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r ^^ *\ ^^p^ / \ A \. J" AN ELEMENTARY ALGEBRA : DESIGI^ED AS I AN INTRODUCTION TO A THOROUGH KNOWLEDGE OF ALGEBRAIC LANGUAGE, AND TO GIVE BEGINNERS FACILITY IN THE USE OF ALGEBRAIC SYMBOLS. CHARLES S. VENABLE, LLD., PfiOPESSOR OF Mathematics in the University o? Yirginia ; Author op " First Lessons in Numbers," "Mental Arithmetic," "Practical Arithmetic," and "Higher Arithmetic." UNIVERSITY PUBLISHING COMPANY, NEW YORK AND BALTIMORE. 1872. or THE VNIVERRITY urn V4. /^ Entered according to Act of Congress, in the year 186f), By the university .^L'BLISHING COMPANY, m the Clerk's ofScQ rf the District Co'irt of the United States for the Southern i;:slrict of New \ ork. PREFACE. The present Elementary Algebra has been prepared with a view to enable the beginner to obtain a thorough knowl- edge of Algebraic Language, and to acquire an early facility in the use of Algebraic Symbols. The translation of Ex g- lish into the symbolical language of Algebra, and the inter- pretation of Algebraic Symbols by arithmetical operations are made prominent from the beginning. Throughout the work I have endeavored, in the Algebraic operations and solutions of problems, to present examples of elegance and conciseness in the transformation of Algebraic expressions. I am convinced by long observation that the difficulties of students in their more advanced mathematical studies are greatly enhanced by their want of knowledge of Algebra as a Language, and their want of facility m the transformation and combination of expressions in the solution of problems. These tM7igs form tlie dasis of any thorough Icnowledge of Algebraic Analysis, and should he learned ivellin the begin- ning. The demonstrations are, I think, clear and easily rntelligible to the young student. The examples for exercise are numerous. In addition to the fundamental Algebraic operations on Entire Quantities and Fractions, Evolution, Surds, Equa- 4 PEEFACE. tions, Arithmetical and Geometrical Progressions, and Pro- portion, I have treated in an elementary manner the subjects of Fractional Exponents, Permutations and Combinations, the Binomial Theorem for whole-number exponents, Har- monical Progression, Theory of dotation, and Logarithms. I am convinced by long experience that it is important to present these subjects to the young student in a simple and practical manner before he comes in contact with them in their greater extensions and more difficult applications. In the preparation of this book I have consulted many of those works which give a view of the progress and improvement in elementary instruction in Algebra. But three English works — Todhunter's Algebra for Beginners, Colenso's Algebra, and Lund^s Wood's Algebra — are made the basis of the worK. The demonstrations of Wood, (a standard of more than half a century,) are singularly clear and simple, while those of Colenso are models of elegance and brevity. Tod- hunter's illustrations are clear and copious. The examples have been selected mainly from the above authors, many of them having been taken by them from the Cambridge Ex- amination papers. I have also used Lund's Easy Algebra, Bobillier's "Principes d'Algebre," Kitt's Problemes d'Al- gebre, and Wrigley's Collection of Problems. University of Virginia, Aug. I, 1869. CONTENTS. Pagb I. Principal Signs 7 II. Factor — Coefficient — Power — Terms . ii III. Remaining Signs— Brackets . . . i6 IV. Change of Order of Terms— Like Terms 20 V. Addition 24 VI. Subtraction 28 VII. Brackets 33 VIII. Multiplication 38 IX. General Results of Multiplication . 45 X. Division 52 yXI. Factors 61 XII. Greatest Common Divisor . • • . (^^ XIII. Least Common Multiple .... "](> XIV. Fractions 80 XV. Reduction of Fractions .... 84 XVI. Addition and Subtraction of Fractions 88 XVII. Multiplication of Fractions ... 94 XVIII. Division of Fractions 98 XIX. Complex Fractions, and other Results ioi XX. Involution 105 XXI. Evolution . . ' . . . . . no XXII. Simple Equations 128 XXIII. Simple Equations — continued . . 137 XXIV. Problems Solved by Simple Equations . 145 XXV. Problems — continued . . . . 153 6 CO]SrTEis-TS. XXVI. Simultaneous Equations of the First Degree 164 XXVII. Problems Solved by Simultaneous Equa- tions OF THE First Degree . . 176 XXVIII. Indices 184 XXIX. Surds ...'..... 191 XXX. Quadratic Equations 200 XXXI. Equations which may be solved like Quadratic Equations . . . . 212 XXXII. Problems which lead to Quadratic Equations containing One Unknown Quantity .... ... 217 XXXIII. Simultaneous Equations involving Quad- ratics . . . . . . .221 XXXIV. Ratio 230 XXXV. Proportion 233 XXXVI. Arithmetical Progression .... 240 XXXVII. Geometrical Progression ... 245 XXXVIII. Harmonical Progression .... 252 XXXIX. Permutations and Combinations . . 254 XL. Binomial Theorem . . . . . . 263 XLI. Scales of Notation 269 XLII. Logarithms . . 275 Answers to Examples . . . • 284 ELEMENTARY ALGEBRA. I. The Peincipal Signs. 1. Algebra is the science in which Ave reason about num- bers with the aid of letters to denote the numbers, and of certain signs to denote the operations performed on the numbers, and the relations of the numbers to each other. These letters and signs are called Algehraic Symbols, 2. Quantity signifies anything which admits of increase, or diminution. The word quantity is often used with the same meaning as numler, 3. The sign + placed before a number denotes that this number is to be added. Thus a ^1) denotes that the num- ber represented by h is to be added to the number repre- sented by a. If a represent 9 and t) represent 3, then a + 5 represents 12. The sign -f is called the ])lu8 sigti, and a + Z> is read " a plus b." 4. The sign — placed before a number, denotes that the number is to be subtracted. Thus a — b denotes that the number represented by Z>, is to be subtracted from the num- ber represented by a. If a represent 9 and b represent 3, then a — b represents 6. The sign — is called the minus sign, and a ~ ^ is read thus, " a minus b." 5. Similarly, a + b ■\- c denotes that we are to add b to a, and then add c to the result ; a-\-b — c denotes that we Define Algebra ; Algebraic Symbols ; Quantity, How is Addition indicated ? Subtraction ? 8 ELEMENTARY ALGEBRA are to add h to a, and then subtract c from the result ] a — l) + c denotes that we are to subtract 1) from a, and then add c to the result ; a — h — c denotes that we are to subtract h from a, and then subtract c from the result. 6. The sign = denotes that the numbers between which it is placed are equal. Thus a = l denotes that the number represented by a is equal to the number represented by l. And a -{-h — c denotes that the sum of the numbers repre- sented by a and h is equal to the number represented by c; so that if a represent 9 and ^ represent 3, then c must represent 12. The sign — is called the sign of equality, and a = Z> is read thus, " a equals b," or a is equal to h, 7. The sign X denotes that the numbers between which it stands are to be multiplied together. Thus a Xl) denotes that the number represented by a is to be multiplied by the number represented by b. If a represent 9 and b represent d, then a X b represents 27. The sign X is called the sign of multiplication, and ^ X ^ is read thus, " a into b/^ or " a multiplied ly b.^^ Similarly, aXbX c denotes the product of the numbers represented by a, h, and c, 8. Sometimes a point is used instead of the sign X . Thus a. I) instead of a X ^. Both of these signs are, however, often omitted for the sake of brevity ; thus a h is used instead of aXh, and has the same meaning. So also, ahc i^ used in- stead of aXb X c, or a.b.c, and has the same meaning, Nor is either the point or the sign X necessary between a numbei expressed by a figure and a number expressed by a letter ; so that 3 a is used instead of 3X a, and has the same meaning. The sign of multiplication must not be omitted between numbers expressed in the ordinary way by figures. Thus 45 cannot be used to represent the product of 4 and 5, because the meaning, forty-five, has already been assigned to 45. Nor can the point be used between figures to express a How is Equality indicated ? Multiplication ? PKIIS^CIPAL SIGKS. 9 product, witliout prodiiciDg confusion, as, for example, to 4 . 5 has already, in Arithmetic, been assigned the meaning 4 + Y^Q- ; still, 4 . 5 is sometimes written for 4x5. 0. The sign -i- denotes that the quantity which stands defore it is to be divided by the quantity which follows it. Thus a-^h denotes that the number represented by a is to be divided by the number represented by I. If a represent 12, and 1) represent 4, then a -^l represents 3. The sign -^ is called the sign of division, and « -f- Z> is read thus, " a divided hy b," or briefly, " a ly b." But most frequently to express division the dividend is placed over the divisor, with a line betw^een them. Thus -^ is used for a-^l, and has the same meaning. 10. A number or quantity expressed by Algebraic Sym- bols is called an Algebraic exinession, or briefly, an expres- sion, 11. We shall now give some examples as an exercise in the use of the symbols which have been explained ; these exam- ples consist in finding the numerical values of certain Alge- braic expressions, and in finding the Algebraic expression for certain quantities expressed in ordinary language. Suppose a = 1, l) — 2, c — 'd,d — 6, e = 6, f= 0. Then 7« + 3Z'-2 6?+/=74-6-10 + = 13-10 = 3. 2 ad + Shc-ae + df =4: + 4.8 -6 + = 62-6 = 4^6. 4:ac , lObe de 12 ,120 30 ^ , _ ,^ b cd ac 2 lo 3 14 - 10 = 4. 4:c + 6e _ 12 + 30 _ 42__ d-b "" 5-2 ~ 3 ~ ' How is Division indicated? What is an Algebraic Expression ? 10 eleme:n^tary algebra. Examples — 1. li a=l, b^ Vf c=3, cl=4:, e=5,f = 0y find the numerical values of the jllowing expressions : 1. 9a+2I)+3c-2f. 2. 4:e-3a-dI}-\-5c. 3. 7ae-{-3I)c+dcl-af. 4. 8adc-hcd+9cde-def. 5. ahcd-habce-habde + acde+dcde, 6. -7- h h-r . c d e 4:ac 8bc 6cd _ 12a 6b 20c 7. —J j J . O. -y 1 ---H —• ode be cd de ^ cde , hbcd 6ade ^. ^.77- ^^de 9. — 7-H = — . 10. 7e-hbcd — ^ — . ab ae be 2ao 2a+6b Sb-h2c a+b+c + d b+c-{-3e c d 2e e+c—d -Q a + c b-\-d . c+e ^. a+b+c + d-\-e c — a d—b e—c e—d + c—b-\-a 15. What is the difference between ~-\ — and -j-' when be be a = 4, 6 = 5, c = 10 ? Ans. 5^. 16. A person who possessed a fortune, expressed by x, re- ceives by inheritance two sums, a and b. Express his whole property. 17. An estate is divided among three heirs: the second ob- tains a dollars more than the first, and the third b dollars more than the second. Express the value of the estate, tlie share of the first heir being x. 18. A man who possessed x dollars,* lost a of them. How many has he left ? FACTOR. — COEFFICIEKT. — POWEK. — TERMS. 1 1 19. The Slim of two quantities being represented by s, if one of these quantities is expressed by x, what will be the other ? 20. A debtor set out to pay his creditor a part, a^ of his debt X ; but on the way he met another person, to whom he gave a sum Z> and carried the remainder to his creditor. Ex- press what remains of the debt. 21. The three figures of a number are such that the tens figure exceeds the units figure by 2, and the hundreds figure exceeds the tens figure by 3. What is the sum of the figures, X being the units figure ? 22. Express 45.6 by means of algebraic signs. 23. The figures of a number in the hundreds, tens, and units place respectively are x, y, z. Express the number. Express the number with the figures reversed. 24. A w^orkman whose wages were x dollars a day, receives during n days an increase of wages amounting to h dollars a day. Express the amount of his wages for this time. 25. Express the number which divided by 11 gives 5 for a remainder, x being the quotient. 26. A man travels uniformly a distance s in a number t of hours. AYhat is his rate of travel per hour ? 27. Two fountains fill a reservoir, one in a number of hours represented by x, the other in 3 hours more. What part of the reservoir does each one fill in one hour ? 28. A number N divided by d gave for remainder the number r. Express the quotient. II. Factor — Coefficient — Power — Terms. 12. When one number consists of the product of two or more numbers, each of the latter is called a factor of the product. Thus, for example, 2 X 3 X 5 = 30 ; and each of Define Factor. 1^ ELEMENTARY ALGEBRA. the numbers 2, 3, and 5 is a factor of the product 30. Or we may regard 30 as the product of the two factors 2 and 15, or as the product of the two factors 3 and 10. So also we may consider 4 a Z> as the product of the three factors 4, a, and ^, or of the two factors 4 a and b; or as the product of the two factors 4 and ad; or as the product of the two factors 4 d and a, 13. When a number consists of the product of two fac- iTors, each factor is called the coefficient of the other factor ; so that coefficient means co-factor. Thus, considering 4: ad as the product of 4 and a d, we call 4 the coefficient of a h, and ab the coefficient of 4; and considering 4a^ as the product of 4: a and d, we call 4 a the coefficient of b, and I? the coefficient of 4 a. In practice, the name coefficient is ap- plied to the factor which precedes the other, and usually the first factor is called the coefficient. When this first factor is a number expressed by a figure, it is called the oiumerical coefficient. Thus 4 is the numerical coefficie7it of ab in the expression 4:al). Since 1 is a factor of every product, when no numerical coefficient is written before an algebraic ex- pression 1 is always understood as its 7iumerical coefficient. Thus 1 is the numerical coefficient of the expressions, a h, a, abc, Wlien we use one of the letters of a product as a coef- ficient it is called a literal coefficient. Thus in the expres- sion a X, a is the literal coefficient of x, 14. When all the factors of a product are equal, the product is called a poiver of that factor. Thus 7 X 7 is called the second poiver of 7; 7 X 7 X 7 is called the third foioer of 7 ; 7 X 7 X 7 X 7 is called the fourtli power of 7, and so on. In like manner, « X <^ is called the second poiver of « ; aX aX a\s called the third power ofa; aXaXaXa is called ihQ fourth power of a, and so on. And a itself is called the first p)0wer of a. Coefficient, numerical and literal. Power. FACTOE — COEFFICIEXT — POWEE — TEEMS. 1 3 15. Instead of writing all the equal factors, we express a power more briefly by writing the factor once, and placing over it, and a little to the right, the number which indicates how often the factor is to be repeated. Thus c^ is used to de- note ^ X « ; ci^ denotes a X aX a] a^ denotes a X a X a X a; 3a^ b^ c^ d denotes oaaaahl)h ccd, ^ndi so on. And a^ may be used to denote the first power of a, that is, a itself; so that a^ has the same ^meaning as a. The number thus placed over another to indicate how many times the latter occurs as a factor in. a power, is called an index of the jpoiver, or an ex- ponent of the i^oiver^ or briefly an index or exponent. Tlius, for example, in a^ the exponent is 3 ; in a"" the exponent is n ; in 2"" the exponent is m, 16. The second power of a, that is a^, is usually called the square oi a oy a squared ; and a^ is often called the cuhe of a or a cubed. For the powers higher than these tliere are no similar words in use ; a'^ is read thus, " a to the fourth potver/' or " a to the fourth f a"^ is read " a to the 7nth." 17. The student must distinguish carefully between a coefficient and an exponent. Thus 3 c means three times c, and m a stands for m times a. Here 3 and m are coefficients. But c^ means c times c times c ; and c"^ stands for c times o times c times c and so on ... . times c. 18. If an expression contain no parts connected by the signs + and — , it is called a simple expression or monomial. If an expression contain parts connected by the signs + and — , it is called a compound expression ; and the parts connected by the signs + and — are called terms of the ex- pression. Thus ax, 4: be, and 6 a"" c"^ are simple expressions; a^-^-F—c^ a^-{-?ta^b'\-c^, are compound expressions, of which a^, F, and c\ and ct\ 3 «^ b, and c^ are the tenns respectively. 19. Let the student distinguish carefully between tenns Index or Exponent, a^ . ^3 • ^.,1 Difference between Coefficient and Exponent. Monomial Terms. Terms and Factors. 14 ELEMEKTARY ALGEBRA. and factoids, recollecting that factors are those parts of an expression which are connected by Multiplication, and terms are those parts of a compound expression which are connected by Addition and Subtraction. Thus 5, a^y d, and c are factors of the expression 6a'^bc; while 6cr, h, and c are terms of the expression ba^ -^-1)— c» 20. When an expression consists of tivo terms it is called a Uiiomial expression, or briefly a binomial ; when it consists of three terms it is called a triyiomial ; any expression con- sisting of several terms is called Si multinomial or jwli/nomial Thus 2«^ -\-dabc is a binomial expression ; a — 2b -h 6c is a trinomial expression ; and a"^— ¥+ c^— 4tab — e is a polynomial. 21. Each of the letters or literal factors of a term is called a dimension of the term, and the number of these literal fac- tors is called the degree of the term. Thus 2a^b^c on: 2 a abb be has six dimensions and is said to be of the sixth degree. The numerical coefficient is not counted; thus 9a^^* and a^ V are of the same dimensions, namely, seven dimensions. The degree of a term, that is, the number of its dimensions, is evidently the sum of the exponents of its literal factors, provided we remember that when no exponent is expressed the exponent 1 must be understood. 22. An expression is said to be Jiomogeneous when all its terms are of the same degree. Thus '7 a^ -hSa^ -i- 4:abci8 homogeneous, for each term is of three dimensions, that is, of the third degree. We shall now give some more examples of finding the nu- merical values of algebraic expressions. Suppose a = 1, b = 2, c = d, d = 4:, e = 5, /= 0. Then b' =4, b' = 8, h* = 16, b' = 32. Binomial. Trinomial, Polynomial. Dimension and Degree of Term. When is an expression homogeneous? VN{Y£RSrTY I 15. a'^^^z^lxS^S, 3^V=3X4X9 = 108. ^ + c'-7aZ»-f-/'=: 64 + 9-14 + = 59. 3c^-4^-10 27-12-10 5 . — = 0. c'-2c' + 5c-23 27-18 + 15-23 1 6^ + rr g^-a^ _ 125+ 64 27-l _189 26_ _ e+^ c-a~ 5 + 4 3-1 ~ 9 2~ ■^^~^' Examples — 2. If a = 1, Z> = 2, c = 3, rf = 4, e = 5, / = 0, find the numer- ical A'alues of the following expressions. 1. «'+Z/' + c'+^+e'+/'. 5. a'-\-?>a^l^Zah^^l\ ^V ^_32 9 ^' + ^' ^' + ^' e'-cl" 6. 8. 2. e'-J^ + c'-JHa'. 4. c^-"2c^+4c-13. e*-4e'^+6e*J'-4eJH^*. 2g + 2 3g-9 e--l e-d e—2 e+3 8a^+3^^ 4:c''+6b\ c' + ^ A v. o , 7 9 "1 11. a' + b' 28 1 + 12 + - a^+^^ + 6''^rf-^^-^*^^V + 6^-c'-6?'* 12. a'-V^fn-V^crb^-V^ab^-\-b'' a^-^?>d'b-\-Zab^V ' 13. d' 14. ^ + Z>'' 16 ELEMENTARY ALGEBRA. III. Kemaining Signs — Brackets. 23. Tlie sign > stands for is greater tlian, and the sign < denotes is less than, Tims a> h denotes that the quantity a is greater than the quantity h, and h <a denotes that the quantity h is less than the quantity a. In this sign the opening is always turned toward the greater quantity. 24. The sign .*. denotes then or therefore; the sign •.* de- notes since or because, 25. The square root of a quantity is that quantity whose square or second potver is equal to the giyen quantity. ' The cube, fourth, &c. root of a given quantity is that quantity whose cube, fourth, &c. power is equal to the given quantity. Thus since 49 == T, the square root of 49 is 7; also the square root of a^ is a. In like manner, since 125 =: 5^ the cube root of 125 is 5 ; and so if a = c^y the cube root of a is c, 26. The symbol used to denote a root is \/ (a corruption of r, the first letter of the word radix), which with the proper number as index on the left side of it, a little above, is set before the quantity whose root is expressed. Thus V«' ^ a, V64 = 4, V3125 = 5, VI = 1, VI = 1, &c. The index, however, is generally omitted in denoting the square root ; thus v^a is written instead of V«. Examples — 3. 1. v/4"+2v/25 + 3v/49-v/ 64 = 25. 2. 3n/i6-4v/36 + 2v/9'-\/8T=-15. 3. V8"+2Vi25-4Vr4-V64 = 12. 4. vr+3Vi6-2V32+3vr=6. If a=2D, b=9, c=4:, d=l, then 5. y/a-{-2^/l; + 3^/'^ + 4:^/d=2l. Explain the signs > < .*. •.• Square root of a quantity ? Cube root? Th« Bymbol used to denote a root. KEilAlXIXG SIGXS. — BRACKETS. 17 7. 3Va+2VAb-4:Vrc+Vm=7. 8. V5^+2V3b-V2c-h4:Vd=13. 10. Vbc+dVacd — 4:Wd+V^^ — 4:, 27. |/^ means that the square root of the fraction % is to be taken ; but — — means that the square root of a is to be divided by t. Examples. 1. What is the difference between 2\/'^ and 2+ \/^^ when X is 100 ? . Ans. 8. 2. What is the difference between 3 \/"^, and V^, when x is 64 ? Ans. 20. 3. What is the difference between \/ a-^h and v/"^ + Z>, when a stands for 1, and 1) for 8 ? Ans. 6. 4. What is the difference between \/%- and -— ^, when a stands for 16 and Z> for 4? Ans. 1. 5. What is the difference between v/ « + \/y and >/^+^^ when a = 16 and Z> = 9? Ans. 2. 28. Brackets, (),{},[], are employed to show that all the quantities within them are to be treated as though forming but one quantity. It is of great importance to notice care- fully the effect of using them. Thus a—{b—c) is not the same as a—b—c; for, in this last, both b and c are subtracted, whereas in the former it is only the quantity b — c which is subtracted. The use of Brackets. 18 ^ ELEMEKTAEY ALGEBRA. Hence, if a = 4:, b = 3, c = 1, we liaye a — d — = 4.-3 — 1 = 0, a—{b — c)=4 — 2 = 2; 2a-'3h + 2c=S-9+2 = l,2a-{3b + 2c) = S-ll = -3, 2a-^b-c=S+3-l = 10,2{a+b)-c=U-l = 13, 2{a + I)-c)=12. If we wisli to denote that the sum of a and d is to be mul- tiplied by Cy we write {a+b)Xc or {a + I)}Xc, or simply {a-}-I))c or {a + d}c; here we mean that the whole of a-\-b is to be multiplied by c. Now if the brackets were omitted we would have a + be, which denotes that b only is to be multi- plied by c and the result added to a. Similarly {a + b—c)d denotes that the result expressed by a + Z* — c is to be multi- plied by d. So also {a — b + c)x{d + e) denotes that the result ex- pressed by a — J + c is to be multiplied by the result ex- pressed by d + e. This may also be denoted briefly thus, (a — b + c) {d-\- e); just as a X ^ is shortened into a b. So also V{a + b-\-c) stands for the square root of the result expressed hj a + b-\-c. So also \/{a+b-\-c) denotes that we are to obtain the re- sult expressed hj a+b+c, and then take the square root of this result. So also {aby denotes abX ab; and (a by denotes abXabXab. So also {a+b—c)-^{d-\-e) denotes that the result expressed by a -\- b — c is to be divided by the result expressed by d-\-e. 29. Sometimes instead of using brackets a line is drawn oyer the numbers which are to be treated as forming one number. Thus a—b-\'CXd+e is used with the same mean- Vinculum. kemai:n'ikg sigxs — brackets. 19 ing as {a—bi-c)x{d+e), A line used for this purpose is called a vincuhun. So also {a+b—c)-^{d-{-e) may be de- noted thus, — 9— — ; and here the line between a -\- b — c and d-{- eh really a vinculum used in a particular sense. Thus, too, a vinculum from the top of a radical sign is fre- quently used, and \/ a + h + c has the same meaning as V{a-\-b-\'c). 30. We have now explained most of the signs used in Algebra. It is well to observe that the word sign is applied specially to the two signs + and — . Thus the expressions "changing the signs^^ "like signs," and "unlike signs" refer exclusively to + and — . 31. We shall now give some more examples of finding the numerical values of expressions. Suppose ^ = 1, 5 = 2, c = 3, ^ = 5, e = 8. Then V(4c-2^>)=V(12-4)=V(8):ir:2. eN/(25+46')--(26Z-^)V(4c-2^)=:::8x4~8X 2=32-16 = 16. v/{(6-^)(26-5Z^)}-v/{(8-2)(16-10)} = v/(6x6)3=6. {(e-^)(^4-c)-(c?-c)(c+a)}(fl^ + ^)^{3x5-2x4}6 = 7X6=42. V(c'' + 3c'^5 + 3c5'^ + &')-^v/(^^+Z>^-2a^) = V(27 + 54 + 36 + 8)--v/(l + 4-4)=V(125)-^l=5. Examples — 4. If a = 1, 5 = 2, c = 3, J = 5, e = 8, find the numerical values of the following expressions. The word Sign^ how applied? 20 ELEMEXTARY ALGEBRA. 1. a{h + c). 2. b{c + d): 3. c{e~d). 4. b^a'+e'-c'). 5. c'ie'-F-c'), 6. cr+b' 7. ^"^^-. 8. n/(3&c^). 9. V{2b+U+6e). 10. (r^+25 + 3c+56— 4<^)(6e-5tZ-4c~36 + 2a). 11. {a'' + b' + c''){e'-d'-c'). 12. (3^-7c^)^ 13. ex/(rr-36) + ^v/(^+3e). 14. e-{V{e+l)+2}+{e-Ve)V{e-'4:). If a = 5, 5 = 3, c = 1, show that the numerical values are equal. 15. Of a'-b' and (a-b) {a'+ab + b'). 16. Of b'-c' and (b + c) {b-c) {b' + c'). 17. Of a' + a'b'+b' and (a^+a5 + ^'^) (a'-^&4-^>'). 18. Of b'+4.c' and {Z>^+2(^ + ^)4{^'-2(5-c)4. lY. Change of the Order of Terms — Like Terms. 32. The terms in an expression which are preceded by the sign + are called positive terms, and the terms which are preceded by the sign — are called 7iegative terms. 33. We must now extend the meaning and use of the sign — beyond the strict application of ordinary arith- metical notions. If in the expression a — b -\-c, « =4, 5=7, and c = 8, then by our first definition of the sign — we should have to subtract 7 from 4, which is impossible. In this case we subtract the 4 from the 7 and write the remain- der with the sign -. Thus -7 + 4= -(7 -4) = -3. Then Positive and Negative Terms. Algebraic meaning and use of the Sign — . CHANGE OF THE ORDER OF TERMS — LIKE TERMS. 21 we consider —3 + 8 to be the same as 8 — 3 = 5, and 5 is the numerical value of the expression a—b-{-c=7—4z-\-8, 34. It is then indifferent in what order the terms of an Algebraic expression be written. This is clear from the com- mon notions of Arithmetic, and from the convention that — 5 + ^ is the same as a — b. Hence, if a term is preceded ' hy no sign, the sign + is to be under stood, and such a term is counted loith the positive terms, ^ Thus, 7+8-2-3 = 8 + 7-2-3=:~3+8-2+7, &c. a-]rb—c=b-\-a—c=b—c-\-a=—c-\-b-\'a, &c. 35. Terms are said to be like when they do not differ at all or differ only in their numerical coefficients ; otherwise they are said to be unlike. Thus a, 4a, and 7«^ are like terms; a^b c, 5 a^b c, and 7 a^b c are like terms ; ba^, 6a b, and 6b^ are unlike terms ; 4a* and b c are unlike terms. 36. An expression which contains like terms may be sim- plified. For example, consider the expression 6a—a+3b + 6c—b + Sc—2a. This expression, by Art. 34, is equivalent to 6a—a—2a+db—b + 6c^dc. JSTow 6a— a— 2a— da. For a from 6a leaves 5a; and taking 2a from 6a we have da left. Similarly db — b = 2b ; and 6c-\- dc = Sc. Thus the expression may be put in the form 3a + 25 + 8c, Again, consider the expression a— 3^— 4^. This is equal to a — 7^. For if we have first to subtract 35 from any number a and then to subtract Ab from the remainder, we shall obtain the required result in one operation by subtract- riie Terms, in what order to be written. Like and unlike terms. Simplification of expressions containing like terms. ^2 ELEMENTARY ALGEBRA. ing ^ib from a; this follows from the common notions of Arithmetic. Thus « - 35 - 4^ = a - 75. 37. The statement — 35 — 45 = — 75 is explained thns : if in the course of an Algebraic operation we have to sub- tract 35 from a number and then to subtract 45 from the remainder, we may subtract 75 at once instead. It will be seen that by an easy extension of this, the expression — 75 has a meaning when standing by itself. 38. It may be noticed (as we have proved in Arithmetic) that it is immaterial in what order the factors of a quantity are arranged. Thus 7X5 is the same as 5x7; 2x6x9 the same as 6X2X9 or 9x2x6, &c.; abc the same as cab or cba, &c. ; 6 x^a and a x^ are like terms. It is usual, however, to arrange literal factors and terms as much as possible in the order of the letters of the alphabet. 39. The simplifying of expressions by collecting like terms is the essential part of the processes of Addition and Sub- traction in Algebra. Ex. 1. Group together like quantities, with their proper signs, from 5a — 35, 4a + 75, and — 8a — 55. Ans. + 5a + 4a ~8a —35 Here the quantities in each column + 75 are lilce, but the two columns are —55 unlike. Ex. 2. Group together like quantities, with their proper signs, from a'+ 3a'5 + 3a5^+ 2a'+ 25^4- 5a5^- Sac''- a^b - b\ -Sac' Ans. + a' I -f-3a'5 + 2a^ I - a^b + 5a5' +25' — b^ Ex. 3. Group together lilce quantities, with their proper signs, from 2a - 35 + 75c + 5'c - 5a5c + 2.Ty -3:c' + 55" + Order of arraK<rement cf Factors. CIIAKGE OF THE ORDER OF ' TERMS. — LIKE TERMS. 23 Wc - 9a - W + 6^ 4- 10a - hx'^ - xy ^ x^ -\- ahc - 2bc + c' -b- dc\ Ans. + 2a -3b -{•7bc\+ Fc —oabc\-\-2xy -^x" ■VW — 9a -4-6^ -2bc -\'Wc + abc — ^y -bx' -2b' + 10a - b + x" 40. We shall close this chapter with some more examples of the conversion of ordinary language into Algebraic lan- guage. Ex. 1. A person makes a mixture of three sorts of wine. The second costs a dollars more per gallon than the first ; and the third b dollars more per gallon than the second. There are m gallons of the first sort, n of the second, and p of the third. What is the price of the mixture, x being the price of the first sort ? Ans. mx ■{- n (x -\- a) ■\- ;p {x ■\- a ■\- b), Ex. 2. Express algebraically a number of five figures a, b, c, d, e, taken in their order from left to right in the decimal system. Ans. lO'Xa + 10'X^ + lO'Xc + lO^+e. Ex. 3. Three fountains run successively into a reservoir, the first during a hours, the second during b hours, the third during c hours ; the second fountain supplies m gallons per hour more than the first, the third n gallons more than the second : how many gallons of water did the three foun- tains yield, x being the number of gallons per hour which the first yields ? Ans. ax + b (x + m) -\- c{x -\- m + n). Ex. 4. A merchant sells a certain number of yards of cloth for a dollars per yard. A second merchant sells b more yards of the same cloth at c dollars more per yard. If a; is the number of yards sold by the first, express the difference in the amounts received by the two. Ans. {x-\-b){a-\-c) — ax. 24 ELEMENTARY ALGEBRA. Ex. 5. A mixture is made of 4 substances, A, B, C, D. It is composed of a gallons of A, costing m dollars per gallon, of h gallons of B at n dollars a gallon, of c gallons of C at 2) dollars a gallon, and of d gallons of D at g dollars a gal Ion. Express tlie price of a gallon of the mixture. , am + bn -\- cp + da Ans. • ;-, — 7-—. a + -\- c + a Ex. 6. Three laborers paid at the same rate worked, the first, m days, the second n days, and the third q days. They received altogether a dollars. Express the daily wages of each. Ans. — ■ -— . m+n+q Ex. 7. A sum a produced in b years c dollars at simple in- terest. Express the rate per cent. 100c Ans. -7 — . ba Ex. 8. A sum a placed at simple interest amounts in n years to b dollars. What is the rate per cent. ? Ans. m^-^. an V. Addition. 