3) X s + (2 + 3) x + 2 x 3 = as 3 - (2 + 3) as + (- 2 x Ex. 2. (as - 2) (x - 3) = as 2 - 5as + 6. Ex. 3. (x + 3) (as - 2) = x 1 + (3 - 2) x f 3 x - 2 = as 2 -4- ^ — G. DIVISION. Ex. 4. (x - 3) (x + 3) = s 9 + (2 - 3)» 4- 2 x : .T 2 - Z - 6. Examoles — 9. Write the squares of 1. m + n ; m — n. 2. m — 2n ; 3# + 46. 3. 4« - 6 ; 5« - 36. 4. 7x — 4y ; # 2 J r 3##. 5. 1 + 2« 2 ; 3:r - 2a 2 . Write the products of G. (a + 2b) (a - 26) ; (2a + 6) (2« - b). 7. (z + 3#) (x - 3y) ; (3a + 4c) (3« - 4c). 8. (8x + 9y) (8x - 9y) ; (z a + if) (.*" - ?/ 2 ). 9. (3ax + 6) (3az — b) ; (ma; + 2«#) (mx — 2a?/). 10. (4x 2 + 1) (4z a - 1) ; (x* - 4) (x* + 4). Write the products of 11. (x + 5) (x + 3) ; (x - 6) (x - 4). 12. (x - G) (z + 2) ; (x - 8) (x + 10). 13. {x + 7) (a - 0) ; (a + 6) (a - c). SECTION VII. DIVISION. 29. + a x + b = -h ab ; hence + fl5 fa x — 5 = — ab ; hence — ab + a = + b. + a — — b. a x + b — — a3 ; hence — ab -i a = -h b. a x b = + ab ; hence + ab a — — b. 24 DIVISION. Hence, for signs in division, we have the Rule. — Like signs give +, and unlike signs give — . a 6 30. « 4 x a 1 = « 6 ; hence, — = a\ a Therefore, to divide two powers of the same letter, we subtract the exponent of the divisor from that of the divi- dend. 31. Hence, to divide one monomial, or single term, by another, we have the Rule. — Divide the coefficients, observing the rule of the signs. To this quotient annex the letters, subtracting the exponents of the like letters. Ex. 1. - Cja'F ~ 3a'b = - 2a 2 b. Ex. 2. Wb ~ ±ab = 2a. Ex. 3. - lOcrbc -f- 5a = — 2abc. Note.— It is plain that if the coefficient of the dividend is not exactly divisible by the coefficient of the divisor, or if a letter enters the divisor which is not in the dividend, or if a letter in the divisor has an exponent greater than the exponent of the same letter in the dividend, the division is impossible, since in all these cases the divisor has factors not contained in the dividend. In such case, the operation is expressed after the manner of a fraction in arithmetic. Examples — 10. Divide 1. 8.?; by 8 ; 8a by — a ; abc by — a ; — axy by y. 2. 12xy by 3x ; — Saxy by + Sax. 3. - 15ab 2 c by - Sab ; - 20« W by 5abc. 4. — 4cV£V by — d'bc 2 ; J4«V?/ by lay. DIVISION. 25 32. To divide a polynomial by a monomial. Rule. — Divide every term in the dividend by the di- visor. Ex. 1. Ex. 2. Divide Sx ) 40z a - 2±ax ; Quotient, ox — 3a ox ) 12.r 3 - 21ax' i + 3a*x Quotient, 4uv* — lax + a* Examples — 11. 1. 2ab + 3ac — kad by a. 2. ax -f bx 2 — cxy by — x. 3. 6«V - Sabx + 2ax s by - 2«z. 4. Uc + 35a£c 2 - 10£V by 5bc. Division of Poly nominate. 33. Definition. — The terms of a polynomial are said to be arranged according to a given letter, when beginning with the highest power of that letter we go regularly down to the lowest, or when we begin with the lowest and go up to the highest. Thus, 4c 5 — 3x* + 2x- — 1 is arranged with reference to x. 34. To divide one polynomial by another. Rule. — 1. Set down the divisor and dividend as in long division in arithmetic, taking care to arrange them both according to the same letter. 2. Divide the first term in the dividend by the first term of the divisor, for the first term of the quotient. Tfien 2 26 DIVISION. multiply the divisor by the first term of the quotient, and subtract the product from the dividend. Divide the first term of the remainder by the first term of the divisor, and proceed as before, continuing the process with the terms that remain. Ex. 1. Divide x~ + kax + 3rr by x + 3a. x + 3a ) x" + &az + 3« 2 (x 4- a, quotient. x' 2 4- 3«£ ax + 3«' 2 a:c 4- oa 2 Ex. 2. a — b)a" — 2ab 4- & 2 ( « — b, quotient a' 2 — ab -ab + b' 2 -ab + V Ex. 3. Ix- 3 ) 7z 8 - 24z a 4- 58a; - 21 ( z 2 - Bx 4- 7, quotient. 7;<: 3 - 3.1' 2 -2Lr + 58.£ — 2Lr4- 9.?; Divide 4-49.C-21 + 49a; -21 Examples — 12. 1. a? 4- 6x + 6 by ay'+ 3. 2. 8a 2 4-17^ + 9 bv 8a 4- 9. DIVISION. 27 3. 21a 2 + ax - 2.r by 7« - 2x. 4. x 2 - ldx + 40 by x - 5. 5. x 2 + y 2 + z 1 + %xy + %zz 4- 2yz by x -f ?/ +2. G. a 2 — b 2 — c 2 — 2bc by a — b — c. 7. a 2 — a — 30 by « + 5. 8. a 4 + a 2 b 2 + 5* by a 2 + «& + & 2 . 9. « 3 — 3« 2 6 + oab 2 — tf by a 2 — 2ab + £ 2 . 10. x* + 7.r + 2.r + 15 by x 2 - x + 5. 11. .ry -f 2.*r — 3?/ 2 — 4:y z — £2 — 2 2 by 2x + 3y + z. 12. 15« 4 + 10a 3 x + 4aV + 6rw 3 - 3.i- 4 by 3« 2 - x 2 + 2«£. 13. ab + 2« 2 — ob 2 — 4fic — ac — c 2 by 2a + ob + c. 14. 3z 4 + llr 5 + Ox + 2 by a 2 + 5.« + 1. 15. 3x* + kabx 2 - 6a 2 b 2 x - 4aW by 2tf£ + x. 35. Remark. — Incomplete Polynomials. — Sometimes some of the powers of the letter are wanting in the dividend or divisor, or in both. In such case it is best for beginners to leave a space for the wanting terms. Ex. 16. Divide a b - 32 by a ~ 2. a -2) a" - 32 (a* + 2a' + la 2 + 8a + 16. a b - 2« 4 + 2a 4 -32 + 2a* - 4« 3 + 4a 3 - 32 + hC - 8a 2 + Sa 2 -32 + 8a 2 - lGrt + I60 - 32 + 16a - 32 28 FACTORING. Ex. 17. Divide Go; 4 - 96 by 2a; 3 + 4a; 2 + 8a; + 16. 2x 3 + 4a; 2 + Sx + 16 ) Gx* - 96 (3a? - 6 6.t 4 + 12a; 3 + 24a: 2 + 48a: - 12a; 3 - 24a; 2 - 48a; - 96 - 12a; 3 - 24a; 2 - 48a; - 96 Divide 18. 9rt 2 6 2 - 16a; 2 by 3ab + 4a;. 19. 8a; 3 - 2W by 2x - 3a. 20. 16« 4 - 81 by 2a - 3. SECTION VIII. FACTORING. 36. By reversing the formulas in Section VI., we may determine, by inspection, the factors of certain polynomials. We have seen that (x + a) (x — a) — x* — a\ (Art. 27.) Hence, the factors of a; 2 — a 2 are x -\- a and x — a. Thus, also, the factors of a; 2 — 25 are x + 5 and x — 5 ; of 9a; 2 - 4 are 3a; -f 2 and 3a; — 2. 37. (x + a)- = x 2 + 2ax + a' ; hence, the factors of x~ + %ax + « 2 are a; + # and x + a ; of ar + 10a; + 25 are x + 5 and a; + 5. (Art, 25.) 38. (x — af — x* — 2ax + a 2 ; hence, the factors of x"' — 2«a; -f- a" 2 are x — a and a; — a. (Art. 26.) Ex. The factors of a; 2 — 8a; + 1 6 are a; — 4 and x — 4. FACTORING. 29 39. We have seen (x + a) (x + b) — x- + (a + #)# + «& ; hence, the factors of a; 1 + (a + #)# + «& are a; + a and a; + J. (Art. 28.) Ex. The factors of a; 2 + ox + 6 are « 4- 3 and a; + 2. 40. (a; — a) (x — 5) = a: 2 — (« + #)# + a& ; hence, the factors of x 2 — (a + #)a; + ab are x — a and x — b. Ex. The factors of or 2 — 7a; + 10 are x — 5 and a; — 2. 41. (a; + a) (x — b) = a; 9 + (a — #)a; — rt& ; hence, the factors of x 1 + (rt — b)x — ab are a; + a and a: — b. Ex. 1. The factors of x* + f>x — 6 are a: + 6 and a: — 1. Ex. 2. The factors of a; 9 — 5x — 6 are a; — 6 and a? 4- 1. Examples — 13. Find the factors of 1. x 2 + 4a: + 4 ; 2? 9 - 12a; ■+ 36. 2. 4a: 2 - 12a; + 9 ; or + 2«c + c\ 3. a;' 3 — 4 ; 4a 2 — 4«c + r. 4. 4a; 2 - 9 ; IQcrV - 9c 2 . 5. 16ft 2 - G4 ; 9mV - 25. 6. a; 4 - a 4 ; 36#V - 25&y ; 1 - 4a; 2 . 7. a; 2 + 9a; + 20 ; x 2 + 4a; + 3. 8. a; 2 - Gx + 8 ; a; 2 + a; - 6. 9. s 2 — a? — 6 ; x' - 8a; - 20. 10. a: 2 - Gx - 7 ; ar + 6a; - 7. 11. x 2 - x - 72 ; a-' - 2a; - 99. 12. x- — 12ax + 32« 2 ; «V — 4dxbx + 4S 2 . 13. x 2 -ex — 110c- ; x n - - %lcx + 110c 2 . 30 GREATEST COMMON DIVISOR. SECTION IX. GREATEST COMMON 1)1 VI SOB. 42. To find the greatest common divisor of two mono- mials. Rule. — Find the G. C. I), of the coefficients by the rule in arithmetic, and of the letters separately by inspection, and multiply the results. Ex. 1. Find the G. C. D. of l§a b Vc and 45« 4 5V. The G. C. D. of 18 and 45 is 9 ; of a' and a b is a* ; of ¥ and b* is b- ; of c and c 4 is c. Hence, the required Gr. C. D. is da^b-c. Ex. 2. The G. 0. D. of 45aV# and GOafy 9 is IZx'y. Ex. 3. The G. C. D. of 54&W and 72¥c i x i is 18J s cV. 43. To find the greatest common divisor of two poly- nomials. Rule.— Proceed by the rule for numbers in arith- metic : Divide the greater polynomial by the less, then divide the less by the remainder, and then the first re- mainder by the second remainder, and so on till there be no remainder. The last divisor will be the Greatest Common Divisor. Note.— We may take out a numerical factor from any divisor, or multiply any dividend by a numerical factor, if necessary, to make the division possible. Any factor common to all the terms of both polynomials is a part of the G. C. D. GREATEST COMMON DIVISOR. 31 Ex. 1. Find the G. 0. D. of Gar - llz + 4 and 2z 2 - bx + 2. 2x i -5x + 2)Qx' i -llx + 4:(3 6rc 2 - 15a; + 6 4x -2 or 2 (2a; -1). 2x- -l)2a; 2 2a; 2 -5a* + 2(a?- — X -2 -4r + 2 -4a; + 2 Hence, 2a; -lis the G. C. D. Ex. 2. Find the G. C. D. of 8x 2 -f 7z — 1 and Ox' + ?# + 1. 7a; -1 &K 2 + 7x 4 + 1 8a; 5 + ) 24c 2 + 28a; + 4(3 24k 2 + 21a; -3 7x + 7 or 7(a; +1). a; 4- l)8a;* + 7a;- 1 (8a;- 1 82;* + 82; - a;-l - x -1 Hence, a; + 1 is the G. C. D. Examples — 1 4. Find- the G. C. D. 1. Of 64«W and 48« 4 ^. 2. Of 78«-^ 3 and 52«s\ 32 LEAST COMMON MULTIPLE. 3. Of 9GcV# 2 and 108cy. 4. Of x" + 2x + 1 and x* - bx - 6. 5. Of x 1 — 4# + 4 and a; 2 - 5a: + 6. 6. Of a? + 8z - 9 and x" + 17.t + 72. 7. Of x 2 — %x + 1 and 3a 9 - 5# + 3. 8. Of 3ar 2 — Kto + 8 and O.r- - 5# - 4. 9. Of 3x°- - 20# + 32 and 15x 2 — 64z + 16. SECTION X. LEAST COMMON MULTIPLE. 44. To find the least common multiple of two or more monomials. Rule. — Find the L. C. M. of the numerical coefficients by the rule in arithmetic, and of the letters separately by inspection, and multiply the results. Ex. 1. Find the L. C. M. of 4«V and 6ax\ Result, 2)4, 6 a ) ax*, a-x" 2, 3 12. x ) x % , ax" x ) x-, ax x, a Result, axxxa — atx 3 , Hence, L. C. M. = 12a*a 3 . Note.— The rule in arithmetic applies to polynomials also. REDUCTION OF FRACTIONS TO LOWEST TERMS. 33 Ex. 2. Find the L. C. M. of gf - 4 and o& - 2x + 1. x — 2 ) x' 2 — 4 x" 2 — 2x + 1 x + 2 L. C. M. = (a? - 2) (a + 2) (rz - 2) (x - 2)(z 2 - 4) = re" - 2z 2 - Ax + 8. Examples — 15. Find the L. C. M. of 1. 2a, V2ab, and Sab. 2. a*, b\ and 2bc. 3. 16« 2 , 12a 3 , and 30« 4 . 4. a#, «c 2 , arc, and 5c. 5. 8z 4 , IWy, and 12^y. 6. 7a 5 , 42«% and 63« 6 . 7. 18a£ 3 and 12« 3 Z>. 