41. It is conyenient to make three cases in Addition, namely : I. When the terms are all like terms and have the same sign. II. When the terms are all like terms, but have not all the same sign. III. When the quantities to be added consist of both like and unlike terms. ■ 42. I. To add like terms which have the same sign. Add the numerical coefficients, prefix tJie common sign to the sum, and annex the commo7i literal factors. For example : 6a + 3a + 7a-{-6a = 21a. - 3b'c - bb'c -10b' c = - ISb'c. Three cases in Addition, Rule for Case I. ADDITION". 25 43. II. To add like terms which haye not all the same sign. Add separately the positive numei^ical coefficients, and the negative numericcd coefficients ; take the difference of these tivo sums, prefix the sign of the greater to this difference, and annex the common liter cd factors. For example : 7rr - 3^f + 11«' +a' - oa' - 2a' = 19a' - 10a' =9a'. %l)0 _ ^ic - Zlc + ^hc + hhc - Q>hc = llhc—l^hc = - 6bc. 44. III. To add expressions which consist of both like and nnlike terms. Add together the like terms by the rule in Case II, Affix to the sums thus obtained the unlike terms^ each preceded by its proper sign, For example : add together 4a-f 5^-7c + 3f/, 3a-& + 2c+5^, ^a-^b-c-d, and —a + 'db-\-4:C—M^-e, It is conyenient to arrange the terms in columns^ so that like terms shall stand in the same column; thus we haye 4a+5^-7c+3f? 3a- b-\-2c-VM 9a— 2b— c— d -a-^Zb^4.c-M-\-e lDa-\-bb—2c^^d-\-e. Here the terms 4a, 3a, 9a, and —a are all like terms; the sum of the positiye coefficients is 16 ; there is one term with. a negative coefficient, namely —a, of which i\\Q coefficient is 1. The difference of 16 and 1 is 15 ; so that we obtain -f- 15a from these like terms : the sign + may, however, be omitted. Similarly we have bb — b — 2b -\- ^b = 6b. And so on. Rule for Case II ; for Case III. 2 26 ELEMENTARY ALGEBRA. 45. In the following examples the terms are arranged suit- ably in columns. 4:x' + '7x''+ x-9 — 2x'+ x'- dx + S -3x'- a;' + 10.T-l dx'- x-1 a'-h ab+ b'-c a^-'Zab-W In the first example, we have in the first column x^ + 4zX^ —2x^ —Zx^ , that is, bx^—bx^, that is, nothing; this is usually expressed by saying the terms ivhicii involve x^ cancel each other. Similarly, in the second example, the terms which involve ab cancel each other; and so also do the terms which in- volve z>^ Ex. Add together a + ^b — c, a—6e-h2c, and x + y + 3e. Here a and a are UJce, — 6e and +3e — c and +2c the rest are imlike. a + 2b— c a—5e-\-2c 3e + x + y Sum =: 2a + 2b + c—2e + x-\-i/, Ex. Add together Za^ — bc, 2b^—ac, 4cG^ — ab, and a^ + b'^ ^c\ Here 3a^ and a^ are lihe, 2^^ and ^-b^ 4c^ and — c' the rest are unlike. 3a^-bc 2b' -ac ^&-ab a'+ y- c' Sum = 4ca^ + 3b' -{-3c'-ab-ac - be. ADDITIOX. 27 Ex. Add together xy—1, vc'-f 2, and ^'' + 3. Here the terms are all x^ + 2 icnlilce, except —1, +2, and + 3. ^^ + 3 Sum = x^ + xi/-^y^ + 4:. 46. The Kules aboye given for the Addition of like and imliJce algebraical quantities are in no wise different from those employed in Arithmetic. For, suppose we have to add together 3 hundreds and 4 hundreds, we combine these lihe quantities by taking the sum of the coefficients 3 and 4, so as to make 7 hundreds. But if we have to add together 3 hundreds, 5 tens, and 6 units, these, being unlike quantities^ cannot be added in the same sense, but are merely collected together in one line, 3 hundreds +5 tens +6 units, which, for convenience, is written shortly 356. It will be observed, however, that algebraical Addition involves the processes both of arithmetical Addition and Subtraction. Examples — 5. Add together 1. a + 5 and a^-h, ^, a^h and a—h, 3. G^— 6 and a— ^. 4. « — S + c and a4-^— f-. 5. a—h-\-c^VL^a-\-h-^c, 6. 1 — 2m + 37^ and3m— 2;i + l. 7. 5m + 3 and %m —4. 8. Zxy—%x and xy^Qx, 9. 4;9-25' + l and 7-3^?-!-^. 10. 5aZ> — 2^c and ah -{-be, 11. 3a~2^, 4a-55, 7a-115, a+95. 12. 4a;' -3?/', 2x^-by\ ~x^-Vy\ -22:'-^4y^ 28 ELEMEXTARY ALGEBRA. 13. 5a-i-3I) + Cy da + 3b + 3c, a + db + oc. 14. 3X + 22J-Z, 2x-2ij + 2z, -x-\-2y-^3z. 15. "/a-U + c, 6a + 3b-6c, -12a + 4.c. 16. x—4:a + b, 3x + 2I), a—x—hl, 17. a-\-l—c, h + c—ay c^-a—l, a^l—c. 18. a4-2Z> + 3c, 2a— I— 2c, l—a—c, c—a—K 19. a— 2J + 3c— M ?>l}—4.c-\-Dd—2a, 5c-6^+3a— 45, 20. x'-^x''-\-bx-Z,2x'-W-li:X-\-^, -x'-^-^x^'-Vx-^S. 21. a;'-2:^' + 3z',ix;' + a;'^ + :?;, 4:^;' + 5:c^ 2a;'^ + 3:?;-4, -32:^ -2.-^-5. 22. a'-'^cn-{-'daV-d\ 2a' + 6a'I)-6ad'-7b\ a'-ab' + 2I?\ 23. x'--2ax' + a''x + a% x' + dax\ 2a^-ax^-2x\ 24. 2a'b—Zax^ ^2(j^x, \2db-\-V)adi^ — ^(^Xy — Sal? + ax^—5a^x, 25. 2;^ + y* + ;^^ -4.x'-6z% Sx'-lif + lOz', 6t/-6z\ 26. 32;' - 4:?;?/ 4- ^' + 2:^; + 3?/ -7, 22;'-4?/'-h32;-5y + 8, lOa;?^' 4- 8^' + 9y, 5a;' - 62;?/ + 3/ + 72^-7^ + 11. VI. Subtraction. 47. Suppose we have to take 5 + 3 from 14. The result is the same as if we first take 5 from 14, and 3 from the re- mainder. This result is denoted by 14 — 5 — 3. SUBTRACTION^. 29 That is, 14 -(5 + 3) = 14 — 5 —3. The brackets mean- in 2: that the whole of the 5 + 3 is to be taken from 14. In like manner, suppose we have to take 'b-{-c-{-d from a. The result is the same as if we first take h from a^ and then take c from the remainder, and then d from that remainder; that is, the result is denoted by a—h—c—d. Thus a~i^^c^-d)=a—'b—c—d. We see in these cases the positive terms of the expression to be subtracted have all been changed to negative terms in the result. 48. l^ext suppose we have to take 5 — 3 from 14. If we take 5 from 14 we get 14—5; but we have taken too much from 14, for we had to take, not 5, but 5 diminished by 3. Hence we must increase the result by 3. The result is then denoted by 14—5 + 3. Thus 14-(5-3) = 14-5 + 3. In like manner, suppose we have to take h—c from a. If we take l from a, we obtain a—h'^ but we have thus taken too much from a, for we had to take, not Z>, but Z> diminished by c. Hence we must increase the result by c. Thus we obtain a— 1)^-0, That is a—{h—c)—a-''b^-c. By the same reasoning a—{h—c—d)=a—l)-\-c-\-d. Here the positive term of the expression to be subtracted is negative in the result, and the negative terms are posi- tive. 80 ELEMEKTAKY ALGEBRA. 49. Hence, we have tlie following rule for Subtraction : Change the signs of all the terms in the ex^jression to be stihtracted, and then proceed as in Addition, For example: from 4:X—?>y -\-2z subtract 2x—y-\-z, Change the signs of all the terms to be subtracted ; thus we obtain —^x + y—z'y then collect the terms, and simplify as in Addition. Thus 4:X—'dy + 2z—'dx + y—z=x—'^y-\-z. From ^x' + bx'-Qx^-'lx + b, take 2.x*-2aj' + 5cc'-6a;-7. Change the signs of all the terms to be subtracted, and proceed as in Addition. We thus have 3aj' + 5cc'-6x'-7cc + 5 -2a3^ + 2aj'-5x' + 6aj-f7 «j' + 7i«'~llaj'-cc + 12. The beginner will find it best at first to go through the process fully as above ; but he will soon learn to put down the result without actually changing all the signs, but merely doing it mentally. 50. We often have a single negative term to be subtracted from another term or expression. Thus, from a subtract — c. Here we can reason thus: Since a=a + 'b—'b, if we sub- tract — h from a, the result is a + h, the same as if we add + h to it. Or we can apply our rule, at once considering the result to have a meaning in connection with some other parts of an algebraical operation. Examples. 1. From ^a 2. From la 3. From a take a take 6a take a Ans. 2a a Rule for Subtraction. SUBTRACTION. 31 4. From 3a 5. From 7a 6. From a take —a take —6a take —a Ans. 4a 13a 2a 7. From ~3a 8. From —7a 9. From —a take Ans. a ~4.a • take 6a take a -13a -2a 10. From —3a 11. From —7a 12. From —a take Ans. — a take —6a — a take —a -2a 13. From a-hb 14. From a—b 15 . From y + ax take Ans. a-b 2b take a + b -2b take y—ax 2ax 16. From 3a-U + 6c 17. From 7a- -2b-\-4:C-2 take Ans. a— 2Z> + 9c take 6a' -6b-\-4.c-l 2a-2b- -3g a + ^b-1 18. From 2a— 6a^— ac + 5 19. From 3cc?/— cc'— y^' + a take 5a~8aJ— 2ac— 1 take 2xy + x^-\-2y'^ — b An s. — 3 a + 2a^ + ac + 6 xy—2x'' — 3y'' + a + b 20. From a^ + 2aJ-36'' take 2a'^ — Dab—7c^ Ans. -a' + 7a2>+4c' 21. From bx"^- xy+ y^ take — a3^ + 4i^^ + 3^^ 6x'-6xy-2y^ 32 ELEMENTARY ALUt:i3EA. 22. 'FiomSa'+x'-bb'-Dc'' 23. From ^'- 3a;' + 6:^;- 10 take x'-{-2F-6c'' tske x'-4.x'-^8x- 9 Ans. 8a' -n' x^-%x-\ 24. From a-\-\h-\-\ 25. From \x^—\xij^\if take 2^ + ^ + 2 ' ^^^^ '~3^'~i^^~2^' 111 Ans. -ir^-17^4-^ x^—xy-\rm^ z z z Examples — 6. 1. From 7a + 14^ subtract 4a + 10Z>. 2. From 6a--2J— c subtract 2a— 2^— 3(?. 3. From 3a— 2Z> + 3c subtract ^a—lh—c—d. 4. From "Ix^—^x—l subtract bx'—Qxi-3. 5. From 4.x' -2>x' -2x^-^1 x-^^ subtract x' — 2x^ — 2x^ + 7x—9, 6. From 2x^ — 2ax+3a^ subtract x^—ax + a^. 7. From x'—3xy—y''-]-yz—2z' subtract x^ + 2xy + bxz — 3 ?/' — 2^;'. 8. From hx'' + Qxy-12xz-4.y''-^2jz-bz^ subtract 2a;' - 7a;?/ + 4.T;a;- 3?/' + ^yz— ^z\ 9. Froma'-3a'Z^ + 3aZ>'-Z^'' subtract -a' ^-^cn-^alf + 1)\ 10. From 7a;' - 2a;' + 2a; +2 subtract 4a;' -2a;' -2a' -14, and from the remainder subtract 2a;' — 8a;'-f-4:c+16. VII. Brackets. 51. On account of the extensive use of brackets in the algebraic language, it is necessary that the student should observe very carefully the rules respecting them. Since the sign + or — preceding a bracket means that the whole included quantity is to be added or subtracted, if we wish to remove the bracket, we must actually perform the operation indicated by means of it; i. e., we must add or subtract the quantity in question. ISTow when a quantity is added, the signs of its terms are not altered ; but when it is subtracted, the signs of its terms are changed. Hence, When an expression is ivitliin a pair of brackets preceded ly the sign -\-, the hrachets may he removed, the signs of the included terms 'being unchanged. When an expression is ivithin a pair of hracJcets preceded ly the sign — , the hraclcets may le removed if the sign of every term luithin the hraclcets he changed. Thus, for example : a—'b-\-{c'-d-\-e)=^a—'b-\-c—d-\-e a—I? — {c—d-\-e) = a—h—c-\-d—e, Kemember, that if the first term within the brackets has no sign, the + sign is understood before it. ^2, In particular, the student must notice such statements as i\\Q following: These are immediate consequences of what we have said of the addition and subtraction of single terms. Eules for removing Brackets. 2^ 34 ELEMENTARY ALGEBRA. 53. Expressions may occur with more than one pair of brackets ; these may be remoyed in succession by the pre- ceding rules, beginning with the inside pair. Thus, for example : a+{l)-{-{c—d)}=a+{b + c—d]=a + 'b-\-c—d, a-v{b—{c—d)]=a^-{'b—c-^d]—a^-'b—c-\-d, a— ' {!)-{- {c—d)}=a—{'b^- c—d]-=ia—'b — c-{- d, a— {l)—{c—d)} ^a— {l)—c-[- d) =a—d -{- c—d. Similarly, a—[b—{c—(d—e)}'\=a—[h—{c—d+e]'\ =a—[b—c-\-d—'e]=a—I) + c—d + e, It will be seen in these examples that, to prevent confu- sion between various pairs of brackets, we use brackets of different shapes ; we might distinguish by using brackets of the same shape but of different sizes. A vinculum is equivalent to a bracket ; see Art. 30. Thus, for example : a-\h-{c-{d-^)]l^=a-\l-{c-{d-e-^f)]\ =a-\h—{G-d-\-e-f]^=a-\l-c + d-e+fl^ ^a—l + c—d-^-e—f. 54. The beginner is recommended to remove brackets in the order shown in the preceding article, {i. e.) by removing first the innermost pair, next the innermost pair of those which remain, and so on. We may, however, vary the order, by removing first the outermost pair, next the outer- most pair of those which remain, and so on. Thus, for example : a-\'{h-\-{c-d)}=a-hh-\-{c-d)=a + b + c—dy BEACKETS. 35 a+ [b— {c— d)} = a + b— {c— d) -—a + I)—c + d, a—{b + {c — d)}=a—I) — {c—d)=a — I) — c + d, a—{b — {c—d)}=a—b-{-{c—d) = a--b-\- c—d. Also, a—[b-{c-{d—e)}]^a-b+{c—{d-e)} =a—b + c—{d—e)~a—b-{-c—d+e, 55. It is often convenient to take up in brackets any given terms of an expression. Tlie rules for thus introducing brackets follow immediately from those of removing brackets. Any oiiwiber of terms 'i?i an expression may be put loitJmi a pair of braclcets, and the sign + pUtced before the brachet, the signs uf the terms being unchanged. Any number of terms in an expression may be put icithm a pair of braclcets, and the sign — placed before the brachet, 'provided the sign of every term put loithin the brackets be changed. In applying this rule, we shall for convenience take the sign of whatever term we choose to set as first term within the brackets, as the sign to be placed before the bracket. Thus -\-a—b — c, collected in a bracket with -\-a SiS first term, will be +{a—b—c); but, with —b as first term, it will be —{b — a + c), and with —c as first term, it will be — {c'-a + b); and now, if we resolve again these last two brackets, the sign ( — ), preceding each of them, will correct the changes we have made, and the quantities will be repro- duced, as at first, —b + a—c, —c-i-a—b. So also we might use an inner bracket, and write the quantity -\r {{a~b)—c}, or + {a—{b + c)}, or —{{b — a) + c}, or —{b—{a—c)}, &c. Rules for intv^d'^clng Bracketg. '56 elementary algebra. Examples — 7. Eeduce to their simplest forms : 1. {a-x)-{2x-a)~{2-2a) + {3~2x)-{l-x). 2. {a' - 2a' c + 3ac') - {a'o - 2a' + 2ac') + {a' - ac' - a'c). 3. {2x'~2jf-z')-{3if + 2x'-z')-{3z'-2tf-x'). 4. {x' + ax' + a'x)-{i/-I?y'-hh'i/) + {z' + cz' + c'z) - i^'-y" +-^") + {cix' + hf + cz') - {a'x-Fij + c'z). 5. a"- {W-c') - {b'- {c'-a')] + [c'- ifi'-a')}. 6. {2a'-{dad-F)}-{a'-{4.ad + W)]-\-{2h'-{a'-al))]. 7. {x'^f-{dx\j-^3xif)}-{{x'-3xhj) - Cdxif-if)}. 8. {'^x-{3y-z)}-{y-\-{2x-z)}-\-{3z-{x-2y)] ^{2x-{y-z)}. 9. l_{l-(l-4r^)} + {2a;^(3-5:r)}-{2~(-44-5a;)}. 10. {2a-{3I)-\-c-2cl)]-{{2a-3h)-{-{c-2d)] > '\-{2a-{3h-\-c)-2d)}-{{2a-3'b-\-c)-2d}. Express by brackets, taking the terms (i) tivo together, (ii) three together : — 11. a-2l-{-3c—d-\-2e—f. 12. —2'b-\-3c—d-^2e—f-\-a. 13. 3c-d^2e-f-\-a-21). 14. -d-^2e-f-\-a-2'b-Y3c, 15. 2e-f-]-a-21)-\-3c-d. 16. ^f^a-21)^-3c-d-\-2e 56. In Addition and Subtraction we have spoken hith- erto only of numerical coefficients; but as any one of the factors of which a term is composed may be considered a coefficient, we often have to apply the rules to these literal Rules for Brackets, as applied to literal coefficients. BRACKETS. 37 coefficients. Thus, when any terms of a quantity contain some common factor, brackets are often employed to collect tlie other factors considered as its literal coefficients into one expression, which is set before or after the common factor. Thus, just as dx-{-2x—x=4:X, that is, ={3-\-2—l)x, so, likewise, ax-]-dx—x={a+I)—l)Xf 2a-4.ax + 6ay=2a{l-2x-\-3i/), {a+2h)x'-{2d-c)x'={{a+2I))-{2d^c)}x'={a+c)x\ Ex. 1. Add {a-2jo)x''+{2c-3r)x {22J-{-a)x^ —x '—{p—a)x^ —{c^l)x ^x^ —{c—2r)x Ans. {^a—p—Vjx^ —rx Ex. 2. From ax' -M +x take —px' ^qx^ -\-rx Ans. (a-\-p))x^ — (b—q)x^ + {l—7')x Examples— 8. 1. Collect coefficients in ax^ — lx'^ — cX'—lx^-\-cx^~-clx-\-cx^ —dx^—ex, 2. Add together ax—ly, ^-\-y, and {a'~l)x—{l)-\-l)y. Jl. Add together {a-\-c)x^—3{a—'b)xy-\-{ib—c)y'', and {l)-c)x'-\-2{a^l)xy + (a-'b)y\ 4. Add together {a + h) x + {I) + c)y and {a—'b)x—{h~c)y, and subtract the latter from the former. 38 ELEMEKTAEY ALGEBRA. 5. Add together (i) the first two, (ii) the last two, and (iii) all four together, of '^{a+'b)x-^^{l) + c)y, -'3{a-I))x + 2{a-c)t/, —(2b-^c)x+{a-2J))y, and {a-2l)x-{h + %c)ij. 6. In (5) (i) subtract the second quantity from the first, and (ii) the fourth from the third, and (iii) add the two results together. YIII. Multiplication. 57. It is convenient to make four cases in Multiplication. I. To multiply one positive single term by another ; II. To multiply a quantity consisting of two or more terms by a positive single term ; III. To multiply one quantity by an- other when both consist of two or more terms ; IV. To mul- tiply one negative single term by another, or by a positive single term. 58. I. Suppose we have to multiply da by 4Z>. The prod- uct may be written thus, ZaX^l)] or, since the product of any number of factors is the same in whatever order the fac- tors may be taken, we may write it dX^XaXh', and it is therefore equal to V^ah, Here we multi2:)ly together the nu- merical coefficients and put the literal factors after this product. Thus, for example : laXdhc^^lalc. Similarly, 4.aXhl)X'dc=ma'bc, 59. Poivers of the saine numler are multiplied together hy adding the exponents. Four Cases in Multiplicatiou. Rule for Case I. Powers of the same luim ber, how multiplied? MULTIPLICATION. 39 Thus, a'^Xa^—a''] for a'—aa, and a^~aaa, ^'.a'Xa^ =aaX ciaa = aaaaa, or a^ In the same manner it maybe shown that a^Xa^"=a'"; and so on for other powers, always taking the sum of the exponents. To prove this generally, viz., that oJ^XdP'^dr-^'^, whatever positive whole-numbers m and n may stand for, we have, by definition, oJ^=za.aM, &c. to m factors, and a^'^aM.a. &c. to n factors, :, a'''Xa''=a.a.a, &c. to m factors x a.a, &c. to n factors, —a, a, a. &c. to 7n + n factors, — a!^^'^, by definition. The reasoning and the rule are the same, if for a we wnte a-^l), or a-\-'b^c, or any other quantity ; that is, the poioers of such quantities are multiplied together by adding the exponents of the powers together. Thus the 2d power of a^h multiplied by the 3d power of the same quantity will produce the 5th poAver of that quantity. Ex. 1. 2vT'x3.^•'=2x3X.^•V=:6.^^ Ex. 2. "^ax X ^axy == 7 X 2 X aaxxy — UaVy. Ex. 3. ha%Xa'bc = oa\iUc=^^a''h''c, Ex. 4. ?>x'y'z' X 4.xYz = 3 X 4 X xVyYz'z = 12xyz\ Ex. 5. 7n7ix^yXpy=ninpx^yy—mnpx^y^, Ex. 6. 2r/"' X 3a' = 2 X 3 X a'^a' = 6a"^+^ Ex. 7. ax'^Xix'' = abx'''x''=aI)x'''+\ Ey 8. ax'"Xbx''Xcx''=:abcx'^x''x'' = ccI)cx''''^''+^. 60. 11. Suppose we have to multiply a-f Z> by 3. We have, 3[a + b)=a + t?-\-a + b+a+b=:3a-{-Sb. 40 ELEMENTARY ALGEBRA. Similarly, In the same manner we have, 3{a—b) — da—3b c{a—l) — ca—cb Tims, to mnltiply an expression consisting of two or more terms by a single positive term: Multiply each term of the expression hy the single term, and put 'before each product the sign of the term tvhich produced it j then collect these results to form the complete product. Ex. 1. a-^-b—c multipl] Led by 2=2a + 2J-26*. Ex. 2. a—b-^-c d=ad—bd + cd. Ex. 3. ax-\-by c—acx-\-bcy. Ex. 4. ax-\-by-'Cz 2p = 2apx + 2 b^jy — 2 cpz. Ex. 5. 2a + db—4:C .. 2x=4:ax+6bx—Scx, Ex. 6. ax-\-by ax=aV-{-abxy, Ex. 7. ax^by by=abxy-\-Fy^, Ex. 8. 7a;-4y+6 ... 3:r = 2l2;'- 12^:^ + 18:*. Ex. 9. ^X^'-\?>X-\-\ ... 6 = 30x'-Gox+5. Ex. 10. x^—px-\-q^ px —px^ —p^x^ -\-pgx. Ex. 11. \ah^\cd . 4.ab=2a'V-VGal)cd. 61. III. Let it be required to mnltiply a-\-b by c-\-d\ this means that a-\-b \^ to be taken c-\-d times, that is, g times and d times. Kow a-\-b taken c times produces, by rule of Art. 60, ac-\-bc', and a-\-b taken d times produces, Rule for Case IL MULTIPLICATIOIS". 41 by tlie same rule, ad-^-hd; .*. a-{-b taken c times aiid d times, that is, c-{-d times, produces ac-^-hc-^-ad-^-ld, wliicli is the product required. Or, if the quantities ho, a-\-'b and c—d, a+h multiplied by c—d means that a+i is to be taken d times less than c times. Now a+b taken c times produces ac-\-bc; but this is too much by d times a+b, that is, by ad+hd; .*. ad+M is to be subtracted from ac-\-bc. Hence the product re- quired is ac+bc—ad—bd, following the rule of Subtrac- tion. Or, if the quantities be a—b, and c—d, the product of these is, as in the last case, c times a—b wanting d times a—b, that is, ad—bd siMracted from ac—bc^ which leaves ac—bc—ad-\-bd (changing the signs in the quantity to be subtracted, according to rule). Collecting these results, we have, (a + b) (c-\-d)—ac-\-bc-\-ad+ bd {a + b){c—d) = ac + bc—ad—bd {a—b){c—d) = ac—bc—ad^-bd. Considering these results, we see, for example, that cor- responding to + « in the multiplicand, and + c in the multi- plier, there is a term -\-ac in the product; corresponding to the terms +a and —d there is a term —ad in the product; corresponding to the terms —b and +c, there is a term —bo in the product; and corresponding to the terms —b and — ^ there is a term + bd in the product. These observations are briefly collected in the following important rule in Multiplication: Lihe signs produce +, and unlike sig7is — . This rule is called the Enle of Signs. 62. IV. Let it be proposed to multiply 2a by —4zb, or The Rule of Si^us. 42 ELEMENTARY ALGEBRA. — 4:C by da, or -4c by -4^. We apply the Eule of Signs, aboye establislied, to these single terms. Thus, we have, 2aX-^h='-Sab -4cX 3a=—12ac '-4:cX—4:b=+16bc, We attach a meaning to these operations on single terms, after the same manner as in Addition and Subtraction. Thus, the statement — 4cX~45= + 16^c, means that if —ic occur among the terms of a multiplicand, and —4^ among the terms of a multiplier, there will be a term i-lQic in the product corresponding to them. As particular cases of examples of this sort, we haye, 2aX-4=-^8a, 2X-4=~8, -2X-l=+2. Eemark. — If several single terms are to be multiplied together the product will be -f or — , according as the number of negative factors is even or odd. Thus, 4aX-25x Zc is -^idbe AaX-2hX—oc is -\-24:abo 4aX-2bX-ScX-2d is -iSabcd. 63. The rules for Multiplication may now be conveniently presented thus : To multiply single terms: Multiply together the numeri" cal coefficie7its, init the literal factors after this product, and determine the sign hy the Rule of Signs, To multiply quantities of two or more terms: Multijjly each term of the multiplicand hy each term of the multiplier, according to the ride for single terms, and the sum of tJicse separate products will be the product required. The process is generally conducted as in the following examples. Two General Rules of Multiplication. EXAMPLES. 48 Ex. 1. Multiply -2a'b''+ 5ab^ - 7b* by — 4«& Prod. 8a'b''-20a''b^+2Sab\ Ex. 2. Multiply 2a+db-ie by a+ 5— c Prod, by a = 2a''+Sab-4:ac by +5= +2ab + Sb''^Abc by -c = —2ac-dbc-\-4:G'^ Whole prod. = 2a^ + 5a6 - 6ac + db'' - 7bc + 4c^ rix. 3. a + b Ex. 4. « + & Ex. 5. cJ-5 a+b a—b a-b a' + ab a' + ab a'-ab + ab + b'' -ab-b^ -ab + P a'' + 2ab + b\ «' ^^ -b\ a^~2ab + b\ Ex. 6. x + a Ex. 7. x^ + {a + b)x + ab x + b x + c y^ x^ + ax x^ + {a + b)x^ + abx + bx + ab + cx'' + (ac + bc)x + abc Ans, x^ + {a + b)x + ab x^ + {a+b-\- c) x^ -^{cib^roc-^ be) x + abc Ex. 8, x^—ax'^+ bx — c x^ + mx + n x^—ax'^+ bx^ — cx^ + mx'^ — amx^ + bwtx^ — cmx + ?zrt^ — aiix^ +bnx—cn Ans. ic'^ — (<x — m) x"^ + {b— am + ?i)«""* — (c — 6//i + a») a?^ — {cm—bn)x — en . 64. We arrange the terms of the partial products so that like terms may stand in the same column. This enables us to collect the terms easily, in order to get the final result. With the view of bringing the like terms of the product into The order of Arrant^jcment of Terms of Multiplicand and Multiplier. 44 ELEMENTAEY ALGEBKA. the same column, we arrange the terms of the multiplicand and multiplier in a certain order. "We fix on some letter which occurs in many of the terms, and arrange the terms according to the poivers of that letter. Thus, taking the last example, we fix on the letter x ; we put first in the multiplicand the term containing the third power of X ; next we put the term which contains the next power of X ; next the term which contains the first power of x\ and last we put the term which does not contain x at all. The multiplicand is said now to be arranged according to de- scending poicers of x. We arrange the multiplier always in the same way as the multiplicand. It would haye done as well to arrange them both according to asce7iding poivers of X. Examples — 9. 1. Multiply ax?\f by Ixy ; mx^ by —nx^ ; —acx by —2axi/; ale by Ic, —ale by —ac] x^y by —xy"^, 2. Multiply x^—xy-^-y"^ by x, and a^ — ax-{-x^ by —ax] x^ — ax-\-l) by — alx ; x^ — ox^y + ^xy"" — %f by xy, . 3. Multiply 2« + Z' by a-^-U, and "la-l by c—M, 4. Multiply "^x-V^ by 2:^+3?/, and 3^5+4^' by 2al-U\ 5. Multiply x' + ^x—^ by ^+3, and x''—4:X-\-^ by x—2, 6. Multiply a'^-^a-l by a'-a-\-l, and by a'-3a-L 7. Multiply 2W-\-^x\j-\-^xy'' + y'hjdx-y, 8. Multiply a'-2a'l-^^a''l''-%al)' + Ul'hy a-^^l. 9. Multiply x"" -\-%ax-{-'^a^ hj x'^—2ax-\-a'. 10. Multiply 9a'-3«J + Z?'-6a-2^+4by 3r«+J + 2. GE]!s^EHAL EESULTS IX MULTIPLICATION. 45 1 1 . Multiply x'^-\-y'^+z'^ + xy — xz + yz by x—y-]-z. 12. Multiply a'-\-2a''-h2a + lhy a'-2a''-{-2a-l. 13. Multiply a' + W-i-9o''+2aI) + dac-6I)chj a—2b-3c, 14. Multiply a'-2a'b + 3a'b'-2aF + I)'hja' + 2ad + I?\ 15. Multiply x^ — ax + b hj x—c, and by x^ + ax—c, 16. Multiply l — ax={-I?x^ — cx''hjl+x—x''. IX. General Eesults in Multiplication. 65. N.B. — The rules for the management of BracTcets, given in YII., apply only to the addition and siiMractmi of quantities so enclosed. If a collection of quantities within brackets is to be ^nultijolied or divided by any quantity or collection of quantities, the brackets must not be struck out until the multiplication or division is actually performed. Thus {a-\-'b)x{c-\'d) signifies that « + Z> is to be taken c-\-d times, and is obviously not the same as either a-\-'b{c-\-d), or {a-\-'b)c-{- d. Again, [a+'b)-^{c-{- d) is not equivalent to either a^-'b^{c-{-d), OY\a-{-'b)-~c-{-d] but it may be written -^, the line which separates the numerator and denominator serving as a vinculum to hoth. The learner would do well to practise multiplication of quantities by means of braclcets as early as possible. Thus, Ex. 1. {a—l){c—d)^{a—'b)c—{a—'b)dy =ac—'bc—{ad—'bd), - ac —be— ad + bd. The Management of Brackets In Multiplication. 4G ELEMENTARY ALGEBRA. Ex. 2. (x-\-a){x+b)=^{x + a)x+{x-\-a)I?, = x^ + ax + Ix + ad, ^x' + {a + b)x + ah* Ex. 3. {x+l){x-\-2){x+3) = {x' + 2 + l.x + 2){x+3), = {x' + dx+2)x+{x' + dx-\-2)3, =x' + 3x' + 2x + 3x^ + 9x + 6, =x' + 6x'' + llx + Q, Ex. 4. {a-]-b—c){a + b—c) = {a + h—c)a + {a-^h—c)h—{a + I?—c)c, =a'^ + ah—ac+al) + d^—l?c--ac—dc + c^, =:o^-\-2al)-\-V-2ac-21)c-\-c\ 66. The student should notice some results in Multiplica- tion, so as to be able to apply them when similar cases occur, and write down at once the corresponding products. Ex. 3, Art. 63/giyes {a-\-'b){a-\-'b) or {a^-iy =:a'+2a$4-Z>Xi). Thus, The square of the siwi of two qicantities is equal to the sum of the squares of the two quantities ijicreased ly tiuice their product, Ex. 5, Art. 63, gives {a—l){a—l), or {a—iy =.a^-2db^V (ii). Thus, Hie square of the difference of two quantities is equal to the sum of the squares of the two quantities diminished dy twice their product, Ex. 4, Art. 63, gives {a + d){a-i)=a'-'b' (iii). Thus, The product of the sum. and difference of two quantities is equal to the difference of their squares. The sqnare of the sum of two quantities. The pquare of the difference of two quantities. The product of the difference of two quantities. MULTIPLICATION. 47 67. General results expressed by symbols, as in the equa- tions (i), (ii), and (iii), are called formulas. In these formulas, a and h indicate any quantities or ex- pressions whateyer. Bemark. — We may express the two formulas, {a + hf=a'' + Uh + ¥ ; and {a-hf-=a'' -2al) + b\ in one formula. Thus, {a±bf=a''±2ab-{-b\ where ± indicates that we may take either the sign + or the sign — keeping throughout the vpper sign or the lotcer sign. a±b is read thu» " a plus or minus Z>." ± is called the double sign. As applications of these rules or formulas, we have, (x + yy=x'-]-2xij-\-y% {x-2y=x'-4.x + 4:, {2x + yy=4:x'' + 4.xy+y\ {2ax-3hjy=4:aV-12aI?xy+9by, {29y={30-iy = 900-60 + l, (54)^^= (50 + 4:)^=:2500 +400 -f 16=2916, {x-]-2){x-2)=x'-4:, {2ax-3hy) {2ax-\-Uy) =4aV-9Z>y, (127)'-(123)^=:(127 + 123)(127-123)=250X4 ==1000. 68. Ex. 6, Art. 63, gives (x + a) {x -{■l)=:x^-\-{a-\- h)x + ab (iv), where the coefficient of x is the sum of the two latter terms of the factors, x-ha, x-\-b, and the last term, -i-ah^ is their product. In like manner, we shall have, (x^a) (x—h) =x^ — (a + d)x+ abf (x—a) {x + h) —x^ -f {l)—a)x—ab, The use of Fo!*mulas. The double sign. 48 ELEMENTARY ALGEBRA. Thus, {x-5){x + 2)=x'-{-{2-d)x-10=x'- 3.T-10, {x-Q){x-6)=x'-{6 + 6)x-hdO=:x'--llx + 30, {xi-2){x-2){x + d){x-d)=z{x'-4:){x'-9), =x'-{9 + ^)x'' + 36=x'-ldx'-\-3(j, 69. By a little ingenuity the formulas (i), (ii), (iii), and (iv), may be extensively applied to lighten the labor of Mul- tiplication. Suppose we require the square of x + y + z. Denote x -{-y by a. Then x + y-\-z=a-\-z] and by the use of (1) we haye, (a^- zy = 0" + 2az+ z^=:{x + ijy + 2{x-\-^j)z^- z" ^x'^ 2xy +rf + 2xz + 2yz + 2;^ Thus, {x-^ y + zy ^x" -^ if + z" + 2xy + 2yz + 2xz, Suppose we require the square oi p — q + r—s. Denote p—q by a, and r—s by l] then^ — ^ + r— 5 = a + Z>. By the use of (i) we have, {a^iy:=:a' + 2al) + ¥={p-qy + 2{p-q){r-s)-\-{:r-sy. Then by the use of (ii) we express {p—qy and {r—sy. Thus, {p-q + r-sy =p'' -'2p)q^- q" -\-2{pr—ps—qr + qs) -\-r'^ — 2rs-\- s" —p" + g^ + r"" + / + 2pr + 2qs —2pq—2ps—2qr— 2rs. Suppose we require the product of p—q-\-r—s and p—q—r + s, ljeip—q=a, and r—s^h; then p—q-^-r—s — a + h, and p — q—r + s=a—b, Wliat are the Formulas (i), (ii), (iii), and (iv)? multiplicatio:n'. 41) Then by the use of (iii) we have, and by the use of (ii) we have, {p—q-\-r—s) {2y—q — r-\-s)=p'' — ^pq + q^—{r^—2rs-[-s^) =2o'' + q'-r'-s'-22Jq + 2rs, As the student becomes more familiar with the subject, he may dispense with some of the work. Thus, in the last example, he will be able to omit that part relating to a and Z>, and simply put down the following process : (p — q + r—s) {p — q—r + s={p — q + {r—s)) {p—q—{r—s)), =:{p~qy-{r-sy —p"" — 2pq -\-q^— (r- — 2rs + s-) —p^ — 2pq + (^^ — r- + 2r5 — 5I 70. Ex. 1. {ax + l) -^ cy + dy={ax-\-h -\- cy + ciy, = {ax-^I)y+{cy + dy + 2{cfx-{-b){cy + d), = a^x^ + ^' + 2al?x -f- (fy^ + <^ + 2cdy + 2acxy + 2adx + 2I)cy + 2M. Ex. 2. (a^—ax-hx^) {a'^—ax — x'^)=^hj (ii) {a^—cfxy-^x* = a^— 2a^x + a-x^ — x*. Ex. 3. {a^-i-ax—x-){a'^—ax—x'^) = { («^ — x") -f ax} { (a^ — x"^) — ax} =z{a'^— ^-) - — a^x^ =a^— 2a^x" -\-x^~ a'x^ = a'* — 3a'x'^ + x\ N. B. — The formula here em])]oyed, {a-{-b)X{a—b)z=a'^ — b'^, may be alwaj^s applied, whenever it is seen that the two quantities to be mul- tiplied consist of tirms which differ only (some of them) in sign, by taking for a those terms which are found with their signs unaltered in each of the given quantities, and the others for b. Thus, in Ex. 3, a? -■x^ appear in both the given quantities, whereas in the one we havo -i-ax, in the other —ax\ hence, the product required is {a^—x^f — a\-;^, as a Dove. 3 V ele:.iextary algebra. Ex. 4. (a' + ax + x') {a'-ax+x') = {cr+xy-a'x' = a^ + a^x^ -f x^, Ex. 5. {a'^+ax—x-){a-—ax + x^) = a*~{ax—xy =^a*--a'x'^+2ax^—x\ Ex. 6. {ar—ax-\-x^^) {ax + x^—a")=x^ — {a^—axy =x^—a^-\-2a^x—a^af, Ex. 7. (a + b + c + d) {a + b-c-cl) = {a + by-{c-{-dy =za' + 2al) + ¥- & - 2cd- d\ Ex.8. {a^2h-?>c-d){ci-%b-^U-d) = {a-dy-{:2b-^cy =a''-2ad+d'-4:b' + 12bc-9c\ Examples — 10. 1. Write down the squares of a—x, l + 2ic'', 2a* + 3 3x—4:i/, 2. Write down the squares of 3 + 2x, 2x—3y, a^—3axw bx^ — cxy, 3. Write down th© products of {2a-\-l)x{2a—l), {3ax-{-b)x{3ax-b), {x-~l) {x^l) {x'-^l). 4. Write down the products of (a: + 3) X (^ + 1), {x''-\-4)x(x'-l), {ab-3) {ab-\-2y (2ax—3b) {2ax-b), 6. Find the continued product ofx-{-a, x—a, x+2a, and x—2a. EXAMPLES. 51 n. Obtain the product of mx + 2ny, mx~27iy, mx—Sny, and mx-\-3ny, 7. Simplify 3{a-2Q:y + 2{a-2x) {a + 2x) + {3x—a) {3x + a) — {2a-3xy, 8. Multiply x'' + 2xy + 2^' by x^ — 2xy + 2y\ and 2a^ - 3ah + 6' by 2a' 4- 3ab + Z>'. 9. Multiply a-\-h-{-c by a-^h — c^ by a— 5 + c, and by a—h—c. 10. Multiply a— ^ + c by a— 5--C, by 5 + c— flj, and by c—h—a. 11. Multiply 2a+5— 3c by 2a— ^ + 3c, and by h + 'dc~2a. 12. Multiply 2a — h—3c by 2a + ^> + 3c, and by Z* — 3c— 2a. 13. Multiply a + 5 + c+c? by a—h-^c—d, by a—h—c-\-d, and by b-^c—d—a, 14. Multiply a— 2^ + 3c + 6^by a + 2^-3c+<^, by 25 — a + 3.c + <^, and by a + 25 + 3c— ^. 71. There are other results in Multiplication which are of less importance than the four formulas given in Art. QQ, but which are deserving of attention. We place them here in order that the student may be able to refer to them when they are wanted ; they can be easily verified by actual multi- plication. {a'-b){a'-\-ab-hb')=a'-b% {a+by=z{a + b) (d'+2ab + ¥) = a''\-3a'b-i-3ab' + b% (a-by={a-b){a'-'2ab-{-b')^a'-3a'b-{-3ab'-b\ other results in Multiplication. 52 ELEMEXTAIIY ALGEBKA. X. DlYi:,ION. 72. Division, as in Arithmetic, is the inverse of Multipli- cation. In Division we have given the product and one of the factors, and we have to determine the other factor. The factor to be determined is the quotient, 73. I. The rule for the division of simple expressions follows at once from the corresponding case in Multiplica- tion. For example, we have. therefore — - therefore therefore therefore l^ahc 4.^^"'^ 12ahc ■Aab. Also ^abX —3c = — 12al)c; -12al?c 12abc — = — dc, 4,ah Also — 4:011) X -VZahc _ Also --4a^X— 3cj= ^iat. =^-4.ab. I2abc_ -iab = -3c, 12ahc_ -3c = -^ab. Hence, we have the following rule for dividing one singte term by another : Divide resjjedively the coefficient and literal parts of the dividend ly those of the divisor ; and then, if the tivo quantities have like signs, prefix to the quotient the sign +, if different, the sign — . What is Division? Rale for dividing one single tenn by another. DIVISION. 53 This division is the familiar process of cancelling like fac- tors in Arithmetic. Hence the rule may be given briefly thus : Strike out from the dividend the factors which occur in the divisor ; the rest of the dividend is the quotient, ivhose sign is deternmied hy the Rule of Signs, Viz, \ Like sigjis give +. unlike sig7is — . Thus, -7^-v-5=-7, -ax-^a^x, Uab-^n=2a, 7b^n=l, abc~ab=c. 74. One poiuer of a quantity is divided by another i)oioer of the same quantity by subtracting the index of the latter from that of the former. For example, suppose we have to divide a^ by a^ a' a\a' . '=a'-\ Or we may ■ show the truth of the rule thus : a'Xa'. =a\ Therefore, a' 3 And generally, if 7n and n be positive integers. and m > n fl^^'-f-a"- ^a'''-^ Similarly, Za'l)' = 2a'b% xhfz' z^x'yzK xyz ha'b\ d'bc (=5a'b'c% a^'b' _^m-pyn-q^ Division of one power of a quantity by another power of the same quantity. 54 ELEMENTARY ALGEBRA. 75. It may happen that the factors of the divisor do not occur in the dividend. In this case we can only indicate the division. Thus, if 6a is to be divided by 3c, the quo- tient is indicated by 6a-^3c, or by — . oC Again, it may happen that some of the factors of the di- visor occur in the dividend, but not all of them ; or that a power of a quantity occurs in the dividend, and a higher power of the same quantity in the divisor. In this case the indicated quotient is a fraction, which can be simplified by striking out common factors, as in Arithmetic. Suppose, for example, 16a^b is to be divided hjQbc; we have, loa'b _ 5a'xSb _5a^ 6bc ~ 2cX3b ~ 2g' striking out the common factor 3b. Again, if 4tab'^ is to be divided by 3cb% the quotient is indi- cated by ^r-n.- Eemove the factor b"^ which occurs in both dividend and divisor, ^aW _ ia '3cb'~'3cb^' 76. II. To divide a quantity consisting of two or moro terms by a single term : Divide each term of the dividend by the divisor, and collect the results to form the complete quotient. For, since fl^+^— c + &c. multiplied by m produces ma-hmb—mc-{-&c., /. ma-\-mb—7nc+&o. divided by m, gives a + b—c-i &c. Hence, the rule is as above stated. Rale for Case n. DIYISIOX. 55 Ex. 1. — h— =4:a'-3Z>c4-a. a a a a a^x'^ — 6aJ)x^ + 6ax^_a^x'^ 6abx^ (5ax* aa; ax ax ax = a'-6I)x + 6x\ Ex. 3. (a+I)+c)-~abc=:-T-+-Y~-\--j-^Y--h — h- 1^. ^ ^ aoc ado abc be ac ab ^ , a'c'-2abc' + dac' a'c' 2abc' 3ac' Ex. 4 T-r-a = — 7-T-2 + - -4:abc^ ^abc^ 4:abc^ 4iabc^* __^ l_3c 77. III. To divide one expression by another when tne divisor consists of two or more terms^ we must proceed as in the operation called Long Division in Arithmetic. The fol- lowing rule may be given : Arra7ige both dividend and divisor aecording to ascending poiuers of some common letter, or both according to descending poiuers of some common letter. Divide the first term of the dividend by the first term of the divisor, and ][tut the result for the first term of the quotient ; multiply the luhole divisor by this term and subtract the product from the dividend. To the remainder join as many terms of the dividend, tahen in order, as may be required, and repeat the whole operation. Continue the process until all the terms of the dividend have been tahen doivn. The reason for this rule is the same as that for the rule of Long Division in Arithmetic, namely, that we may break the dividend up into parts and find how often the divisor is contained in each part, and then the aggregate of these re- sults is the complete quotient. Rule for Case III. 56 ELEMENTARY ALGEBRA. 78. We shall now give some examples of Division arranged in a convenient form. Ex. L 1— .t) l-2x-i-x' {l~x 1— X — x + x^ — x + x^ Ex. 2. 3x ^ %) Gx' - 1 7x'ij + 1 (jy\2x' - dxi/ - 4.f - 9x'y-j-l2x?/ -12xf-{-lGy' Ex. 3. a—x) a^—x'' {a^^ax^x^ a^—ct^x d\v- -x' cc'x- - ax^ a/jy — ■x^ ax'-^ x' Ex. 4. a-^x) d-\-'^ {a^—ax + x'^ a^ + a'^x —cC'x-^x^ ^a/x—ax^ ax^ + x^ aaf + x^ Ex. 5. -2^^4-3^=) na*-10a'b-22aW + 22aF-{-lDl?' {da' -4.ab + 5d' 3a'- 6a'b+ 9a'b' - 4.a'b + rda'b'-22ab' - 4a'b+ 8aW-12ab' baW-lOab' + lob' da'b'-lOab'i-lDb' Consider the last example. The dividend and divisor are both ar- ranged according to descending powers of a. Tiie first term in the dividend is 3a*, and the first term in the divisor is a"^ ; dividing the former by the latter we obtain Sa^ for the first term of tlie quotient. We then multiply the whole divisor by 3a- , and place the result so DIVISION. 57 lliat each term comes below the term of the dividend which contains the same power of a; we subtract, and obtain — 4a^b + lda^b'^ ; and wo bring down the next term of the dividend, namely, — 22a&^. We di' vide the first term, —Aa'^b^ by the first term of the clivisor, a'^ ; thus we obtain — 4:ab for the next term in the quotient. We then multiply the whole divisor by — 4«Z> and place the result in order under those terms of the dividend with which we are now occupied ; we subtract, and obtain 5a'^b'^—10ab^ ; and we bring down the next term of the dividend, namely, 15^^. We divide 6a'^b'^ by a\ and thus we obtain 5^^ for the next term in the quotient. We then multiply the whole divisor by 55^, and place the terms as before ; we subtract, and there is no remainder. As all the terms in the dividend have been brought down, the opera- tion is completed; and the quotient is Sa"^ — 4:ab-hob'\ It is of great iinj^ortance to arrange ioth dividend and di- visor according to the same order of some conamon letter ; and to attend to this order in every part of the operation, 79. It may happen that the division cannot he exactly performed. Thus, for example, if we divide <^^+2«^-f2Z>^ by a-^1) we shall obtain a+J in the quotient, and there will then le a remainder, V, This result we place, as in Arith- metic, in the quotient over the divisor, in the form of a fraction, thus indicating that If remains still to be divided by a+h. Thus, -T — =a + Z>H — -y. Ex. 6. a+x) c^^rx^ {a—x^ a+x a^ +' ax —ax + x —ax—x^ 2x' 2x^ Ex. 7. a-x) a^^x^ {a^x-\- a—x -ax ax+ x^ ax— x^ 2? Important principle in Division. Division with a remainder. 58 ELEMENTARY ALGEBRA. 80. We give some more examples : Ex, 8. 1-^)1 {li-x+x^ + x^ + &c,+ -^~ ^ -X + x + x—x^ + X'' + x''-x' + x'-x* + x' &c. Ex. 9. Divide x'-5x' + '7x'-{-2x'-6x-2 by l + 2x—3x' + x\ Arrange both dividend and divisor according to descend- ing powers of x. x'-3x''-\-2x-]-l) x'-bx' +7x'-^2x'-6x-2 (x''^2x--2 x''-3x' + 2x'+ x' --2x'-2x' + 6x' + 2x'-Gx —2x' +6x'-4:x'-2x -2x' +6x'-4.x- -2x' +6x''-4:X- -2 -2 Ex. 10. Divide 64-^" by 2- -a. 64-32a 33a- -a' 32a- -16a' 16a'- 16a'- -a' -8a' 8a'- 8a'- -a' -ia' 4a*- 4a'- -a' -2a' 2a'- 2a'- -a' -a' DIYISIOIS^ 59 Ex. 11. Divide a^ ^h^-^-c^—^^aU by a + ^ + c. Arrange the diyidend according to descending powers of «, a + 'b + c)a' -?>abc-\-¥ ^c\a' -ab-ac-^F -U-^G^ a^-^a'h + a'c —a^h—a^G —?>abG —c^G ■\-a¥ — 1abG —a^G — obG—aG^ aW— abc + aG^ + h^ a^ +h' + l)'G — abc^a& —Fg — abc —Wg—W a? '\-M-\-g'' ac^ +Ig''-\-g' The above is the easier method in such a case, but the fol- lowing, in which the coefficients of the different powers of a are collected in brackets, is the neater and more compendi- ous: — {b + G)a'-dI)Ga -{b + G)a'-{b'+2bc+G')a + {b'- bG+G')a-^{b' + G') + {b'- bG+G')a+{b' + G') Examples — 11. Divide 1. 16x' by ^x\ 2. 24a° by -Sa\ 3. 182:y by 6<y. 4. 2^.a'b'G' by -da'b'G\ 5. 20a'b'xY by 5Z»Vy. ^^ ELE31EOTAKY ALGEBRA. 6. 4^^- 8:^^ + 16:. by ^x, 7. 3^^-12^^ + 15a^ by ~-U\ 8. xhj-3xY + 4:xy'hjxy, 9. -15^^Z*^-3a^Z^^+12a^ by ^3a^. 10. 60r.^^V-48a^^V + 36«T.^~20«5^^ by ^ahc\ 11. ^^^-7^ + 12 by :.-3, 12. ^^ + r.^72 by .'. + 9. 13. 2x'-x' + ^x-^hj2x~d. 14. Qx' ^Ux'^ 4.x + 24 by 2x + 6. 15. 9x' + 3x' + x-~lhj3x~l. ^^' 7x'~24.x'-{-58x~21hj7x~3. 17. ^«-l by x~l. 18. a«-2a^^+^^ by a~^. 19. ^'^-81^^by:r-3y 20. x'~-2x\j-\-2x\f-xf by x~y. 21. ^^_y^ by a;-y. 22. a^+32^^ by a + 25. 23. 2a'-V%7aF-%Wh^a^3-b. 24. ^^ + :^V + ^y + 2;y + :^3/^ + ,y^ by r^^/. ^h. x' + 2x^y+3xY^xY-2xf-~3f by o.^-.^^ 26. ^'-5a5^ + lla3^-12a; + 6byi«^-3^ + 3. 27. x' + x'^^x''-^Ux-4.hjx'!-\-4.x + L 28. aj*-13a3H36 hj x'-^^x-\-Q. 29. i«* + 64by^H4.T + 8. 30. x' + 10x' + 3bx' + b0x + 2i by ^^^ + 5^ + 4. 31. ^'-(a + 5 + £?)a3^ + (a5-l-^^ + ^^)^_^^^^ 32. ^'~{a+p)x''+{qj^ap)x-aqhj x-a, 33. v' - ^?z?/^ + ny"" — ny^ ^my-lhj y-^1. FACTOKS. 61 34. a^—lf + c^ + ^ahc by a—h-{-c, and a'—F—c^-dadc by a—h—c. 35. a by 1 + 33, and l-f2a; by 1— 3;^', each to four terms in the quotient. 36. 1 by 1— 2i;c+.T^, to four terms. XI. Factors. 81. We shall now notice some general results in Division, m connection with those already given in Multiplication; and we shall apply some of these to find what expressions will divide a given expression, or, in other words, to resolve algebraic exjjressions into their factors, 82. For example, by the use of formula (iii) of Art. 66, we have, a'-Jj'-^^a' + h') {a'-h'')^{a'' + h'') {a-^h) (a-h) a'^I?'=--{a' + h') {a'-b') = {a' + b') {a' + b') (a^b) {a-b). Hence, Ave see that a^—b^ is thQ product of the four fac- tors a^-\-b\ a^-^y, a + b, and a — b. Thus, a^ — b^ is divisible by any of these factors, or by the product of any two of them, or by the product of any three of them. Again, in Art. 70, we have, {a'^ab + b'){a'-a])^b') = {a'^-by^{aby=a'-^a''b'-Vb\ Thus, a^^a^b'^-^b" is the product of the two factors €^-\-ab^-W and a''—ab-\rb'', and is therefore divisible by either of them. 83. The following results in Division may be easily veri- fied, and will enable us to write out with ease the quotients in many similar cases. x—y -y -1, How to resolve Algebraic quantities into their Factors. 62 ELEME^^TARY ALGEBRA. X^ — lt x-y ^' Also, x-y x'-y' x-\-y ^' — =x — X y + xy — y\ x + y ^ iJ J ^ — ^ —x''—x^y-^x^if—x^y^-\-xy''—y^j x-\-y and so on. Also, x-\-y ' "^^^.x'-x'y + xY-xf^f, and so on. The student can carry on these operations as far as \w pleases, and he will thus gain confidence in the truth of the statements which we shall now make, and which are strictly demonstrated in larger works on Algebra. The following are the statements : x'^^y^ is divisible by x—y if n be any whole number; jc"— ?/" is divisible by cc + ^ if w be any even whole number; iK" + ^" is divisible by a; +^ if ^ be any odd whole number. FACTORS. G3 "VVe might also put into words a statement of the forms of the quotient in the three cases; but the student will most readily learn these forms by looking at the above examples, and, if necessary, carrying the operations still farther. We may add that x'^^'if is never divisible by x-\-y or x—y^ when n is an even whole number. 84. The student will be assisted in remembering the re- sults of the preceding Article by noticing the simplest case in each of the four results, and referring other cases to it. For example, suppose we wish to consider whether x^—y'^i'^ divisible by x—y or \i^ x^y\ the index 7 is an 06?^ whole number, and the simplest case of this kind is a:— y, which is divisible by x—y, but not by x-{-y\ so we infer that x'—y'' is divisible by x—y and not by x + y. Again, take x*—y^\ the index 8 is an even whole number, and the simplest case of this kind is x'—y'', which is divisible both by x—y and x-{-y\ so we infer that x^—y^ is divisible both by x—y and %-\-y, Now, in every case the quotient will consist (as above. Art. 83) of terms in which the exponents of x decrease, and of y increase continually by 1 ; but when the divisor is x—y, these terms are all plus; when it is x-^-y, they are alter- nately + and — . We shall now apply these results to some examples. Thus, 'lax — l x + oy x'-lQ r=x'-3xy+9y\ x-2 = x'-2x^-h^x- Law for the esponents and signs of the quotient 64 ELEMEXTAliy ALGEBRA. Examples — 12. Divide 1. a^—x^ by a-\-x, a^—x^ by a—x, and a^—x^ by a-^x. 2. 9^'-- 1 by Zx-1, 252:'-l by hx-\-\, and 4a;'-9 by 2:^+3. 3. 9wV — 25 by 3??^?^+5, and 16m*— 7^* by Am^ + n^. 4. 1 + 82;' by 1 + 2^;, 27^'-l by 3:r-l, and l-16a;* by 1+2:?:. 5. :r*-8l3/' by x-Sy, a''+32h' by «+2Z>, and x''~-t/^ by :^;' + ^^ 6. ia^ + h^ by i^+5, and x*y*—z' by 0^^ + ;^. 7. (« + &)' — <?' by a + h—c, and a^—(l)'-cY by ^— Zi + (7. 8. (^+^)'+^^ by .':c + y+^, and x^ — {ij—zf by a;— ?/+^. 85. The above results and those of {^^) may also be ap- plied to resolve algebraical quantities into their elementary factors, a process which is often required. Ex. 1. ^x^-if^{%x^y) {^x-y), Ex.2. .TH8 = (a; + 2)0'?;'-2:r + 4). Ex. 3. {2a-'by-{a-21)Y=^{U-'b^a-'21)){2a-l-a + 2h) Ex.4. x'-a'={x' + a'){x'-a') =z{x-{-a) (x^ — ax-ha^) (x—a) (x^ + ax + a^), Ex. 5. {a'-xy={{a-x) {a'+ax + x')y =:{a-xyx{a'i-ax + xy. FACTORS. Go Examples — 13. Eesolve into elementary factors 1. l-4a;^ a'-dx', 9r}f-hi\ 25aV-4< 16xy~2Dxy. 2. x' + y% x'-y\ l + x'y\ x'-l, a'xy'-x'y, 2a'b'c-SabV. 3. 25x'-a'x% a'-9a'b% 8^^-27, a'-Sb% aVy-{ 27a;y. 4. x' + 32, aV + 27x% 8x'-{-y% a'h''-c\ a'bc + 2a'I)c'' + abG' 5. 81:?;*-1, x'-G4:, x'-2baf + bV, x'-2a'x'-\-a'x\ 6. (3.T-2)^~(2;-3)^ (« + ^)'-4Z^', (4a3 + 3^)^-(32: + 4^)^ 7. {x' + yy-^xY, c'-ia-by, {2a + by-{2a-by. 8. a;' + '?/' + 3a:?/ (x + y), m'— if —in {m^—n'')+ 72 {m—iiy, a'-ab + 2 {b'-ab)+3 {a'-b')-4. {a-by. 9. 5 (x'-y')-\-'^ {x + y)\ 3 {x'-y')-b {x-yy, (^4.^)^ + 2 (:.^+:r^)-3(a:^-y^). 10. 2{a' + a:'b-{-ab')-{a'-F), a'-b'-3ab {a-b), a'-b' + {a'-by-2a' + 2a'b\ 86. So, too, Ave may often apply (68, iv) to resolve a trino- mial into factors when it is of the form ax"^ -{- bx + c. We repeat formulas (iv) : x''-\-{a + b)x-\-ab = {x + a){x-^b); x'^—{a-\-b)x + ab—{x—a){x—b); x'^ + {a—b)x—ab={x + a){x—b). Ex. 1. x''+7x+12 = { x + 3){ X+4-). Ex. 2. x'-dx + U^i x-2){ x-^iy Ex. 3. x'-DX-U^i x-'7){ .T+2). Ex.4. 6.r'4.^-12 = (3.7;-4)(2:^-f3). Resolution of Trinomials into factors. 66 ELEMENTARY ALGEBIIA. The student may notice that, if the lasfc term of the given trinomial be positive (Ex. 1, 2), then the last terms of the two factors will have the same sign as the middle term of the trinomial ; but if negative (Ex. 3, 4), they will have, one the sign +, the other — . In Ex. 4, it is clear that the first terms of the two factors might be 6x and x, or '^x and 2.^•, since the product of either of these pairs is 6x^, and so the last two terms might be 12 and 1, 6 and 2, or 4 and 3 ; it is easily seen on trial which are to be taken, that is, which serve also to produce the middle term of the trinomial. Examples — 14. Eesolve into elementary factors 1. a;' + 6:?: + 5, 2;' + 9:r + 20, x'-hx-^Q, x'-^x-Vl^, x'' + ^x + 'l,x'-l^x-\-^. 2. ir'+^-6, ^'-a;-6, a;'~22;-3, x^-\-2x-lb, x'^-lx-^, x'-^x-^. \ 3. 4a;' + 82; + 3, 42;' + 13a: + 3, 42;' + lla;-3, 4:x''-4:X-3, dx' + 4^x-4., 6x'' + 6x-4c. 4. 12^-'-5:z;-2, 12:i;'-14:z; + 2, 12a;'-a;-l, a;'-f ^-12, dx^-2x-6, 6. aV-3a'x + 2a\ a'-a^x-6ax\ 3a*I)+a''b^-2ad\ 12a' + aV-x\ 6. 2x'y-^bxy-j-2xf, 9xYSxf-6y\ 6aV + a'a;-a% (jb'x'-7bx'-dx\ 87. We shall now give a few examples of multiplication and division of expressions in which factors or terms occur with letters for their exponents. GREATEST COMMOX DIVISOR. 07 Ex. 1. ]\Iultiply a'-'^'F^+'o' by w"''-'¥c, Ans. a'"'Z>'^+V. Ex. 2. Multipljof^ + fl^ii;'^ -a':^^ by a"^^^^. Ans. a"^a;'^ + a"^+'a;'^--a"^+V» Ex. 3. Multiply a'-'l-d!'--lf^aV'-^ by a J. Ans. oJ'W-aJ'-^l^^a^lf. Ex. 4. Find the continued product of Ans. a"+^Z»^^+V+^(^"+^. Ex. 5. Multiply a"*— 2c'* by a"*— c^ Ans. a-'**— 3a"'c"4-2c-". Ex. 6. Divide 115a'"^V^+'^J-'-' by -69a"^>V+'^d Ans. -\dr-''c''dr'-\ Ex. 7. Divide a"'5-^"^-*^- + a"^--^^'-«'^-'^' + a"^-'Z>' by fl^Z». Ans. oJ^-^ - a^'-l + oT-^V' - oT'-'W + ^'"-'Z^'. Ex. 8. Divide aj'^+^ + a:"^^^.^"^^ + /'*+« by a;" + ?/^ Ans. a;"'^-?/". Ex. 9. Divide c^-"^-3a'"c^ + 2r« by fl^'^-c^ Ans. a'"-2c". XII. Greatest Common Divisoe, 88. In Arithmetic, a whole number which divides another whole number exactly is said to be a divisor or measure of it, or to divide or measure it. A whole number which divides two or more whole numbers exactly is said to be a co7nmon divisor or commmi measure of them. Divisor or Measure. Common Divisor or Measure. CS ELEMENTARY ALGEBllA. In Algebra, an expression wliicli divides another expres- sion exactly is said to be a divisor or 77ieasicre of it. An ex- pression which divides two or more expressions exactly is said to be a common measure or common divisor of them. ISToTE. — The English use the word measure ; the French, the words divisor and divide in the same sense. We shall use the latter, as they have been generally adopted in this country. - 89. In Arithmetic, the greatest common divisor of two or more whole numbers is the greatest whole number which will divide them all. The expression Greatest Common Di- visor, in Algebra, must be understood as applying, not to the numericcd magnitude of the quantity, but to its dimen- sions only; on which account it is sometimes called the Highest Common Divisor, The expression Greatest Common Divisor is, however, re- tained in accordance with established usage, and we shall use the letters g.c.d. for shortness, to indicate it. 90. The following is the rule for finding the g.c.d. of mo- nomials : Find ly Aritlimetic the g.c.d. of the Numerical Coeffi- cients ; after this number put every letter which is common to cdl the monomials, giving to each letter respectively the least exponent which it has in the monomials, 91. T'or example : required the g.c.d. of IQa^Vc and 20a^b\L Here the numerical coefficients are 16 and 20, and their g.c.d. is 4. The letters common to both the expressions are a and I) ; the least index of a is 3, and the least- index of b is 2. Thus we obtain 4a^^^ as the required g.c.d. Again : required the g.c.d. of Sa^Fc^'x^yz^, 12a'^l?cx^y% and 16a^cVy\ Here the numerical coefficients are 8, 12, and 16 ; and their g.c.d. is 4. The letters common to all the expres- Rule for the Greatest Common Divisor. GREATEST COMMON DIVISOR. 69 sions are a^ c, x, and y ; and their least indices are respect- ively 2, 1, 2, and. 1. Thus we obtain ^a^cx'y as the required G.C.D. 92. Tlie folloAving statement gives the best practical idea of what is meant by the term greatest common divisor, in Algebra, as it shows the sense of the word greatest lic^e. When ttuo or more exjjressions are divided dy their greatest common divisor, the quotients have no common divisor. Take the first example of Art. 91, and divide the expres- sions by their g.c.d. ; the quotients are 4ac and. bhd, and these quotients have no common measure. Again, take the second example of Art. 91, and divide the expressions by their g.c.d. ; the quotients are 2Fcx^z^, Zc^ly''^ and 4«cy, and these quotients have no common measure. 93. The idea which is supplied by the preceding Article, with the aid of the Chapter on Factors, will enable the student to determine in many cases the g.c.d. of compound expressions. For example: required the g.c.d. of AiC^(ci-\-l)Y and ^ab {a^ — lf). Here 2a is the g.c.d. of the factors ^a"^ and Q>ah ; and 6^+^ is a factor of {a-\-hy and of a^—lf, and is the only common factor. The product 2a{a + I)) is then the G.C.D. of the given expressions. The rule in this case is similar to that given in Art. 90 : Put doto7i every factor common to cdl the expressiojis, giving to each factor respectively the least exponent luhich it has in the exp)ressions. The product of these factors iviU be the Great- est Common Divisor of the expressions, Ex. 1. The G.C.D. of 15.?;' and l^x\ is ^xJ", Ex. 2. The g.c.d. of SG.r'^V and A:%x'y'z\ is 12x\fz\ Ex. 3. The g.c.d. of SoaWx't/ and ^^a^Vx'y\ is Ha'b'xy. Ex. 4. The G.C.D. of oax'' — 2a"x and a'^x^—dabx, is ax. The G.C.D. of Compound Expressions. 70 ELEMENTARY ALGEBRA. Ex. 5. The G.c.T>.of 6x'y-12xY-\'3xy' and 4:ax'^ -}- 4:axy + 4:a^x, is x. Ex. 6. The g.c.d. of 6aV {a'-x') and 4^\r (a+^)\ is 2a'^x {a + x), Ex. 7. The G.C.D. of a\a'x' -Sax' + 2x') and x^a'-iaV)^ that is, of 0^x^(0^— 3ax + 2x'^) or aV(a— 2a;) (a— a;) and aV(a'--4a;''), is aV(a~ 2:c). Examples — 15. Find by inspection the g.c.d. of 1. Ax' {a+xy and 10 {a'x-x'y. 2. x' {a'-xy and {a'x+axy. 3. {a'b-aby and a^ (a^-5^)^ 4. 6 (:^;'-l) and 8 (2;^-3.r+2). 5. {x^+xy and a;' (:r'-2:~2). 6. 4 (ic'+a') and 6 {x'-2ax-3a''). 7. a' (^^+12^+11) and a'x'-lla'x-12a\ 8. 9 (aV-4) and 12 (^V + 4(^:?;+4. 9. :?;' — 9a;+14 and a:'— lla;4-28. 10. a;'' + 8a;+15 and ic' + 9:r+20. 11. a;'+2:r-120 and a;' — 2a;-80. 12. 4 (x'-x-hl) and 3 (a;'+a;' + l). 13. x'-xy-ny' and a;'+5a:?/+6^'. 94. The G.C.D. of two polynomials cannot be generally fonnd, however, thus by inspection. Hence, for more com- plex examples it is necessary to adopt another method — the . same given in Arithmetic for two numbers. GREATEST COMMOIS^ DIViSCR. 71 95. Let there be given then two algebraic quantities of which it is required to find the g.c.d. EuLE. — Arrange the quantities according to poivers of some common letter, and divide the one of higher dimen- sions dy the other ; or, if the highest exponent happen to he the same in each, taJce either of them for dividend. Take now, as in Arithmetic, the remainder after this division for divisor, and the preceding divisor for dividend, and so on until there is no remainder ; then the last divisor tvill ie the g.c.d. of the tiuo given quantities, Ex. Find the g.c.d. of 2;' - 7^ -f 10 and 4x'~25:z;'+20:r+25 a;'-7ii;+10)4a;'-25a;' + 20:2;-f25(4:?;+3 4a;^-28:^:'' + 40:y 3.^'- 20a; +25 x-h)x^'-nx^-\^(x-% x^ — hx -2^+10 — 2a;+10 Ans. ic— 5. Examples — 16. Find the g.c.d. 1. Of 32)'+ ^-2 and 3.T'+4a;--4. 2. Of 6:?;' + 7a;-3 and 12a;' + 16rr-3. 3. Of 9a;'-~2o and 9a;'' + 3:r-20. 4. Of 8a;' + 14.T-15 and 8a;' + 30a:' + 13a;- 30. 5. Of 4a;'+3a;-10 and 4a;' + 7a;' -3a; -15. 6. Of 2a;* +a;'- 20a;' -7a; + 24 and 2a;* + 3a;' -13a;' -7a; +15. Rule for the g.c.d. of Polynomials. 72 ELEME^s'TARY ALGEBRA. 96. In order to prove the Eule above given, it will bt* necessary to show first the truth of the following statement If a quantity c he a common divisor of a and b, it Krill also divide the sum or difference of any multiioles of a and b, as ma ± nb. For let c be contained p times in a and q times in I ; then a—pc, h—qc, and ma ± nh — mi^c -ihnqG— {mp ± ncfjc ; hence c is contained mp ± nq times in ma ± nh, and therefore c divides am^ ± nh. Thus, since 6 will divide 12 and 18 without remainder, it will also divide any number such as 7x12 + 5x18, 11x12 — 3x18, 12 (or 1x12) + 7x18, 5x12—18, &c., /. 6., any number found by adding or subtract- ing any multiples of 12 and 18. 97. To prove the Rule for finding the Greatest Common Di- visor of tioo quantities. First, let the two given quantities, denoted by a and 5, have neither of them any monomial factor. Let a be that which is not of lower dimensions than the other; and suppose a divided by Z>, with quotient j!9 and re- mainder c\ bhj c, with quotient q and remainder d, &c. b) a {p 546) 672 (1 2)b_ 546 c)l){q 126) 546 (4 ' qc_ 504 d)c{r 42)126(3 rd 126 Then, by (96), all the common divisors of a and Z>.are also divisors of a—ph or c, and are therefore common divisors oT J) and c; and conversely, all the common divisors oib and o are also divisors oi p)b-{-c or a, and are therefore common di- visors of a and h. Hence it is plain that b and c have pre- cisely the same common divisors as a and b. Troof of tlie Rule for the g.c.d. of Polynomials. GREATEST COMMOI^ DIVISOR. ^73 111 like manner it may be shown that c and d have the same common divisors as h and c, and therefore the same as a and h. And so we might proceed if there were more remainders, the quantities* a, d, c, d, &c. getting lower and lower, yet still being such that a and 1), h and c, c and d, &c. have the same common divisors. But, if d divides c without remainder, then d is itself the greatest quantity that divides both c and d\ that is, d is the greatest of the common divisors of c and d, and therefore is the Greatest Common Divisor of a and h. Thus, in the numerical example, the common divisors of 546 and 672 ax-e precisely the same as those of 126 and 546, and these again are the same as those of 42 and 126 ; but 42 is the g.c.d. of 42 and 126, and is therefore the g.c.d. of 126 and 546, and also of 546 and 672. (See Venable's Arithmetic, Arts. 82, 83.) 98. If the original expressions contain a common fiictor F, which is obvious on inspection, then this factor F will be a factoi of the g.c.d. We strike it out from both the quanti- ties and apply the rule to the resulting quantities. The G.c.D. ihus found must be multiplied by F to get the g.c.d. of the original quantities. 99. If either of the quantities contain a factor which is obviously not a factor of the other, this must be struck out, and the g.o.d. of the resulting quantities is the g.c.d. of the original quantities. So, whenever we take a Eemainder for a Divisor in apply- ing the rule, we may strike out any simple factor it may contain. 100. Again, if after having thus prepared the divisor, at any step of the process we find that the first term of the divi- dend is not exactly divisible by the first of the divisor, then, other features of the process of finding the g.c.d. 4 74 ELEMEXTAET ALGEBRA. in order to avoid fractions in the quotient, we may multiply the whole dividend by such a simple factor as will make its first term so divisible. In general, we may divide the divisor hy any cxjjression wliicli has no factor commo7ito the hoo quantities luhose G.C.D, we are seeking ; or tue may multiply the dividend hj any ex- pression luhich has no factor common to the divisor. Ex. Find the g.c.d. of ^x'-^x^'-^l^x'-^x' + ^x and dx'-Qx'^-^x. Here, striking out of the first the factor 2x (which is com- mon to all its terms), and of the second the factor 3:r, we re- duce the quantities to x^—4:X^-\-Gx^—4,x+l and ^^— 2^?;^+!; but as 2x and 3x have themselves a common factor x, it is plain that the original quantities have a common factor x, which these latter quantities have not; hence the g.c.d. of these, when found, must be multiplied by x to produce that of the given quantities. x''-2x' + l)x'-4:x'+6x'-4.x+l{l -ix\- - 4a;' + 8a;' -4a; x' -3a;+l x"- -2x+l)x*- -3a;' + l(a;'-l-3a; + l X*- -2a;' ^-x' %x'- ^3a;' + l 3a;'- -4a;'+3a; a;'— 2a; + l a;'-3a;+l In this example, the first remainder is reduced by dividing it by —4a:; and, the g.c.d. of these two quantities being x^— 2a; 4-1, that of the two given q? antities will be x (.t' — 2a;+l) or x^—2x^ + x. GREATEST OOMMOK DIVISOll. 75 Ex. Find the g.c.d. of Qx'y + ^xif — ^y^ and ^x^-^4cX^y—4:X7f, Stripping them of their simple factors 2y and 4a; ( ind noting that these contain the common factor 2), we } ave ^x^-{-2xy—y'^ and ^x^ + xy—y"^, and proceed with these qi en- tities as follows : 2 2x' + xy'-y')^x^ + 4.xy—2y\^ 6^jy3xy-3f y)xy+y^ x + y)2x^ + xy—y^{2x—y 2x^ + 2xy —xy—y"^ The G.C.D. then will be 2 (a; + y) ; it being plain that the G.C.D. of 2{^x'-[-2xy'-y^) and 2x'-\-xy—y'' will be the same as that of dx^-^^xy—y"^ and ^x^ + xy—y"^, because the 2 intro- duced into the first is no factor of the second quantity. Examples — 17. Find the g.c.d. 1. Oilox''-x-Q and ^x'-Zx-^, 2. Of 6a;'-a;-2and21a;'-26a;' + 8a;. 3. Of 2a;' + 6a;' + 6a: 4-2 and 6a;'' + 6a;' -6a; -6. 4. Of 2/- 10?/' + 12?/ and 3?/*- 15/ + 24?/' -24. 5. Of x' - 6ax' + 12^'a;— 8^' and x' - 4a'.a;'. 6. Of 2x' + 10a;' + 14a; + 6 and x' + x'-}- 7x + 39. 7. Of 3a;^ + 3a;'-15a; + 9 and e3a;^ + 3a;^-21a;'-9aj. 76 ELEMEJS'TARY ALGEBRA. 