8. ax + #7/ and ax — #?/. 9. 2(« + b) and 6(a 9 - b 2 ). 10. a; 2 - 2a + 1 and x* — 3x + 2. SECTION XL REDUCTION OF FRACTIONS TO LOWEST TERMS. 45. To reduce a fraction to its lowest terms. Rule. — The same as in arithmetic : — Divide the numer- ator and denominator of the fraction by their greatest common divisor. 2* 34 REDUCTION OF FRACTIONS TO LOWEST TERMS. Ex. 1. Reduce =-=-=-* to its lowest terms. 15a 2 x Cancelling out like factors, we have 4 Ex. 2. Reduce ^-^ ^tt to its lowest terms. 3ab — 2o~ 2ab - b°- (2a - b)b 2a dab - 2¥ ~ (3a - 2b) b 3a - 2b X' — 9 Ex. 3. Reduce — ^ . x- + 6x + 9 x* -9 (x — 3) (x + 3) x - 3 x- + Qx + 9 ~ (z + 3) 9 _ x + 3 Examples — 16. Reduce to their lowest terms ax abc bx ' bed * 3a - 35 a 3 - afl 36 " ' ab 25a&V «5^_ 15a V ' 2a£ 2 c 4 ' x* + xy 4:a*b — 5W x* — xy' %i)abc REDUCTION TO COMMON DENOMINATOR. 35 x* — y- x* — 2x -f 1 5. 6. x* + xy ' x~ — 3x + 2 ' x- — a" x- — 4 x' + 2ax + «-' ' ar + 4a; + 4 SECTION XII. REDUCTION OF FRACTIONS TO A COMMON DENOMINA TOR. 46. To reduce fractions to a common denominator. Rule. — The same as in arithmetic : — Multiply each numerator by all the denominators, except its own, for new numerators, and the denominators together for the common denominator. Or, Multiply each numerator by the quotient resulting from dividi?ig the least common multiple of the denominators by its denominator, and ivrite the L. G. M. of the denomina- tors for the common denominator. XXX Ex. 1. Reduce -r- , — , — to a common denomi- 4 o o nator. The least common denominator is 24. Hence the result, 6x 8x dx 24' 24' 24* Ex. 2. Reduce ^- , 0j and — — to a common de- nominator. 36 REDUCTION TO COMMON DENOMINATOR, 36ic 3 is the L. C. M. of the denominators. Hence the result, 6acx 4:bc abx*- 36^' 36a? ' 36a? ° Examples — 17. Seduce to a common denominator X X -, X L %' T' and 15 a¥ b + a , _a?_ Z ' Tex' "W 16gs° 3. a and a — x 4. — , — , and B 5ar + 4 , 10a: + 17 5. --^- and -_-, n 5x — 1 -. 27 + 2 6. — = and -£«— 4a 26a 6a , ha 7. r and a? — 4 a; — 3 4a 3a , 6a 8. , , and -= — -. x—Z'x + Z' x* — 4 5 6x 4 a; ' a; — 1 ' x + 1 ADDITION OF FRACTIONS. 37 SECTION XIII. ADDITION OF FB ACTIONS. 47. To add fractions. Rule. — The same as in arithmetic : — Reduce the frac- tions, if necessary, to a common denominator, then add the numerators and tvrite their sum over the common denomi- nator. Ex. 1. Add -, -r, and -. O 4: The L. C. M. of the denominators is 60. 20x 15a; 12a; _ TO" + "60 + 60 — 20a; + 15.r + 12a; 47a; 60 _ 60 5 10 Ex.2. Simplify -^ + -^ . The common denominator is x 2 — 1. 5(a+l) 10 (s-l) Hence, we have tf — \ + ~~& — 1 5.r + 5 + lOz - 10 15a; x- - 1 x- - 1 Ex. 3. Simplify ^ + y + 26. The L. C. M. of the denominators is 12. _ 9a 105 245 Hence, we have ^ + -^- -h -^ = 9a + 105 + 245 _ 9a + 345 12 " 12 38 ADDITION OF FRACTIONS. Examples — 18. Add 1. -, -, and -, a a a 2. and — x 3a; n X X , X 3. $, c p and - 4. ^-, — , and j- %a da 4a OX OX -, i X 5 - T' T' and 30 „ a a , « 3c 66c 86c „ a + b n a — 6 7. -y- and -j- . 3a; + 1 , 4iB - 5 8. — s and 24 4a; - 5 , 3a; - 4 9. a, — £ — , and — g — n J — Grt 10. 2rt and — ^ . 11. a and - . c SUBTRACTION OF FRACTIONS. 12. x and — ~ . _, n a — x x — y , y — a 13. , — , and * . 5 10 20 14. Simplify T H — r + ^ T 1 - x — 1 a; + 1 a? — 1 15. Simplify 8 + ^— — 16. Simplify — „ — + — ^ — SECTION XIV. SUBTRACTION OF FRACTIONS. 48. To subtract fractions. Rule. — The same as in arithmetic: — Reduce the frac- tions, if necessary, to a common denominator. Subtract the numerator of the subtrahend from that of the minuend and place the result over the common denominator. Ex. 1. From — take ^ . The common denominator is GO. 8a! 7x S6x 35# 36a; - 35a _ x Hence, 5 12 60 "CO 60 60 40 SUBTRACTION OF FRACTIONS. Ex. 2. From a take — = . 4 We hava 4a 4a — b 4a — (4a — b) 4a — 4a + b b Note.— The - before -^-r — belongs to the whole numerator, and hence 4« — b must be put in brackets with — before it when the common denominator is written. Examples — 19. i -n 5X , , 9X 1. From -y take yj Sx 2. From x take -^ . 3. From x take — ^ — Bx + 9 , , 4.^ + 3 4. From — ^— take — g — 5. From — 5 — take — ? — 6. From T take « -f # a — b 2 5 3 4 7. From - + - take - + -- 8. From - — r take — —^ x + 1 a + * 9. From - take x-2 x*—±' 10. From — take x + y x 111? a + b . « — 6 11. Jbrom — ^ take — T . « — 6» « + o 12. Prom - + 3 - 5 take -gj- 13. Simplify ac . 14. Simplify - h — - 1 " x + 1 z — 1 ar-1 SECTION XV. MULTIPLICATION OF FRACTIONS. 49. To multiply fractions. Rule. — The same as in the arithmetic : — Multiply the numerators together and the denominators together, after cancelling out factors whicli are the same in the numera- tors and denominators. Ex. 1. v- x 5 = t- . o b 42 MULTIPLICATION OF FRACTIONS. a c _ a x c _ ac Ex. 2. g- x j = y7^ -Yd* Examples — 20. 1. Multiply J-j-| by 10. 2. Multiply — by 2a. Q r K 3. Multiply 1( , by 9(3. 4. Multiply -- by — O A Q 5. Multiply -^— by T 6. Multiply T - by w 7. Find the continued product of ma? - 27ml? oa Multiply 1 + — by 1 -- — 9. Multiply r ^ + j-?- by T DIVISION OF FRACTIONS. 43 10. Multiply 1 ~ by 1 + ~— . rj a + I J 1 — a -.i Tv/r 14.- i x ~ — %x + 4 z 2 — 7a; + 10 11. Multiply ^^^— r5 by g8 _ 3 ^ + 2 SECTION XVI. DIVISION OF FBACTIONS. 50. To divide fractions. Rule. — Tlie same as in arithmetic : — Invert the divi- sor and proceed as in multiplication, cancelling when pos- sible. Examples — 21. 1. Divide by a. cy J 2. Divide "— by b. by J 3. Divide °~ bv 7. 4 4. Divide -.- by bx. 5. Divide Sab by — - J hm U SUBSTITUTION. n .p.. . , 2a — 6ac , _ 6. Divide by 2a. 7. Divide - — by — r - a; ^ ox _ _. ., «":r?/ . ay 8. Dmde - ^JL „ y _ ^ 9. Divide 2# H bv £ # * a; 1n -p.. ., 10(z + ?/) . 2{x + y) 10. Divide -£ ±1 by -^ ^~ 11. Divide 5 — by — 7 — . 12. Divide by = K — , x — bx + 6 J x 1 — 9 SECTION XVII. FINDING NUMERICAL VALUES BY SUBSTI- TUTION. 51. We find a numerical value for an alge- braic expression by substituting numbers for the letters in the expression and performing the operations indicated by the signs. Ex. 1. If a = 6 and b = 2, then a — J =? 6 — 2 = I J « a - £ 2 = 36 - 4 = 32. SUBSTITUTION. 45 Ex. 2. If a — 2, b = 3, c = 4, then 2c 3 - ftfo = 2 x 64 - 2 x 3 x 4 = 128 — 24 = 104. Examples — 22. Find the numerical values of the following expressions when a = 1, 5 = 2, e = 3, d — \, / = 5 : 1. ft - 5. 2. - ft - 5. 3. — ft — # — c. ^ 4. a — b — c — d. 5. a — b — c + d — f. 6. c + ft 1 + 5 — ft — /• 7. ftfo ; r/^ctZ ; ab + ac — be — 2bd. 8. 5ft 2 + 6 2 + c- ; 3ft' J + 5 2 +' c 1 - 6«T - 5/. Find the numerical values of the following expressions when ft = 10, J = 4, - (3c - 2/). 11. a - (i - e - d) ; 2a + (b - c - df). C — ft ft — n a a ft , ; 35d '6 c 5c ' 4ftc 14. (« + by -a"- V ; (ft 4- 5) 2 - (ft - b)\ 46 SIMPLE EQUATIONS. 16 e A- c l-fl 10 ' a" d* c 3 ' 16. a(ft + c) 4- b(a - c) - c(f- a) ; (a - 2) (b - c] 17 - 5 ~ -^- ; t + y " 6 18. 4 3 8(« - 1) 8(« + 1) SECTION XVIII. SIMPLE EQUATIONS. 52. A principal object of algebra, as of arith- metic, is to find from numbers which are known other numbers which arc unknown. 53. To do this, we put a letter for the unknown number, then make an equality from the given conditions ; this we call an equation, and from it find the value of the unknown letter. For example : If x 4- 4 = 9, then x = 9 — 4, or x = 5. 54. Note.— All expressions with the sign = between them are not, however, equations in the above sense, but are sometimes identities, that is, equalities in which both sides arc the same. Thus, 3+5 = 8 is an identity : x h Sx = 4x is an identity. So, too, (x + of = x' 2 + 2ax + a? is an identity, and is also called a formula. In such expressions x may have any value. SIMPLE EQUATIONS. 47 55. If x + 4 = 9, x can have but one value, 5. Definition. — A simple equation with one unknown letter consists of two expressions ivith the sign = between them, in which the unknown letter has a determinate value. 5ti. To solve an equation is to find the value of this un- known letter. 57. The equation is satisfied, or the solution is verified, when this value put for the unknown letter in the equa- tion makes it an identity. Thus, x + 4 = 9, gives x = 5 ; and 5 put for x in the equation gives 5 4-4 = 9, an identity, and the equation is satisfied. 58. The expression on the left-hand side of the sign == is called the^rs^ side. The expression on the right-hand side is called the second side of the equation. 59. Axioms concerning equations. Axiom 1. — Both sides of an equation may be multiplied or divided by the same number, and the equality still sub- sists. Axiom 2. — The same number may be added to or sub- tracted from both sides of an equation, and the equality still subsists. The equality still subsists when we change the signs of atl the terms on both sides, as this is simply multiplying both sides by — 1. 48 SIMPLE EQUATIONS. 60. Transposition. — To transpose a term is to change it from one side of an equation to the other. Rule. — When we transpose a term, ive must at the same time change its sign. This is the same as adding the same number to, or sub- tracting the same number from, botli sides. For example, if we subtract G from both sides of the equation x + 6 = 15, we have x + 6 — 6 = 15 — G, or x — 15 — G. Thus, 6 has been transposed from the first side to the second, and its sign changed from -f to — . So, also, in Qx = 2 — 3x. To transpose 3x to the first side is the same as adding 3x to both sides. Thus, Gx + 3x = 2 - 3x + Sx, or G.c + 3x = 2. So ox is transposed and its sign changed. SOLUTION OF EQUATIONS. Equations without Fraction*, 61. If the equation has no fractions, it is solved by the following Rule. — 1. Transpose the unknown letters to the first side, and the numbers or known terms to the second side. 2. Apply the rule of addition to the terms collected on the tivo sides. 