9. Oi2a' + a'h-4.a'h''-Uh' and W ■\-a'h-2a''lf + ab\ 10. OiM'-\-16a'h-'da'h''-lba'h' and 10a'-^0a'h-lQa''¥-\-'d0al\ 11. O^x'-^x'y + ^xy^-y' and x'-2x'y^^x''y''-'^xif + y\ 12. Of a;'+6a;' + ll^'+42;-4and2;'+2a;'-5:?;'-12a;-4. XIII. Least Common Multiple. 101. When one Q\ndin.i\ij contains another as a divisor with- out remainder, it is said to be a multiple of it ; and a common multiple of two or more quantities is one that contains each of them without remainder. Thus, Qx^y is a common multiple of 2a^^, Zxy, 6.2;^, &c., and any quan- tity is a multiple of any of its divisors. 102. The Least Common Multiple of two or more alge- braic expressions is a term not appropriate if we use it in the arithmetical sense. We must understand it to mean the quantity of lowest dimensions which is exactly divisible by these expressions. As in Arithmetic, we will use the letters L.C.M. for shortness. 103. To find the l.c.m. of simple expfressions or monomials : Fi7id by Arithmetic the l.c.m. of the nwnerical coefficients ; after this number put every letter luhich occurs in the expres- sions, and give to each letter respectively the greatest exponent lohich it has in the expressions. Ex. Find the l.c.m. of IQa^'bc and 20aWd. Here the l.c.m. of 16 and 20 is 80. The letters which occur in the expres- sions are a, b, c, d; and their greatest exponents are 4, 3, 1, and 1. The required l.c.m. is, therefore, 80a*b^cd. Meaning of Least Common Multiple in Algebra. Rule for l.c.m. of Monomials, LEAST COMMO]Sr MULTIPLE. 77 Ex. Find the l.c.m. of ^a^h'c'x'y^, l^a'hcxY and IMc'x'if, Here the L. c. m. of the numerical coefficients is 48. The letters which occur are a, h, c, x, y, and z] and their greatest exponents are, respectively, 4, 3, 3, 5, 4, and 3. Thus we obtain ^^a^lfc'x^y^z' as the required l. c. m. 104. We shall now show how to find the l.c.m. of two compound expressions or polynomials. Let a and h represent the two quantities, d their G. c. D. : and let a~pd, h—qd, so that jt? and q will have no common factor. Then the least quantity which contains p and q will be pq, and therefore the least quantity which contains pd and qd will \}q pqd, which is consequently the l.c.m. required of a and d, Bince pqd=- — --^=— — -, it appears that the l.c.m of a and b may be found by dividing their product iy their G.c.D. ; or, which is more simple in practice, by dividing either of them by their g.c.d., a7id multiply i7ig the quotient by the other. For example: required the l.c.m. of 2;"^— 4x+3 and42;'-9:c'^-152;+18. The G.c.D. is x—'d] see Art. 95. Divide x'—4zX-\-^ by x—o\ the quotient is x—1. Therefore the l.c.m. is {x—l){4:X^ — ^x'^—lhx-\-l'^)\ and this gives, by multiplying out, 4.x'-l'dx'-Qx''-\-?,dx-lS, It is, however, often convenient to have the l.c.m. ex- pressed in factors, rather than multiplied out. "We know that the g.c.d., which is ^—3, will divide the expression -ix^'—^x^ — lbx + l^] by division we obtain the quotient. Hence the l.c.m. is {x-Z){x-l){4.x^-dx-Q). 105. The principle of the rule in Art. 103, with the aid of L.C.M.. by Inspection. 78 ELEME]S"TAEY ALGEBRA. tlie Chapter on Factors, will enable ns in many cases to de- termine, by inspection, the L.C.M. of polynomial expressions; as we liaye only to set doiun the factors luMcli compose them, each affected with the highest exponent ivhich it has in the ex- pressions ; and the product of these is the Jj.cm, required, Ex. 1. Find the l.c.m. of ^hx, Qahxy, dacx. Here the factors are 2bXf day, c ; and the l.c.m. is 6abcxy, Ex. 2. Find the l.c.m. of 2a\a \-x), 4:ax {a—x), 6x''{a+x), Here the l.c.m. of the simple factors is 12aV, and that of the coniponnd factors is a^—x'^; therefore the l.c.m. required is 12a'x'{a'-x'), 106. Every common multiple of two quantities A and B, is a multiple of their l.c.m. For, let M denote the l.c.m. of A and B, and ]N" some other multiple. Suppose that if possible when !N" is divided by M there is a remainder K ; let ^^ denote the quotient. Then, Isr=M^ + E, and E=K-M^. J^ow since A and B divide both M and IST, they divide, also^ IST—Mg or K Therefore R, which from the nature of di- vision is of loioer dimensions than M, is a multiple of A and B less than the l.c.m. This is absurd. Therefore there can be no remaiiider E. That is, N is a multiple of M. 107. Hence to find the l.c.m. of several expressions, we may find the l.c.m. of two of them ; then find the l.c.m. of this first L.C.M. and the third expression, and so on. Examples — 18. Find the l.c.m. 1. Of ^a%c and Qali^c', of ^x^y and 12xy^ \ of axy and a{xy—y'^) ; of ab + ad and ab—ad, L.C.M. of more than two expressions. ( VNIVERSITY ^LE. 79 2. Of ^a% l^a% and 12a'Jf', of a% ba% lQa'h\ 10d'b% bah\ and h" ) of ^x", 6ax, 8a^ 36x% dax", 60a% and 24a'. 3. Of 2(a + Z>) and S{a'-b'); of 4(a'-a) and 6{a' + a)\ of 6(:^;' + :^•^), S{xy-y'), and 10(cz;'^-/). 4. Of 4(a'-aZ»^), 12(aZ)^ + Z^'), S{a'-a'b); and of 6{x''y + xy^), 9{x^—xy''), 4z{y^+xy''). 5. Of a;'-3a;-4, :i;'-a;-I2. 6. Of i^H5.^•H7^^-2, ic^ + 6cc + 8. 7. Of 12a;^ + 5^-3, Qx'+x'-x, 8. Of a;*-' - 6a;' + 11:^-6,2;'- 92;' + 26a;-24. 9. Of x'-7x-6, a;''+8a;' + 17a; + 10. 10. Of x' + x' + 2x' + x + l, x'-l. 11. Of 4a^Z>'c, 6«^V, -^Sa'M. 12. Of 8(a'-^>'), 12(a + ^)', 20{a-'^)\ 13. Of 4(a + ^), 6(a'-^'), 8{a' + ¥), 14. Of 15(a'Z>-a^'), 21(a^-a&'), 36{ab' + b'). 15. Of i?;'-l, c?;^+l, :?;^-l. IG. Of x'-l, x'+l, x' + l, x'-l. 17. Of x'-l, x' + l, x'-l, x' + l. 18. Of a;' + 3a; + 2, x' + 4:X-i-d, x' + 5x + 6. 80 ELEMENTARY ALGEBEA. XIV. Feactions. 108. Algebraical fractions are for the most part precisely similar, both in their nature and treatment, to common arithmetical fractions. Hence, the student will find the rules and demonstrations in the chapters on Fractions are little more than a repetition of those with which he is already familiar in Arithmetic. 109. The expression -7-, we have agreed shall denote that a is divided by h. We now say -j- means that the unit is di- vided into 1) equal parts, a of w^hich are taken : a is called the numerator, and h the denominator, and the expression -j- is a fraction. (We shall show, as in Arithmetic, that a frac- tion does also express the quotient of the numerator divided by the denominator.) Every integral quantity may be considered as a fraction whose denominator is 1. Thus, a—-, o + c=—-— 110. llo multiply 2b fraction by an integer: Either multi- ply the numerator, or divide the denominator hy the integer ; and conversely, to divide a fraction by an integer : Either divide the numerator, or multiply the deoiominator ly. the in- teger. Thus, -Xx=— ; for in each of the fractions ^, -^, the 00 ho unit is divided into h equal parts, and x times as many of them are taken in the latter as in the former ; hence the lat- Algebraic Fractions. To multiply or divide a fraction by an integer. FRACTIONS. 81 ter fraction is x times the former, that is, — =— -Xcc; and, ax by similar reasonino:, -^-^ccr=- A^ain, —-^x—-i-\ for in each of the fractions -r-, t-» the ^ b bx' b bx^ same number of parts is taken, but each of the parts in the latter is -th of each in the former, since the unit in the lat- X ter case is divided into x times as many parts as in the former ; hence the latter fraction is -th part of the former, that is, -—-—--^x\ and, similarly, -—Xx=-, 111. If any quantity be both multiplied and divided by the same quantity, its value will, of course, remain unaltered. Hence, if the numerator and denominator of a fraction be both multiplied or divided by the same quantity, its value loill remain unaltered, ^, a ax a^ ,, , a3 a ac ^ ilius, -T- =-7-=-T =<!^c., and -— - = -— — =&c. b bx ab a be c c This result is of great importance, and many of the op- erations in Fractions depend on it. 112. To reduce an integer to a fraction with a given de- nominator: Multiply it by the given de^iominator, and the product will be the numerator of the required fraction. The truth of this is evident from (110). Thus, a expressed as a fraction with denominator a; is — ; .,, , • . 7 . ctb—ac or with denominator /9— c, is -^ . b — c 113. Since a — —, and therefore a divided ]dy b =~-^b=--r 1 -^ 1 b (109), it follows, as stated in (108), that a fraction represents To reduce an integer to a fraction with a given denominator. 4« 82 ELEMENTARY ALGEBRA. the quotient of tlie numerator by the denominator. In fact we get -y-th of a units (or a-^h) by taking — th of each of the a units, and this is the same as a such parts of one unit which we haye expressed (108) by -^. 1 3 Thus, in Arithmetic, — of $3 is the same as — of $1. III. The demonstrations giyen in the preceding Articles are based on the assumption that eyery letter denotes some 2)ositive luliole numler. By the Rule of the Sigjis established in Multiplication and Diyision, we haye the following : c^' (^ CIO ^ ... ^ ^ ^ a —a bmce — = — , by puttmg —1 tor c we haye — = — -, Hence, we may change the signs of all the terms in loth the numerator and denominator of a fraction without altering its value, ^ x^—^ax—a^ . ., ^. , .,, a^-\-2ax—x^ Jix. — ^— is identical with ^ — . dax—x X —3ax ^ ^ a ^ —a a So also, -^=+-^-=-^5 —b ~ h ~ h' In like manner, by assuming that -^-X^? is equal to -j-, whateyer be the sign of c, Ave obtain such results as the fol- lowing : 115. If the numerator of a fraction be of lower dimen- sions than the denominator, the fraction may be considered in the light of a proper fraction in Arithmetic : if the nu- Changing the signs of tlie terms. FRACTIOI^fo. 83 merator be of liiglier dwiensions than the denominator, it may be considered in the light of an improper fraction, which (Art. 112) in Algebra, as in Arithmetic, may be ex- pressed as a mixed quantity by the rule : Divide the nunierator hy the denominator as far-as the di- vision is possible, and annex to the quotient a fraction having the remainder for numerator and the divisor for the denom- inator. Thus, 24a „ , M - — aJr^O- = — — M. -r -vt/ 7. x'-dx-V^ x'-dx + 4:' x'-3x-\-4: This last step the student should particularly notice, as an example of the use of brackets, namely : + (-x + 2) = -{x-2). Examples — 19. Express the following fractions as mixed quantities. 25x 36ac+4.c. Sa^' + Sb, 7 9 4flj , nx'-hy ^ a;'4-3.'c+2 ^ 2x'-6x-l 6x x+3 x-3 ^ x'-\-ax'-3a:'x-3a' ^ x'-2x'' x—2a ' ' x^ — x + 1' To reduce improper fractions to whole or mixed quantities. 84: ELEMEKTAEY ALGEBRA. x—1 x + 1 Multiply ■,•, 4a' ^, ,„ 8(0.' +b'') , „, ,, 11. g^ by 35. 12. A^_^by3(a-5). Diyide 15. -^— by 2a;. 16. —r- by 3a— 2^. 18. ^^hjx^^x+1. XV. Eeduction of Feactions. 116. The result contained in Art. 110 will now be applied to the reduction of a fraction to its lowest terms, and the re- duction of fractions to a common denominator. 117. Eule for reducing a fraction to its lowest terms: Divide the numerator and denominator of the fraction hy their Greatest Common Divisor, For example : reduce ^7—^,7-^ to its lowest terms. %i — ■ ■ To reduce a fraction to its lowest terms. REDUCTIO^q^ OF FRACTIONS. 85 Dividing both numerator and denominator by AiCi^h'^, which 4i:ClC is their g.c.d., we obtain for the required result -^j-. That is, -^j-j is equal to KTrrfJl {\^^)y ^^^ i^ is expressed in a more simple form. Ao:ain : reduce -j—, — — ,\^ — — -r to its lowest terms. Dividing both numerator and denominator by x—'d, which x—1 is their g.c.d,, we obtain for the required result j-ti-q^ — F ^X ~\~ ijX — o. 118. In many examples we may apply the results ol the chapter on Factors, and strike out the common factors from the numerator and denominator without using the rule for finding the g.c.d. ; or rather, we may ly mere in- sjpedion find the g.c.d. (Art. 93), and strike it out from the 7mmerator and denominator. Ex. 1. o^x^y'^ _ a^x^y^ _^ axy a^xy + axy"^ axy {a + y) a + y ^ a^-{-x^_{a-\-x) (a^—ax+x^) a^—ax^-x^ a —X {a+x) {a—x) a—x Ex 3 t±^±lJ^^li^J^J^±l a;' + 5:^ + 6 (2: + 3) {x±2)'~x-{-'-Z' x'^x''-\-^x-^ _ {x-l) {x'-\-2x^b) _ x'' + '^x- \-5 ^^' • x'-^x-\-'d ~~ {x-1) {x-d) ~ a;-3"~"* -p, ^ {a—hy—c^_{a—h-\-c){a—h — c)_a — h-^c a'—{h-\-cy~\a-^b + c){a—b — c)~a + b-\-c 119. Kule for reducing fractions to a common denominator : Multiply the numerator of each fraction ly all the denom* To reduce a fraction to its lowest terms by Inspection. To reduce fractions to a Common Denominator. 86 ELEMENTARY ALGEBEA. inators exceipt its own for the numerator corrcsjjonding to that fraction ; and muUijply all the denominators together for a common denominator. The truth of this rule is eyident; since the numerator and denon\inator of each fraction are hoth multiplied by the same quantities (viz. the denominators of the other fractions), its Taluo will not be altered, though all the fractions will now appear with the same denominator. For example : reduce -r ? -^ ? and -r , to a common denom- mator. a_adf c _chf e__ebd iThdf 2~dbf'f~fkV Thus, T-j-^, -7^, and -7^-7 are fractions of the same value, odf dof fbd respectiyely, as -tj-^? and — , and they haye the common denominator hdf. We often wish to reduce fractions to their lowest common denominator, which the aboye process will not effect if there ai-e any common factors in the denominators. It is there- fore often conyenient, as in Arithmetic, to use another rule. 120. Eule for reducing fractions to the lowest common denominator Find the l.c.m. of the denominators. This ivill he the Lowest Common Denominator. Then for the new numerators multi- ply the numerator of each of the given fractions hy the quotient ivhich is obtained by dividiyig the l.c.m. by its denominator. For example: reduce — , — , — , to-their lowest common yz zx xy denominator. The l.c.m. of the denominators is xyz ; and, To reduce fractions to their LoweBt. Common Denominator. EEDUCTIO]^ OF FEACTIOKS. 87 a ax 1) hj c cz ^, , _ ^ — = — , _— -^ _— — . 'ii-^Q numerator and denom- yz xyz zx xyz xy xyz inator of each fraction has been multiplied by the same , quantity. Examples — 20. Eeduce the following fractions to their lowest terms. l^a'lfx ^ a^ah a^ -\- ah ISa'b'y T^* '^F^^b' ^ l Oa'x ^ 4:{a+I)y ^ a' + F 6a'x-15ay'' b(ce^}f)' d'-V' x^-VZx ^l a;'^ + 10a;+21 22;' + ^^'— 15 x^^-{a-^l:))x-\-db 2.r' — 192; + 35* ' ' x^-\- {a^c)x-\-aG x^-{a^h)x-\ -db 3^^3^_36 • x^-V{c-a)x-ac 4a;^+33i^;-27* \x-\-hY-ia-VcY x'-^x^\^' ^^ x^—l^x-^'^X ^^ :z;' + 92; + 20 Id. —^ — — — -— . 16. -^ x^-\JoX'-V: ' x^ ^Ix^ ■\-\^x-\-'S x' + aV + a* x'^-y ^' - « . 1«. ^2«y+l- 19. X —a x^—l x^-a"" a^ — V x^ — lfx d^ — al-Vax—lx ax-^a' x'—aV d'-b'' x" -{•^hx + b'"' d'-\-ab-\-ax-\-hx Eeduce the following fractions to their lowest common denominator. 3 4 5 ^ 9. rr 20. ^, ^,^. 21. ^x' 6.^^' 12x'' x + V 4.'r4-4' :r'-l* 88 ELEMENTARY ALGEBRA. fl^ X a^ ax x—a' a—x :a a l ah 24. — 1 a; 3 4 25. 26. (x-iy {x-iy x+v {x+iy x'-i' a a-\-x ax x—a' x^-^-ax + a'"' x^—a^' x^—ax-^a"' x''-\-ax + a'" ic^ + aV+a*' XVI. Addition and Subtraction of Fractions. 121. To add or subtract fractions: Reduce tliem to a com^ men denomi7iator (if necessary), a7id add or subtract the numerators for a neiu numerator and retain tlie common de?iominator, Ex.1. Add ^ and ^. TT ,1 . a-\-c-\-a—c 2a Here the sum is ^ =-7-. Ex. 2. From take . c c ^a-3h da-'4 :I?_ 4.a-3h-{3a-U) _ 4:a-3b-Sa + ^ b c c ~~ c ~~ c a + b To add or subtract fractions. ADDITIO:^" AISTD SUBTRACTIOiq" OF FRACTI0:N^S. 89 Ex. 3. Add -^ and "^ Here the common denominator will be the product of a-^h and a — h, that is, a^^lf, c __c(a—b)^ c _c{a + h) Therefore, ^^ + -^='A^3^±^^ a + b a—b a -o" _ca—cb-\-ca-\-cb__ 2ca ^ . _ a + b . , a — b Ex. 4. From 7 take a—b a+b' The common denominator is a^—b^, a + b_{a+b)\ a-b_{a-by a-b~ a'-b''' a-^b" a'-b'' Therefore, ^"^^ a-^b_{a + by-{a-by a—b a + b oi^—V _ y + 2ab+F-{a'-2ab + b') _ iab ~ a'-b' ~a'-r Ex. 5. Add -4^4-,+ ^ ^"^ , . l-{-x + x l—x-\-x {1 + x) {l-x^-x'')-\-{l-x) {l-^-x^-x"") __ 2 Ex.6. From . "^"^^ , take "^"^ l + x-\-x' 1 — x+x^' {1 + x) (l-x-\-x')-{l-x) (1 + ^; + :^:') _ 2x' {l + x + x'){l-x + x') '~l + x' + x*' 90 ELEMENTARY ALGEBRA. 122. AYe haye sometimes to reduce a mixed quantity to a fraction ; this is a simple case of addition or subtraction of fractions. For example : h a h ac I ac-\-h c 1 c c c c h __a h _ac h _ac—h c~ 1 c~ c c~ c Hence, Multiply the entire part ly the denominator of the frac- tion, add to or suMract from this the numerator of the frac- tion, and place the result over the denominator, -, ^ . 2«Z> a 2«Z> aia + i) 2ab a^ + 3ab Ex. 1. a-\ 7=-T-H r-7=— r^ H 7= t— • a + o 1 a+o a^-0 a-\-o a + o x—2 x+3 x-2 Ex. 2. x + 3 x''-3x + 4. 1 x'-3x + 4: _ {x + 3)(x'-3x + 4:) x-2 '~ x''-3x-\'4: x''-3x-\-4. _ x^'-^x-\-12-{x-2) _ x^—bx + 12'-x-^ 2 _x^ -^x + l ^ Ex 3 1 ^'+^'-^ '_ ^^^ + ^' + ^' -^' _ ('^+^)'-^' 2ah ~~ 2ah ~' 2ab _ {a + d + c)(a + h—c) "~ 2ab ^ . ^, XT, 4. -, a^' + h^-c^ {a~d + c)(I)-a + c) Ex.4. Snow that 1 ^r-^ ==^ -zr^ : 2ao 2ao To reduce a mixed quantity to a fraction. ADDITION AXD SUBTRACTIO:^" OF FRACTIOXS. 91 Ex. 5. Show that a''^]f-c'\\_{a-^l) + c){a^-l)-c){a-\-c-l) {b + c-a w The attention of the student is again called to the fact that the line which separates the numerator from the de- nominator of a fraction is a vinculum or bracket. Hence, he will apply the rules of brackets, Arts. 51 and 55. 123. Expressions may occur involving both addition and subtraction. Ex. Find the value of 2 + ^^^,'-^. Ans 2(a^-~^^) + (^'4-y)-(a-Z>)(^-^) ^ci'-^'^al-W Examples — 21. Find the value of a {a—l) a (a + I?) ^ Sa—4:b 2a—h—c 15a— 4c Yb'^oTb) ' Yb'^3 (a-h) ' ~"2 3 '^~T2~"' a a b a b a—b ah ^' a-b ^' a + b'^ a-b' a-b a + b' a + b'^ a'-b' a (ad—bc)x^ a^ + b"^ a—b 2x'^—2xij+i/^ x c c{c+dx) ' c^—b^ a-\-b^ x^—xy ^—y' 1 1 ^ . ^ — ^ C'—a b—c •• 2{a-x)'^2{a-\-x) '^ a'-^-x" ' ~W^~^^'^~bG~' ^^ 1 1 1 _1_ 1 3a 2{x-l) 2[x-\-l) x'' ' 2a + b'^2a~b ^a^-V 92 ELEMEKTARY ALGEBRA. a {a!' — lf)x a(a^ — l'')x^ ^ \_ 1 x —\ x^—y x—y x + y x + y x ~y^ x -j-y 11 ^_ ^+^ 12. 2 ^'"" ^ ' • ^^^' 13. -^ 7 r. 14. ^ ; V-f j. a a(a—x) y x+y xy—y^ ^^ X x^ x? 15. -r, ri- + - 1-x {1-xy "^ (1-^)'* 16. ^-v+. ' '-^ 8(1-^) ' 8(1+^) 4(1 + ^')' Bemarlc. — In the preceding examples we have combined two or more fractions in a single fraction. On the other hand, we may if we please break up a single fraction into two or more fractions. For example : 8Z>c— 4ac + 5a5__35c 4<xc ^ab 3 4 5 abc ~abc abo abc a b c' but the beginner must not confound -; — with -r . b—c be 124. The addition and subtraction of fractions can often be much simplified by observing closely the factors of the denominators, and avoiding unnecessary multiplications in reducing the fractions to a common denominator. Ex. Find the value of a l c ^ {a—d){a — c) {b~c){b—a) {c—a){c—'b)' Here the beginner is liable to take the product of the de- nominators for the common denominator, and thus to ren- der the operations laborious. How tlie addition and Bubtraction of fractious may often bo simplified. ADDITIOIs^ AKD SUBTEACTIOiT OF FBACTIOXS. ^3 The second fraction contains tlie factor l—a in its denom- inator, and this factor differs from the factor a — l), which occurs in the denominator of the first fraction only, in the sign of each term ; and by Art. 110 : 1)_ I {b-c){b-a)~ {b-c){a-l))' Also, in the denominator of the third fraction, by the Eule of Signs we have, (c — a){c—'b) — {a— c) {b—c). Hence, the given expression may be written, a b + - {a-b){a-c) {b-c){a-by {a-c){b-c)' And in this form we see at once that the L.C.M. of the de- nominators is {a'-b){a—c){b—c). By reducing the fractions to this lowest common denomi- nator, we get a{b—c) — b{a—c)-\-c{a—b) ab — ac'-ab + bc^-ac—bc_ {a—b){a—c){b—c) ~~ {a—b){a—c){b — c) Examples — 22. Find the value of a 1. {x—a){a—b) {x—b){b—ay b' 3, {x-a){a-b) ^ {x-b){b'-a) 1 1 {a-b){a-~c)'^ {b-a)(b-cy ^4 ELEMEIS-TARY ALGEBEA. a ^' {a-~b){a-c) "^ {h-a)(})-G) ' 6. . .^. . -f- ^ 7. a{a—b){a—c) h{b—a){h—c) abc {a-b){a-c) "^ {b-a) {b-c) "^ (c-a) (c-b)' XYII. Multiplication of Feactions. 125. To multiply fractions together; Multiply the numerators together for a neiu numerator^ and the denominators for a neiu denominator. Suppose that we have to multiply -r hy -r : let — =0?, /> -j—y\ :. a—bx, c—dy, and ac=bdxy\ hence (dividing a ac a c each of these equals by bd), j-^^xy, but xy—~y,—^ and ac aXc product of numerators , ,. , ,, T-i=Ti — i^-^^—j — T-i-^ — 7 — y whence the truth of the bd bXd product of denominators rule is manifest. Similarly we may proceed for any number of fractions. a^b a-b 3_ 3(a+^)(o^~^) dja'-V) ^^' c + d^c-d^2~2{c-\-d){c-d)'~'2{c'-d')' 126. The Kule of Signs (Art. 61) gives the following re- sults in the multiplication of fractions : To multiply fractions together. MULTIPLICATIOI!^ OF FRACTIO^^S. ^ft a c _a — c — ac_ ac a c ~ a c — ac ac h^ d~ b ^ d~ bd ~ bd' a c — a — c_ac b d~ b d ~~bd' 127. We shall now give some examples. Before multiply jng the factors of the new numerator together, and the factors of the new denominator together, examine if any factor occurs in both the numerator and denominator; in which case it may be struclc out of both, and the result will be more simple. Art. 116. (See method of cancelling in Arithmetic — Vulgar Fractions.) Ex. 1. Multiply a by — . c a a b ah Hence a— and — are equivalent; so, for example, . X 4:X T 1 /^ 0\ ^^ — 3 4— =r-^; and — (2x— 3) = — - — . 5 5 4^ ^ 4 Ex. 2. Multiply - by ^. X X _xXx_x^ ^ y y'vy^'f' thus (— ) =-a. ^yJ y Application of the Rule of Signs. Cancellation, 96 eleme:n'taey algebka. Ex.3. Multiply I? by |. Sa Sc_3aXSc__2cXl2a_2c 4:b 9a~4:bx9a~dbxi2'a~3b' Ex. 4. Multiply , , ,,, by -^—z.-^- (a+^)'^ 3ab ~ b {a + b)Xda {a + b ~b{a-\-b)' ^ ^ a ex a c ac Ex. 5. tX-t=-tX-j=j--,* bx a b a bd ^ 5fl^.T xy-^-if^ _J)a {x-^ij) _ hax 4- 5^-?/ 3c;z/ x'^~xy~'3c{x—y)~ 3c{x~yy -^ 4fl^a; a^ — o;^ ^c + bx _ 4:r ((7- + x) _ 4rir.T + 4^^ 3bt/ c^—x^ a^—ax~Sy{c—x)~oy{c—xy Bemark. — The student slioulcl leave the denominators of fractions ■ with their factors unmultipUed^ as in Ex. 6 and 7, unless they combine very simply. The convenience of this will be found in practice. Ex. 8. Multiply 4 + -+1 by -^+--1. ^ '' b a '^ b a b a ~ab ab ab~' ab ^ a b -, _^' ^' ab ^a^-^-b"^ —ab b a ab ab ab~ ab ^_±^±ab a'-\-b'-a b_{a''-\-b'' -\-al)) (a' + b'-ab) ab ab "~ a'^b^ __ {a' + by -a'b'_ a' + b'-^a''b' "" a'b' - a'b' • MULTIPLICATION OF FRACTIONS. 97 Or we may proceed thus: fa h ^\ f a h ^\ fa hy" ^ therefore, fa h ^ ^\ fa I , \ a\ ^ h^ , a" h\ ^ \ a J \h a J a la The two results agree, for -p,-\ — ^ +1 = 2 7,2 Examples— 23. Find the yalue of the following : M ^hc ^ h^ c^ ob oa 00 ac ah a^h Vc c'a . x-{-l x-h2 x—l 0» 2 X ^ X "^ • ^» T X 2 -1 X V ■ L-i \ i» i?;i/ y z zx ^—1^—1 (a;4-2j cc + a Va xy \ b / \ a/ '■ (»+^J (*-^)- cc(6i^— cc) a(a + x) 2X- a^ + 2a:?; + ^^ ^^ — 2ax + 2;^ ^'— «/• ^' + ^' ^ + y x'—(a-^h)x-}-ah x^—c^ J.l/. -~ '^ I r - X 2 72' X —{a + c) x + ac x—b 5 98 elemejs^taky algebra. X -\-y \x—y x-\-yJ 13. f« i_4_i)xfi — ¥_). \oc ac ab a) \ a + o + cj ^^ fx"" a" X a . \ fx a\ 13. (-T+— + l)x( . \a X a X J \a X J ^ . f X a tf h \ ( X a y b\ 14. ( + ^ )X — T + -)- \a X b y J \a x by) x^—%x-\-\ x''—4x_j-^ x^ — 6x+9 ' x'-6x-{-6^x'-4.x+3^x'--3x-}-2' XVIII. Division of Feactions. 128. Eule for diyiding one fraction by another : Invert the divisor, and proceed as in Multiplication. The following is the usual demonstration of the rule. Suppose we haye to divide "t- by -7 ; put -j-=x, and -T=y'y then, a = bx, and c=dy'y and, ad=bdx, and bc—idy\ therefore. ad bdx X bc~~bdy~ y* -n , X a c y -^ b d^ ., « a c ad a d therefore, -r-f--7= r=irX— • b d be b c To divide one traction by another. diyisio:n" of feactioxs. 99 129. The results giyen in Art. 125 give us the following in connection with Division of Fractions : o- ^ c ac ^ a c ac b d M Id bd , ac c a ^ ac c a bd d b bd d b • T . a c ac , ac c a Also, since — 7- x — 7=^-75 we have :r-,-. -— — —. b d bd bd d b 130. The student should, in the division of fractions, en- deavor to simplify the operations as much as possible by striking out factors which occur in both numerator and denominator. Ex. 1. Divide a by — . __ a a b _a c __ac Ex. 2. Divide ?f by I?. 3a_^9^_3^ 8£_2^ 4.b'^SG~^b^'^a~W Ex.3. Divide ^^, by ^' {a-{-by ^ a'-b'' ab-y W ab-b' a' {a+by ' a'-b'~{a + by^ b' _ b (a-b) (a+b) {a'-b _ {a-by "" b\d^by ~b{a'\-by x^^xy ^ x'-if jc^^-xy ix-yy _ x x-y (x-yy x-y x -y' x +/ 100 ELEMENTARY ALGEBRA. E] 5:amples — 24. Divide 4.xyz' ^ 3xYz' 1 1 6{ab-I?') W x'-y' ^ x-y a{a + bY ^^ a{a'-b'j* , a'-4:x' , a'-^ax ^ 82;' ^ Ix" a^4-4a:c "^ ax + 4a;^* * ^'— ^ x' + xy-{-y'^' iz;^ + ^^ -^ x^—xy^y^' o^-\-(a-k-c)x-^ac ^ x^—a^ x'+{h + c)x^c ^ ?^'' 10. ?!i:^I&: by ^'-^^ a; 4-^ x—xy-\-y* I. (i.|)(.-£)b,J^.. 13. 5x^-4 by :r + -^. 14. a'-\hj a---. 15. — 5 by . 16. Sa-\ ^ by a; . a X '' a x a x ^ x x" 1 a: 1 1 _ a;\ ^ ^r' x ^ a 17. — , by-jH — . 18. -^+l + -,by 1+— . y X ^ y^ y X a'^ ^ x^ ^ a ^ x COMPLEX i^RACTIOXS. 101 XIX. Complex Fractions and other Eesults. 131. Hitherto we have supposed, in the chapters on Frac- tions, that the letters represented whole numbers, but when we come to interpret the multiplication of fractions we must extend the meaning of the term, as we have done in Arith- metic. Thus, to multiply "t- by -7, the fraction -j- is divided into d equal parts, and c such parts are taken. Now if -7- be divided into d equal parts, each of these parts is y-^; and etc if c such parts be taken, the result is j-y Then, too, to divide a c ■j-^J -J "ineans to find a quantity such, that if it be multi- C Cb plied by -^ the product shall be — . 132. N^ow with our extended definitions we can easily prove that all the rules and formulas given are true when the letters denote any numbers ivliole or fractional. Take, for example, the formula — =— , and suppose we wish to show that this is true when a——. 0—-, and c=-, 71 q s ^ ^ a _m ^ p __m q _mq on q n p np Also ac=^ — , and hc~ —, ns qs rpi ac _mr pr _vir qs mrqs wq he ~ ns ' qs ~ ns pr ~ nsjjr ~ np ' Thus the formula is proved to be true. -1 2-x 2 2- 2-x 4a; - ~ Ax ' ~ 2 ~ 8x " 102 ELEMEKTAKY ALGEBRA. 133. Complex fractional expressions may be simplified by tbe aid of rules respecting fractions which have now been given. Ex. 1. Hence obserye that, when a complex fraction is pnt into the form of a 7; — -r — , the simple expression for it will be traction ^ ^ found by taking the product of the upper and lower quanti- ties, or extremes, for the numerator, and that of the two mid- dle ones, or means, for the denominator ; and that any factor may be struck out from one of the extremes, if it be struck out also from one of the means. Ex. 2. Ex. 3. 2x 20-ri; 2x 1 6x 6-ix 4 60 -3a: ^-3 ~3aj-l~"3^-r x + l^~~dx+4.~4:{3x + 4:y 3 3 Ex. 4. a+h a-h {a + hy+ia-hy 2a'+2h' a-\-b a-b {a+by-{a-by Aab 2ab ' a-b a + b a'-b' a'-b^ Ex. 5. n 1 . 0—a Simplification of Complex Fractions. COMPLEX FRACTIOJ^S. 1U3 3 — a~~d-~a 3—a~ d—a 3-a' 3 — a 4a 3— a 3-\-3a a + --r- ^-r-] T— = -. . , 3 + 3a 1 4 _ 4 • 4 ■~1^3+3a~3+3a' Ex. 6. Find the value of ^ when x= Here a—x=a~ and l—x—1)- ab a^ + ah—ah a^ ab "aVb a-\-b a+b' b^ = ab\-V—ab=^ r. a-^b Hence, a—x a-hb _ a^ l^" b' ~J'' a + b Simplify 1 1. l-\-x 1 1 ' 1 + X 1 X Examples — 25. ':+.=" 2. l + iz; 1— a; 1— ^ l + a; 3x x—1 5. 2"*"" 3 -(.H-l)---2i 1-. 3. 1+- a;-l + x—6 X-2 + x-6 104 ELEMENTARY ALGEBRA. Find the yalue of -, x—a x—b , a^ 7. — ^ when x= ~. o a a—h TT when x=--}- -^. a-\-b b{b + a) 8. -+T^ a o—a 9, a'x-hh'ij . 2,^2 ■ — ~ when a— — and h= — . i?: + .^ 3 3 10. -— — — ^ ^— „ when t/ = — . ic+y x—y X —y ^ 4 ^^ x + 2a x—2a 4:ab . ^5 J-J-. ^77 +?rr"^ TU "2 when x= -. 2b — X 2b+x ^W—x^ a + b ^^ fx—aV x—2a + b , a + b l/c. I — -1 ; — when 0:=— -— . \x—bj x + a—2b 2 x+y—1 a-hl , ab + a lo. ■ — — -— when xz= , ^ , and y=—, — -. x-y + 1 ab-\-V ^ ab-hl 134. The following results should be noticed. If —==—-. then b d ^ , ci c b d ,.. abbe a b ,... ^--b=^-d' ^^ ¥=7«' 7X7=7X7' '' 7=7 (">' a c a + b c-\-d ,.... CI' ^ ^ a—b c—d ,. . a:^b b c^d d a:^b r hence -7-X— ^-^-X— , or = /; a an a iXYOLUTio:Nr. 105 , a-^l h c-\-d d a + b c{-d, .^ and — T— X -j= — j-X 7' ^^^ 7= 7(^1): a—o d c—d a—o c—d^ ^ and any of tliese last may be inverted by (i), or alternated , ,... ,^ ct c a a^h a + b a—h . by (n); tlms, -^^=y^^j, -=^-^, ^=^=5' &'^- So that, If any tivo fractions are equal, ive may comhin^ hy addition or subtraction, in any ivay, the numerator and denominator of the one, provided that ive do the same with the other. XX. Involution. 135. The jorocess of obtaining the poivers of quantities is called Involution, A poiver has been defined to be the p)roduct of two or more equal factors. All cases of Involution, then, are merely examples of multiplication, where all the factors are the same; and the rules given in the present chapter follow immediately from the laws of Multiplication. 136. Any even p)Oiver of a negative quantity is positive. Any odd power of a negative quantity is negative. This is a simple consequence of the Eule of Signs. Thus, —■aX—a—-\-a^', —aX—aX—a=-{-a''X-'a——a^'y — aX—aX —aX —a=—a''X —a — -}-a*; and so on. Let the student notice : 1. That any eve^i power of a quantity is the same whether that quantity be negative or positive. Thus ( + «)^ and { — ay are each=+^^; and { — {a + b)y and { + {a-\-b)y are each= + (a + ^)\ 2. ISTo even power of any quantity can be negative, 3. Any odd power of a quantity will have the same sign as the quantity itself. 137. The expo7ient of any poiver of a power is equal to the product of the exponents of the tivo poivers. Involution— Power— Signs of Powers. 5* 106 ELEME^^TAEY ALGEBRA. Thus, tlie cube of a% that is, {ay=za'i foY,{ay = a'Xa'Xa* Similarly, {ay = a''; {-ay=-a''; {-ay=z-a'', {a^Y 138. Eule for obtaining any power of a monomial ex- pression : Multiply the exponent of every factor in the expression hy the index of the required power, and give the proper sign to the result. Thus, for example, {a''l)y=a'h'] {-aWy=-a'b'; {abVy=a'bV'; {-a'bYy=-a''b'V'', {2ab'cy=2Vb'Y'^64:a'b''c'\ It is usual to raise the numerical coefficient at once to the required power, instead of first writing it with an exponent. Thus, ( - 2xy'z') ' = - 8xYz\ 139. Eule for obtaining any power of a fraction : Eaise both the numerator and deiiominator to that potver, and give the proper sign to the result. This follows from Art. 122. For example, 140. Some examples of Involution of binomial expressions have already been given. Thus, {a-\-by=a'' + 2ab + b\ (a-'by=a'-2ab+b\ By (137) we may shorten the operation, finding the 4th power of a quantity by squaring its square ; and similarly, to find the 6th, 8th, &c. powers, we may square the 3d, 4th, &c. powers. Rule for obtaining any power of a monomial exprebsion ; — of a fraction. In- volution of binomial expressions. IXYOLUTIOX. 