3. Then divide loth sides by the coefficient of x. (Ax- iom 1.) SIMPLE EQUATIONS. 49 Example. Given 12a; - 27 = 37 - 3x + 41, to find x. Transposing, 12z + Sx = 37 + 27 + 41. Adding collective terms, 15# = 105. Dividing by lo, x = 7. Examples — 23. Find the value of a; in each of the following equations : 1. x +'4 = 10 ; x + 24 = 20. 2. 2a? + 5 +J* = a; + 14 + 2 ; 9x = 72. 3. 2z + 3a; = 55 ; 16a; — 2x — 6x = 25 + 4a;. 4. x - 12.5 = 13. G ; 12a; = 104. 5. 1 + 3a; + 3 + 5x = 5 + 7a; + 7 + 9a\ Equations with Letters for the Known Numbers. 62. We often have equations in which the known num- bers are represented by the letters #, b, c, etc. (first letters of the alphabet), as the unknown are represented by x, y, z. etc., the last letters. Ex. 1. 4a; + 3« - 25 = Via + x - Sb. Transposing, 4z — x — 12a — 3a -f 25 — Sb. Collecting, 3x = da — 6b. Dividing by 3, x = 3a - 2b. Ex. 2. mx — nx + c. Transposing, nix — nx = c. Collecting, (m — n) x — c. Dividing by coefficient (co-factor) of x, we have _ c ~ m — n* 50 SIMPLE EQUATIONS. Examples — 24. Find x in the equations : 1. x + a — b ; x — a = c. 2. 2x — 2a -{- c = b ; ax — c. 3. ax — bx -f jp = m — ft. 4. 3a; + J — a = 5x + c. Equations with Terms in Brackets, 63. When the equation contains terms in brackets, the brackets must be removed, as in Art. 20. Ex. 1. Qx - (2x - 18) = 22. Removing the brackets, we have Qx - 2x + 18 = 22. Transposing, Qx - 2x = 22 - 18. Reducing, 4x = 4, and x = l. Examples — 25. Solve the equations : 1. hx - (3 + 2x) = 12 ; a - 9 = 5(x - 5). 2. 5(a - 6) = - 40 ; 30 - 2x = 6x- (24 - z). 3. 1 4- Sx - (2x - 7) = 10. 4. bx + 3(4a - 15) = 50 - 2x. Equations with Fractions. 16. If the equation has fractions, we first get rid of the fractions. This is done by the following Rule. — Apply the process for reducing all the terms on SIMPLE EQUATIONS. 51 both sides to the least common denominator, dropping the common denominator. „ ., x bx 3x . The L. C. D. is 24. Hence we have 6x + 20? — 9x = 96. 173 = 9G, 96 KIA and a; = — = 5H. „ _ 2# SB — 2 . x — 3 Ex. 2. — + — ^— = 4 - — ^ r - Clearing of fractions, Ax + 3(3 - 2) = 24 - 2(a! - 3), or 4a; + 33 - 6 = 24 — 23 + 6. Collecting, 9a; = 3G, and 3 = 4. Ex.3. '- 1 to " a; - 2 4a; - 7 Clearing of fractions, (a — 1) (4flj — 7) = (43 — 5) (3 — 2). Removing brackets (Art. 21), we have Ax 1 - lis + 7 = Ax" - 13.? + 10. Striking out the common term Ax" from both sides, we have - llr + 7 = - 133 + 10. Transposing and reducing, 23 = 3, 3 and 3 = g . 52 SIMPLE EQUATIONS. Examples — 26. Find x in the following equations : 2a; 4a; 99 . x x _ 2 „ 2. ^ + T = ^ 4 5 3. |. + | + | = 156. *• a 9 + 6 „ 2x Zx ix „, 5 ' "3 + 5 " 15 = ** a; a; _ 3a 2x a_ 6 * a + 3~ ~~ ~9 ~ T 5 + 5 x + 12 a; - 10 = 1_ 7 * 7 10 " 2 ' n 2a? a? - 2 a? - 3 8 - y + -2- = 4 -~3-- ■ 3a; + 4 7a; - 3 _ x - 16 9 ' —5 2~~ - — i io. K* + 6) - T v(ifi - to) = H. ii A_±=±-l "' a; 4a: 5a; 20 PROBLEMS IN SIMPLE EQUATIONS. 53 21 42—5' 12. 6 5 x - 2 ~ x - 3 ' 12 2x +3 13. 2x -3 %x - 4 3# — 4 ~ 3z — 5 ' 14. 6a: - 4 a? - 2 21 ' 5^-0 2a; 7 " SECTION XIX. TRANSLATION OF ORDINARY LANGUAGE INTO ALGEBRAIC EXPRESSIONS. 65. As an introduction to the solution of prob- lems by algebra, we give some examples of translations from ordinary language into algebraic expressions. Ex. 1. A man has x dollars, and gains two dollars. How much has he altogether ? Ans. x + 2 dollars. Ex. 2. A man has x dollars, and loses 20. How much has he left? Ans. x - 20. Ex. 3. A man has x dollars, what is one-fourth of it ? one-third of it ? etc. A x x Ans. —r, -=-, etc. Ex. 4. If x is the price of a dozen oranges, what is the price of nine ? . 9x Ans. j2 . Ex. 5. A number exceeds x by 7. What is it ? Ans. x + 7. 54 PROBLEMS IN SIMPLE EQ UA TIONS. Ex. 6. x exceeds a number by 7. What is that number ? Ans. x — 7. Ex. 7. A man has x dollars, and loses 3 of them, and then loses ^ of what is left. How much has he after both losses ? Ans. f (x - 3). Ex. 8. x dollars at G per cent, interest will yield how much in one year ? . Gx Ans.- m . (jX Will amount to how much ? Ans. x + ^ . Ex. 9. A man having x dollars gave away ^ of it, \ of it, and ^ of it ; how much had he left ? fx x x\ VLx . x Ans . a _ (j + w + ¥z ) or x -—, t .e. B . Ex. 10. 500 is divided into two parts, one of which is x, what is the other ? Ans. 500 — x, Ex. 11. If a man goes x miles in 8 hours, how many miles per hour does he travel ? . x 1 Ans. 3- . o Ex. 12. If a man goes 15 miles in x hours, how many miles per hour ? .15 1 Ans. — . x Ex. 13. If a man buy x yards of cloth for 810, what is the price per yard ? .10 X Ex. 14. If a man pay x dollars a yard for 815 worth of cloth, what is the number of yards bought ? 15 Jlns. - — - . PROBLEMS IN SIMPLE EQUATIONS. 55 Ex. 15. A man goes x miles at the rate of 5 miles an hour ; what is the number of hours ? x Ans. =-. o Ex. 1G. If a man does a piece of work (working uni- formly) in x hours, how much of it does he in 1 hour, in 2 hours, in 3 hours, in 6 hours, etc. ? 12 3 6 Ans. — , — , — , — , etc. JD £ 'Jb JO Ex. 17. If a man does a piece of work in 10 days, how much of it does he in x days ? . x Ans. -. Ex. 18. A pipe fills a cistern in 8 hours, what fraction of it does it fill in x hours ? A x Ans. — . 8 Ex. 19. What number bears to x the proportion of 3 to 4 ? Ans. — . 4 Ex. 20. If two numbers bear to each other the ratio of 4 to 5, and one of them is 4%, what is the other ? Ans. 5x. Note.— Consecutive numbers are numbers each of which is greater by unity than the preceding one ; thus, 2, 3, 4 are consecutive numbers. Ex. 21. Write four consecutive numbers of which x is the smallest. Ans. x, x -f 1, x + 2, x + 3. Ex. 22. Write three consecutive numbers of which x is the greatest. Ans. x — 2, x — 1, x. Ex. 23. If £ is the tens figure, and y the units figure of a number, what is the number ? Ans. lOx -f y. 56 PROBLEMS IN SIMPLE EQUATIONS. Ex. 24. If x be the number of minute spaces moved over by the minute hand of a watch in a certain time, what number of spaces does the hour hand move over at the same time ? ^ m x_ ^ Ex. 25. A man travels 25 miles in 6 hours, how many miles does he travel in x hours at the same rate ? . 25# Am. — — . o SECTION XX. PROBLEMS IN SIMPLE EQUATIONS. 66. To form an equation we follow this Rule. — Represent the unknown quantity by x ; form the expressions and the equation according to the conditions of the problem. The equation thus formed is solved as explained in Section XVIII. Ex. 1. What number is that to which if 5 be added J of the sum will be 25 ? Let x = the number. Adding 5 to it, we have x + 5, and J of this is $(x + 5). By the condition, {(x + 5) = 25. Hence, x + 5 = 75, and x — 70. Verification. £(70 + 5) = 25, or 25 = 25, an identity. PROBLEMS IN SIMPLE EQUATIONS. 57 Ex. 2. What number is it of which the third and fourth parts together make 21 ? Let X = the number. . Then X 3 = its third part, and X 4 = its fourth part. Then by the questior X 1 3 X + r = 81. Reducing, 4x + 3a; : = 21 x 12, or 7X: = 252, and X : = 36. Ex. 3. Find two consecutive numbers such that J of the smaller added to ^ of the greater is equal to 5. Let x = the smaller ; then x + 1 = the greater. x x + 1 Hence we have -r -\ 5— = 5. 4 3 Reducing, 7x = 56, and x = 8. Hence, a; + 1 = 9, and the two numbers are 8 and 9. Ex. 4. Divide 54 into two parts, one of which shall be to the other as 4 : 5. Let 4x = one part, and 5x — the other part. Then 4ic + 5x = 54. .'. x = 6 ; 4x = 24, one part ; and 5x = 30, the other part. 3* 58 PROBLEMS IN SIMPLE EQUATIONS. Verification. 30 + 24 = 54. ?i _1 30 " 5"' Ex. 5. Divide eight dollars and a half into the same number of dimes, half-dollars, and quarters. Let x — the number of each. Then 10^ = the value of the dimes in cents, 50.r — "of the half-dollars in cents, 25x — "of the quarters in cents. Therefore, lOx + 50x +■ 25a; = 850 ; whence X = 10. Hence, 10 dimes, 10 half-dollars, and 10 quarters make eight dol- lars and a half. Examples— 27. 1. What number is that which multiplied by 7 is greater by 12 than 51 ? 2. What number is it, ^ of which is 3 greater than 15 ? 3. A train has 15 more freight cars than passenger curs, and 33 cars in all. How many of each sort in it ? 4. A garrison of 3,280 men has 3 times as many artil- lerists as cavalry men, and 4 times as many infantry as artillerists. How many of each of these men ? 5. What number is it J of which added to T 1 ¥ of it is equal to 20 ? PROBLEMS IJY SIMPLE EQUATIONS. 59 6. The difference between J and J of a number added to | of it is 22, what is the number ? 7. Find two consecutive numbers of which \ of the greater subtracted from J- of the lesser leaves 1. 8. Divide 45 into three parts which shall be consecu- tive numbers. 9. Find two consecutive numbers of which the lesser diminished by 8 is one-half the greater. 10. The sum of two numbers is 24, and 9 times the one is equal to 3 times the other. Find the numbers. 11. The difference of two numbers is 12, and 4 added to twice the smaller gives the greater. Find the two num- bers. 12. I paid £34 in half-dollars, quarters, and dimes, and used the same number of each of these coins. What was this number ? 13. A father is now 4 times as old as his son. Five years ago lie was 5 times as old; what is now the age of each ? 14. A company consists of 90 persons ; the men are 4 more than the women, and the children 10 more than the grown persons. Find the number of each. 15. The \, \, and | of a certain sum of money are to- gether £4 more than the amount itself. What is it ? 00 PROBLEMS— CONTINUED. SECTION XXI. PROBLEMS— Continued. 67. Ex. 1. A can do a piece of work in 4 days, and B can do it in 3 days. In what number of days will they both together do it ? Let the work be 1, and x = the required number of days. x In x days A does j of the work. x In x days B does ~- of the work. o Hence, _ + — 1 4 6 or 7x ■— 12, and x = ly days. Ex. 2. A, B, and C divide 700 acres ; A taking 4 acres to B's 5, and 3 acres to C's 2. How many acres did each get? Let x — A's number of acres; then \x - B's and \x — C's " " Hence, x + f x + \x — 700. Solving, x — 240 = A's acres. \x = 300 = B's " lx = 160 = C's " Ex. 3. A man had £2,000, a part of which he lent at 4 per cent, per annum, and a part at 6 per cent. The an- nual income from the whole was $92. Find the two parts. PROBLEMS— CONTINUED. 61 Let x = the number of dollars lent at 4 per cent. This produces Ax zr^. dollars per annum. 2000 — x = the number of dollars lent at 6 per cent, and this 6_ 100 yields — — (2000 — x) dollars per annum. 4r fi Therefore, m + m (2000 - x) = 92 whence x = 1400 dollars, and 2000 - x = 600 dollars. Examples — 28. 1. Divide 102 into 4 parts which shall be consecutive numbers. 2. A cistern is filled by one pipe in 8 hours, and by an- other in 3 hours — in what time will it be filled if both pipes run at the same time ? 3. Find a number such that if the half of it be taken from 3G, ^ of the remainder will be equal to f of the original number. 4. A and B being on the same road 21 miles apart, they set out at the same hour towards each other, A walk- ing 3 miles an hour and B at the rate of 4 miles an hour. How many hours will elapse ere they meet, and how far will each have walked ? 5. A can do a piece of work in 5 days, B in 6 days, and C in 8 days. In what time can they do it all working to- gether ? 62 PR0BLE3IS-C0JSTINUED. 6. A farmer had two flocks of sheep of the same num- ber. He sold 39 from one flock, and 93 from the other, and found he had remaining in one flock twice as many as in the other. How many were in the flocks at the be- ginning ? 7. A sets out for a town 12 miles off, and walks at the rate of 4 miles an hour. Half an hour afterward B sets out from the same place, in the same direction, running 5 miles an hour. How far from the town will B overtake A? 8. The denominator of a certain fraction exceeds the numerator by 2, and if 2 be subtracted from the numerator and added to the denominator, the new fraction thus formed is equal to J. What is the fraction ? 9. Seven maidens met a boy who was carrying a basket of apples. One maiden bought f of the apples ; the second, -fa ; the third, -J- ; and the fourth, ^ of them ; the fifth bought 20 apples ; the sixth bought 12, and the seventh bought 11, and this left the boy one apple. How many had he at first, and how many did the first four maidens take ? 10. Polycrates, the tyrant of Samos, asked Pythagoras the number of his pupils. Pythagoras answered him : The half of them study mathematics ; one-fourth part study the secrets of nature ; the seventh part listen to me in silent meditation, and then there are three more, of whom Theano excels them all. This will give you the number of pupils whom I am guiding to the boundaries of immortal truth. TWO UNKNOWN QUANTITIES— ELIMINATION. 63 SECTION XXII. SIMPLE EQUATIONS WITH TWO UNKNOWN Q UANTITIES.—ELI3IINA TION. 68. Simultaneous Equations are those winch are true for the same values of the unknown quantities which they contain. If x — y = 5, or x = y + 5 . . . . (1). Then, when y = 1, x = G ; y = 2, x = 7 ; y ■— 3, x = 8, etc., indefinitely. Now suppose, at the same time, » + y = » (^). Then among the values of # and ?/ in the above list, only x = 2, y = 7 will satisfy equation (2), and, therefore, I are the only values which belong to both of the x — y — 5 ) equations [■ taken together, or simultaneously. x + y = 9 ) J Hence, 2V;o simultaneous simple equations with two unknowns give one value for each of the two unknowns. 69. Elimination.— To find these values, we first re- duce the two equations to a single (qua Hon containing only one of the unknown letters. This process is called elimination. That ie, we first eliminate, or get rid of one of the unknown letters. Three methods of elimination are usually given. 70. First Method. — Multiply the given equations by 64 TWO UNKNOWN QUANTITIES— ELIMINATION. such numbers as will make the coefficients of one of the un- knowns the same in both. Then add or subtract the equa- tion thus obtained, according as these equal coefficients have contrary or the same signs. We thus get a simple equation with one unknown letter. Ex. 1. Given %x + 3y = 9 (1) ) , , r to find x and ?/. 4z - by = 7 (2) ) J To eliminate y, multiply equation (1) by 5, and equation (2) by 3. We have this. lOz + 15y = 45 12:r - 15y = 21 Adding, we get 22# = 66 .-. x - 3. We may find y by eliminating x, or more simply thus : Since x ■= 3, 2x — 6 ; and hence, substituting 6 for 2x in (1), we get 6 + Sy = 9, or y = 1. Therefore, * = 3 i . are the required values of x and y. y = 1 ) Ex. 2. Given 2x + 3w = 13 (1)' ) x a , 4z + 2t/ = 14 (2) 5 9 To eliminate x, we have 4,r + Qy = 26 ix + 2y = 14 Subtracting, we get Ay = 12 or y = 3. And putting 3 for y in (1), we have 2* + 9 = 13, or x — 2. TWO UNKNOWN QUANTITIES— ELIMINATION. 65 71. Second Method. —Find the expression for one of the unknown letters, as x, in terms of the other, in one of the equations, and substitute this for x in the other equation. Ex. 3. Sx - y = 6 (1). 5x + %y = 32 (2). The expression for x in (1) is x = *—= — . o Putting this for x in (2), we have 5 (^f 8 ) + 2 V = 32, or 5y + 30 + 6y = 96, lly = 66, and y = 6. Again, putting 6 for y in (1), we have 3x — 6 = 6, or 3a; = 12 and x = 4. 72. Third Method. — Express one of the unknown letters in terms of the other in each equation, and put these expressions equal to one another. Ex. 4. Given 4a + Sy = 19 . . . . (1). 2- X + 3x- 4?/ = = 14) =4 m 8a; + 12a;- 3y, 9y = = 113) X 4 + a; 2 + 3 " 4 " K8). = 7 \ 16a; + 24a* -1 17y = 05y = = 274/ 7*-3 2 , = ,■ 2x + 5y = 41 f l ; * 5(z+2)-3(y + l)=23| 3(a;-2)+5(7/-l) = 19i (12) - 3 + 10 ~ G7 ° 3a; W13). - y = 1250 3x = 2y + 14 J5_ _ 200 x — y~" x PROBLEMS— SIMULTANEOUS EQUATIONS. 67 — 3 -+ 8y= 31 1 1.7*-a.2y=-7.9) HIS). , K 1 M 16 )- y —-^ + 10*; = 192 ^ 4 J a - 1 y + 2 _ 2(x - y) } x ) "~3 T~ 5~" 3~ 7J=Z °l H^)- His). ?^5 _ lSl^ - 2v -x x - t - 10 SECTION XXIII. PROBLEMS PRODUCING SIMULTANEOUS EQUATIONS. 73. Ex. 1. The sum of two numbers is 21, and their difference is 5. Find the numbers. Let x — one number, and y — the other number. Then X + y = 21 1 x — y — 5 ) Adding, 2z = 26, and B= 13. Putting 13 for s», 13 + y = 21, From which we find y = 8, 68 PROBLEMS—SIMULTANEOUS EQUATIONS. Ex. 2. The sum of two numbers is 44, and if \ of the less be added to J of the greater, the result is 12. Find the numbers. Let x = the greater number, and y = the less number. Then x + y — 44 \x + \y = 12 From which we find # = 32, y= 12. Ex. 3. A certain number consists of two digits. The sum of the digits is 7, and if 45 be added to the number, we get a number the digits of which are the digits of the first number reversed. Find the number. Let x = the tens digit, and y — the units digit. Then 10# + y = the number, and we have x + y = 7 ) 10a + y + 45 = lOy + x ) which give X = 1, y= 6; or the number is 16 ; and 1G + 45 — 61, the digits of 16 reversed. Examples — 30. 1. Find two numbers such that -J of the one added to £ of the other shall be 20, but ^ of the latter added to J of the former shall be equal to 22. PROBLEMS— SI3WLTANE0US EQUATIONS. 69 2. On adding 18 to a certain number of two digits, we get a number in which these digits are reversed, and the two numbers added together make 44. What is the first number ? 3. Divide 46 into two such parts that when the greater part is divided by 7, and the smaller part by 3, the quo- tients together make 10. 4. A said to B, give me f of your money and I will have $100. B said tc A, give me J of yours and I will have 8100. Find how much A and B had at first. 5. The difference between two sums put out at interest for one year is $2,000. One sum is put out at 5 per cent, and the other at 4 per cent. , and the incomes from them are equal. Find the two sums. 6. A merchant has his house and goods together insured for $36,000. His house at 1| per cent, premium, and his goods at 2 per cent. The difference of the two premiums amounts to $6.75. What was the amount of insurance .on the house, and what on the goods ? 7. What fraction is that which is equal to T V when 2 is subtracted from its numerator, and equal to -J- when 4 is added to its denominator ? 8. If the number of cows in a field were doubled, then there would be 84 cows and horses together in the field. If the number of horses be doubled, and f of the cows be taken out there would be 80 horses and cows in the field. What is the number of horses and cows in the field ? 70 EQUATIONS OF THREE UNKNOWN QUANTITIES. 9. The sum of two numbers is 30, and their quotient is 4. Find the numbers. 10. In the second class of a school there are 1-J-J times as many pupils as there are in the first class. From the first class 7 go away, and 6 enter the second class, and then the second has 2f as many as the first. How many in each class ? SECTION XXIV. SIMULTANEOUS SIMPLE EQUATIONS OF THHEE OB MOliE UNKNOWN QUANTITIES. 74. If three equations are given, with three unknown quantities, we may eliminate one of the unknown quan- tities, and thus obtain two equations with two unknowns, and then from these two we may obtain one equation with one unknown, by the methods of Arts. 70, 71, 72. Example. Given 7x + 2tj + Zz = 20 (1) ^1 3x — fy + 2z = 1 (2) )■ , to find %, y, and z. -2x + 5y + 7z = 29 (3) J First, to eliminate y between (1) and (2), we have 14a; + 4?