107 So also to find the cube or 3d power, vre may take the product of the quantity itself and its square ; to find the 5th, we may take that of the square and cube, &c. Thus we shall have, (a-by={a'-2ab + b'){a-b)=a'-3a'b + 3ab'-b'; {a+by = {a' + 2ab+b'){a' + 2ab + ¥)=a' + ^a'b+ea'b* + 4.ab' + b'; {a-by={a'-2ab + b'){a''-2ab + b')=za*-'4.a'b + 6aV ^4.ab' + b'; {a+by={a + by{a-\-by=a' + 6a*b + 10a'b'-hl0a'b'-^6ab^ (a-by={a-by{a-by=a'-6a*b+10a'b'-10a'b'-\-5ab* -b\ The student should remember the above results, though the higher powers of binomial expressions are best obtained by the Binomial Theorem, which we shall give subsequently. It will be noticed in the above examples that any power oi a—b can be immediately obtained from the same power of a-\-b by changing the signs of the terms which involve the odd powers of b. 141. The results of Art. 140, can readily be applied to tri- nomial expressions, Ex. 1. {a + b^cy=a'-Y2a{b^-c)^{b+cy =a'' + 2ab + 2ac-{-b'-\-2bc-{-c\ Involution of trinomial expreseions. 108 ELEMEKTARY ALGEBEA, Ex. 2. {a-\-l) + cy={a-\-{'b^-c)Y Ex. 3. (a-^-c)^={^-(Z» + c)}^ ==a'-3^'^-3aV + 3a5' + 6rtJc + 3r^c'~Z»'-3/;'c -3Z^c^-6'^ Or thus : (a-l-cy={{a-l)-cY^{a-l>y-^{a-'byc-{-^a-l)c^-c\ which, of course, when expanded, would give the same re- sult as before. Ex. 4. {^x-dy={2xy-4. . 3 . {2xy + 6 . 31 {2xy-^ . 3^(2a:) + 3* Examples — 26. 1. Eind the yahies oi {2aiy, (-3a^^V)', (-^T. Write down the expansions of 2. {x^%)\ 3. {x-'^y, 4. (^+3)^ 5. (l + 2a;)*. 6. (2m-l)'. 7. {3.r+l)\ 8. {9.x-ay. 9. (3.'?; + 2a)*. 10. (4a-3Z>)'. 11. {ax-yy. 12. (ax + rr^)'. 13, {%am-my, 14. (a-Z> + c)«. 15. (l-a:-f .t'*)'. IXYOLUTIOX. 109 142. The square of imy 2)oIl/nG}mal expression may be ob- tained by either of two rules. Take for example, {a + b + c+dy. We will find, {a + b + c-hcl}'' = a' + Z>' + c' + cr + 2al) + 2ac + 2ad +2bc + 2bd+ 2cd. We see from this — the square of any polynomial may be found by setting down the square of each term., and then the doiihle 2^rodiict3 of all the terms, talcen two and two, iVgain, we may put the result in this form, {a^l^c + dy ^a''-\-2a{l) + c + d)-Vl)'^2'b{c-Vd)+c'^2ci: vd^. and this may be obtained by the following rule : The square of any multinomial expression consist i df the square of each term,, together ivith twice the product \ f cadh term by the sitm of all the terms ivhich folloiv it. Ex. 1. {l + 2x-\-^xy^l-\-2{2x + 'dx')+^x^-\-ix{^x'')-^.&J = l-\-4cX-\-10x'-\-l2x' + ^x\ Ex. 2. {l-2xy=[{l-2xyY^{l'-^,x-\-12x''-^x'Y =:l-12x-\-2ix''-Ux^ + 144^^-1920;* f 64^" = l-12.^' + 602;^-1602;^4-240.^'^-192a;* V^^x\ The square of polynomial expreseiona,— two rules. 110 elementary algebra. Examples — 27. Find 1. {a + h-^c-\-dy-{a-'b + c-d)\ 2. {a + d-^c + dy+{a-b-\-c-dy. 3. {1 + x-hxy. 4. {l--x+xy. 5. (l+2;-ic^)\ 6. {l-\-3x-h2xy. 7. {l-3x+dxy. 8. (2 + 3a;+4a;^)^+(2-3a; + 4:c^)^ 9. (l_:c+a;^+a;y. 10. {l + 2x+3x'+^xy. XXI. Eyolution. 143. Evolution is the inverse of Involution. Evolution is, then, the method of finding the roots of quantities. It is usual in this connection to use the word extract in the same sense as find. Thus to extract the square root is to find the square root, 144. It follows from (136) that— 1. Any even root of a positive quantity will have the douUe sign db. Thus the square root of a^ is ±a, the fourth root of a* is 2. Any odd root of a quantity has the same sign as the quantity itself Thus, for example, the cube root of a^ is a, and the cube root of —a^ \^ —a, 3. There can he no even root of a negative quantity. Hence the indicated even root of a negative quantity is called an impossible quantity or imaginary quantity, ^ —a^, ^—a, "^ — 1, are imaginary quantities. Evolutiou. Three Rules for the Signs of Roots. Imaginary quantities. EVOLUTION-. Ill 145. Eule for finding any root of a monomial integral expression. Extract the required root of the numerical coef- ficient, divide the expo7ient of each literal factor hy the index of the root, and give the proper sign to the result. Since the cube power of a^ is a^, therefore the cube root of a^ is a^, and so on. Thus, for example, V {Ua^h')= V {^^a''h') = :^4.a'b\ V{-Sa'b'c'') = V{-2'a'bY') = -2aWc\ V{266xy) =V{4:'xy) = ±:4.xy\ 146. To obtain any root of a fraction: Find the root of the numerator and denominator, and give the proper sign to the result. For example, \/ {^)=\^ {-^) 2a 147. Suppose we require the cube root of a^. In this case the exponent 2 of the quantity is not divisible by the index 3 of the root ; then we cannot find the root of it, but can only indicate that the root is to he extracted by writing it thus, V^. Similarly, \f~^, v/^, V^, indicate roots which we cannot extract. Such quantities are called surds, or irrational quantities; the difference between surds and imaginary quantities being that surds have real values, though we cannot find them exactly, while there cannot be a quantity, positive or negative, an even power of which would produce a negative quantity. Examples — 28. 1. Find the square roots of ^a^h'c\ 4t9xyz% 100a'b'Y\ Eule for finding any root of a monomial integral expression; of a fraction. Surds, or Irrational quantities. 115> ELEMENTARY ALGEBIIA. _ ^. , ., . . ^a'x\f 4.9xy mxY' 2. Find the square roots of ---4- , -777-^- , -ttt-ott- 27^;^'' ^125a^'^ ^ 343 * 3. Find v/-^^4^, V-^,, V^,T., V *■ - v(w> i/(5^.> ♦/(^:-> 148. To find the square root of a polynomial: We hnnw that the square of a + ^ is a^-\-2ah-\-lf. Let us observe, then, how from a'^ + 2ab-hh^ we may deduce its square root a + b. This will lead us to a general method of finding the square root of polynomial expressions. a'+2ah + I)'{a+b a^ 2a + b)2ab + b' 2ab+b'' Arrange the terms according to the powers of one letter, a; then the first term is a^, and its square root is a. Subtract the square of a, that is, a^, from the whole expression, and bring down the remainder 2ab + b^. Diyide 2ab by 2a and the quotient is b, which is the other term of the root; lastly, if we add this b to the 2a, multiply the 2a+b thus formed by b, and subtract the product from 2ab + b'^, there is no re- mainder. Now we may follow this plan in any other case, and, if we find no remainder, we may conclude that the root is exactly obtained. Ex.1. Ex.2. 9x' + exy+y\3x + y 16a'-56ab + 4:9b'{4.a-'7b i)x+y) Qxy-Vy'' 8a-7Z>)-5G^/Z» + 49Z'' ^xy-^y'' — 56«Z> + 49Z>' To find the square root of a polynomial. evolijtio:n'. 113 Ex. 3. ^a^-^ah-F^^a-l) ^4.ab + b' -2b\ Here we find a remainder —25"^ ; we conclude, therefore, that 2a— b is not tlie exact root of Aa^ —Aab—b'^ which is a surd, and can only be written V^a'_4^ab-b-'' 149. If the root consist of more than two terms, a similar process will enable iis to fir 4 it, as in the following example, where it will be seen that the divisor at any step is obtained by doubliiig the quantity already found in the root, or (which amounts to the same thing and is more convenient in practice) by douhling the last term of the ])receding divisor, and then annexing the neiu term of the root, Ex. 162,-'-24^'+252;*-20a;'-fl0a;'-4::c+l(42.-'-32;^ + 2a;-l —24:x'-{- Qx' 8x'-6x'' + 2x) 16x'-20x'-\-10x^ 16x'-12x^ji^_i^ 8x'-Gx'' + 4:X-l)- Sx'+ i5x''-4:X + l - 8x '-\- 6x''-4:X-\-l 150. It has already been remarked that all even roots have double signs. Thus, the square root of a'^-\-2ab + b^ may be — {a+b), that is, —a—b, as well as a-\-b; and, in fact, the first term in the root, which we found by taking the square root of tt^ might have been —a as well as a, and b}^ using this we should have obtained, also, ~~b. So in (148) Ex. 1, the root may also be —3x—y; in Ex. in (149), —4:X^ + dx'^~2x-{-l; and in all such cases, we should get the two roots by giving a double sign to the first term in the root. When the root consists of more than two terms. Double Signs. 114 ELEMENTARY ALGEBRA. 151. As the 4tli poiver of a quantity is the square of its square, so the 4tli root of a quantity is the square root of its square root, and may therefore be found by the preceding j*ule. Similarly, the 8th root may be found by extracting the square root of the 4th root. Thus, if it be required to find the 4th root of a' + 4a'ic+ 6«^V + 4«a;' + x\ the square root will be found to be a^-\-^ax-\-x^, and the square root of this to be a+ic, which is therefore the 4th root of the given quantity. Examples — 29. Extract the squnre root of 1. a;'+2a;' + 3a;' + 2a; + l. 2. 1— 2^ + 5a;''-4a;'' + 4ct . 3. x'-V^x^ + '^bx^^-^^x+U. 4. x'^4.x'-V%x-\-4:. 5. l-^x^l()x^- 12x^ + 9a;*. 6. 4x'--4x''-7i«* + 4x' + 4. 7. x'-2ax'-\-6aV-4.a'x+4.a\ 8. x'-2ax' + {a' + 2I}^)x''-'2ab'x + b\ 9. ic*-12a;'+60^*-160:z:' + 240:r'-192a;+64. 10. x' + ^ax'-10aV-\-4:a'x + a\ 11. l-2x-{-dx'-4.x' + 6x'--4:x' + 3x'-2x'' + x\ ^x" X l^x^ 9f_ 6xy 16^ 9p~7~l5y^^l6?"^ 57"^25;2^' The fourth root, etc. EVOLUTIO:^'. 115 Extract the 4tli root 13. Of l-4:X-{-6x''-4.x' + x* and of a'-8a'-f 24a'-32«4-16. 14. 0£16a'-96a'b + 2Wa'b'-216ah'-i-SU\ Find the 8th root 15. Of {x'-2x'y-{-3xY-2xy'-{-t/}\ 152. The observation of the square roots of trhiomial ex- pressions enables us to find the square root of complete (/. e.), exact squares of these terms very easily, without going through the entire process of Art. 148. EuLE. — Arrange the terms according to the powers of some one letter. Find separately the square roots of the extreme terms, and take their sum or difference accordingly as the sig?t of the middle term is -h or — . Thus, a^ + 2ax -f 2;* is a complete square arranged according to powers of a, and its square root is ^a^~^ ^x^, or a-\-x, .*. a-{-x squared produces a^ + 2ax + x^. The square root of a^—2ax + x^ is a—x, for the same reason. Ex. 1. ^¥~+l-{-2a='^a' + 2a-hl=^^a^-{-^l=a-hl. Ex. 2. ^¥~-{-9-6x= y/x''-6x + 9= v^?- >/d=x—S. Ex. 3. v^4Ty^-4^= s/y^-4:y + A=: n/^- \^I=y-2. Ex. 4. \/x^-px+^= v/i^_f^^^_|. Ex. 5. \/x' + 3x+\=^x'^]/\=x-{-%. To find the square root of complete trinomial squares. 116 ELEME^'TARY ALGEBRA. Ex. 6. ^ ni'jf + '^iniix + n^ — "^ i)fx' + ^ if ~ mx + n. Ex. 7. v'9^'7^to7/ + a~'== ^9xy- ^7'r=dxy-a, Ex. 8. v/i,rZ^^4-^^^;-|-c-^ =: \/I^^" + V? = J-«Z^ 4- c. Find the square roots of the following expressions : Ex. 9. 16a^ + 40aZ>+25^^ 10. 49^*-84a'^^ + 36^>^ Ex. 11. 36:^;° + 12i^' + l. 12. 64a' + 48a/;c + 9Z^V. -g .o 25^^^^«^4^^ 9 x^-242^^ + 16 ^"^^ * 2ba' + 20ac-\-^o^ ' S^~12:z: + 9* 153. By observing the terms of a complete trinomial square arranged according to one letter, we see that the mid- dle term is twice the product of the square roots of the two extreme terms. Hence, the quantity which must be added to an expression of the form, x^ + 22jx, in order to form a complete trinomial square, is the square of one-half of the CO factor, or coef[icient, 2p of x\ that is, ]f. Observe that x represents the square root of the first term. Thus, m^x^ + ^mnx requires the square of the half of 2?^, or 7f^ to complete it, giving m^x^ + 2in7ix + ^^^ dx^y^—Qaxy requires the square of the half of 2a, giving 9x'^y^ — 6axy + a^. Complete the squares in each of the following cases : (1.) x'-12x + - (3.) x'+llx+- (5.) x'- ix-h- (7.) ■ x'-i- px-h- 7t (9.) --fo+- (2.) x=-|+- (4.) x''— X +- (G.) 36:^;H24:?;-|— (8.) 16:?;'-56a; + — (10.) 4«V + 4«^^+- To complete the square of expressions of the form of a;2 + 2/xc. EVOLUTIOIS^ 117 154. The method of finding the square root of numbers is derived from the methods of Arts. 148 and 149. (See V^enable's Arithmetic — Square Eoot.) The square root of 100 is 10 ; the square root of 10000 is 100 ; the square root of 1000000 is 1000, and so on. Hence, it follows that the square root of any number between 1 and 100, lies between 1 and 10, that is, the square root of any number haying one or hvo figures is a number of one figure ; ISO, also, the square root of any number between 100 and 1000, that is, having three or four figures, lies between 10 and 100, that is, is a number of hvo figures, and so on. Hence, if we set a dot over every other figure of any given square number, leginning ivitli the units figure, the number of dots will exactly indicate the number of figures in its square root. Thus, for example, the square roots of 256 and 4096 consist of two figures each, and the square roots of 16384 and 6il524, of three figures each. 155. Find the square root of 3249. 9^00 Set the dots accordinsr to the rule. The ^^^ ZZ^TT^ . , . , . X . T ^ 100 + 7 749 root must consist oi two figures. Let ^aq a-^-h denote the root, where a is the value of the figure in the tens place, and h of the figure in the units place. Then a must be the greatest multiple of ten, whose square is less than 3200, that is, a must be the square root of the greatest exact square contained in 3200. Now, as 25 is the greatest square in 32, 2500 must be the greatest in 3200 ; hence, a is 50. Subtract a^ — that is, the square of 50 — from the given number, and the remainder is 749. Di- vide the remainder by 2a — that is, by K'O — and the quotient is 7, wiiicli is the value of h. Then {2a-{-b)b — that is, (100 + 7)7, or 107x7 = 749—18 the number to be subtracted ; and as there is no remainder, we conclude that 50 + 7, or 57, is the required square root. If the number be such that its root consists of three places of figures, let a represent the 118 ELEME:^?TARY ALGEBRA. value of the liuDdreds figure, and h of the tens fignie; then having obtained a and h as before, let the hundreds and tena together be a new yalne of a, and then as before find a ne«»' yalue of h for the units. Example. 186624 (400 + 30 + 2 160000 800 + 30 =830) 26624 24900 800 + 60 + 2=862) 1724 1724 Here the number of dots is three, and therefore the num- ber of figures in the root will be three. ISTow the greatest square-number contained in 18, the first period (as it is called), is 16, and the number evidently lies between 160000 and 250000, that is, between the squares of 400 and 500. We take therefore 400 for the first term in the root, and proceeding just as before, we obtain the whole root, 400+30 + 2=432. 186624(432 16 The ciphers are usually omitted in practice, and it 83)366 will be seen that we need only, at any step, take down __ the next period, instead of the whole remainder. 862)1724 1724 156. Kule for finding the square root of any given number : Set a dot over every other figure, beginning Ex. 1. with that in the units' place, and thus divide 3249(57 the whole number into periods. Find the 25 greatest number whose square is contained in i07) 749 the first period ; this is the first figure in the 749 root; subtract its square from the first period, and to the remainder bring down the next period. Divide this quantity, omitting the last figure, by twice the part of the root Rule for finding the square root of any given number. EVOLUTION. 119 already found, and. annex the residt to the root and also to the divisor ; then midtiply the divisor as it noio stands by the part of the root last obtained for the subtrahend. If there be more periods to be brought doivn, the operatio7i must be re- peated. Ex. 2. In Ex. 2, notice (i) that the second remainder, 49, 77841 ( 279 i^ greater than the divisor 47 ; this may sometimes 4 happen, but no difficulty can arise from it, as it 47)378 would be found that, if instead of 7 we took 8 ^^^ for the second figure, the subtrahend would be 549)4941 384, which is too large. And (ii) that the last figure, 7, of the first divisor, being doubled in order Ex. 3. to make the second divisor, and thus becoming 14, 10291264 (3208 ^^^^^es 1 to be added to the preceding figure, 4, 9 which now becomes 5. In fact, the first divisor 62)129 is 400-i-70, which, when its second term is doubled, 124 becomes 400-f 140, or 540. 6408) 51264 In Ex. 3, we have an instance of a cipher oc- ^^^^^ curring in the root. 157. If the root have any number of decimal places, it is plain (by the rule for the multiplication of decimals) that the square will have twice as many, and therefore the number of decimal places in the root will be half that number. Hence, if the given square number be a decimal, and one of an even number of places, we set, as before, the dot over the units' figure, and then over every other figure on both sides of it. The number of dots on the Uft of the decimal point tvill indicate the number of integers in the root, and the number of dots to the right, the number of decimal places in the root. For example : The square root of 32.49, is one-tenth of the square root of 100x32.49; that is, of 3249. So, also, the square root of '003249 is one thousandth of the square root of 1000000 X -003249, that is, of 3249. If the number have decimal places, how do we proceed? 120 ELEMENTAEY ALGEBPtA. Tlnis 10.201264 would be dotted 10.291264 the dot being first placed on the units-figure 0; and the root will have one integral and three decimal places, that is, would be (Ex. 3 above) 3.208. If, however, the given number be a decimal of an odd number of places, or if in any case of finding the square root there be a remainder, then there is no exact square root ; but we may approximate to it as far as we please, by dotting, as before (rememhering ahvays to set the dot first over the U7iits figure), and then annexing ciphers (which by the nature of decimals will not alter the value of the number itself), and taking them down as they are wanted until we have got as many decimal places in the root as we desire. Ex. Eind the square root of 2 and of 259.351, to three decimal places. 2 (1.414 &c. 1 24)100 96 259.3510 (16.104 &c, 1 26)159 156 281)400 281 321)335 321 2824)11900 11296 32204)141000 128816 Examples — 30. Eind the square roots 1. Of 177241, 120409, 4816.36, 543169, 1094116, 18671041. 2. Of 4334724, 437.6464, 1022121, 408.8484, 16803.9369. 3. Extract to five figures the square roots of 2.5, 2000, .3, .03, 111, .00111, .004, .005. evolutions". 121 158. To find the cithe root of a polynomial expression: AYe know that the cube root of a^ -\-da'h-\-'dah'' -\-lf, is a + ^; and we shall be led to a general rule for the extraction of the cube root of any polynomial by observing the manner in which a^-h may be derived from a^ + ^a^h + dalf + ^^ Arrange the terms according to the a^ + Za^h-\-Zab'^ + 1}^ {a-\-b dimensions of one letter, <?-; then tlie first a^ term is a^, and its cube root is a, which 3^2 \ 3^2^ ^ g^^a ^ ^s is tlie first term of the required root, ^d^b-\-^aJ? -^-h^ Subtract its cube, that is, a\ from tlie ^ whole expression, and bring down the remainder, 3a^& + 3a5^ + 6". Divide ^a^b by 3a^, and the quotient is Z>, which is the other term of the required root ; then form ( 3<x'"* + ^ab + b"^) 6, (i. e.) '6a^b + ^ab"^ + ^^, and sub- tract it from the remainder, and the whole cube of a + 6 has been sub- tracted. This finishes the operation in the present case. If any quantity be left, proceed with a + b a^ a new a-^ its cube, that is, c^' + 3a'^5 + 3<3!&"-^ + 6^, has already been subtracted from the proposed expression, so we should divide the remainder hj Z{a + bY for a new term in the root; and so on. That the rule may be thus extended will be obvious from comparing the form of the cubes 0^ a-\-h-\-c, a + 5-f c + ^, &c., with that oi a + b, from which the rule was deduced. For, {a + b + cY={a + bf + ^{a + bYc + ^(a^b)c'' + c\ ==d' + {W + ^ah + b'')b+\^{a + bf + ^{a^-b)c-^c']c, Similarly, {a-^b + c-vdy=d'-v{Za''-^%ab + b'')b+[^{a + bY + ^{a + b)c-^c''\c ■¥\^{a + b + cf-\-Z{a + b + c)d+d'^d; and so on. Pursuing the same course as above in any other case, if there be no remainder, we conclude that we have obtained the exact cube root. ^x^ + 12x''y + Q,xy'^+y^{2x + y Here the quantity corresponding to ^x^ the trial-divisor ^a? is 3 ( 2x f— 12a;\ that X^x^) 12x'^y + 6xy^+y'^ to Sd'b is 12^'^y, that to Sab"" is 6xy'', and 12x^y + 6xy'^-]-y^ that to b^ is y^ ; so that the w^hole sub- ■ trahend is 12x^y + Qxy^ + y^. To find the cube root of a polynomial expression. 6 ELEMENTARY ALGEBRA. By attending, however, to the following hint, the subtrahend mav be more easily constructed. da -f- b 8a' {Sa'}-b)b dai + dab + W Sa'b + Sab'' + b^ Wb + Sab'' + W Set down first 8a, some little way to the left of the first remainder, and then, multiplying this by a, obtain Sa'^ as before ; by means of this trial-divisor find Z>, and annex it to the 8a, so making 8a + i ; multiply this by Z), and set the product {Sa-\-b)b or Sab-\-P under the 3a'^, and add them up, making Sa' + 'Sab + b'' ; then, multiplying this by 5, we have da''b-{-Sab''-{-b'\ the quantity required. The value of the above method, in saving labor, will be more fully seen when the root has more than two terms, or, if numerical, more than two figures.. Ex. 8^H '^2x'y + 6xy' -hy^2x + y 6x + y 12^;' -\-fjxy + y'' 12X'''}' Qxy^y'' nx'y + Qxy'' + y-' 12x''y + Gxy' + y^ Examples — 31. Find the cube roots 1. Ofx' + Cx'y + 12xtf + Stj\ 2.0ia'-9a'+27a-27. 3. Of a;^ + lXV+48a3+64. 4. Of Sa'-36a'b + 64.ab'-27b\ 5. Ofa'+24:a'b + ld2aI)' + 612h\ 6. Of Sx' - Ux\j + 2^4.xtf - 343^^ 7. Of 771' -12m'nx + 4.SmnV - 64:7i'x\ 8. Of aV-loa'bx' + 75abV -126Fx\ 9. Oi a' + (^a'-\-lDa'-{-20a' + 15a' + 6a + l. 10. Ofa;"-12:?;^ + 54:r*-112a;^ + 108.'r''-48:r + 8 11. Of a'- da'b + 6a'b' - 7a'b' + ^a'b' - Sab' + b\ 12. Ofa'-b' + c'-3{a'b-a'c-ab''-ac'-b'c + bc')-(jabc. EYOLUTIOJS^ 123 159. The method of finding the cube root of an algebraic expression suggests a method for the extraction of the cub^ root of any number. The cube root of 1000 is 10 ; the 'cube root of 1000000 is 100, and so on ; -hence, it follows that the cube root of a number less than 1000 must consist of only one figure; the cube root of a number between 1000 and 1000000, of two places of figures, and so on. If, then, a point be placed over every third figure in any number, beginning with the figure in the units^ place, the number of points will show the number of figures in the cube root. Thus, for example, the cube root of 405224 consists of two figures, and the cube root of 12812904 consists of three figures. Suppose the cube root of 274625 required. 180 + 5 10800 274625(60+5 925 216000 11725 58625 58625 Point the number according to the rule ; thus it appears that the root must consist of two places of figures. Let a-^-h denote the root, where a is the value of the figure in the tens' place, and h of that in the units' place. Then a must be the greatest multiple of ten which has its cube less than 274000 ; this is found to be 60. Place the cube of 60, that is 216000, in the third column under the given number and subtract. Place three times 60, that is 180, in the first column, and three times the square of 60, that is 10800, in the second column. Divide the remainder in the third column by the number in the second column, that is, divide 58625 by 10800 ; we thus obtain 5, which is the value of h. Add 5 to the first column, and multiply To find the cube root of any number. 124 ELEME^^TARY ALGEBRA. the sum thus formed by 5^ that is, multiply 185 by 5 ; we thus obtain 925, which we place in the second column and add to the number already there. Thus we obtain 11725 ; multiply this by. 5, place the product in the third column, and subtract. The remainder is zero, and therefore 65 is the required cube root. The ciphers may be omitted for brevity, and the process will stand thus : 185 108 925 11725 274625(65 216 58625 58625 It will be seen by the following example, where the root has more than two figures, how the numerical process cor- responds to the algebraical. The ciphers are omitted, ex- cept that in the numbers corresponding to 3a^, 3a' ^ &c., it is better to express two at the end: thus 'a is really 4000, and therefore oa"^ is 48000000 ; but, as in the first remainder, we only need the figures of the first and second periods, cor- responding to 43 in the root, we may treat the a as 40, and thus So" will be 4800, and 3a will be 120, so that 3a + h will become 123. Ex. 80677568161 (4; 321 64 3^4-^ = 123 3a'^=4800 16677 a' = 43 {3a-i-b)b= 369 a"=432 3a'-i-3ab+F = ^169 15507 3a' + d=1292 3a'^ = 554700 1170568 {3a'-\-b)b= 2584 da" + 3a'b-{-I)'=657284. 1114568 12961 55987200 56000161 12961 56000161 56000161 EYOLUTIOK. . 125 XoTE. — Our trial-divisors may frequently give figures too large for the next figure of the root. In such case try the next less figure, and if necessary, the next less, until we get the right one. 100. If the root have any number of decimal places, it is plain by the rule for the multiplication of decimals, that the cube will have thrice as many ; and therefore the number of decimal places in every cube decimal will be necessarily a midti]jle of three, and the number of decimal places in the root will be a third of that number. Hence, if the given cube number be a decimal, and consequently have its num- ber of decimal places a multiple of three, by setting as be- fore the dot upon the units-figure, and then over every third figure on loth sides of it, the number of dots to the left will still indicate the number of integral figures in the root, and the number of dots to the right the number of decimal places. If the given number be not a perfect cube, we may dot as before (always setting the dot first upon the units figure), and annex ciphers as in the case of the square root, so as to ap- proximate to the cube root required, to as many decimal places as we please. Example. Extract the cube root of 14102.327296. 641 1200 14102.327296(24.16 »\ 256' 8 721' 2/ 1456 6102 16, 5824 7236 172800 278327 721' 173521 173521 104806296 1 104806296 1742430 4341 6 1746 771 6 126 ELEMEKTARY ALGEBKA. Note. — A careful examination of the two columns of figures on the left will disclose a much abbreviated process of finding the divisors. In the left-hand column, adding to 64, 721, etc., twice the units-figure gives the same result as multiplying the root already found by 3. In the second column, adding the three numbers enclosed by a brace {i.e.^ the last true divisor, the number above it, and the square of the last root figure), and annexing two ciphers, gives the next trial-divisor. The examples and explanations above furnish us the follow- ing rule, given also in the Arithmetic : I. Place a dot over the U7iits-figu're of the nuwher, and over every third figure to the left, and also to the right tuhen the number contains decimals {altvays tahing care in this latter case to mahe the number of decimal figures a multiple of 3). II. Find the greatest cube in the nuniber which forms the first period on the left, and place its root after the manner of a quotient in division. This root is the first figure of the re- quired root. Subtract its cube from the first period, and to the remainder bring doivn the figures of the second period for a First Dividend. III. Multiply the square of this first figure by 3, annex two ciphers, and find hoio often this Trial-Divisor ^6' con- tained in the first dividend ; place the quotient as the second (trial) figure of the root. Then to three tiines the first figure of the root annex this second figure, and multiply^ the result by the second figure ; add the product to the Trial-Divisor, and call the sum the First Divisor. IV. Multip)ly the First Divisor by the second figure of the root ; if the product be greater than the First Dividend, use a lower figure for the second figure of the root, and thus repeat the process III. until the product be less than the First Divi- dend ; subtract this product from this dividend, arid to the remainder bring doiun the figures of the third period for a Second Dividend. Rule for extracting the cube root of any number. EYOLUTIOISr. l:i? V. Mulfiphf the square of the tioo figures of the root hy 3; annex tc^o cipher ^^, and proceed as in III. and lY. Proceed in this manner until all the 'periods have heen drought dotv7i. iNOTii. — In extracting either the square or cube root of any number, w2ieo a certain number of figures in the root have been obtained by the iommoii rcle, tiiat number ma?/ be nearly doubled by dicidoii only. 1. In the e.xtraciion of the square root, when n + 1 figures are found in the root, n more may be found by merely dividing the last remain- der by the trial-divisor. For, let N be tlie number whose square root * is to be found, conoi&ting of 2/i + 1 figures. Let a= the part ai/eady found (consisting of n + 1 figures, and n ci- phers after them, that i3, altogether of 2/2. + ! figures). Let x-= required rema.uh\g part of tlie root, consisting of n figures. So that V j.Y~—a + x\ Then i7 ^ vj'-' + 2ax + x" ; J^'—a"^ x^ ?.'.i 2a Now J^^—a^ is the remainder, after n + 1 figures of the root are found, x^ and 2a the trial-divisor; if, then, we can show that — is ^proper frac- fja tion^ it will follow that the integer obtained by dividing N—a^ by 2a will be X., the remaining part of the root. But as X contains n figures, it must be <10'^, which has n + 1 figures, and x^ <W-'^ '^ and since a contains 2?i + 1 figures, it cannot be <W'^ (which is the smallest number of 2n + l figures). rj^ 10-" 1 Hence, ,r-< ^ ^^, <^, and is therefore a proper fraction. That is, if 2a 2.10-'^ 2 ' N—d^ the quotient of —z be taken for the n remaining figures of the root, the siim is less than 1. 2. In the extraction of the cube root, when n + 2 figures are found in tlie root, n more may be found by dividing the last remainder by the trial-divisor. For let N=^ the number ; a= the part of root found (consisting of n + 2 figures followed by n ciphers, that is, of 2n + 2 figures altogether) ; a'= the required part of root (consisting of n figures). . Then N=a^ + Za^x + ^ax^ + x\ and -~--^—=.v, + — + -^ ; and here da^ a ?ia^ 128 ELE3IE:N^TAIiy ALGEBEA. 2;<10«; and a^ since it contains 2n + 2 figures, cannot be <10-"+^-, — + ^r-T, <1. That is, if tlie quotient of -tt-^t- t)e taken for the n re« a Sa^ Sa^ maining figures of the root, the error is less than 1. Now iV^—f^^-*— re- mainder after n-\-2 figures of root are found, and da^ is the Trial- Di- visor for the next figure. Hence the rule as above. * Examples — 32. Find the cube roots of 1. 9261, 12167, 15625, 32768, 103.823, 110592, 262144, 884.736. 2. 1481544, 1601.613, 1953125, 1259712, 2.803221, 7077888. 3. Extract to 4 figures the cube roots of 2.5, .2, .01, 4. XXII. Simple Equations. 