/ + 6z = 40 3x — 4// + 2z = 1 Hence, 17a; + Sz = 41 (4) EQUATIONS OF THREE UNKNOWN QUANTITIES. H Second, to eliminate y between (1) and (3), we have 35a + 1% + I02 = 100 - 4a + lOy + 14s = 58 Hence, 39a + z = 42 (5). Third, to eliminate z between (4) and (5), we have 312a + Sz = 336 17a + 8z = 41 Hence, 295a = 295, or a = 1. Then, from (4) 17 + Sz = 41, or Sz = 24, and z = 3. And from (2), 3 — 4y + 6 = 1, or 4y = 8, and y = 2. Examples — 31. Solve the simultaneous equations bx 4- 4y - 2z = 14 ^ 3a: + 2y + z = 16 J. (1). I a; - 9// + 82 = 7 J # + y + 2? = 1 # — # + 2= 4 J> (2). 5# + y + z = 20 72 EQUATIONS OF THREE UNKNOWN QUANTITIES, x + y = 8} y + z = 6 K3). x + z = 10 J y-X = iz z - x — \y ►(*)■ x + % = 2(y - 1) , 2 + 3= 9 ,(5). z x r 5 + 2= 5 > 3a + 4y + z -- 14 "| 2a; + y + 52 = 1!) ►(6) 5z + %y + 3z = 18 , 40a: - 1G# + 25s = 6 1 12a; + 4ij + 20z = 02 V (7). 36a; + 17?/ — 152; = 5G ^ INVOLUTION OR RAISING TO POWERS. 73 SECTION XXV. INVOLUTION OB RAISING TO BOWERS. 75. Involution is the process of raising quantities to powers. It is multiplication, therefore, in which the fac- tors are all equal, and requires no rules different from those already given (Section IV.). 76. We will notice some results, however, which will be of use in shortening the process. (1.) Rule of the Signs. — Any evenpoiver of a minus quantity is plus. And any odd power of a minus quan- tity is minus. Thus, the square of — a is — a x — a = -f a~. The fourth power of — a is — a x — a x — a x — a = + a\ The third power of — a is — a x — a x — a = — a\ The fifth power of — a is —ax —ax —ax —ax — a = — a\ (2.) Rule of the Exponents. — To raise a quantity to a power, multiply its exponent by the exponent of the re- quired power. Thus, (a") 2 —a' x a* = a* = a*** ; (a 3 ) 2 = a 3 x a 5 — a 6 = rr* 2 ; (a*)* = a 1 x a* = a* — a**\ 74 INVOLUTION OB RAISING TO POWERS. 77. Powers of Monomials. — To raise a monomial to a power, Rule. — Raise the coefficient to the required power, ob- serving the rule of the signs, and multiply the exvonent of each letter by the exponent of the power. Thus, the square of a~b z is a 4 b\ The third power of - 2ab is - SaW. The fourth power of 2ab°~c 2 is 16« 4 &V 2 . 78. Squares of Monomials. — As we will have mainly to do with squares in this book, we will particular- ize for this case. Therefore, to square a monomial, Rule. — Square the coefficient and multiply the exponents by two. Thus, the square of 5a*b*c is 25# fi #V 2 . 79. Powers of Fractions. — To raise a fraction to a power Rule. — Raise the numerator and denominator to the re- quired power, observing the rule of the signs. . 2 V 2 2 4 Thus, (_ )= ¥ x T = y a \ _ a a' _ a TV ~ T x T 3 ~ T HI) INVOLUTION OR RAISING TO POWERS. 75 80. Squares of Fractions. — As a particular case, to square a fraction, Rule. — Square the numerator and denominator. 81. Powers of Polynomials. — To raise a polyno- mial to any power, we simply perform the multiplications indicated by the exponent of the required power (always one less than the exponent). Thus, (a + by = (a + b) (a + b). (a + b) z = (a + b) (a + b) {a + b). (a + by = (a + by x (a + 6) 2 = (a + J) (a + 6) (a + b) (a + b). 82. Squares of Polynomials. — For the squares of polynomials, we repeat the two important results of Articles 25, 26. (a + b) 2 = a 2 + 2ab + b* = (- a - b)\ (a - by = a' - 2ab + b* = (b - «) 2 . Also, by performing the multiplications required, we find the following useful results, (a + b + cy = a" + 2ab + 2ac + b- + 25c + c 2 . (« - b + c) 2 = a" - 2ab 4- 2«c + b* - 2fo + c 2 . Examples— 32. 1. Find the third power of 3ab\ 2. Find the third power of — 2a 2 bc. 76 EVOLUTION OR EXTRACTION OF ROOTS. 3. Find the fourth power of ^ . Square each of the following quantities : 3 ex 2ay ' 2x 6 4. -Sab. 5. 8crbc\ 6. |--. 7. - ^ Write the squares of 8. a + 2. 9. 2bc + 1. 10. 3m — 5«. 11. #C£ + #• 12. — + c. o 13. Find the square of 2a + 3b — c. 14. Find the third power of a — £. 15. Find the fourth power of a + b. SECTION XXVI. EVOLUTION OB EXTRACTION OF ROOTS.— SQUARE ROOT. 83. Roots of Quantities. — The root of a quantity is the quantity the involution of which produces the given quantity. Thus, the square root of a* is a, because a squared gives a 2 ; also, the cube root of 8a 3 is 2a, because (2a) 3 = 2a x 2a x 2a = $a\ The fourth root of 16a* is 2a, because (2a)* = 2a x 2a x 2a x 2a = 16a 4 . EVOLUTION OR EXTRACTION OF ROOTS. 77 84. Evolution is the process of finding the roots of quantities, and is the reverse operation of involution ; and the rule of evolution, or extraction of roots, is found from the results of raising to powers. 85. The signs of evolution, or of roots to be extracted, are placed on the left of the quantities, and are as follows : V for " the square root of," y' - for " the cube root of," J/ for "fourth root of," etc., etc. 3, 4, etc., being called indices of the roots (the index 2 being understood when none is written). 86. We will notice some results which follow directly from the rules of involution. (1.) Any even root of a + quantity may be either + or —, and must, therefore, be written with the double sign ± (plus or minus). Thus, \/9 = ±3, V4^ = ± 2a, ^/a T = ± a. (2.) Any odd root of a quantity has the same sign as the quantity itself Thus, \/a 3 — + a, and V— « 3 — — a. (3. ) There can be no even root of a minus quantity. Thus, the square root of — a 2 cannot be extracted, for 78 EVOLUTION OR EXTRACTION OF ROOTS. (+ af = + a% and (— a)' 2 = + a' ; and, therefore, such expressions as V— d 2 ? are called impossible, or imaginary quantities. SQUARE ROOT. 87. Under evolution we shall only give the rules for finding the Square Root of algebraic quantities. 88. Square Root of Monomials. — The square of 2« 3 Z> 2 is 2« 3 & 2 x 2a 3 b' 2 — Aa 6 b* ; hence the square root of 4« 6 Z> 4 is ±2a*b\ (Art. 86 (1).) Therefore, to find, the square root of a monomial, we have the Rule. — Take the square root of the coefficient, and di- vide the exponents of the letters by 2. Write the sign ± before the result. Note.— A minus quantity has no square root. (Art. 86 (3).) 89. Square Root of Fractions.— The square of 2a .4a 2 ,, , - 4« a . , 2a ± — is op ; hence the square root oi ^ is ±^-, or, to extract the square root of a fraction, Rule.— Take the square root of the numerator and de- nominator. 16a 6 x 2 4:a*x Ex. 1. The square root of _ ,, , is ± Ex. 2. Find Vf + *#■• First add these fractions : f + a 3 a Hence, V§ + ¥ = W = ± I- EVOLUTION OR EXTRACTION OF ROOTS. 79 Exampl es— 33. ind the square root of the following quantities : 1. 25a 2 u\ 2. lOOftVj/ 2 . 3. 49a?b\ 4. 9ftV 4*y * 5. Wtfb* 6. $xy ±9xy ' ~Ja* ' 7. 19ft 2 3a- ~16" + ~8~* 8. 6 2 25 + y 9. 4 1 9 3' 10. 37 3 16 7" 90. Square Root of Trinomials. — Since (« + by— a 2 h- 2ft# -f- & 2 , the square root of a 2 + 2ab -f b 2 is ± (ft -f &). Since (a — b) 2 = ft 2 — 2ab -f Z> 2 , the square root of « 2 - 2ft£ + b 2 is ± (ft - J). Therefore, to find the square root of a trinomial, we have the Rule. — Arrange it with reference to one of its letters, then take the square roots of the first and last terms, and put the sign of the middle term between them. Ex. 1. The square root of x x + Gx + 9 = a/s? 4- a/9 = ± (a? + 3). Ex. 2. The square root of 4ft 2 - 12 ab + W = Via* - VW - ± (2a - 3*). 80 EVOLUTION OR EXTRACTION OF ROOTS. 91. Note 1. — Any number or algebraic expression is called a per- fect or complete square when its square root can be exactly found. Note 2. — A trinomial is a complete square if, when arranged by one of its letters, the middle term is twice the product of the square roots of the first and last terms; that is, when the square of the middle term is four times the product of the first and last terms. Note 3. — Since a monomial squared gives a monomial, and a bino- mial squared gives a trinomial, a binomial cannot be a complete square. Examples — 34. Find the square root of the following trinomials : 1. a 2 + %a + 1. 2. of + 4 — 4=x. 3. x 2 + 5x + -\K 4. 9a*b 2 - Qabx 4- x\ 5. 16a 4 - 2±aW + W. 6. 64a; 4 + f 6 2 - 2Ux\ 7. a 2 + i + a. 8. 16V -8a; + 1. 9. x 2 - 3x + f, 10. Are x 2 - 6x + 4, a 2 b 2 - 2abx + x\ a 2 - \a + f complete squares ? 92. Completing the Square.— In equations we have often to make such algebraic expressions as x u + px, and EVOLUTION OR EXTRACTION OF ROOTS. 81 x 2 — px, complete trinomial squares by adding the right term. This process is called Completing the Square. Examining the trinomial squares given in Art. 90, yiz., a~ + 2ab + ¥ , a 2 — 2ab + b 2 , we see that in order to make any binomial expression of the form x 2 + px, or x 2 — px a complete trinomial square, we have the Rule. — Add the square of half the coefficient or factor of x in the second term. Ex. 1. To x 2 + 4:X add 2% and we have x 2 + 4# + 4, the square root of which is ± (x + 2) . Ex. 2. To x 1 — Sax add (-«-)> and we have x 1 — 3ax + — , the square root of which is ± Ix — x- Ex. 3. To x 2 — %x add (f ) 2 , and we have x 2 — fa; + $, the square root of which is ± (x — |). E x ample s — 3 5 . Complete the squares of the following expressions, and find the square root of each result. 1. x 2 + 6x. 2. x 2 - 12a. 3. x 1 - lias. 4. a; 2 — a. 5. f + 3y. 6. a; 2 — px. 7. a 2 - fa. 8. x 2 — 4#2. 9. y f + 4*A 10. a: 2 - }x. 1. /v 2 9 /y 12. x 2 + fa?. 4* 82 EVOLUTION OR EXTRACTION OF ROOTS. 93. Square Root of Polynomials.— The rule for the square root of polynomials is similar to the rule in arithmetic. We know that the square root of a 2 + 2ab + b 2 is a + b. Hence, writing down the terms, and proceeding as in arithmetic, Ave have the following arrangement : a 2 + 2ab -f b 2 \a + b a 2 2a 2a + b ) 2ab + b 2 2ab + b 2 That is, 1. Arrange the polynomial with reference to one of its letters. 2. Find the square root of the first term, and subtract its square from the polynomial. 3. For a divisor double the first term of the root. Di- vide the first term of the remainder by this divisor ; the quotient will be the second term of the root. 4. Place this second term in the root, and lo the right of the divisor. Multiply the divisor, thus increased, by the second term of the root, and subtract the product from the dividend. 91. If the root contain more than two terms, a like pro- cess continued, doubling each time the root already found, will find all the terms. Thus, we know (a + b + a)' = a 2 + 2ab + b 2 + 2ac 4- 2lc + c\ EVOLUTION OB EXTRACTION OF HOOTS. 8£ Hence, to find the square root of this latter exjn'ession, we proceed as in the following arrangement : a 2 + 2ab -f ¥ -f 2ao -f 2bc -f c 2 1 a + b 4- c a 2 2a + b | 2ab + b' 1 2ab + 6 2 2a +2b + c | 2ae -f 2Z>6* + c 8 2ac + 2bc + c- 2a Examples— 36. Find the square root of 1. 64a 2 + lUab + SlbK 2. a; 4 + 4a# s + Qxfx 1 + 4a 8 # + a\ 3. a 4 - 4^ s + Sx + 4. 4. 9a 2 - 12afl 4- 4£ 2 -f Gac - 4fo + c\ 5. 9a 4 - 12a r ' f 10a 2 - 4a + 1. 6. 10a 4 + W 4- & - lGcrb -f 8aV - 4#c*. 7. 9c 4 — 6ar + a- + 12c 2 - 4a + 4. 8. a 2 + Z> J -- 2ab + 4ac — 4fo + 4c 2 . 9. x k - 2x % - x- + 2a; + 1. io. f-i^ + i. 11. a 4 - 10a 3 + 37a 2 - 60a + 36. 84 QUADRATIC EQUATIONS. SECTION XXVII. QUADRATIC EQUATIONS. £5. A Quadratic Equation is an equation of the second degree, that is, it contains the second power of the unknown letter. 96. There are two sorts of quadratic equations : 1st. Pure Quadratics, which contain # 2 and not tf. 2d. Affected Quadratics, which contain both # 2 and x. Thus, 5z 2 - 2 = 6x> -5, x" = 9, _ + — = 1 are pure quadratics ; x 1 While 3# 2 + 5x — 6, 9x = 5, ax* 1 + Ix = c are affected quadratics. 97. Solution of Pure Quadratics.— To solve a Pure Quadratic, Rule. — Find x 2 , as in simple equations. Then tafce the square root of both sides, putting the sign ± before the root of the second side. Ex. 1. Given x> - ~ = 1. 4 Clearing of fractions, 4a;' 2 - 3x* = 4, x 2 = 4, x = ± 2. Note 1.— This result might be written ± x — ± 2 ; but this would not be different from x = ± 2. QUADRATIC EQUATIONS. 85 Note 2. — If we have x° = a, and a is not a perfect square, we write x= ± Va. ,., _ x x* x- 8 Ex. 2. _-- T+ -= T . Clearing the equation of fractions, 4z' 2 - dx°- + x- = 32, 3a; 2 = 32, x> = 1G, x = ± 4. Examples — 37. Solve the following pure quadratics 1. 9a; 2 - 4 = 3z 2 + 5. 2. 2a; 2 - 12 = 36 - 8a; 2 . 4r 2 3. ~ = C25. iS-» a; + 1 a; —J. _ 13 a? - 1 + aTTT ~ = T a;- - 1 a;- %x l 6 ' -7~-T = X- 23 - 7. (3a? + 1) (3a; - 1) = 81a: 2 - 33. 86 SOLUTION OF AFFECTED QUADRATICS, x-2 __ _30_ 2 ~ x~+~2 9. (x + 2)2 = 4:X + 20. 10. * -£- = 6 — - SECTION XXVIII. SOLUTION OF AFFECTED QUADRATICS. 98. To solve an equation which has both x 2 and x in it, we bring it to a simple equation by taking the square root. To do this we have the following Rule. — 1. Reduce the equation to the form x 2 + px = q, in which x 2 has + \ for coefficient. 2. Then add the square of half the coefficient of x to both sides; thus making the first side a perfect square, and pre- serving the equality. 3. Take the square root of both sides, putting the sign ± before the root of the second side. Then find x in tli is simple equation. Ex. 1. 3z 2 - \%x = 36. Dividing by o, wc have x- - Ax — 12. SOLUTION OF AFFECTED QUADRATICS. 87 Adding the square of 2 to both sides, we get x- — 4ic + 4 = 12 + 4, or x- — 4x + 4 = 16. Taking the square root of both sides, x-2 = ±4; whence, a = 2 + 4 = 6, or a; = 2 — 4 = — 2. Note. — When the second side is not a perfect square, we put the sign ± V over it. Ex. 2. x°- + Gx = 2. Adding the square of 3 to both sides, we get x? + 6x + 9 = 11. Taking the square root, x + 3 = ± Vll, whence, x = — 3 -1- Vll, or ic= — 3 — VTT, and we can only find x approximately by getting the approximate square root of 11. Ex. 3. Sx 2 + 15s = 18. Dividing by 3, x- + 5z = 6, x' + 5x + *£ = 6 + '*$■ = 4 4 9 x + |.= ±h x = — 1 + 1 = 1, or x~ — f — |=— 6. 88 SOLUTION OF AFFECTED QUADRATICS. Ex. 4, x* + -V-£ = V 3 -. « 2 + ^ + (f) 2 = ¥ + ¥ = ¥, x + $= ±|, and a> = -£ + f=-l, or a; = - | - § = - ± 3 \ Ex. 5. a 2 - \x = 34. » 2 -^ + (i) 2 = 34 +s ^ = -4F, * - i = ± ¥, and a; = £ + ^ = 6, or x = £ — \* = — Y. Examples — 38. 1. 2x* - 12x + 40 = a 2 + Gx - 5. x — Z x + 2 d 3. a 2 - 4# == 5. 4. x l — 3a; = 2a; 2 — x — £. 5. a; 2 — Ga; = — 55. 6. x 2 -x = 42. 7. # = f # — ^j. 8. 2a: 2 - 9x = 110. 9. 100z 2 + 80a; = 9. 10. 9z 2 - 7x = 16. 11. 11a; 2 - 3a; = 14. PROBLEMS-QUADRATIC EQUATIONS. 89 x +1 x 3 13 . -JO + _40 =9 _ # + 1 x — 1 u. - 1 L_ = A x + 3 36 # 3^+^_3^-2_ 3a7-^~~ 3*7+2 2 * 16. — t-? - ^Zli = 7. 5 — a; 6 — x 17. (a; - 2) 2 + (x - 3) (»-!)=«■- 8. SECTION XXIX. PROBLEMS GIVING RISE TO QUADRATIC EQUATIONS. 99. Ex. 1. Find a number such that its square is equal to twice the product of two numbers, one of which is greater by 3, and the other less by 4. Let x == the number. Then x + 3 = the number greater by 3, and x — 4 = the number less by 4. Then x 2 = 2 (x + 3) (x - 4), that is, x 2 = 2x 2 -2x- 24, reducing, x 2 — 2x =24, completing the square, x 2 — 2x + 1 = 24 + 1 = 25, taking the square root, x — 1— ±5, whence # = 1 + 5 = 6, or a; = 1 — 5 = — 4. 90 PROBLEMS— QUADRA TIC EQ UA TIONS. Examples — 39. 1. Find a number which multiplied by its excess over 21 gives 196. 2. Find the number whose square increased by 4 times the number is equal to 117. 3. Find the number, 5 times the square of which di- vided by 3 is 135. 4. Find the number whose increase by 60 multiplied by its excess over 60 gives 6400. 5. Two numbers are in the ratio of 5 to 3, and the difference of their squares is equal to 144. Find the num- bers. 6. Find the fraction which exceeds its square by f. 7. Find two numbers the difference of which is 5, and the product of the greater by their sum is equal to 150. 8. Find two consecutive numbers the product of which is 5 times the sum of the numbers increased by 5. 9. Find a number such that 12 divided by the number, added to 12 divided by the number increased by 9, is equal to 5. 10. A man bought $100 worth of sheep. He lost 5 of them, and sold the rest for $100, and gained $1 a head on those sold. Find the number of sheep which he bought. SIMULTANEOUS EQUATIONS— QUADRATICS. 91 SECTION XXX. EASY SIMULTANEOUS EQUATIONS SOLVED BY QUADRATICS. 100. We will consider the case of two equations and two unknowns, when one of the equations is of the first degree, or a simple equation, and the other of the second degree, or a quadratic equation. For this case we have the Rule. — From the equation of the first degree find the expression of one of the unknown quantities in terms of the other, and then substitute this expression in the second equation. Ex. 1. Given x + y = 10 (1) ) to find the values of x 1 + if = 52 (2) ) x and y- From (1), y = 10 - x. Putting this in (2), x- + (10 - xf = 52. Expanding (10 - xf, x 2 + 100 - 20z + x 1 - 52. Uniting terms, etc., 2x- — 20.r = — 48. Dividing by 2, x- — 10^ = — 24. Completing square, x lJ — 10.E + 25 = — 24 + 25 — 1. Taking the square root, . x — 5 — ± 1. Transposing, x = 6 or 4. Substituting value of x in (1), y — 4 or 6. 92 SIMULTANEOUS EQUATIONS— QUADRATICS. Ex. 2. 5x - 2ij = 4 (1) to find the values of x and y. dxy - 4:X> =2 (2) ' From (1), x = i±Jl Putting tins for a in (2), 3y (—J^) - 4 (— -J^)*= 3 » 12y + faf (64 + My + % 2 ) -2. Reducing, 7y 2 -2y = 57, y 77 2y _57 19 and y — o, or — . The first value of y, substituted in first equation, gives x = 2. Examples— 40. Solve the simultaneous equations X + y ;i' [I)- xy — 6 ) X x l + V = ::s (2). x 1 X + y : 100 14 j(3). SIMULTANEOUS EQUATIONS— QUADRATICS. 93 x — y = 5 Bx + 4# =? 2zy - 12 xy = 432 ^1 a; :3 J xy = 5x = :48C : 6^ (6). ') a; 5 3 1 *y = : 6 J z 2 - + # + 8 = 14, to find x. Transposing, V6 + x = 6. Squaring both sides, 6 + x = 36 ; and x = 30. 113. If the equation contains two radicals of the second degree with the unknown letter under them, we must re- peat the above operation for the second radical ; that is, two transpositions and two squarings will be necessary. 100 RADICALS OF THE SECOND DEGREE. Example.— Given a 6 a + a — a — 4o. Add T . T , and -, r x + 1 x — 1 a; 2 — 1 ,„ -r, # + 8 ,, , # — 7 46. From r subtract - — s a: — 2 # — 2 120 MISCELLANEOUS EXAMPLES. 47. From • take x — 1 x + 1' JO -, a + 1 , , ft — 1 48. From take T . a — 1 « + 1 7ft - 4 lift - 7 49. Simplify 5 50. Simplify 1 - -^- and 1 + ~- 1 J 1 + ft 1 —ft (Sections XV-XVII.) 51. Multiply __ + _ I by s . 52. Multiply 2a ~ X by 17. ro tvi- 14.- i ^ 2 + 2^ + 1 . ,-r —3x + 2 53. Multiply ^_ 5 ^ +6 by ^ + 4 ^ + 3 K . r „ ft 2 6 2 c 2 54. bimplily -7- x — x — r . r J oc ac ab KK D . ,.„ ft + 1 ft + 2 ft — 1 55. Simplify — - x — — r- x ft - 1 ft 2 - 1 (ft + 2) 2 K _ ... 4ft 2 ^ . %a¥ 56 - Dmde 6^ b yi5^- 57. Divide -= 7 - by - — T . a~ — o~ J ft — o 58. Divide -^- by — — -^ . 2^2 2 MISCELLANEOUS EXAMPLES. 121 59. Divide — ^ by 1 ^r. a + 1 J a + 1 GO. Divide a + 1 by ^±1 + 1. _. _. ., a' 2 — 4a + 4 , a' — 3a -f 2 61. Divide -^ _ — by —, -. — . a J - 6x + J x l — 4a + 3 62. Find the numerical value of 3a — 25 -f 2c — (45 — (5c — M)) when a = 4, 5=1, c = — 1, d = 0. He 2c -35 63. Find the value of — — l 7 tt when 5 = 3, c = 7. , -r,. -, xi • i ! . 2« + 2 %a - 9 . 64. Find the numerical value of ^- H ~ when a — 3 rt — 2 a = 4. 65. Find the numerical value of 3a — £(35 — 7(c — J)) jvhen « = 15, 5 = 2, c = 3, d = 5. M _. •. ,. . , , . « 2 + 5 2 c 2 -rf , 66. Find the numerical value of ■— -\ - n — when c a a — \, 5 = 2, c = 3, d = 4. 67. Find the numerical value of — + -^ when a o a a ■= 1, 5 = 4, c = 6. 68. Find the numerical value of 6«5 a + 10^ 2 5 — 5c 2 when a = 1, 5 = 9, c = 8. 69. What is the difference between 4« and « 4 when a = 2? 70. What is the difference between a 5 and 8a 2 when «= 8? 6 122 MISCELLANEOUS EXAMPLES. (Section XVIII.) Solve the equations 71. 12(3 - 3) - 3(23 - 1) + 5x = 22. 72. ~ + 12 = ^ + 7. 73. ^x — \x = 5 1 — \x. 74. 0(3 + 5) = 8(57-3). 75. 104- 10(2^ -1) = 54. 76. 3 5 1 7x ~ 14 ~" 2 ' 7^ + 4 83-2 K/ 77. 