161. The statement of the equality of two algebraical quantities which differ only in form, is called an " Identity, ^^ An Identity is true for any value whatever of the letters which enter it. Thus, 2.'c + 5a;=7a;; 2{a-{-x)='Za + 2x', {x-\-ay=x^-\-2ax + a'^] {x + a) (a;— a) = 0;^ — a', are Identities. Up to this point we have been using Identities — especially to express general facts — by means of letters. Our formulas heretofore given are Identities. 16.2. An equation, however, is the statement of the equality of two cliff event algebraical expressions; in which case the equality does not exist for all values, but only for some par- ticular values, of one or more of the letters contained in it. Thus the equation x—b=i, will be found true only when An Identity. An Equation. SIMPLE EQUATIONS. l^/Q we give x the value 9; and x'^=z3x—2, true only when we give X the value 1 or 2. In equations, the question always is, what value of the letter or letters not already known will verify or satisfy, (i. e.) make true, the expressed equality. The finding of such value or values is called solving. the equatio7i. 163. The Iwo expressions connected by the symbol = are called sides of the equation, or memlers of the equation. • The expression to the left of the sign of equality, is called the first side; and the expression to the right is called the second side. 164. Those quantities to which particular values are to be given in order to satisfy the equation, are called the unhnoiun quantities. The last letters of the alphabet, x, y, z, &c., are usually employed to denote these quantities. 165. An equation is said to be satisfied by any value of the unknown quantity which makes the values of the two sides of the equation the same, {i. e.) which makes the Equation an Identity. This includes the case where all the terms of an equation lie on one side and on the other, as in x^—^x + '^ — Q, which is satisfied by 1 or 2, either of which being put for x makes the first side also 0. Those values of the unknown quantities by which the equation is satisfied, are called the roots of the equation. Thus, '7 is the root of x—d=z4:; 1 and 2 are the roots ofx'-dx + 'Z^O. 168. An equation of one unknown quantity, when cleared of surds and fractions, is said to be of as many dimensions as there are units in the index of the highest power of the un- known quantity. 1^0 Thus, cc— 5=4 is an equation of one dimension, or, of Solving the Equation. Sides or Members of the Equation. Unknown Qua?: titles. Satisfying an Equation. 130 ELEMEKTARY ALGEBRA. the first degree, or a simple equation; x^ = 'dx—^, is of tiuo dimensions, or, of the second degree, or a quad- ratic equation; x^ — h^=.^oi^ is of three dimensions, or of the tliird degree, or a cuhic equation; x^ — ^x^=.V6, is of four dimensions, or the fourth degree, or a hiquadratic equation, &c., &c. 167. In the present chapter we shall show how to solye simple equations. We have first to indicate some operations which we may perform on any equation without destroying the equality which it expresses. 168. If every term of each side of an equation he multiplied ly the same quantity, the ttvo sides ivill still he equal. For, if equals he multiplied by the same quantity, the results are equal. This principle is chiefly used for clearing an equation of fractions, if they stand in the way of solving it. Thus, taking the equation ^x—^ — -^, multiplying every o term by 3, the denominator of the fractional term, we have 21:?; — 18=3X-^, or 21:^^—18 = 5^, in which no fraction ap- o pears. An equation of several fractional terms may be cleared of fractions by multiplying every term by any common multiple of all the denominators. If the L.o.M.of the denominators be employed, the equation will be expressed in its simplest terms. Take, for example, — + — + — = 9. o 4 D Multiply every term by 3 X 4 X 6, or, 72 ; thus, 72a; 72a; 72a; '^^ that is, 24a; + 18a:+12a:=648, cleared effractions. RootR of an Equation. Difi'firont kinds of Equations. SIMPLE EQUATION'S. 131 Instead of multiplying every term by 72, we may multiply eyery term by 12, the l.c.m. of 3, 4, and 6. We would \2x VHx 1.2x thus liaye — — H — -- + —- = 108; that is, 4tx + dx+2x=108, O ~c K) expressed in simple terms. -^ ^T ,. ,. 6x + 4: 7x + 5 28 x—1 „^ hx. Clear the equation — ttt"— ~^~~o~"^ ^^ ™^'" /C 10 o /^ tions. The L.C.M. of the denominators is 10. Multiply by 10. Thus, 6{6x + 4:)-{7x + 5) = b6-6{x-l); that is, 26x-{-20^'7x-6=z66-6x + 6. The beginner should write the operations out in full, as above, using brackets, in order that he may attend to the . . . 7.T + 5 , x—1 signs 01 such expressions as — r— , and — . 10 Zi 169. Any term may he transferred from one side of a7i equation to the otlter side ivitliout destroying the equality, j^ro- vided 2ve change the sign of the term. This transference is called transposing. Suppose, for example, x—a — h—y. Add a to each side (which of course will not destroy the equality) ; then, x—a-\-a—'b—y-^a\ that is, x='b—y-\-a. NoAV subtract h from each side ; thus, x—'b—lj^a—y—'b\ that is, x—t^za—y. Here we see that —a has been removed from one side of the equation, and appears as +a on the other side; and -\-'b has been removed from one side and appears as — Z> on the other side. Transposition of Terms. 132 eleme:n^tary algebra. 170. If the sign of every term of an equation he changed the equality still holds. This follows from Art. 169, by transposing every term. Thus, suppose, for example, that x—a=t)—y. By transposition, y—h—a—x'^ that is, a—x=:y—ib. And this result is what we shall obtain if we change the sign of every term in the original equation. It is also clear that if the same quantity occur with the same sign on both sides of the equation, it may he erased from hoth sides. For, by the erasure w^e either subtract equals from equals or add equals to equals, according as the sign of the term erased is + or —. 171. Every term of each side of an equation may he divided hy the same quantity luithout destroying the equality expressed hy it. For, if equals be divided by the same qua;ntity, the quotients are equal. Thus, the equation 12a; + 5.^=1 3 6, or 17a? = 13 6, gives 17^_136 _1^_Q 17 — 1 7 ' ^^ '^ — 17 — .. .n -, ct^ ^ 'b Ai^o,\iax=o. — = — , or x=:—, • a a a Again: if 24:?;+18:?; + 12:^: = 648 be divided by 6, we get 4a;+32: + 2a-108, or 92;=108. 9ix 108 108 ,^ Hence, ~a—~K~^ ^"^ x——=V^, ^ then Again: M ax-\-hx—cx^^d^(dY (a-\-h--c)x—d*y {a-\'h—c)x d _ ^ a-\-h—c a + h — c^ a-\-h- Chaflge of the signs of an equation. Division of the terms of an equation. SIMPLE EQUATIOXS. 133 172. The operations indicated in the preceding articles may be performed upon equations of any degree, and con- taining any number of unknowns ; for they depend on principles true of every equation. Of course all these opera- tions can be performed on Identities, or identical equations, as they are sometimes called. 173. To solve a simple equation of one unknown quantity : Eule. Clear the equation of fractions, if necessary. Collect all the terms involving the unknown quantity on one side of the equation and the known quantities on the other side, tra^is- posing them, when necessary, with change of sign. Add to- gether the terms of each side, and divide both sides dy the coef- ficient or sum of the coefficients of the unknoiun quantity ; and thus the root required will he found. Note I. — Erase terms by Art. 170, or simpUfy the equation by di- vision, Art. 171, at any stage of the process. Note II. — It is usual to collect all the unknown quantities on the first side, and the known on the second side of the equation. 174. We shall now give some examples. Ex. 1. Solve 7:?; + 25 = 35 + 5.T. Here there are no fractions; by transposing we have 7:2;-5:z:=35-25; that is, 2<c=10; dividing by 2, x=—- = o. We may verify this result by putting 5 for x in the' original equation ; then each side is equal to 60. Ex. 2. 4.x + b = 10x-U. Here Wx-4.x=b-\-l(j; ,\ Qx=2l, ^ndi x^zz^ z=z^=^, Ex. 3. b{x-\-l)-2 = ^{x-b). Tlule for solving a Simple Equation. Note I. Note II. 134 ELEMENTARY ALGEBRA. Here^ remoying the brackets, 6x+6—2 = dx—16; .-. 5x—3x=:—16—6-i-2, or 2^=— 18, and /. ^=— 9. Ex. 4. Solye4:{3x--2)-2{4.x-3)-3{4:-x)=0. Performing the multiplications indicated, 12x-8-{Sx-6)-{12-3x)=0. Eemoving the brackets, 12a;-,8— 82: + 6— 12 + 3^=0; collecting the terms, 7a;— 14=0; transposing, 7^== 14 ; 14 dividing by 7, x=—=2. The student will find it a useful exercise to yerify the cor- rectness of his solutions. Thus, in the aboye example, if we put 2 for X in the original equation, we shall obtain 16 — 10—6, that is 0, as it should be. Ex. 5. dx-^2x—a=3x-{-2c. Transposing, I)X + 2x—dx=a + 2c; or, dx—x=a + 2c; collecting the coefficients, (^—1) x=a + 2c; a -{-2c dividing by Z> — 1, b-1 Examples — 33. 1. 4.x-2 = 3x-{-3. 2. 3^ + 7=:9a;-5. 3. 4.x-\-9=zSx-3, 4. 3-{-2x=7—6x. 5. x—7 + 16x. 6. 7nx-{-a = nx-{-d. 7. 3{:?:-2)+4=4(3-.t). 8. 5-3 (4-^-) +4 (3-22:)r=0. -J 9. 13:i;-21 (a;-3) = 10-21 (3-a;). 10. 6{a + x)-2x = 3{a-5x), SIMPLE EQUATIOXS. ' 135 11. 3{x-3)-2{x-2)+x-l:=x-{-3 + 2{x + 2) + 3{x+l). 12. 2.i;-l-2 (3:?;"- 2) +3 (4a;-3)-4 (5:?;-4):=0. 13. {2-{-x){a-3) = -4.-2ax. 14. {7)1 +n) {m—x)=7n {?i—x). 15. 5.'?;-[8a:-3{16-6a;-(4-5^)}]=:6. 175. The following examples will illustrate the solution of equations containing fractions. Ex.1. If |-|:- 1=^-3, find:.. Multiplying by 2 X 3, or 6, 3:?;-10a;-8 =8x-lS; transposing, 3.^—10:^—82:=: 8 —18; combining, — 15a;=— 10; dividing by —15, x= — — =— . — J o Ex. 2. ix-^x+lx=ll+^x. Here we first clear the equation of fractions, by multiply ing every term by 24, the l.c.m. of the denominators, and (observing that in the first fraction ^^ = 12, in the second ^^ =8, and so in the others) thus we get 12:?;— 8x2:^:+6x3a; = 264 + 3a:, or 12x-iex-\-lSx=264:-{-dx; .-. 12.i;-16r^ + 18.T-3.T=264; .-. lla;=:264, and ^^.-^^V^^^- Ex. 3. If x + —^:=:12-—,:p^, find X. To clear of fractions, multiply by 2x3, or 6, and we have 62^ + 3 (32:-5)=.72-2 (2x-4) ; or 62;+(9a;-15) = 72-(42:-8); .-. C):j^ + 9.7:-15 = 72-4.t4-8. 136 ELEMEKTARY ALGEBRA. Transposing, 6x-{-9x + 4:X=l!2-{-8-{-16; combining, 192;= 95; 95 dividing, 0:=— =5. Ex. 4. Solve h6x+3)-^{16-5x)=31!-4.x. This is the same as 6x^-3 16-5x ^^ , —- ^ — =37-4:X. Multiplying by 21, 7 {6x + 3)-'d (16-52;) = 21 (37-42;) that is, 352; + 21-48 + 152;=r777-842;; transposing, 352^ + 152; + 842;= 777 -21 + 48; that is, 1342; = 804; therefore, ^= ttt: = 6. 134 Ex.5. i{x-\-l)+i{x-^2)=16-i{x + 3). Multiplying by 12, we have 6(2; + l)+4(2; + 2) = 192-3(2;+3), or 62;+ 6 + 42; + 8 =192 — 32;— 9; .•.62;+42; + 32;=192-9-6-8; .•.132;=169, and x=\%^z=:13. Examples— 34. 1. ^2;+-j2;=2;— 7. 2. ix—^x=lx—l. o 2r (T 2 5. y+-^=a;-4 6. i(9-3a;)=|-TV(7a.- 18). 7. x+\{U-x)=^{%l-x). 8; 2a;-^=|(3-2a;)+ia;. 9. ^(2a:+7)-yT(9.'P-8)=i(a;-ll). SIMPLE equatio:n"S. 137 ,^ x-a 2x—3b a—x ^ ^^ 6x—7 2x+'7 .. -, a;-2 «— 3 „ ,„ x+3 x+4: x+5 ,„ 13. ^_i_ -^+^=0. 13. _ + _+_=:16. 14. !^=7 + .-^-i^i. 15. ^i-^i.£^. 3a;-5 5a;-3 „„ „ ,„ 7:r-4 „„ 4-7a; 7 12' IC. -3 ^+2|=0. 17. ^^- + 3| + -^=a;-^. ,„ 3— a; 3— a; 4—2; 5— a; 3 18.— + — + —+— + ^=0. ,„ 5a;-3 9-a; 5x 19, XXIII. - Simple Equations CoNTrNUED. 176. We shall now give some examples which are a little more difficult than those in the preceding chapter. In many of these examples the common multiple of all the denomina- tors is too large to be conveniently employed. In such a case we may see whether two or three of the denominators have a simple common multiple, and get rid of these fractions first, observing to collect the terms and simplify as much as pos- sible after each step. ^ ^ 2.^ + 3 :i;— 12 32:4-1 ^, . 4:X + 3 ^-^- ^- ^T 3-+-^=^*+-]^- Here the l.c.m. of all the denominators would be 132 ; but as 12 will include three of them, multiplying by it (having iirst changed 5^ to ^3^), we get ^^?-^tl)-4(a;-12) + 3(3a; + l) = 64 + 4a;+3; .\\^{2x-^d)-4cX+4:8 + 9x + 3 = 64:+4.X'\-3; When the Common Multiple of all the denominators is too large for coa venience, what course may you pursue? 138 ELEMENTARY ALGEBRA. hence, collecting terms and simplifying, we liaye m2x+d)-4.x-{-9x-4:X=64.-\-3-4:8-3, ovii{2x + 3)+x=:16; .-. 12 (2:2;+3) +11^=176, or 24^; + 112;= 176 -36; /. o6x=:^14:0, and 2;=V¥=^- 177. It will often happen that the icnlcnoion quantity is found in the denominator of one or more of the fractions. x^ooi 24352 Ex.2. Solye —+—=_-] _. X X x X 11 Since the first four fractions haye a common denominator, by addition, _=,___; 8 6 2 transposmg, ^~"^""l7' 2 2 combining, '^=17' .-. a; =17. Multiplying by 3^;, 9 — 2 = 5 + :?; ; :. x=2. If any of the denominators which contain the unknown quantity consists of two or more terms, it will generally be adyisable to follow the method of Art. 176, and clear the equation of the simplest denominators first, leaying tho others to be dealt with afterward, when, by transposing, collecting the terms, &c., the equation has been reduced to feiver terms. Or, if all the denominators consist of two or more terms, then they may be cleared off singly, one hy cne, till all haye disappeared. How may you proceed when the unknown quantity is found in the denora Inator? When any of the denominators coneists of more than two terms? Ex. 4. Solve SIMPLE EQUATIONS. 139 6x-\-13 3.T+5 2x 15 5:^;— 25 5 Multiply by 15 to clear away the simjyie denominators first, and we have ., . ^^ 15(3:^4-5) y, and transposing, 13= \ ^^ ? erasing, ana transposing, 16= k _<^- or, dividing numerator and denominator of the fraction by 5, ^^^ 3(3a; + 5) ^ x—b Multiplying by a;— 5, 13:?;— 65 = 9^; + 15 ; transposing, 13^;— 9^=65 + 15 ; combining, 4^ = 80 ; 80 dividing, x=——20, Ex.5. Solve— g ___^_-_. To remove first the denominators 18 and 9, multiply the whole by 18, and we have 10^_,17_-j^_^.10:.-8; erasing, and transposing, 216r.+36. , . . ^^ 2162: + 36 combining, 2o=-— r— ; Wx — o multiplying by 1Lt-8, 25 (ll:r— 8)=216:?: + 36; or, 275:^:-200=:216a; + 36; transposing, 275a;— 2l6x==200 + 36; combining, 5 9a; rr: 23 6 ; dividing, 0;:=— ^=4. 140 ELEMENTARY ALGEBRA. -^ ^ ^ , 2x-hS ^x + 6 3x + d JbiX. 6. 'bolve — r^^z 7 + ?; T. x+1 4.X + 4: Sx + l Here it is convenient to multiply by ^x-\-4:, that is bv 4(0^+1); . /^ ox . . . 4 (i?; 4- 1)3(^+1) thus 4(2ic + 3) = 4a;+5+-^-— -^^--^--^; 12 (a^ + l)' ^ therefore, 8a; + 12— 4^— 5 = that is^ 42^+7= 12 {x-\-iy dx-\-l Multiplying by 3a; + l, (3:2: + l) (42^+7) = 12 (a:+l)'; that is, 12:c'+25^ + 7=12a;'+24x+12. Here l^x^ is found on both sides of the equation. Ee- move it by subtraction, and the equation becomes a sim2)le equation; that is, 25^ + 7=24:^;-fl2; or, 252J— 24^=12-7; Ex. 7. Solve x—1 x—2 x—4: x—5 x—2 x—3 x—5 x—6' Eeducing the terms on the first side to a common denom- . ^ ^ x^~Ax + 3-(x'-4:X + 4.) 1 mator, we get ^, — , or — -, -^, — -^, " {x—2) [x—d) {x^2) (x—d) Eeducing the terms of the second side to a common de- nominator, we have for this side .^'-rl0:c + 24-(.'r'-10:r+25) 1 (x-b)(x-Q>) ' (x-b){x-Qy Thus the proposed equation becomes 1 1^ ^ "" {x-2) {x-^)" (x-5) (.T-6) ' SIMPLE EQUATIONS. 141 1 1 Changing the Signs, l^_^)j^_^^-^',j(^^Ze)^ clearing of fractions, (x— 5) (x—Q) = {x—2) 0^—3) ; that is, x^--llx + 30=x''—6x + 6; whence, — lice + 5a; =6— 30; that is, — 6.T = — 24 ; therefore, a: =4. A5x-,76 1.2 .dx-.6 Ex. 8. Solve .5^;+- .6 ~ .2 .9 To insure accuracy, it is advisable to express all the deci- mals as common fractions ; thus 6x 10/45^ __75^\_10 l^_10/3^_j6\ Simplifying, | +|(|-|) -=6- (-|-|-) ; ,, , . a; 3cc 5 ^ a; 2 that IS, _ + _^_:.6--+-. Multiplying by 12, 6x+9x-16=:'72-4:X + S ; transposing, 19a;=72+8 + 15 = 95; therefore, x=—=5. 178. Complex fractions in an equation should first be re- duced to simple ones, by the known rules. T, ^ 25-4:?; 16.^ + 44 . 23 ^"^•'- -^+-3-^+?=^+^T Hei e, first simplifying the complex fractions, we get ' 75-x 80:r + 21 _ _23_^ 3(a;+l)"^5(32: + 2)~ ^^+1' Complex fractions in an equation. 142 ELEMEKTARY ALGEBRxV. ,^. , . ^ ,. 375-5.-?; 240:^+63 ^. , 345 then, multiplying by lo, —— + -———— = 7o + x + 1 ^ 32^+2 ' x + 1' whence, multiplying hj x+1, o^. r. . 2402;'+303.T+63 ,,^ ^. . o... 875— 5:^;^ ^-— -^ =75iz; + 75+345; OX -J- /i or, simplifying, J— — r = 752;+5aj + 75+345— 375=80:^+45: 3a;+2 /. 240^;' + 303:^ + 63 = 240;^'+ 295a; + 90, and 8a;:=27, or x=3%. 179. We will now solve three more equations, in whi^'^ letters are used to represent known quantities. Ex. 10. Solye - + ^=^. a Multiplying by ^Z>, Ix-^-ax—dbc^ that is, {a-^'b)x=dbc\ dbc ,\ x=-—-. a-\-o Ex. 11. Solve a— ^ — yd —X, a Multiplying by ab, a^ {x—d)-\- If (x—l) = abx ; that is, a'^x—a^-\-l'^x—lf=:abx] transposing and collecting, a^x-\-lfx—abx—a^ + V ; that is, {d^—ah-\-¥^ x^cc" -\-W \ dividing by c^-\-db-\- Z>^ x = ,_ ^ ; .•. x=a+b, -n -.n CI 1 ^— ^ (2x~ay Ex. 12. Solve 7=77^ r(?. x—o {2x—o) Clearing of fractions, (x-a) {2x-hy = {x-b) {2x-ay; that is, (x-a) {^x''-ihx-^b')=z{x-b) (4a;'-4«a; + a'), SIMPLE EQUATIONS. J 43 multiplying, we obtain 4.x'-4.x\a^h)^x{4.a'b-\-l)'')-ah'' = 4a;' - 4a;' {a -\-i)-\-x {4.ah + a') - a'Z> ; whence, lfx—ah'^=a^x^a^h\ /. {a'—V^) x—a^l—aV—ab (a—V) ; __ah{a—l)_ ah Examples — 35. 12 _1__29 _iL__££_ 3 l^B _ 216 '^"^12a;~24* ' a;-2~a;-3* ' 3a;-4~5a;-6' 2^T3~4^^' ^' 2 ~ 3 "^ 4 6~^'^ ' 1 3 --a;— 3 -a;— 10 . ^_ 2 ,4 , 4:—x 10— a; 5 2 4 6 5a;~8 ^ 3a;-8 . a;-2 , 1 2a;-l ^. ^ ^ a;^+3 ^ 9. _+_ = ^____. 10. a;+l-^^=2. a;-l_7a;-21 7a;-4 7a;-26 * a;-2~7a;-2G' * a;-l ~" x-3 ' X dx 71_ 3a; + l . . 2a;-6 _ 2a;— 5 7 ~ 2 "^ 7 ~ 2 "^ ■^^* 3a;-8~3a;-7' 15. a;--3~(3-a;)(a; + l)=:a;(a;-3)+8. IG. 3-a;-2(a;-l) (a; + 2)^(aj-3) (5-2a;). 17. I+^_i+^=,7^^. 18. (a; + 7)(a:+l) = (a-4-3r. 144 ELEMEKTARY ALGEBRA. 19. ^{2x-10)-^{3x-4.0) = 1d-^{d^-9:). 2x+l ^+12 ~ 21. ^i^{2x-3)-i{dx-2)=i{^x-3)-d^^^. 22. 5(^__9)+_T(^_5)^9(^_7) + l|. 23. ,-V(2^-l)-3J^(3^-2)=^V(^-12)-^VG^+12). 24. ^(^x+20)-^\{3x + 4.)=^^-^{3x + l)-M^9-Sx). ^^ dx-1 4x-2 1 ,,^ 2 1 6 20. rr T — n 7^ = 7r- ^6. 5 + 27. 2:^-1 3:z;-2 6* ' 22;-3^a;-2 3:z; + 2' x—4: x—6 x—7 x—8 x—b x—^ x—d> x—9 28. JL, + ^=^ + ^' x—2 x—l x—1 x—6 29. ^-l±^.- ~=:1. 30. ,6x-2=--.26x+,2x-l. 3 — X 2 — x l-\-x 31. .5:z; + .6a3-.8=:.75.'?; + .25. .135aj-.225 .36 Mx-.IS 32. .15a; + - .6 ~.2 .9 • fl^— a: ^ J+a: ^, x^ — a"" a-x 2x a 33. a~^f =x, 34. —j tt—t • h a ox X 35. x{x—a)-{-x{x — 'b)=2{x—a){x—h). 36. ix-a){.-l) = {.-a-iy. 37. ^-^j=^. 28 -1 ^=-^^. * x—a x—h x^ — ab 39. 40. SIMPLE EQUATIOJS'S. 145 111 x—a x—a-\-c x—b — o x—V m X —a — h mx —a — c nx—c~d nx—h—cV 41. (ci-l) {x-c)-{b-c) {x-a)-{c-a) {x~b)=:0. x—a x + a 2ax 42. a — b a + b a^ — b''' ,_ x—a x—a—X x—b x—b—1 /rj t^ ^ 43. r -= J — ~ r~^. (See Ex. 6, x—a—1 x—a— 2 x—b—1 x—b— 2 ^ Art. 177.) XXIV. Peoblems solved by Simple Equations. 180. We shall now see tlie practical application of the above methods in the solution of many arithmetical problems. In these problems certain quantities are given, and another quantity, Avhich has certain given relations to these, has to be found : the quantity which has to be found is called the unhioivn quantity. The relations between the given quanti- ties and this unknoiun are expressed in the enunciation of the problem in ordinary language, -and these are to be translated into algebraical expressions, to be used in the solution of the problem. The method of solving the problem may be given in general terms, as folloAvs : Put X to represent the unhnoion quantity. Set doivn, in algebraical language, the statements made in the problem, and the relations betiueen the unhnoiun quantity and given quanti- ties derived from these statements, using x tvhenever the un- hnoion quayitity occurs. We shall thus arrive at an equation from luhich the value ofx may be found. Solution of problems by means of Simple Equations. State in general terms the steps to be taken. 7 146 ele:>ii:xtaky algekha. Ex. 1. What number is that, to which if 8 be added, one- fourth of the sum is equal to 29 ? Let X represent the number required. Adding 8 to it, we have x + 8, and one-fourth of this is \ {x 4- 8) ; Ave have, therefore, the equation i(a; + 8)=29; whence it; = 108. Ex. 2. What number is that, the double of which exceeds its half by 6 ? Let a'= the number ; then the double of x is 2.r, and the half of x is \X', hence, 2x—\x=^%\ whence a;=4. Ex. 3. The ages of 3 children together amount to 24 years, and they were born two years apart : what is the age of each ? Here we have Known quantities. 1. The sum of the ages of all three, 24 years. 2. The difference between the ages of any two of them. Unknown and required. 1. Age of youngest. 2. Age of next. 3. Age of the oldest. But, in reality, we have only one unknown quantity to find^ because, when we know the age of one of the children, the ages of the two others immediately follow. So that we say, let x be the age of the youngest ; then X + 2= next ; and « + 4= oldest. Thus far we have expressed, algebraically, 07ie of the two known coftditions of the problem. There still remains to notice, that the sum of the ages is 24 years. Now this sum is ^x + 6, adding together x^ a;+2, and x-\-^; :. 3.-?? + 6=24, an equation from which to find x. Transposing, 3a=24— 6, or 18; dividing, ^="5" » ^^ ^' o .'. the age of the youngest is 6 years, next . . 8 . oldest . . 10 SIMPLE EQUAT]0]S'S. 147 Ex. 4. A cask, wliicli held 270 gallons, was filled with a mixture of brandy, wine, and water. There were 30 gallons of wine in it more than of brandy, and 30 of water more than there were of wine and brandy together. How manv were there of each ? Let ^=: number of gals, of brandy ; .*. x + dO= wine; and2aj + 30= wine and brandy together; ... 2a,' + 30 + 30 or 2x + 60=gals. of water ; but the whole number of gallons was 270 ; .-. a^ + (aj + 30) + (2,2; + 60)=270; whence x= 45, the number of gals, of brandy, 25 + 30= 75, wine, 22^ + 60=150, water. Ex. 5. A sum of £50 is to be divided among A, B,.and C, so that A may have 13 guineas more than B, and C £5 more than A : determine their shares. Tjet x=B^s share in sliilUngs : .'. « + 273=^'s, and .(;r + 273) + 100 or ^ + 373=6''s ; .-., since £50=1000s., {x + 27S) + x + {x + d7S)=Sx + 64.Q=1000; .-. 3^r=354, and 0^=118, :c + 273=391, aj + 373=491, and the shares are 391s., 118s., 491s., or £19 lis., £5 18s., £24 lis., re- spectively. Ex. 6. A, B, C divide among themselves 620 cartridges, A taking 4 to ^'s 3, and 6 to (7's 5 : how many did each take ? Let .Tr=^'s share ; then |a;=jB's, ■|.c=(7's ; .-. aj + |aj+ 5,^=620; whence ir=240, |.t'=180, |-a^=200. We might have avoided fractions by assuming 12x for J.'s share, whea we should have had 9aj=^'s, and 10.2^=: C^ ; .'. 12a^ + 9^ + 10i=620 ; whence a'=20 ; and the shares are 240, 180, 200, as before. 148 ELEMEXTARY xVLGEBRA. Ex. 7. A line is 2 feet 4 inches long ; it is required to di- vide it into two parts, such that one part may be three- fourths of the other part. Let X denote the number of inches in the larger part ; then — will denote the number of inches in the other part. The number of inches in' the whole line is 28 ; therefore . + 1-28; Whence, ^ + 3a?=112 ; that is, 7^=112 ; and, x=10. Thus one part is 16 inches long, and the other part 12 inches long. Ex. 8. A grocer has some tea worth half-a-dollar a lb., and some worth 87^ cents a lb. ; how many lbs. must he take of each sort to produce 100 lbs. of a mixture worth 62^ cents a lb. ? Let x= the number of pounds of the first sort; then 100— a? will de note the number of lbs. of the second sort. The value of the x lbs. is ix dollars, and the value of the (100— 3j) lbs. is KlOO— .r) dollars; and the whole value is to be | X 100 dollars. Therefore, § X 100= Jc + 1(100 -x) ; multiplying by 8, 500=4a; + 700- 7^ ; whence, 3x=200 ; Thus there must be 66| lbs. of the first sort, and 33^ lbs. of the sec- ond sort. Ex. 9. A person had $5000, part of which he lent at 4 per cent., and the rest at 5 per cent. ; the whole annual in- terest received was $220 : how much was lent at 4 per cent. ? Let x= the number of dollars lent at 4 per cent. ; then 5000— a;= the number of dollars lent at 5 per cent. ; SIMPLE EQUATIOXS. 149 and T7m~ ^^^^ aimual interest from the former; 5(5000 x) and ' — the annual interest from tli£ latter; 4x 5(5000 -.'?) ^^^ therefore, ___ + 1-__J=220; whence, 4x + 5(5000-.t)=22000 ; that is, 47?-f 25000-52=22000 ; /. -a;=-8000, or 2=3000. Thus $3000 was lent at 4 per cent. Ex. 10. Divide 42 into 4 parts which shall be 4 consecu- tive numbers. Let X be one part ; then ic-{-l^ ^+2, x-{-S, are tlie other parts ; and X + (2+l)+(2+2)+(2+3)=42, by the question ; combining, 42j-|-6=42, or, 4^=36 ; .-. 2=9, and 2+1=10, 2 + 2=11, 2+8=12 ; .*. 9, 10, 11, 12, are the required parts. 181. The great difficulty which the beginner finds m solv- ing these problems is in translating the statements of the enunciation into algebraical language. In this, practice alone can give readiness and accuracy. The teacher will find it advantageous to train the student orally in such transla- tions, by means of examples like those given in Art. 40. 182. The student should always read carefully and con- sider well the meaning of the question proposed ; and in or- der to avoid error, he should observe that x represents an unJcno2vn number of dollars, poimds, feet, miles, liours, or in general, an iinhnoivn nwnher of tilings or units, and both the land and denomination of the units of x should be dis- tinctly noticed in the statement. What caution is given to the student in Art. 181? 150 ELEMEJ^TARY ALGEBRA. 183. Many of the problems given below may be solved readily by Arithmetic, but the student will soon perceive the superiority of the method of solution by Algebra, in power and generality and easy application. Examples — 36. 1. What number is that which exceeds its sixth part by 10? 2. What number is that, to which if 7 be added, twice the sum will be equal to 32 ? 3. Find a number, such that its half, third, and fourth parts shall be together greater than its fifth part by 106. 4. A bookseller sold 10 books at a certain price, and after- ward 15 more at the same rate, and at the latter time re- ceived $8.75 more than at the former : what was the price per book ? 5. What two numbers are those, whose sum is 48 and dif- ference 22 ? 6. At an election where 979 votes were given, the success- ful candidate had a majority of 47 : what were the num- bers for each ? 7. A spent 62J- cents in oranges, and says, that 3 of them cost as much under 25 cts., as 9 of them cost over 25 cts. -■ how many did he buy ? 8. The sum of the ages of two brothers is 49, and one of them is 13 years older than the other : find their ages. 9. Find a number such that if increased by 10 it will be- come five times as great as the third part of the original number. 10. Divide 150 into two parts, so that one of them shall be two-thirds of the other. SIMPLE EQUATIONS. 151 11. A child is born in November, and on the tenth day of December he is as many days old as the month was on the day of his birth : when was he born ? 12. There is a number such that, if 8 be added to its double, the sum will be five times its half. Find it. 13. Divide 87 into three parts, such that the first may ex- ceed the second by 7, and the third by 17. 14. Find a number such that, if 10 be taken from its double, and 20 from the double of the remainder, there may be 40 left. 15. A market-woman being asked how many eggs she had, replied : If I had as many more, half as many more, and one egg and a half, I should have 104 eggs : how many had she ? 16. A is twice as old as B ; twenty-two years ago he was three times as old. Required ^'s present age. 17. Divide $64 among three persons, so that the first may have three times as much as the second, and the third, one- third as much as the first and second together. 18. A workman is engaged for 28 days at 62^ cts. a day, but is to pay 25 cts. a day, instead of receiving anything, on all days upon which he is idle. He receives altogether $13.12.j : for how many idle days did he pay ? 19. A person buys 4 horses, for the second of which he gives £12 more than for the first, for the third £6 more than for the second, and for the fourth £2 more than for the third. The sum paid for all was £230. How much did each cost? 20. A person bought 20 yards of cloth for 10 guineas, for part of which he gave lis. 6d. a yard, and for the rest 75. 6d. a yard : how many yards of each did he buy ? 152 ELEMEXTARY ALGEBllA. 21. Two coaches start at the same time from A and B, a distance of 200 miles, travellirng one at 9^ miles an honr, the other at 9^: where will they meet, and in what time from starting? 22. A father has six sons, each of whom is four years older than his next younger brother ; and the eldest is three times as old as the youngest : find their respective ages. 23. ^ is twice as old as B, and seven years ago their united ages amounted to as many years as now represent the age of A: find the ages of A and B, 24. Two persons, A and B, are travelling together; A has £100 and B has £48: they are met by robbers, who take twice as much from A as from B, and leave to A three times as much as to i? : how much was taken from each ? 25. Find two consecutive numbers, such that one-half and one-fifth of the first, taken together, shall be equal to one- third and one-fourth of the second, taken together. 26. A cistern is filled in 20 minutes by 3 pipes, the first of w^hich conveys 10 gallons more, and the second 5 gallons less, than the third, per minute. The cistern holds 820 gallons. How much flows through each pipe in a minute ? 27. A garrison of 1000 men was victualled for 30 days ; dfter 10 days it was re-enforced, and then the provisions were exhausted in 5 days: find the number of men in the re- enforcement. 28. In a certain weight of gunpowder, the saltpetre com- posed 6 lbs. more than a half of the weight, the sulphur 5 lbs. less than one-third, and the charcoal 3 lbs. less than one- fourth : how many pounds were there of each of the three ingredients ? 29. A general, after having lost a battle, found that he had left, fit for action, 3600 men more than half of his army; 600 men more than one-eighth of his army were wounded ; SIMPLE EQUATIONS. 153 and the remainder, forming one-fiftli of liis army, were slain, taken prisoners, or missing: what was the number of the army ? 30. A tradesman starts v/ith a certain sum of money : at the end of the first year he had doubled his original stock, all but £100; also, at the end of the second year he ha\ doubled the stock at the beginning of that year, all but £100; also in like manner at the end of the third year; -and at the end of the third year he found himself three times as rich as at first : what Avas his original stock ? XXV. Problems — continued. 