2 - — y— = 5(40 - 23). ^43-6 83 - 10 78. 79. 80. 33 — 4 63 ■ 60 30 3+2 3—1 32 54 33 — 4 53 — 6 81. |(53 - 1) - 6(22 - 33) = 2x - 3. Q _ 343 - 56 73-3 7ft -5 , OI 82. -3^- - T ___ r --+2^ /*3 3 — O J ,_. 4 < MISCELLANEOUS EXAMPLES. z+_l 2(x + 2) _ 9(s - 3) 84# 2 + 3 4 ' 85> -3~ + 2- 12 3~" 123 86. 5x- 6 2x - 13 x + 7 (Sections XXII, XXIV.) Find # and ?/ in the following simultaneous equations 87. 3x - 4?/ = 25 ) 5x — 2y — 7 ) 88. X 4- * ?/ - 8 2 + 3 ~ 8 « y _i 4 12 ~ > . 89. 22: — 3 = y J 90. 4:x + 9// = 12 j 6a; - 3y = 7 ) * 91. 82: - ty = 12 # — 2;/ 2a 4 ^ 124 MISCELLANEOUS EXAMPLES* 92. 5 7 1 i-M 93. iri =1 ' 2 6 J 94. a; + | = 19 Find x, y, and 2 : 95. x + y = z x — y 4- z = 4 5z + # + 2 = 20 (Sections XX, XXI, XXIII.) 96. Eight times a certain number added to 16 is equal to 16 times a number one less. Find it. 97. Find a number such that 45 times the number in- creased by 60 is equal to 500 diminished by 10 times the number. 98. What two consecutive numbers are such that J of the larger added to -J of the smaller is equal to 9 ? MISCELLANEOUS EXAMPLES. 125 99. If j\ of the larger of two consecutive numbers taken from \ of the smaller leaves 7, what are they ? 100. Two persons have equal sums of money, but the first owes the second 60 dollars ; when he has paid his debt the second has twice as much as the first. How much had each ? 101. Divide 21 into two such parrs that one of them shall contain the other 21 times exactly. 102. Divide the decimal fraction .07 into two other decimal fractions which differ from each other by .007. 103. Find the number which increased by 5 is contained the same number of times in 45 as the same number dimin- ished by 5 is contained in 12. 104. A purse of eagles is divided among three persons, the first receiving half of them and one more, the second half of the remainder and one more, and the third 6. Find the number of eagles the purse contained. 105. A person possesses $5,000 of stock. Some yields 3 per cent., four times as much yields 3^ per cent., and the rest 4 per cent. Find the amount of each kind of stock when his income is $170. 10G. Divide a yard into two parts such that half of one part added to 22 inches may be double the other part. 107. Divide $120 among three persons so that the first may have three times as much as the second, and the third one-fourth as much as the first and second together. 108. Two coaches start at the same time from two places, 126 MISCELLANEOUS EXAMPLES. A and B, 150 miles apart, one travelling 5 miles an hour, the other 6J miles an hour. Where will they meet, and at what time ? 109. A number is written with two digits whose differ- ence is 7, and if the digits be reversed the number so formed will be f of the former. Find the original num- ber. 110. Divide 200 into two parts so that one of them shall be two-thirds of the other. 111. A is three times as old as B ; twelve years ago he was eleven times as old. What are their ages ? 112. A father has five sons, each of whom is three years older than his next younger brother, and the oldest is four times as old as the youngest. Find their respective ages. 113. Divide 60 into two parts so that the difference of their squares shall be 1,200. 114. Divide 30 into three parts so that the ratio of the first two shall be 1 : 2, and that of the last two 5 : 3. 115. If Gx - 3 : Ax - 5 : : 3x + 5 : 2x + 3, find x. 116. Eight horses and five cows consume a stack of hay in 10 days, and three horses can eat it alone in 40 days. In how many days will one cow be able to cat it ? 117. One-fourth of a ship belongs to A and one-fifth to B, and A's part is worth $6,000 more than B's. What is the value of the ship ? 118. A certain fraction becomes -J- if 2 be added to its MISCELLANEOUS EXAMPLES. 127 numerator, and if 2 be added to its denominator it be- comes £. What is the fraction ? 119. If a certain number be multiplied by 7f, the pro- duct is as much greater than 16 as the product of its multiplication by 2f is less than 110. What is the number ? 120. The highest pyramid in Egypt is 25 feet higher than the steeple of St. Stephen's Church in Vienna, and the height of this last is -j-J of the height of the pyramid. How high is each ? 121. " The clock has struck ," called out the night- watchman. " What hour did it strike ? " asked a passer- by. The watchman replied: "The half, the third, and the fourth of the hour struck is one greater than the hour." What hour did it strike ? 122. Of a swarm of bees the fifth part lighted on a blooming Cadamba, and the third part on the blossoms of Silind'hri, three times as many as the difference between the first two numbers flew to the flower Cutaja, and the one bee remaining hovered in the air unable to choose be- tween the aromatic fragrance of the Jasmin and the Pan- danus. " Tell me, beautiful girl," said the Brahmin, " the number of bees. " 123. A father died, and left to his two sons and his wife $30,000, with the conditions that the share of the elder brother should be to the share of the younger as 4:3; and the share of the mother should be | of the amount left to both brothers. What was the share of each ? 128 MISCELLANEOUS EXAMPLES. 124. A farmer grazes a certain number of sheep and oxen in two fields. One contains 13 animals, but only half the sheep and one-fourth the oxen are in it. The other field contains 21 animals. How many of each sort had he in the fields ? 125. Find two numbers in the ratio of 3 to 4, whose sum is to 1 as 30 added to the second is to 2. 120. If 8 times one number be taken from 7 times another number 3 remains ; and 9 times the first added to 6 times the second is 96. What are the numbers ? 127. A said to B: " If you give me 7 dollars of your money, then I shall have twice as much as will remain to you." B said to A : "If you give me 4 dollars, then I shall have twice as much as remains to you." How much had A and B, each ? 128. The sum of two numbers is 40, and their quotient is 3. What are the numbers ? 129. Find two consecutive numbers whose product diminished by 20 equals the square of the first. 130. The hour and minute hands of a clock are together at 12 o'clock, when will they be together again ? If x = number minute-spaces gone over by minute-hand before they are together again, then, ^x — number minute-spaces gone over by hour-hand before they are together again. And \lx = gain of minute-hand. And, since this gain must be 60 minute-spaces, .-. \\x = GO. MISCELLANEOUS EXAMPLES. 129 (Sections XXV, XXVI.) Find the squares of the following quantities : 131. (1), x* - 3x + 4. (2), 4a - 2b + 3c: (3), x - a + 2b — c. Find the square roots of the following quantities : 132. 49z a + 12Qax + 81a 2 . 133. 121a 2 - 3S0ab + 2256 2 . 134. 400«V - 200abx + 25b\ 135. 4a 2 - 12ab + W + 20ac - SObc + 2bc\ 136. x* -- Ax 3 + Gx" - \x + 1. (Sections XXVII-XXIX.) Find the value of x in each of the following equations : 137. {x + 3) 2 = C>x + 25, 138. ~— + --- = 8. 1 -} -a; 1 — x 139. i(9 - 2x 2 ) = | - T V(7z 2 - 18). - . A 2 - ar 3 - x* 4 - or or - 5 3 140. -— + - 4 — + -g- = _,- - T . 141. 2.t 2 - 12a; + 16 = 160. 6* ]30 MISCELLANEOUS EXAMPLES. 142. 4z 2 - 32^ + 40 = 76. 143. x* - x - 40 = 170. 144. 3a; 2 + 2x - 9 = 76. 145. x 2 - %bx = a" - h\ 146. 16 - %■ = ^ + 7|. 147. a;' - -f- = 14a + 10. o 148. There are three numbers in ratio J : | : J, the sum of whose squares is 724. Find them. 149. Find the number which added to its square gives 182. 150. There are two numbers one of which is f of the other, and the difference of their squares is 63. Find them. 151. There is a rectangular bathing pool whose length exceeds its breadth by ten feet, and it contains 1,200 square feet. Find its length and breadth. 152. The difference of two numbers is 5, and their prod- uct is 1,800. Find them. 153. The product of two numbers is 126, and if one be increased by 2 and the other by 1, their product is 160. Find them. 154. Find two consecutive numbers whose product is 600. MISCELLANEOUS EXAMPLES. 131 (Sections XXX, XXXI.) Find x and y in the following simultaneous equations : 155. x — y — 15 ) 156. z + y :x — y : : 13 : o ) y- + x = 25 J 157. y" - lOz = lOy + 36 ) a; + 2y = 3o J • 158. 3x + 2y = 20 ) zx* - f = n \ ' 159. Beduce a/45 - V%0 + a/50 + yl25 - V180. 160. Square 1 + a/3 . 161. Multiply 4 - V3 by 4 + i/3. 162. Simplify a/512oW 163. Simplify yf • 164. Simplify 165. Simplify 3- V5 4 a/5-1 166. Multiply a/3 + 2a/2~ by 2a/3~ + V% Find a? in the following equations : 167. Vx + Vz - 7 = 7. 132 MISCELLANEOUS EXAMPLES, 168. a/5z + 11 + a/&» — 9 = 10. , . , 2 1G9. ^x + 1 + Vv - 1 = -7==- V# + 1 (Section XXXII.) 170. Compare the ratios 4 : 5 and 15 : 1G ; 14 : 15 and 22 : 23. 171. What is the ratio of a inches to c feet ? 172. Find a fourth proportional to f, f , \. 173. What number is that to which if 2, 4, and 7 be severally added, the first sum is to the second sum as the second is to the third ? 174. What two numbers are those whose difference, sum, and product are proportionate to the numbers 3, 5, and 20, respectively ? (Sections XXXIII, XXXIV.) 175. Find the sum of the progression 1, 7, 13, 19, etc., to 50 terms. 176. Find the sum of f, if, 1|, etc., to 20 terms. 177. Insert five arithmetical means between 12 and 20. 178. The first and last of 30 numbers in arithmetical progression is 2^ and 2J. What are the intervening terms ? MISCELLANEOUS EXAMPLES. 133 179. A certain number consists of three digits, which are in arithmetical progression ; and the quotient of the number divided by the sum of its digits is 15 ; but if 39G be added to it, the digits will be inverted. What is the number ? Let x — the middle digit, and y — the common difference ; then x — y, x and x + y = the digits, respectively ; and the number = 100 (x - y) + 10s + (x + y) = 111 x - 99y. Hence, the simultaneous equations : Ilia; - mj _ 3x Ilia; - ddy + 896 = 100 (x + y) + 10z + (x - y). 180. The population of a town increases in the ratio of -jV annually; it is now 10,000. What will it be at the end of four years ? 181. Find the geometric mean between - — and 182. Insert two geometric means between 5 and — |. 183. Insert 3 geometric means between 12 and 972. 184. Find the value of the recurring decimal .8181.8181 as a decreasing geometrical progression. 185. A farmer sowed a bushel of wheat and used the whole produce, 15 bushels, for seed the second year, the produce of this second year for seed the third year, and the produce of this again for the fourth year. Supposing the increase to have been always in the same proportion to the seed sown, how many bushels of wheat did he harvest at the end of the fourth year ? 134 GENERAL REVIEW QUESTIONS. SECTION XXXVI. GENERAL REVIEW QUESTIONS. I. 1. How are quantities represented in algebra ? Why do we use letters ? 2. What is the chief point of distinction between digits and letters as used in algebra ? 3. What are the signs of addition and subtraction ? 4. What are positive quantities ? Negative quantities ? 5. What are the signs of multiplication ? 6. When may we omit them and still denote multipli- cation ? 7. What is the difference between 345 and abc when a = 3, b = 4, c .-= 5 ? 8. What are the signs of division ? II. 1. AVhat is a factor ? What is a coefficient ? 2. What is a power ? 3. What is an index ? What other name have we for the index of a power ? 4. What is an algebraic expression ? 5. What are the terms of an expression ? GENERAL REVIEW QUESTIONS. 135 6. What is a monomial ? a binomial ? a trinomial ? a potynomial ? Give an example of each. 7. What are like terms ? Unlike terms ? 8. How do we simplify an algebraic expression ? 9. What is the rule for addition ? 10. What is the rule for subtraction ? 11. Show the reason for the rule of the signs in sub- traction. 12. How does algebraic addition differ from arith- metical ? Illustrate by an example. 13. How does algebraic subtraction differ from arith- metical ? Illustrate by an example. III. 1. State and prove the rule of the signs in multiplica- tion. 2. How do we multiply monomials together ? 3. Give the rule for multiplying a polynomial by a monomial. 4. Give the rule for multiplying a polynomial by a polynomial. 5. State and prove the rule of signs in division. G. What is the rule for dividing a monomial by a mo- nomial ? A polynomial by a monomial ? 7. Give the rule for the division of polynomials. 136 GENERAL REVIEW QUESTIONS. 8. What is meant by arranging a polynomial with ref- erence to a certain letter ? Give an example. IV. 1. What is the square of the sum of two quantities equal to ? 2. Express this by a formula, when a and b are the quantities. 3. What is the square of the difference of two quantities equal to ? 4. Express this by a formula. 5. What is the product of the sum and difference of two quantities equal to ? 6. Express this by a formula. 7. What are brackets ? 8. Give the rules for removing brackets. 9. Show by examples how to remove brackets with the plus sign before them ; with the minus sign before them ; with the sign of multiplication before them. V. 1. What is the product of x + a by x + h ? Of x — a by x — b ? Of x + a by x — b ? 2. What are the factors of x" + 5x + G ? Of x 1 ~ 13a + 40 ? Of a? + bx - 6 ? Of x 1 - x - 30 ? 3. What are the factors of x 2 + 4z + 4 ? Of x* - 6x + 9 ? Of 4z 2 - 9a a ? GENERAL REVIEW QUESTIONS. 137 4. What is the least common multiple of two algebraic expressions ? 5. Give the rule for finding this L. C. M. 6. What is the G. 0. D. of two algebraic expressions. 7. Give the rule for finding it. VI. 1. Do the rules for the operations of reduction, addi- tion, multiplication, etc., on algebraic fractions differ from those in the arithmetic for common fractions ? 2. Give these rules. 3. What is meant by the numerical value of an alge- braic expression ? and how do we find it in any supposed case ? 4. What is the difference between a % and 2# 2 , when a = 2? 5. What is the value of ft /> fl OftC 2UC X— if 4ft — bb „ x — y x — 1 _ x — a x — 2 20c ' * # ' 2: — 2 ' a: + ft ' £ + 2 ' Examples — 17, page 36. j 5x Sx x 9 12ab*x 4bc -f kac 3a*x 15' 15' 15* 48c£ 2 ' 48c^ ' 48c.r 3 ^-^ 2ft 2 _6_ _3_ _2_ 10a + 8 a — x ' a — x' 6x ' (Jx ' Qx ' 18 10a; + 17 a 65ftz - 13ft 2x + 4 18 * b * " 52ft 2 ' 52ft 2 ' 14G ANSWERS. Examples — 18, page 38. i % +V + * « 3« +5 „ 7a? . 13 5a: I. — . Z. p: . O. -pr- . 4. r—v . 0. -77- a 3a? 9 12a 3 „ 8«Z> + 7« . 3a +£ 13a? -2 Q 24/; -- 23 246c i 24 10 . » 11.^+1. 12. * 13. 0. 14. 15 ' 3 ' c 3 " x- 1 15. 5 -. 16- 2 T V Examples — 19, page 40. t x x m 3a? -f 5 .3 , 2c '• u- 2 - ft' 3 ~6— 4 - io- 5 y — 4f0C m r\ c — \ 2. -8/rW. 3. ~. 4. Ma*b\ lbc 64« 4 &V. 6. ^V. 7. % . 8. a' + 4a + 4. 4a; 4/,V + 4§/> + 1. 10. 9m a - 30//?;/ f 25 //'■". 11. «Vaj" + 2ffca# + y'. 12. 9 + 3 • + ^ \ 150 ANSWERS. Examples — 33, page 79. I. ±5ay. 2. ±10a?xy. 3. ± 7ab\ 4. ± ~. loy 5. ±£L. 6. ±*£. 7. ±^. 8. ±f 9. ±i . 7« 8 y* 2a 4 3 10. ±f. Examples— 34, page 80. 1. ± (a + 1). 2. ± (« - 2). 3. ± (a; + f). 4. ± (3a£ - &). 5. ± (4a - 3b). 6. ± (8a: 2 - §6). 7. ± (a + i). 8. ± (4* - 1). 9. ± (x - |). Examples — 35, page 81. 1. ± {x + 3). 2. ± (a; - 6). 3. ± (x - V). 4. ± (s- i), 5. ± (y+ {). ti. ±(*-|). 7. ± (a - i). 8. ± (* - 2a). 9. ± (y + f). 10. ± (*-■&). 11. ± («-*). 12. ± (ar + j|. Examples — 36, page 83, 1. 8« + 95. 2. x 1 + 2aa; + a\ 3. a; 2 - 2a; — 2L 4. 3« - 2/y + c. 5. 3«' ? - 2« + 1. 6. 4^ 2 - 2b + c 2 . 7. 3r - « + 2. 8. a — b + 2c. 9. as 2 — x — 1. r 2 11 . a 2 — 5a + G. ANSWERS. 151 Examples — 37, page 85. 1. ±VT- 2. ±V% 3. ±-V-. * ±2. 5. ± 5. 0. ± 6. 7. ± |. 8. ±8. 9. ± 4. 10. ± 2. Examples — 38, page 88. 1. 15 or 3. 2. 4 or - 1. 3. 5 or - 1. 4. - 1 ± V'f . 5. 3 ± V -4^. 6. 7 or - 6. 7. f or 4. 8. 10 or - V. 9. T V or - T \. 10. - 1 or V 5 . 11- It 01 * - 1- 12. - f or 2. IS. - I or + 9. 14. - 7 or + 0. 15. 2 or - f. 16. 4 or %»-. 17. 5 or 3. Examples — 39, page 90. 1.28. 2.9. 3.9. 4.100. 5.9,15. i. \ or f 7. 10, 5. 8. 10, 11. 9. 3. 10. 25. In these answers the negative results are omitted. Examples — 40, page 98. 1. x = 2 or 3) 2. ar=6) 3. x = 8 or 6) y = 'd or2f' y = 5) y = 6 or 8) 4. x = 8 or I ) 5. a?= ±'l2) 6. a- = ± 24 j y = 3 or - 4± ) y = ± 36 ) y = ± SO) 7. a; = ± 3 £ 8. a? = 11 ) 9. * = 4«r-«* y = ±8)'" // = 8 ) ' y = 5 or -Vs 6 - J 152 ANSWERS. ). x = 5 or -9 } 11. x = 5 or 6 T \ y — 2 or - 12 ) " y = 2 or ft Examples — 41, page 98. 1. 2afc 8 \/7a. 2. W% - a. 3. 3^3, 2^3, 20 V^ 4. 2V27 5^27 0a/2T 5. 47 v^ 0. 12 - lOV'C. 7. 3 + V3. 8. yV - ft*. 9. 2. 10. 7 - 4^/3: 11. !a/2L 12. 1^27 13. t^-a/I^ 14. 3V^+ 3. 15. V3-1. 16. -KV5 + V2). 17. 0. Examples — 42, page 100. 1. V - 2bc + c*-a. 2. 144. 3. 36. 4. 5. 5. 25. 6. 3. 7. 5. 8. 2. 9. -16. Examples — 43, page 102. 1# * 5 S" A 4? ; 3"" d * J ; 4ft' 4 * 5ft ; 3^ ' K ft + 5 2ft + b . . w 36 20m 5 -^r ; ~^- •■ !-«;« + * '-s'-ts-- 8. 17 ; 18. Examples — 44, page 105. 2. 7. 3. x : y : : 5 : 4. 4. 11. 5. 4/>m. 6. 8 and 12. 8.24,30. 9.11. 10. 4 + x\ 11.2 or - 12. 12. 21, 22, 23, 24. ANSWUES. 153 Examples — 45, page 110. 1. (1.) 55, 111. (2.) 44, 86. (3.) 131, 27J. (4.) 135, 65. 2. (1.) 590. (2.) T240. (3.) 23G1. 3. 300. 4. 881. 5. 1,700 yards. 8. T 4 F . 9. a. 10 10J, 10f. 11. Com. dif. 2. 12. Com. dif. ^ series, 3||, 3||, etc. 13. 64331. Examples — 46, page 115. 1. (1.) 3. (2). 2. (3.) f. (4>).l. (5.) 2. (6.) 3s. 2. 25. 3. 18, 54. 4. J, |. 5. The arithmetical. 6. 25924. 7. 2 dollars and 55 cents. 8. J. 9. f 10. if 11. 24, 48, 96. Miscellaneous Examples, page 117. 1. - Sa'x + Sax" - 2x\ 2. - Ga 2 b - 4«\ 3. ±a - 7c. 4. 4rr - W. 5. 9« a - b\ 6. ft 2 + 2ab + b* - c 1 - 2cd - d\ 7. a* + artf + &\ 8. 25a' + 1000r^ + lOOOCZr, 19G« 2 - 30Sab + 1216 2 . 9. x. 10. 6rtz - 4.r. 11. 27a; 3 + 9tfy + 3.tt/' 2 + ?/ 3 . 12. a 3 - 2cC + 2a — 1. 13. z 2 + to + 3« 2 . 14. 'Sa - ±x 4- £. 15. 4.f 3 - Gcz 2 + 2c 2 x. 16. 8 + 6 2T= BJ-N s = 10J 96. 4. 97. 8. 98. 15, 16. 99. 65, 66. 100. 180 dollars. 101. ft, 20^. 102. .0315, .0385. 103. 8 T V 104. 30. 105. $800, 83,200, 81,000. 106. 20 inches, 16 inches. 107. 824, 872, $24. 108. 65^ miles from A, 13 4j hours after starting. 109. 81. 110. 80, 120. 111. 45 years of age, and 15 years of age. 112. 4, 7, 10, 13, 16 years of age re- spectively. 113. 40, 20. 114. 7\, 14f, 8f 115. - 2f 116. 150. 117. $120,000. 118. &. 119. 12. 156 ANSWERS. 120. 425 feet, 450 feet. 121. 12 o'clock. 122. 15. 123. Eldest son, $12,467.52; youngest son, 19,350.64; the widow, $8,181.84. 124. 18 sheep, 16 oxen. 125. 9, 12. 126. 6 T \4 T , 7fft. 127- A, $15, and B, $18. 128. 10, 30. 129. 20, 21. 130. 5 T 5 r minutes past one o'clock. 131. (1) x* - 6x* -f 17z 2 - Ux + 16 ; (2) 16a* - 16ab + 2iac + W — 12bc + 9c 2 ; (3) x* - 2ax + ibx - 2cx -f- a* - Aab + 4fo 2 + 2ac - Uc + c\ 132. 7x + 9a. 133. 11« - 156. 134. 20ax - 5b. 135. 2a - 36 + 5c. 136. x°- - 2x + 1. 137. ± 4. 138. ± I. 139. ± 2. 140. ± 2. 141, 12, or - 6. 142. 9 or - 1. 143. 15 or - 14. 144. 5 or - 5|. 145. b ± a. 146. 3 or - ^-. 147. 15or-|. 148. 12, 16, 18. 149. 13 or -14. 150. 12, 9. 151. 30, 40. 152. 40, 45. 153. 14 and 9, or 18 and 7. 154. 24, 25. ANSWERS. 157 155. x — 18 or *&) 156. x = 9 or - 14V y = 3 or — | ) ^ = 4 or — 6 4- 1 ) n 4 ' 157. a = 2 or 78 ) 158. x = 6 or 114 «! # = 14 or - 24 ) 2/ = 1 or - 1G1 159. 5V27" 160. 4 + 2^/37 161. 13. 162. 16ai>cV2a. 163. |a/35\ 164. 2(3 + \/5). 165. ^/5 + 1. 166. 10 + 5V6. 167. 16. 168 5. 169. 1. 170. 15 : 16 greater by || ; 22 : 23 greater by -jfg. a -i ^o 25 12c ' 56 171. =?-. 172. ~. 173. 2. 174. 5 20. 175. 7400. 176. 92i. 177. Common difference f Series 12, 13-J, etc., ...20. 178. Common difference ^ Series 2J, 2 T 3 T • • • 2 i- 1T9 135 - 18 ° 14,;41 - 181. a + x. 182. -I, f. 183. 36, 108, 324. 184. ff 185. 50625. 18%