184. We shall now^ give some examples rather more diffi- cult than the examples of the preceding chapter. Ex. 1. Find a number, such that if | of it be subtracted from 20, and y^ of the remainder from | of the original number, 12 times the second remainder shall be half the original number. Let X = the number ; .-. 20— |.T=lst remainder, and l-a'—-f^{20—^x) = 2d remainder; .*. 12[J:r-y^Y(20— |.?0]=4.r, by the question; whence ^ = 24. Ex. 2. A certain number consists of two digits whose difference is 3 ; and, if the digits be inverted, the number so formed will be ^ of ^^^^ former : find the original number. Let a;=lesser digit, and .*. ^+3==ihe greater: then, since the value of a number of two digits = ten times the lirst digit+the second digit, (thus 67=10x0+7), the number in question=10 ( a'+a ) H-^ ; similarly, the number formed by the same digits inverted=10.2'+(.'c+3); hence, by question, 10c+(a?+3)=f[10(;c+3)+.i'], whence ii=3, ic-\-'S=6, and the number required is 63. Ex. 3. A can do a piece of work in 10 days; but, after he has been upon it 4 days, B is sent to help him, and they 154: ELEMENT AH Y ALGEBRA. finish it together in 2 days. In what time would B have done the whole ? Let a'^u umber of days B would have taken, and IT denote the work : W W .-. — , — , are the portions of the work which A, B would do in one 10 X 4Tr day ; hence in 4 days, A does — -r-, and in 2 days, A and B together ^" 10+^= ••• TO+lO +-^-^^' whence .==0. It is plain that, in the above, we might have omitted W altogether, or taken unity to represent the work, as follows : J., 5 do — , — of the work, respectively, in one day, and therefore, 4 2 2 reasonino^ lust as before, -7:+-^H — = the whole w^ork = 1. [In all such questions, the student should notice that, if a person does — ths of any work in one day, he will do — th of it in — th of a day, and therefore the whole work in — days. Thus, if he does -| in one day, he will do ^ in -J- of a day, and there- fore the whole, or -^ , in 3^=2^ days.] Ex. 4. A alone can perform a piece of work in 9 days, and B alone can perform it in 12 days : in what time will they perform it if they work together ? Let X denote the required number of days. In one day A can perform 1 X — th of the work ; therefore, in x days he can perform ^ths of the work. In one day B can perform r^ of the work ; therefore, in x days he can perform T^ths of the work. And since in x days A and B together perfoi-m the whole work, the sum of the fractions of the work must be equal to unity ; that is, 9^12 Multiplying by 36. 4^'+3:i;=36, that is, 7a?=36 ; 36 therefore, if=— =5-JJ- SIMPLE EQUATIOIs^S. 155 Ex. 5. A cistern could be filled with water by means of one pipe alone in 6 hours, and by means of another pipe alone in 8 hours ; and it could be emptied by a tap in 12 hours, if the two pipes were closed : in what time will the cistern be filled if the pipes and the tap are all open ? Let X denote the required number of hours. In one hour the first 1 X pipe fills — th of the cistern; therefore, in x hours it fills Trths of the 6 6 cistern. In one hour the second i3ipe fills — th of the cistern ; therefore, o X 1 in X hours it fills ^ths of the cistern. In one hour the tap empties — rth X of the cistern ; therefore, in x hours it empties T^ths of the cistern. And since in x hours the tohole cistern is filled, we haye XX ^ _-, Multiplying by 24, 4x + dx—2x=24:, that is, 5x=24 ; therefore, a*==^==4|-. 185. It is sometimes convenient to take x to represent not the quantity which is actually demanded in the question, but some other unknown quantity on which if depends. This we will illustrate by some examples. But experience is the only guide to the best selection of the unknown quantity. Ex. 6. A colonel, on attempting to draw up his regiment in the form of a solid square, finds that he has 31 men over, and that he would require 24 men more in his regiment in order to increase the side of the square by one man : how many men were there in the regiment ? Let X denote the number of men in the side of the first square ; then the number of men in the square is .^''^, and the number of men in the regiment is x"^ + 31. If there were x-hl men in a side of the square, the number of men in the square would be (x + lf ; thus the number of men in the regiment is (:r' + l)'^— 24. 156 ELEMEXTAEY ALGEBRA. Therefore, (^+1)^- -24=. i'' + 31; that is, x' + 2x + l- -24- ,v'' + Sl. From these two equal expressions we can remove x"^, which occurs in both ; thus, 2^ + 1 -24= =31; therefore. 2.r=31-l + 24= =54; or, a: 54_ =27. Hence the number of men in the regiment is (27f + 31; that is, 729 + 31; that is, 760. Ex. 7. A starts from a certain place and travels at the :ate of 21 miles in 5 hours ; B starts from the same place 8 hoars after A, and travels in the same direction at the rate of 15 miles in 3 hours : how far will A travel before he is overtaken by 5? Let a^=the number of hours which A travels before he is overtaken ; then ic— 8=the number of hours B travels. 21 Now since A travels 21 miles in 5 hours, (i. e.) ~ of a mile in one o hour, therefore, ~— = the number of miles which A travels in x hours ; 5 15 and similarly, —(^—8)= number of miles which B travels In x horn's o Therefor^' ^ (aj-8)=^ ; o O 25(«-8)=21a?; 25iz;-21^=200 ; a— 50. 21 V 21 Therefore, -—=— X 50=210 ; so that A travelled 210 miles before 5 5 lie was overtaken. 186. The principles of proportion, as taught in Arithmetic, are often used to form the equations. Ex. 8. It is required to divide the number 60 into three • parts, such that they may be to each other in the proportion of the numbers 3, 4, and 5. SIMPLE KOUATIOXS. 15? Let the number x denote the first part. Then, since 1st part : 2d part : : 8 : 4, therefore — - is the second part; o and since 1st part : 3d part : : 3 : 5, tlierefore — is the third part o Therefore, the sum of the parts x + -x-\--x—m', 3a? + 4aj + 5aj=180; , 12.r=180 ; a:'=15, the first part. 4 5 Hence the 2d part is — Xl5, or 20, and the 3d part is ^Xl5, or 25. o o The preceding mode of solution, and many similar solu- tions involving proportions, may be shortened after the following manner : Let 3.^ denote the first part ; then the second part must be 4a;, and the third part must be 5.r. Therefore, ^x + ^x + ^i =80 ; 12, =60; a =5. . . 3x5=15, the first part; 4x5=20, the second part; 5X5=25, the third part. Ex. 9. There are two bars of metal, the first containing 14 oz. of silver and 6 of tin, the second containing 8 of silver and 12 of tin : how much must be taken from each to form a bar of 20 oz., containing equal weights of silver and tin ? Let a:=number of oz., to be taken from first bar, '^0 — x from second; now ^0" of ^^^ fi^st bar, and therefore of every oz. of it, is silver ; and, similarly, -^ of every oz. of the second bar is silver ; Hud there are to be, altogether, 10 oz. of silver in the compound ; .'. ii^' + ^o (3O-.70=lO, whence .r=r)f, and 20-.?=13i. Ex. 10. Find the time between two and three o'clock, when the minute-hand of a watch is exactly over the hour-hand. Find, also, the time between 2 and 3 o'clock, when the hands are exactly opposite each other. 158 ELEMEXTAilY ALGEBEA. 1st Ciise. — Let x denote the required number of minutes after 2 o'clock. In X minutes the minute-hand moves over x divisions of the watch-face ; and as the long hand moves 12 times as fiist as the short hand, the latter "will move over — divisions in x minutes. At 2 o'clock the short hand is 10 divisions in advance of the long hand ; so that in the x minutes the long hand must pass over 10 more divisions than the short hand. Therefore, ^=T^ + 1^ 5 12^=^ + 120; 110^=120; aj=-— 7- =:10yt minutes. Or more briefly, thus : The minute-hand in every minute gains — of one minute-division on the hour-hand. Hence, in x minutes, it gains -r^r- divisions. Therefore, -3-^=10. :. x=———10\f. 1/^ \Z 11 . 2d Case. — Here the minute-hand must not only overtake the hour- hand, but advance so as to leave it 30 minute-divisions behind. Then let (C= required number of minutes after 2 o'clock ; then the gain of the llx minute-hand, or — ^ =10 -\- 30. llx :. -—-=40, lla?=480 ; x=A:^-^j minutes. Hence the hands are in the required position at 43x\ minutes past 2 o'clock. Ex. 11. A hare takes four leaps to a greyhound's three, but two of the greyhound's leaps are equivalent to three of the hare's ; the hare has a start of 50 leaps : how many leaps must the greyhound take to catch the hare ? Suppose that ^x denote the number of leaps taken by the greyhound ; then Ax will denote the number of leaps taken by the hare in the same time. Let a denote the number of inches in one leap of the hare ; then 3rx denotes the number of inches in three leaps of the hare, and there- fore also the number of inches in two leaps of the greyhound : therefore 3<2 -— denotes the number of inches in one leap of the greyhound. Then ^x leaps of the greyhound will contain ^xX~ inches. And 50-l-4iB leaps of the hare will contain (50-f-4.T) a inches; therefore, ^xa _. . . — -=(oO -I- 4a;) a. SIMPLE EQUATIONS. 159 Dividing by a, i7=50 4- 4c ; therefore, 9.^=100 + 8^; or, . a.'=100. Thus the greyhound must take 300 leaps. Here an auxiliary symbol a has been introduced to enable us to form the equation more easily. Being in every term, it is removed by di- vision Avhen the equation is formed. Ex. 12. If tlie specific gravity of pure milk be 1*03, and a certain mixture of milk and water be found (by means of an instrument for the purpose) to be of specific gravity 1*02625, how much water has been added ? [Definition.— ^By the specific gramty of a substance is meant the number of times which its weight is of an equal bulk of water. Thus the specific gramty of silver is 10"5, or 10 j, which means that any quantity of silver is lOJ times the weight of the same hulk of water. The specific gravity of milk being 1.03, signifies that milk is lyl-g- times as heavy as water ; and so on.] Let 1 quart of water be added to x quarts of pure milk to form the mixture; then, Since the w^ eight of x quarts of pure milk =1 .03 times the weight of x quarts of water, — 1.03X X X weight of 1 quart of water; .-. whole weight of water and milk =(1 + 1.03.r)X weight of 1 quart of water. But there are 1+x quarts of the mixture whose specific gravity is 1.02625; .*. the Avhole weight of this =1.02625(1 +cr)X weight of 1 quart of water; .-. l + 1.03.T=1.02625(l + ir). Therefore, (1. 03 -1.02625>r=l. 02625-1 ; that is, .00375.t;=.02625 ; _. 02625 •*• "^"".00375 Hence, 1 quart of water has been added to 7 quarts of milk ; (i. e.) o7i€-eigWi of the mixture is water. 160 ELEMENTARY ALGEBRA. Examples — 37. 1. Out of a cask of wine of which a fifteenth part had leaked away, 12 gallons were drawn, and then it was two- thirds full : how much did it hold ? 2. In a garrison of 2744 men there are two cavalry sol- diers to twenty-five infantry, and half as many artillery as cavalry : find the number of each. 3. The first digit of a certain number exceeds the second by 4, and when the number is divided by the sum of the digits, the quotient is 7 : find it. 4. The length of a floor exceeds the breadth by 4 feet; if each had been increased by a foot, there would have been 27 more square feet in it : find its original dimensions. 5. In a mixture of copper, lead, and tin, the copper was 5 lbs. less than half the whole quantity, and the lead and tin each 5 lbs. more than a third of the remainder : find the respective quantities. 6. A horse was sold at a loss, for $210; but if it had been sold for $262.50, the gain would have been three- fourths of the former loss : find its real value. 7. A can do a piece of work in 10 days, which B can do in eight ; after A has been at work upon it 3 days, B comes to help him : in what time will they finish it ? 8. There is a number of two digits whose difference is 2, and, if it be diminished by half as much again as the sum of the digits, it will give a number expressed by the digits inverted : find it. 9. A number of troops being formed into a solid square, it was found that there were 60 over; but when formed into a column with five men more in front than before, and three less in depth, there was just one man wanting to com- plete it : what was the number of troops ? SIMPLE EQUATIONS. 161 10. A person has travelled altogether 3036 miles, of which he has gone seven miles by water to four on foot, and five by Avater to two on horseback: how many did he travel each way ? 11. A mass of copper and tin weighs 80 lbs., and for every 7 lbs. of copper there are 3 lbs. of tin : how much copper must be added to the mass, that there may be 4 lbs. of tin for every 11 lbs. of copper? 12. A does I of a piece of work in 10 days, when B comes to help him, and they take three days more to finish it : in what time would they have done the whole, each separately, or both together ? 13. A and B were employed together for 50 days, each at $1.20 a day; during which time A, by spending 12 cents a day less than B, had saved three times as much as B, and 2^ days' pay besides : what did each spend per day ? 14. There are two silver cups, and one cover for both ; the first weighs 12 oz., and with the cover weighs twice as much as the other cup without it ; but the second with the cover weighs a third as much again as the first without it : find the weight of the cover. 15. Find a number of three digits, each greater by unity than that which follows it, so that its excess above one- fourth of the number formed by inverting the digits shall be 36 times the sum of the digits. 16. If 19 lbs. of gold weigh 18 lbs. in water, and 10 lbs. of silver weigh 9 lbs. in water, find the quantity of gold and silver respectively in a mass of gold and silver weighing 106 lbs. in air and 99 lbs. in water. VI, A and B can reap a field together in 12 hours, A and (7 in 16 hours, and A by himself in 20 hours : in what time could, 1st, B and C together, and, 2d, A, B, and G to- gether, reajf it? 162 ELEMEXTARY ALGEBRA. 18. Find two numbers whose difference is 4, and the dif- ference of their squares 112. 19. Divide the number 88 into four parts, such that the first increased by 2, the second diminished by 3, the third multiplied by 4, and the fourth divided by 5, may all be equal. 20. Three persons whose powers for w^ork are as the num- bers 3, 4, 6, can together complete a piece of work in 60 days : in what time could each alone complete the work ? 21. A and B are at present of the same age; if ^'s age be increased by 36 years, and ^^s by 52 years, their ages will be as 3 to 4 : w^hat is the present age of each ? 22. Divide 100 into two parts, such that the square of their difference may exceed the square of twice the less part by 2000. 23. A cistern has two supply-pipes which will singly fill it in 4|- hours, and 6 hours, respectively ; and it has also a leak, by which it would be emptied in 5 hours: in how many hours will it be filled when all are working together ? 24. A market-woman bought a certain number of eggs at the rate of 5 for 2 cents ; she sold half of them at 2 for a .cent, and half of them at 3 for a cent, and gained 4 cents by so doing : what was the number of eggs ? 25. A and B shoot by turns at a target ; A puts 7 bullets out of 12 into the bull's-eye, and B puts in 9 out of 12 ; be- tween them they put in 32 bullets : how many shots did each fire ? 26. Two casks, A and B, contain mixtures of wine and water ; in A the quantity of wine is to the quantity of water as 4 to 3 ; in B the like proportion is that of 2 to 3. If A contains 84 gallons, what must B contain, so that when the two are put together the new mixture may be half wine aud half water? "^ SIMPLE EQUATION'S. ' 163 27. How many minutes does it want to 4 o'clock, if three- quarters of an hour ago it was twice as many minutes past two o'clock? 28. What is the time after 6 o'clock at which the hands of a watch are, 1st, directly o23posite, and, 2d, at right angles to each other ? 29. It is between 11 and 12 o'clock, and it is observed that the number of minute-spaces between the hands is two- thirds of what it was ten minutes previously : find the time. 30. The national debt of a country was increased by one- fourth in a time of war. During a long peace which fol- lowed, $125,000,000 was paid off, and at the end of that time the rate of interest was reduced from 4|- to 4 per cent. It was then found that tfle amount of annual interest was the same as before the war : what was the amount of debt before the war ? 31. Find three numbers, the sum of which is 70, and such that the second divided by the first gives 2 for quotient, and 1 for remainder ; and the third divided by the second gives 3 for quotient, and 3 for remainder. 32. Shells are thrown from two mortars in a besieged city ; the first mortar has thrown 36 shells before the second com- mences its fire, and it sends 8 shells for every 7 sent by the second ; but the second expends as much powder in 3 dis- charges as the first does in 4 : how many shells must the sec- ond mortar throw in order to expend the same amount of powder as the first ? 187. We shall now give a few problems in which the known quantities are represented by the first letters of the alphabet, instead of numbers. Examples — 38. 1. Find a number, such that being divided successively by m and n, the sum of the quotients shall be equal to a. T64 ELEMENTARY ALGEBRA. 2. Divide a number a into two such part3 tV^ai the quo- tient of the one divided by m, and the other divided by n, may be equal to h. 3. Divide a number a into two parts proportional to the numbers m and n, 4. Divide a number a into three parts, such that the first may be to the second as m is to n^ and* the second to the third asj9 is to q. 5. Two numbers, a and I, being given, what number must be added to each one of them in order that the ratio of the '}7l two sums may be equal to — ? 6. 'Three fountains will fill a ce]i;ain reservoir, when each one runs alone, in the times a, t, and c, respectively. In what time will they fill it, all running together ? 7. Two couriers, whose distance apart > when they set out was d miles, travel toward each other, the one moving at the rate of h miles an hour, and the other at the rate of c miles an hour. In what time after starting will they meet ? 8. A can do a piece of work in h days, and B can do the same work in c days. In what time can they together do the work? XXYI. SiMUIiTANEOUS EQUATIONS OF THE FlRST DEGREE. 188. If one equation contain tivo unknown quantities, there are an infinite number of pairs of values of these by which it may be satisfied. Thus in .t=10 — 2?/, if we give any value to y, we shall get a corresponding value for x, by which pair of values the equation will of course be satisfied. If, for example, we take When one equation contains two unknown quantities, what values may they have? SIMULTANEOUS EQUATIONS. 165 t/—l, we shall get :?;r=:10 — 2 = 8; if y^2, x=z6; if ^=3, x=4cf &c. One equation then, containing ttvo unknown quantities (or, as it is expressed, "between two unknown quantities"), admits of an infinite number of solutions ; but if we have as many different equations as there are quantities, the num- ber of solutions will be limited ; for it will be seen that they can always be reduced to a single equation containing a sin- gle unknown quantity. Thus, while each of the equations, 2:= 10— 2^, 4:X=:32 — 6y, separately considered, is satisfied by an infinite number of pairs of values of x and y, we shall find there is only one value of X, and one value of y, which will satisfy both equa- tions; for, multiplying the first equation by 3, dx=30 — 6y; now take this from the second equation 4:^=32 — 6^, and we get x=i2. Thus 2; =2 is the only value of x common to both equa- tions. Put this value of ;c in either of the two given equa- tions — for example, in the first; and we obtain, 2=^10-2^; /. 2y=S; .-. y=^. Thus, x=2, y=^4:, are the only pair of values which satisfy both equations. 189. Equations of this kind, which are to be satisfied by the same pair or pairs of values of x and y, are called simul- taneous equations. In the present chapter we treat of si- multaneous equations of the first degree, (i. e.) where each unknown quantity occurs only in the first power, and the product of the unknown quantities does not occur. If there be three unknown quantities there must be three equa- tions, and so on. Simultaneous EquaUons. 166 ELEMEl^TARY ALGEBRA. 190. These simultaneous equations must all express dijf ev- ent relations between the unknown quantities. Thus, if we had the equation x=10—2y, it would be of no use to join with it the equation 2iz;=20— 4y (which is the double of the former), or any other deriyed like this from the former. 191. There are generally given three methods for solving simultaneous equations of two unknowns; but the object •aimed at is the same in each, viz., to combine the two equa- tions in such a manner as to' expel, or eliminate, one un- known from the result, and so get an equation of 07ie un- known only. 192. First Method. — Multiply fhe equations by the least numbers wliicli will maJce the coefficients of 07ie of the tin Icnoivn quantities the same in both resulting equations ; then adding or subtracting the two equatio7is thus obtained, accord mg as the equal terms have different or the same signs, these terms ivill destroy each other, and, the elimination will be ef- fected. Ex.1. ^x-vZy^ 4 1(1) 3a:-2^=-7) (2) Multiply (1) by 3, 6a; +9^/= 12. Multiply (2) by 2, 62;-4y--14. Subtracting, 13?/=26 and /. ^=2. Then put this value of y in either (1) or (2) ; for example, in (1). We have thus: 2a; + 6=4; .-. 2a;==4-6=:-2; .-. a:=-l. Ex. 2. 8a;+7?/=100 ) (1) 12a;- 5y= 88 ) (2) What is said of the relations expressed by simultaneous equations ? Of the ohject aimed at by each of the three methods of solving them ? First method ? SIMULTANEOUS EQUATI02s^S. 167 We might multiply (1) by 12 and (2) by 8, giving thus : 96x-4:0y= 7 04 Subtracting, 124^= 496 .-.«/ = 4. • But the process is more simple if we multiply equation (1) by 3 and equation (2) by 2. Thus : 24:X-\-21y==300 24.x-10tj=176 3 1^=124; .•.^=4; and, substituting m (1), 8a;+28 = 100; /. Sx=z'72, and x=9. We see here the advantage of multiplying by the leasi numbers which will make the coefficients the same, though we may multiply by any nmnbers which will effect the same object. It is sometimes possible to multiply one of the given equa- tions by some number which will make the coefficient of x or y in it the same as in the other equation. The process is in this case much shortened. Ex.3. 4.x -^y^U) (1) 4y+a;=16j (2) Here multiplying (2) by 4, 16^ + 42^=64; but y+jtx=U; (1) /. subtracting, 16y =30, and ,\y=2-^ and (2) .r=:16-4y = 16-8=8. Ex. 4. 4:x- y= 7) (1) dx + 4:y=29 ) (2) Here 3.T + 4y=29, and, multiplying (1) by 4, 16a;— 4^=28; .*. adding, 19a; —57, and ,\x=d; and (1) ^=4a;~7n= 12-7=5.. 168 ELEMEKTARY ALGEBRA. 193. Second Method. — Express one of the unhnoiun quan- fities 171 terms of the other by means of one of the equations^ and put this expression for it in the other equation. Thus, taking the example 1 in the preceding article, 2.T + 3?/=: 4)(1) • ^x-^^-l) (2) 4—3^ From {l),x^=z — ^r-^; substituting this expression in (2), we obtain whence 3(4— 3?/)— 4^= — 14; that is, 12-9«/-4^=-14; /. -133/^-26; .-. iy=2. 4-3y 4-6 ,,x- ^ _ ^ _ i. Ex. 5. 7^' + i(2y + 4) = 16 ) or reducing, 35:^ + 2;/= 76 ) (1) 3y-i(a: + 2)= 8) 12^- a;=:34j (2) Here from (2), i?;=:12^-34, and from (1), 35(12i/-34)+2«/ = 76; whence y=^, and /. ir=2. 194. Third Method. — Express the same unlcnoivn quantity in terms of the other in both equatio7is, and put these expres- sions equal Thus, taking again Ex. 1, 2x + 3y= 4 ) (1) 3x-2y=-7) (2), (1) gives y=—^; (2) gives y=-^-; ^, ^ 3a; + 7 4.—2X therefore, — - — = — - — . Second method ? Third method ? SIMULTAXEOUS EQUATIOKS. 1G9 Clearing of fractions, 9x + 21=zS-4.x. .\13x=z-r6, x=-^. Ex.6. 5.7;-i(5y + 2) = 32) or reducing, 20.-?;— 5^=130 ) (1) 3?/+t(^ + 2)= 9) 9ij+ x= 25) (2) Here in (1), y=^{20x-ld0), in (2), y=^{2o-x) ; .^|(20.^•-130)3^i(25-^), whence a;=7, y=^2. The first meiliod is to be preferred generally; but the second may be used with advantage whenever either x or y has a co- efficient unity in one of the equations. Note. — It may be well to give, here, an abbreviation of the first method, which saves much trouble when the coefficients are large. Two examples will serve to illustrate it. Ex. 54^-121^=15 ) (1) ^ . y to find X and y. 36a?-88?/=-12;) 36a.'- 77^= 21 ;f (2) Subtracting, 1 8a?— 44y= - 6 ; multiplying by 2, from 2d equation, subtracting, 11^=33 ; And I8.r=44?/-G=132-6=126; .'. a'=7. Ex. 101aj-242/=G3 ) d) 103..-28^=29f^?^^^^"^^^- Subtracting, 2x- 4?/= -34; multiplying by 6, 12^'— 24?/= - 204 ; ) but 101a;- 24y= 63; ) subtracting, 89a; ==267 ; 267 ^ And 4^=2aj + 34=40 ; .-. y=10. Which method is preferable ? 8 170 £Lemp:ktar.y algebra. 195. Ex. 7. Solve —+-=:=8l (1) X y ^ ^^-1? = 3! (2) X y J If we cleared these equations of fractions they would con- tain X y, and could not then be solved by the methods of this chapter. But if we do not clear them of fractions, they may be solved readily by the methods given. Thus, multiplying (1) by 3, '-V?-*=24; X y multiplying (2) by 2, X y ' adding, -=.30; .-.90= X ' .-. (1) gives o therefore, — = 8 — 4 = =4; .-.8=4./; . =2. Ex. 8. Solve a''x-\-Vy:=^c'' (1), ax-^hy^c (2). Multiplying (2) by Z>, and subtracting it from (1), (^x^-h^y—& abx-\-h''y — 'bG G^x—abx—c^ — 'bc\ that is, a(a—h)x—c{C'—y)\ ' ~a(a-by substituting this value of x in (2), acic—V) 7 ~f iT + oy=c\ in therefore, SIMULTANEOUS EQUATIOJfS. hy-~ c{c- -b) c(a- -c) ' («- h)- a— -b :. ij~ c{a—c) Or this value of y might be found in the same way as that of X was found. Examples — 39. Find the values of x and y in the following pairs of equations : 1. 3a:-4y=2, Ix-^y^^t, 2. 7^-5?/=24, 4.x-3y=ll. 3. 3^^+2^=32, 20x-3y=:l, 4. llx-7y=37, 8^+%=41. 5. '7x-i-5y=^60, 13:r-lly=:10. 6. 6:^-7^=:. 42, 7x-6y = 75. 9. V+ 2 =^' 2+V=°- 4~3~' 3 + 6~ „ l-3a; 3?/— 1 „ dx + y 11. -nj^ + -Y-=2' ^^ + .V=9- 13. 3(3a; + 3j/) = 3(2a;-32/)+10, 4a;-3y=4(6y-2a;)4-3. 172 ELEMENTARY ALGEBRA. 13. x{ij^7)=y{x + l)l 14. i.^+i7/ = 13) 2a;4-20 = 3^+l ) :^x + i2j=D) 15. 1.^+^^=43) 16. 3.^+%=2.4, ,21x-My=z.03. 17. ,dx + A26tj=x-6, 3x-.6y^2S—,26i/. 18. ,0Sx-.21tj=.S3, .12x-}-.'7y=3M. x y X y 20. — +-|-=2, Ix—ay—^. 21. ir + ^=:<2+Z>, 'bx-\-ay—2ab, 22. - + 4=1' -^ + -^=1. ah a 23. {a-]-c)x—'by—'bc, x-\-y—a-\-h, a ha 25. x-}-y=c, ax—hy—c{a—h), 26. « (^ + ^)+Z>(^— ^) = 1, a {x~y)+h {x-{-y) = l. 27. ?i:i^+l(z:^=o, ?±i/:i^+?i=lr^=o. 196. Simultaneous equations of three unknown quantities are solved by eliminating one of the unknown quantities by means of any pair of the equations, and then the same un- known by means of another pair; we shall then hare two equations involving only two unknown quantities, which we may solve by the preceding rules. The remaining unknown is found by substituting the values obtained for the other two in any of the given equations. Simultaneous equations of three or more unknown quantities. SIMli/rAXEOUS EQUATIONS. 173 Similarly for simultaneous equatiojis of more than three unknown quantities. Ex.1. x-2y+ 3^=2] (1) 2^-3^+ z=l I (2) dx- y\ ?.z-^\ (3) From (1) 2.^--4y+ 6^=4 (^) "^x-Zy^ z=l •. - y+ 5^=3 (4) Again, from (1) dx—6y+ 9z=6 (3) 3x- y+ 2z=d \ -6y+ 7;2=-3 (5) but from (4) -6y + 25z=15 —18^=— 18, and z=l hence (4), y= 6z-3=2 and (1), x=2+2y-3z=:2 + 4:-3=S, Ex.2. 1+^-^=1 (1) t+± + l=2^ (2) X y z ^ ^ I-l+i=14 (3) X y z ^ ' Multiplying (1) by 2, | + A^| = 2 5 4 - + — ■. X y z ^ + -^=36 (4) Multiply (1) by 3, |+|_|=3 Adding (3) to this, ^ + -+-=24: 174 ELEMENT AKY ALGEBKA. Add (3) to this, 1 X y " = 14 10 X _2_ y" -.11 (i NoTv ' multiply (5) by 4, 40 X 8_ 'y~ :68 Add (4) to this. X y :26 47_ X ~ --M- ,-.47=! __47_1 •''-94-2' Substitute the value of x in (5) ; thus, 20--=17; /. -=20-17^:3; /. v=-|-. y y ' ^ z Substitute the values of x and y in (1) ; thus, Ex.3. Solve -+^=1 (1) From (1) -+-|-=1 subtract (2) — I — — 1; a c to the result, add (3) y z = b c z c =1 ^=1 y-^ Sr^iULTAXEOUS EQUATIONS. IvO Substitute this value in (1) ; thus, Substitute the same yalue in (3) ; thus, 1 z z 1 __c These values of z and x might have been written down at once from the symmetry of the equations, since it is obvious that the values of x and z will be of the same form as that of y, only interchanging a and c, respectively, with h. Examples — 40. 1. 2a;-h3v + 4^=20l 2. 5^ + 3^^:65^ 3a:+4y + 5;^=26 I 2y- z=ll 3:?; + 5^ + 6^=31 J 3^;+ 4;2=57^ 3. dx^-2y- ^=201 4. x^y-^z=b^ '2x-^dy-\-Qz='10 \ x-\-y=z-l X- y-\-Qz—Al\ x-3=y + zj 5. x + 2y=7 ^ 6. xy=x + y y-{-2z=2 I xz=2{x + z) dx+2y=z—i J y^=^(y+^). 7. 2(2;-^) = 3^-2 1 8. ix + iy=12-iz x + l = ^y+z) I iy+iz= S+ix 2.r+3^=4(l-^)J ix + iz=10 9. y + iz=:ix + 5 i(^-l)-i(y-^)-TV(^ + 3) t.-i(2t/-5)=:lj-^^. 176 ELEMEXTARY ALGEBRA. 10. = -, - + - = 3|, - + - = -. X y h y z ^ X y z 11. y-\-z=a, z-\-x=h, x-\-y=c, 12. x-\-y-\-z—a-^h-\-Cy x + a—y^h—z-\-c. 13. y^z—x=a, z^x—y—l^ x + y—z—c. ^ , X y z ^ X y z ^ x y z . a o € a c a c KoTE. — In Ex. 6, divide the equations by xy, xz,, and yz, respectively, and they will then be of the form of those given in Ex. 10. XXVII. Problems solved by Simultaneous Equations OF THE First Degree. 197. Prob. 1. There is a certain fraction which becomes equal to -J- when both numerator and denominator are dimin- ished by 1 ; but, if 2 be taken from the numerator and added to the denominator, the fraction becomes equal to ^ : find it. Let X denote the numerator, and y the denominator of the required fraction ; then the conditions of the problem give, f-i~Y' y + 2~Y' Clear the equations of fractions, — transpose, and reduce. We obtani thus, 2x-y=l (1) dx-y=S (2) Subtracting(l)from(2) we getir=7; .'. (1) gives 14— 2/= 1; .*. 2^=1«>; therefore, the required fraction is J^. Prob. 2. There is a certain number composed of two fig- ures or digits, which is equal to four times the sum of its digits ; and if the digits exchange places, the number thus formed is less by 12 than twice the former number: what is the number? SIMULTANEOUS EQUATION'S. 177 Let X be the digit in the tens^ place, y ". . . . ^m^ts^ ; then lO-r+y is the number (just as 23 = 10x2-f3), /. by the condi- tions of the question, 10.zj + 2/=4(.i' 1-2/), that is, =4:X-\Ay\ transposing, 10x — 4:X=A:y—y\ uniting, ^x=^y\ or, 2x=y (1). Again, if the digits be reversed, 10y-{-x will be the number ; /. by the question, 102/ + ^=2(103:-l-^)-12; that is, =20.2^+2^-12; transposing, l^x—Sy=VZ\ or, [••• 2/=2^', (1)], mc-\Qx=12', (2) uniting, 3.r=12; .'. ir=4; and ^=2.2?= 8. . . the number requu'ed is 48. pROB. 3. A railway train after travelling an hour is de- tained 24 minutes, after which it proceeds at six-fifths of its former rate, and arrives 15 minutes late. If the detention had taken place 5 miles further on, the train would have ar- rived 2 minutes later than it did. Find the original rate of the train, and the distance travelled. Let 5.^' denote the number of miles per hour at which the train origi- nally travelled, and let y denote the number of miles in the whole dis- tance travelled. Then y—t)X will denote the number of miles w^hicli remain to be travelled after the detention. At the original rate of the train this distance would be travelled in -~ — hours : at the increased rate it will be travelled in ^ — hours. Since the train is detained 24 K)X minutes, and yet is only 15 minutes late at its arrival, it follows 8* 1T8 ELEMKJSTAKY ALGEBiiA. that the remainder of the journey is performed in 9 minutes less than it would have been if the rate had not been increased ; and 9 minutes 9 is 7^ of an hour ; therefore, 00 y—6x_y-6x 9^ '~Q^~~5x~ 60 ^ ^* If the detention bad taken place 5 miles further on, there would have been y—6x—6 miles left to be travelled. Thus we shall find that y—6x-5 _y-5x-5 7 Qx ~ 6x 60 ^ ^' Subtracting (2) from (1), Qx~5x 60' therefore, 50=60-2aj; whence, 2a?=10; and x=5. Substitute this value of a? in (1), and it will be found by solving the equation that y=4!7^. Prob. 4. A and B can together do a piece of work in a days; A and can together do it in b days; B and C can together perform it in o days: find the number of days in which each alone could perform the work. Let X denote the number of days in which A alone could perform it, y the number of days in which B alone could perforai it, z the num- ber of days in which C alone could perform it. Then we have, X z b ^ ^' * 1.1=1 (3). y z c Bubtractmg (2) fi-om (1) we obtain, 1_1=1_1 (4) y z a b ^ ' SIMULTAIS^EOUS EOTATIOXS. 170 1 1 _ 1 ^ y z c ' 2_1 1 . 1 y a b c bc + db—ac ~~ aba 2abc bc + ab—ac' 2abG ■■—r- T-, and x= ab + ac—bc' 2abc ac-\-bc—ab' adding (3), Therefore. these latter vahies bemg written out at once hj the symmetry of the equations. Or we might have solved the problem thus : Let ic=the number of units of work performed hj A in one day; Let y= " " " performed by B in one day ; Let z=z " " performed by C in one day. hen, <^+y=^ (1); X + 2=ry (3); y+z=\ (8). These give by eliminating, as before, bc-{-ab—ac 7/ = — -— . ^ 2abc ' Therefore J5's time of performing the whole work, or — =7 7 • ^ ° y bc + ab—ac as before. 198. A problem may often be solved, as readily by a single equation and one unknown quantity, as by simultaneous equations with two or more unknown quantities. The ad- vantage to a beginner in taking several letters to denote the unknowns is that, though he has more equations and longer work, he can more easily follow the steps by which the equa- tions are formed. Thus Ex. 19, Chap. XXV., may be solved by four simul- taneous equations, involving four unknown quantities. 180 ELEMEJs^TARY ALGEBRA. Examples — 41. 1. What fraction is that, to the numerator of which if 7 be added, its yahie will be | ; but if 7 be taken from the denom- inator its value will be | ? 2. There is a number of two digits wliich, when divided by their sum, gives the quotient 4; but if the digits change places, and the number thus formed be increased by 12, and then divided by their sum, the quotient is 8: find the number. 3. A rectangular bowling-green having been measured, it was observed that, if it were 5 feet broader and 4 feet longer, it would contain 116 feet more; but if it were 4 feet broader and 5 feet longer it would contain 113 feet more: find its present area. 4. A person rows on a uniformly flowing stream a distance of 20 miles and back again, in 10 hours ; and he finds that he can row 2 miles against the stream in the same time that he rows 3 miles with it. Find the time of rowing down and the time of rowing up. 5. A and B can do a piece of work together in 12 days, which B, working for 15 days, and C for 30 days, would to- gether complete. In 10 days, working all three together, they would finish the work : in what time could they sepa- rately do it ? 6. Some smugglers found a cave which would just exactly hold the cargo of their boat, viz., 13 bales of silk and 33 casks of rum. "While unloading, a revenue cutter came in sight, and they were obliged to sail away, having landed only 9 casks and 5 bales, and filled one-third of the cave. How many bales separately, or how many casks, would it hold ? SIML'LTAXKOrS IX^UATLO^^S. 181 7. Seven years ago, A was three times as old as B was; and seven years hence, A will be twice as old as B will be : find their present ages. 8. A certain fishing-rod consists of two parts : the length of the npper part is to the length of the lower as 5 to 7 ; and 9 times the npper part, together Avith 13 times the lower part, exceed 11 times the whole rod by 36 inches : find the lengths of the two parts. 9. If the nnmerator of a certain fraction be increased by 1, and the denominator be diminished by 1, the yalne will be 1 ; if the nnmerator be increased by the denominator, and the denominator diminished by the nnmerator, the valne will be 4: find the fraction. 10. A nnmber of posts are placed at equal distances in a straight line. If to twice the nnmber of them we add the number of feet between two consecutive posts, the sum is 68. If from four times the number of feet between two consecu- tive posts we subtract half the number of posts, the remainder is 68. Find the distance between the extreme posts. 11. On the addition of 9 to a certain number of two digits, its digits change places; and the sum of the first number and the number thus formed is 33 : find the digits. 12. A and B ran a race which lasted five minutes. B had a start of 20 yards ; but A ran 3 yards while B was running 2, and won by 30 yards: find the length of the course, and the speed of each. 13. A person has two casks, with a certain quantity of wine in each. He draws out of the first into the second as much as there was in the second to begin with; then he draws out of the second into the first, as much as was left in the first; and then again out of the first into the second, as much as was left in the second. There are then exactly 8 gallons in each cask. How much was there in each at first? 182 ELEMEIN^TAIIY ALGEBRA. 14. The year of our Lord in which the ^ cltangc of style^ from the Julian to the Gregorian Calendar was made in England, may be thus expressed : The first digit being 1 for thousands, the second is the sum of the third and fourth, the third is the tJiird part of the sum of all four, and the fourth is the fourth part of the sum of the first two. De- termine the year. 15. A and B can together perform a certain work in 30 days : at the end of 18 days, howeyer, B is called ofi", and A finishes it alone in 20 more days. Find the time in which each could perform the work alone. 16. A cistern holding 1200 gallons is filled by three pipes, A, B, C, together, in 24 minutes. The pipe A requires 30 minutes more than G to fill the cistern; and 10 gallons less run through C per minute than through A and B together : find the time in which each pipe alone would fill the cistern. 17. Find two numbers, the sum of which is equal to 3 times their difference, and their product equal to 4 times their difference. (See ISTote at the end of "Examples — 40.") 18. The sum of two numbers is 13, and the difference of their squares is 39. What are these numbers? Note. — Here divide the second equafion by tlie first. 19. ^ and B are two towns, situated 24 miles apart, on the same bank of a river : a gentleman goes fi^om A to B m^ hours by rowing the first half of the distance, and walking the second half. In returning, he walks the first half at three-fourths his former rate; but the stream being with him in returning, he rows at double his rate in going ; and he accomplishes the whole distance in 6 hours : find his rates of walking and rowing. 20. Two trains, 92 feet long and 84 feet long, respectively, are moving with uniform velocities on parallel rails : when SIMULTAXEOUS EQUATIOKS. 183 tliey move in opposite directions, they are observed to pass each other in one second and a half; but when they move in the same direction, the faster train is observed to pass the other in six seconds : find the rate at which each train moves. 21. A raih^oad runs from A to C. A freight train starts from A at 12 o'clock, and a passenger train at 1 o'clock. After going two-thirds of the distance the freight train breaks down, and can only travel at three-fourths of its former rate. At 40 minutes past 2 o'clock a collision occurs, 10 miles from C, The rate of the passenger train is double the diminished rate of the freight train. Find the distance from A to C, and the rates of the trains. 22. A certain sum of money was divided between A, B, and C, so that ^'s share exceeded four-sevenths of the shares of B and (7 by $30 ; also 5's share exceeded three-eighths of the shares of A and C by $30 ; and (7s share exceeded two- ninths of the shares of A and B by $30 : find the share of each person. 23. A, B, and C can together perform a piece of work in 30 days ; A and B can together perform it in 32 days, and B and C can together perform it in 120 days : find the num- ber of days in which each alone could perform the work. 24 Express the two numbers whose sum is a, and their diflPerence h. 25. Find two numbers, such that the product of the first increased by a, and the second increased by h, exceeds by c the produce of the two numbers ; and the product of the first increased by m, and the second increased by n, exceeds by h the product of the two numbers. 26. The sum of two numbers is a, and the difference of their squares is h : find the two numbers. 27. Find two numbers, whose sum is m times their differ- ence, and their product is n times their difference. 184 ELEMEJS^TARY ALGEBJIA. XXVIII. Indices. 199. The indices or exponents which we have used hith- erto are positive whole numbers, whicli express briefly the repetition of the same factor in any product. (See Art. 15.) Under this definition we have proved (Art. 59), Also (Art. 74), oT'-^cf—a'^'', 200. Hence it will follow that (a'")^=:«"^"=(<^")^ For {a;''y' = d!'\0r,ar\ &C. n factors =^^+»»+"^+&c. n terms ^^mn^ and {o}'Y=^d!'.d!'.a\kQ>. m factors ^z^'^+^+^+ac- ^ terms _^nm. .-. since a'^''~-=^cr\ we have {(f''Y=or'''—(a!'Y \ that is, tlie n*^ power of the m*^ potver of 2^ — the m*^ potoer of the Vl^ power of 2.\ and either of them is found by multi- plying the two indices. 201. Hence, also, it will follow that Va"^=(V<^)^ For let Va"'=:^^'^ ; then rr = {pf'Y^ {x^'Y by (199) ; hence a — x^, and .*. Va=2:, and {\/aY^^x'^\ but also, by our first assumption, V^"'^-^"'; hence, we have, Va"' = (V^)"' ; that is, the n*^ root of the va^^ poiver of ^ — the m"' poiver of the n*'' o^oot of a. 202. These results refer so far to positive integral indices. But now suppose we write down a quantity with a positive fraction for an index, and agree that the law of multiplica- tion, a^XaP'=a'^^'', shall hold true for m a7id n fractions as well as m and n j^ositive ivhole numbers. What would such a fractional index denote ? For example : required the meaning of aK Indices. Positive fractional indices. INDICES. 185 By supposition we are to liave a'^Xa^—ci'—a. Thus a^ must be such a ii umber that if it be multiplied by itself the result is a ; and the square root of a is by definition such a number; therefore a^ must be equivalent to the square root of a, that is, a7^=\/a. Again ; required the meaning of a^. By supposition Ave are to have, a^Xa^Xa^'--=a^-^^^"'^^=a'=a. Hence, as before, a^ must be equivalent to the cube root of fl^; that is, a'^ = ya. Again ; required tlie meaning of a*. By supposition, a^Xa^Xa'^X a*=a^ ; therefore, a* = Va^. To give the definition in general symbols : 1. Required the meaning of a" where n is any positive whole numler. By supposition, JL - JL _L^ i-^J-_(. ... to n terms a''Xa''Xa''X ... to n factors — a"" "" "" —a =a; _i_ therefore a" must be equivalent to the n^^ root of a, 2_ that is, ar=Va. m 2. Required the meaning of a" tuhere m a7id n are any pos- itive ivhole numbers. By supposition, !!L i!i !!L ^',^.^..... ton terms a'^Xce^Xa^'X ... to ^z factors = «j" " " ==«"*; therefore a'' must be equivalent to the n^^ root of a^ ; VI that is, ^=Va"*. ISG ELEMENTARY ALGEBRA. m Hence, a" means the 7^*^ root of the m^^ power of a ; that is, in a fractional index the numerator denotes a power, and the denominator a root. 203. Again ; if we wTite down a quantity with a negative index, as ar^ (where p is either an integer or a fraction), and agree that this symbol shall be treated by the same law of multiplication as a positive index ; what would this symbol denote ? By this law of multiplication, a^-^^ Xor^—d^''^^—dJ^\ but we have also, oJ^^^-^oF— — — = — '—-=d!^\ oF oF ^ so that to multiply by a~^ is the same as to divide by oF ; and therefore, lXa-^=l-^«^ or or''——-. Hence, any quantity with a negative index denotes the re- ciprocal of the same quantity with the same positive index. Thus a~^=~, a~^=r— , cri = -r= — — , or= >/a~^= V— ; a a a^ Va a ^3 v/2 a Hence, also, any power in the numerator of a quantity may be removed into the denominator, and vice versd, by merely changing the sign of its index. Thus a ^trc ^— 1=^-^= — ^=&c. c cr^c a 204. Lastly, if we write down a quantity with zero for an index, as aF, and agree that this symbol shall be treated as if the index were ai; actual number— what then would it denote ? Since, by this law, a°Xtt"'=flf'+"'=«^^ it follows that «° is only equivalent to 1, whatever be the value of a. Negative indices. Zero as an index. INDICES. 18? In actual practice, such a quantity as a^ would only occur in certain cases, where we wish to keep in mind from what a certain number may have arisen : thus (»" + 2d^-\-'da-\-&c.)-^d^=a-{-2 + 3a~^ + &c., where the 2 has lost all sign of its having been originally a coefficient of some power oia\ if, hoAvever, we write the quotient a + 2«" + 3«-^ + &c., we preserve an indication of this, and have, as it were, a connecting link between the positive and negative powers of a. The quantity a* is still called a to tlw power of — , and similarly in the case of a-^^^ a^ ; but the word power has here lost its original meaning, and denotes merely a quantity with an index^ whatever that index may be, subject, in all cases, to the Law, a"^. a^ = a^+^ . Examples — 42. Express, with fractional indices, 1. Vx' + Vx'-h{^xY+{Vxy; V{a'b')^V{a'b')-{-V{ab') + V{a'b'). 2. aVb' + {s/ay + V{a'b)-i-V{a'b'); V{aW)+a{Vby +V{a'b'')+V{a'b'y Express, with negative indices, so as to remove all powers, 1st, into the numerators, and 2d, into the denominators. 1 2 3 4« 5Z> «^ U' 6a U 2F ^* '^'^b'^'7'^'b^/a'' J^^T^T'-^^'^-^' 3^V a'b' a ^ Zabc' %y c^ ?> sf a'^ ^V {a'bc'y d^^/W Express, with signs of Evolution, JL 3.7 X ^ 1. 3 5. «^ + 2r/ 3 4- 3^ 4 ^ 4^ 6 4. ^4 , _|. . _| — b' 2c ^ db^ J)l c^ b^ c^ 4^^ 6a^ Express, with positive indices, and with the sign of Evo- lution, 188 ELEMENTARY ALGEBRA. +a-H^ + b-^' cr%^ 2a_ 3b-'c-' 1 ^ ^ ^ ^ ^' C-' '^b-'c-'^ a-' '^a-'b-'c-'' ^'k'^a-i'^a-^^ ^~"' 205. From our definitions, {a^ y = a^ said (a^)'* = a's and in general, (a «)"" = {a'"'') n ; then, also, [a'^)^ = a^^ -^ also, a^, bn ,c^ , , , z=i[abc . . . )"^ since each, raised to the n^'^ power, gives abc. It follows, then, that ^vJiatever be the indices, cd"^ 1 It will be observed, also, that since a fraction may take different forms without changing its value, the form of a fractional index may be changed without altering the valuo of the quantity. Thus (3^ 3 — ^(j • for either raised to the sixth power gives a* ; and in general, a^=a^, for either raised to the np power gives a"'^. Hence we can reduce the indices of two quantities to a common denominator without changing the value of the expressions. 20^. Hence, (1) to multiply together any powers of the same quantity we must add the ijidices ; (2), to divide any one power of a quantity by another power of the same quantity, subtract the index of the divisor from that of the divideiid ; and, (3), to obtain sluj poiuer of a poiver of a quantity, we must muIM- ply together the two indices. The signs of the indices must of course be carefully ob- served. Ex.1. Multiply a^'b^ c^hj a^'b^c^ . Reduction of the indices of two quantities to a common denominator. Rule for (1) multiplying together powers of the same quantity ; (2) for dividing one power of a quantity by another power of the same quantity ; (3) for obtaining any power of a power of a quantity. IXDICES. 189 3 "^2 ""6' 4'^8~12' 3'^3~ * therefore, c^ h^ c « x r^^ Z> -^ t^ ^ = ^^ b ^^ ^• Ex.2. Divide x^ y^^ hj x^^ y'^ , 4 2~4' 3 6~2' tlierefore, ir* y-^ -^x' y^ =zx^ y~ . Also, a^Xa-^-:a^~'=^; «^--«-*=:a^+^^ri^; a-^-4-a-^=a-*+^=a-^'-<^ ; Ex. 3. Multiply x-\-x^ -\- x~^ by cc^ + i?;"^ — x~^. X -\-x'^ + x-i x^ ■\-x~^—x-'- • x^+x^ + 1 x^ + 1+x 4 - 1-x -I. -x-i x^^lx- i+l -x^ Here in the first line, x^ Xx=x^^^^ — x^^\ x^ Xx^ = x^ \ x'^ X X '^=x'^ = l] and so on. Ex. 4. Divide x^ - a^ x'-Ux^ ^eJx~ 2a V' by x^ -4.ax^ +2 J • x^ - 4:ax^ + 2a^ ) x^ - a^ x' - 4.ax^ + 6a^ x-2a''x^ (x-a^ x^ x^ — 4:ax^ + 2a^ x -a^x' -h 4:J x-~2a''x^ -a^x' -h 4:a^ x-2a*x^ 190 elemektaky algebka. Examples — 48 Find the value of 1. 9-^. 2. 4~i 3. (100)-^. 4. (1000)^. 5. 81"^. Simplify 6. (a=)-l 7. (a-^)-^ 8. ^/a-^ 9. V^"". 10. aixaixa~^' Multiply 11. X"^ + ^4 ]3y ^f _ ^! ^ 13. x-\-x^' + 2 by iz; + :2;* - 2. 14. .^*+:^' + 1 by ^'c-'-ii;-' + 1. 15. oT^^ar'-^l by a~i^l. 16. «3_2+^-^ by a^-^-t 17^ a + ^^Z** — ii;"^^/^ by a + a^^i+o;*^^. Divide 18. :^;t-3^t byi^;*-^*. 19. a-5bya^-J^. 20. 64^-^ + 27y-^ by 4:?;-^ + Zy-^\ 21. o;^ — x]p + i^^?/ - ^^ by o^ — ^-i. 22.. a^ ^-a^l^ ^ifih^ a^ ^a^lh j^l\^ 23. fl^^ + ^>5_c§ +2^3^* by a^ + ^^ + c^. 24. ^^ - '^a^x^ + a' by x^ - %a^ x^ + «. Find the square roots of the following expressions. 25. a:* -4 + 42;-*. 26. (a: + aj-^)2-4 (i?;~ar-^). 27. a^l-'' + 2a^-^ 4- 3 + ^ar^l + a-^Z>l SURDS. 191 XXIX. Surds. 207. It was stated (Art. 147) that when any root of a quan- tity cannot be exactly obtained it is expressed by the use of the sign of Evolution, as \/5, V{3ab), \/(a' + c''); and such quantities are called Iri^ational quantities, or Surds. It is also stated in (147) that there cannot be any even root of a negative quantity, but that such roots may be ex- pressed in the form of surds, as \/ — 3, V— Z>^ V—{a^-\-lf)y and are then called impossible, or imaginary, quantities. 208. Since every fractional index indicates by its denomi- nator a root to be extracted, all quantities having such in- dices are expressed as Surds. When a negative quantity has the denominator of its in- dex (reduced to its lowest terms) even, the expression will be imaginary. Thus, (—3)^, ( — 9)4, are imaginary quantities; but (— 4)ti is not so, since it is the same as ( — 4)^, where the root to be taken is odd, 209. The operations of the preceding chapter are opera- tions on surds, but we may apply the rules which are de- monstrated in that chapter by the use of fractional indices, also to surds expressed by the sign of Evolution, or Radical Sign, 2i0. In the case of a mmierical surd expressed with a fractional index, should the numerator be any other than unity, we may take at once the required power, and so have unity only for the numerator, and simply a root to be ex- tracted. Thus, 2^ = {2')^ = 4* =V4; 3-^=:(3-^)* = (^)^ =V^S, Surds. Surds expressed by the radical sign. 192 ELEMENTARY ALGEBRA. 211. Quantities are often expressed in the form of surds, which are not really so — i.e., when we ca7i, if we please, ex- tract the roots indicated. Thus, Va, Vh {cc' + ah + h'')^ are actually surds, whose roots we cannot obtain; but Va^, \/27, (4a^-|-4«Z> + ^^)J, are only apparently so, and are respectively equivalent to a, 3, 2a-\-l). Conversely, any rational quantity may be expressed in the form of a surd, by raising it to the power indicated by the root-index of the surd. "For example, 3-= v/3'r:. v/9 ; 4=V4' = V64; a^\/a'\ a + h=:l/{a-\-l)y. 212. In like manner a mixed surd, i. e. a product partly rational and partly surd, or a surd with a rational coefficient, may be expressed as an entire, i. e., complete surd, by raising the rational factor to the power indicated by the root-index of the surd, and placing beneath the sign of Evolution the product of this power and the surd factor. An entire or complete surd is one in which the whole expression is under the sign of Evolution. Thus, 2v/3=v/4Xv/3=v^l2; 3. 23=3^4^ V27x 1/4 rr:Vl08 ; iCa /Ca Conversely, a surd may often be reduced to a mixed form, by separating the quantity beneath the sign of Evolution into factors, of one of which the root required may be ob- tained, and set outside the sign. Thus, v/20=:n/(4x5) = 2v/5; V24=V(8x3)==2V3; y/{^a'h)^^a>/{ZaV) ; \/{^a'h'c')=iahV(2ac'). Rational quantities in the form of a surd. Mixed surds and entire surds. SUKDS. 193 213. A surd is reduced to its simplest form, when the quantity beneath the root, or surd-factor, is made as small as possible, but so as still to remain integral. Hence, if the surd-factor be a fraction, its numerator and denominator should both be multiplied by such a number as will allow us to take the latter from under the root. Thus, /2 /2.3 1 ,^ 5 s /24 ^ ;/3 , f/3.5^ ,,^. /3^ /3X2^ /_^^v^6 r ft r ftv2 y Ifi 8X2 ^ 16 4 ' y 2 _ ;/^X9_ yi8_ V18 ^ 3-'^3x9~'^27~ 3 * 214. Surds which have not the same index may be trans- formed into equivalent surds which have. (See Art. 204.) For example, take \/5 and Vll ; v/5 = 5^, Vll = (ll)3 ; 5^ =5^ = V5^= V125, (11)3 =111 =V(11)'=V121. 215. Similar surds are those which have, or may be made to have, the same surd-factors. • Thus 3 Va and Va, "^aVc and ?>Wc, are pairs of similar surds ; bVa and '^VaJ^ are similar surds, because ^^/a" may be written 2 V^^; and \/8, \/50, \/18 are also similar, because they may be written 2v/2, 5\/2, 3\/2. Examples — 44. 1. Express 4^, 9^, 3~s 2"^, (f)"^, (^)-^, with indices, whose numerator is in each case unity. Eeduction of surds lo their simplest form, —to equivalent surds having a given index. Similar surds. 9 194 ELEME]S^TAIIY ALGEBRA. 2. Express 5, 2^, |«, ^a'^, 2(^^ + ^)^ ^^ surds, Avith indices I and I". 3. Express d~% (3^)"^ a~\ ab~'c~\ with indices ^ and Eeduce to complete or entire surds, 4. 5v/5, 2Vh |.3S fN/li i(|)-S 25(li)-^. 5. dV2, 8X2-S 4X2^ 3x3-^ Kf)"', id)"*. 6. 2v^^, 7a%/(22:), (^ + Z^) (^^-Z,^)-i. ^ a ^ 3x 3b ^ 2a 3 ^ 4a' ^ ^ '^ a-\-x Eeduce to tlieir simplest form, 8. v'45,x/125, 3v/432, V135, 31/432, v/f, 2V|, 3Vi 4V3-|. 9. si, 32% 72*, (H)-^, (20i)-t, (30|)-^ fv/-^^, 5V4^V, 10. Show that x/12, Sv'TS, |V147, fV^. VyV and (144)-i are similar surds. 216. To compare surds with one another in magnitude, express tliem. as entire surds, and then reduce them, if neces- sary, to a common siird-index, and simplify as in Art. 213. Their relative values will then be apparent. Thus, 3\/2 and 2\/3, expressed as entire surds, are \/18 and \/12, and it is at once plain which is the greater. To compare \/5 and VH : v/5-:5^=5^r=V125; V11 = 11^=::11*=n/121. We see now that \/5 is greater than Vll. 217. To add or subtract similar surds, reduce them, luhen necessary, to the same surdf actor, then add or subtract their rational factors or coefficients, and affix to the result the com mon surd-factor. > compare surds with one another in magnitude. To add or suhtract similar surds. SUEDS. • 195 Thus, v/8+v/50-v/18 = 2\/2 + 5v/2-3N/2:rr4v/2; ^aV{a'h')-\-bV{^a'h)-V{nba'h') ^A.a'Wl + WbVh-^ hcCWl^a^Wb. 2' Id 1' 7256 2' /12 1' /64xl2 2' /3 1' /256 2' /12 1' /6 27 __2_V12 1 4V12 _ 2V12 ■~3 2 "^4 3 ~" 3 • Dissimilar surds can be added or subtracted only by con- liccting them with their proper signs. 218. To midtiply simple surds which have the same surd- mdex, multiply separately the rational factors and the surd factors, retaining the same surd-index for the product of the latter. Thus, 3v/2X\/3 = 3v/6; 4\/oX7v/6=:28n/30; 2V4X3V2=:6V"8 = 6X2 = 12; 2v/3x3n/10x4v/6 = 24v/180 = 144\/5. 219. To multiply simple surds which have not the same surd-index, reduce them to the same surd-index and proceed as hefore. Thus, 4v/5x2V11 = 4V125x2V121:=8V(125x121) = 8V15125; 2^3 X 3 V2 =2V27 X 3 V4= 6V108. Compound surd-expressions are multiplied according to the method of compound rational expressions. Ex. 1. (2±N/3)=^=:4dz4v/3 + 3 = 7±4y3. Ex. 2. (2+ v/3) (2- v/3) = 4-3 = l. Ex.3. (2+v/3)(3-\/2).= 6 + 3v/2-2v/2- v/6. To multiply simple surds— two cases. 196 ELEMENTARY ALGEBIIA. 220. To divide one simple surd by another^ reduce hotli surds to the same surd-index, ivlien necessary ; then divide the coefficients and surd-factors separately, retaining the common surd-index aver the quotient of the latter. The result may be simplified by Art. 212. — o f 125 X 121 X 121 4V1830125 "3" 121X121X121 3X11 Ex.3, (8^/2-12^/3 + 3^/6-4)-^2v/6 =:4v/|~6v/| + |--^=tv/3-3x/2+|-ix/6. Ex. 4. (2v/3-6V2)-f:\/6=:2v/|-6V-^t6 = %/2-V864. 221. But, if the divisor be not monomial, the division is not so easily performed. The form, however, in which com- pound surds usually occur, is that of a l)inomial quadratic Burd, i, e. a binomial, one or both of whose terms are surds, in which the square root is to be taken, such as 3 + 2\/5, 2v/3— 3v/5, or, generally, s/a^s/h, where one or both terms may be irrational; and it will be easy, in such a case, to convert the operation of division into one of multiplication, hy putting the dividend and divisor in the form of a fraction, and multiplying loth numerator and denominator by that quantity which is obtained by changing the sign between the ttvo terms of the denominator. By this means the denomina- tor will be made rational: thus, if it be originally of the form \/a± \/Z>, it will become a rational quantity, a—b, when both numerator and denominator are multiplied by Va^i y/b. To divide one Bimple surd by another. To divide binomial quadratic surds. SUKDS. 197 2+v/3 _ (2+v/3) (3-n/3) _ 6+3v/3-2v/ 3-3 • 3+v/3~(3+v/3)(3-v/3)~ 9-3 3+V/3 Ex.2. 6 2\/2+\/3 2v/2+v/3 2v/2~v/3~ 8-3 5 This process is called rationalizing the denominators of the fractions, and the fractions thus modified are considered to be reduced to their simplest form. Examples — i.5, 1. Compare 6v/3 and 4v/7; 3V3 and 2V10; 2V15, 4V2, and 3V5. Simplify 2. 3v/2 + 4:n/8-v/32. 3. 2V4:+5V32-V10« 4. 2v/3 + 3v/(li)-v/(5i). 5. 1 1 V2 ~ V16' Multiply -1 -j 6. v/5+%/(li)--;^by v/3. 7. '^^ Vl6+V3^y'^^- 8. l+%/3-v/2by v/6-v/2. 9. v/3+v/2byj3H--i. 10. Divide * 2x/3 + 3v/2 + V30 by 3x/6, and 2^/3 + 3V2+V30 by 3^/2 11. Eationalize the denominators of __1 4 3 8--5v^ 3 + \/5 2^/2-^/3' Vb-1' \/5 + v/2' 3-2v/2' 3-n/5' 12. Eationalize the denominator of —7 "-4: 77 7. V{a-\-x) — y/{a—x) 222. The squar® root of a binomial, one of whose terms is a quadratic surd and the other rational, may sometimes be expressed by a binomial, one or both of whose terms are Eationalizing the denominators of fractions. To express the square root of a Dinomial, one of whose terms is a quadratic surd and the other rational. 198 ELEMEKTAKY ALGEBKA. quadratic surds. (A quadratic surd is one whose indicated root is the square root.) Since {y/x:^Vyy=^x-±:^Vxy + y, therefore Vx^2Vxy-\-y =z^xdci^y'^ hence if any proposed binomial surd can be put under the form x±^Vxy + y, its root may be found by in- spection to be \/x±^/y. To show how to proceed in any proposed case, let us take the binomial 3 + 2\/2. To place this under the form x + 2y/xy-\-y, we observe that 2\/2 =2v'2X\/l, and the sum of the two numbers under the radical signs 2 + 1=3, the rational term of the binomial; /. 34-2\/2=2 + 2\/2X\/l + l. Hence the square root of 3+2v/2='v/2 + 2V2X\/l + l=:\/2 + l. Ex. 2. Kequired the square root of 7— 2\/10. Here2yiO=2v/5X\/2. Also, 5 + 2=7, the rational term. Hence, 7-2x/10=5~2v/5X\/2 + 2; .-. root required is Vo-V2. Ex. 3. Eequired the square root of 11 — 6\/2. Here 6x/2=2v/18=:2v'9X\/2, or 2v/6X\/3; of which the former answers the condition that their sum 9+2=11, the rational term; .-. ll-6y2=:9--2x/9x%/2 + 2. A the required root is \/9 — \/2; that is, 3 — %/2. These illustrations show the method to be, to put the term loMcli contains the surd into factors, of the form ^VxXVy, in such a manner that the sum, x-{-y, of the numbers under the ttvo radicals may be equal to the rational term. Then the y/x'±.\/y loill be the required square root, Ex. 4. Eequired the square root of 7 + \/13. Here N/13 = 2v'y = ^^^¥X^/i. Also, 1^ +-|-=7, the rational term. %/13+l /. root required is x/V + v^i, or n/2 SURDS. 199 Examples — 46. Find the square roots^ of 1. 4+2x/3. 2. ll^-6^/2. 3. 8-2v/15. 4. 38-12v/10. 5. 41-24v/2. 6. 2-1— v/5. 7. 4|-fv'3. 2222^. It is often required to c/mr an equation of surds. An equation may be cleared of a surd by transposing the terms so that the surd shall form one side and the rational quanti- ties the other side, and then raising loth sides to that poiver which will rationalize the surd. In the case of quadratic surds we square both sides. We shall confine ourselves to clearing equations of quadratic surds. Thus if ^ya^-x—h — c, by transposition, '^« + :i'=:^ + c; and, squaring both sides, a + x—{l)-\-cy. We thus have an equa- tion without surds. If the equation contain tivo surds connected by the signs + or — , then the same operations must be repeated for the second surd. Thus, if "^a + x ^s/x—l, by transposition, '^ a-\-x=l)—\/x\ squaring, a-^-x^y^—^bs/x-^rX^ reducing and transposing, %b\fx—lf—a\ squaring, W'x^iV^ — aY, an equation in which the surds do not appear. Ex. 1. ^b-\-x-\- "^b—x—2\/x\ required^. Transposing, ^b-\-x=-2Vx — ^b—x ; squaring, b-\-x—A:X—4:^bx—x''-\-b—x] To clear an equation of surds. 200 ELEMF.XTAllY ALGEBKA. reduc'g and transp'g, 4:V5x—x^—2x; %y/hx—x^= x; squaring, 20x—^x^=x* ; 20x=bx'', .'. 20= 5a;, and a;=4. Examples— 47. Find X in the following equations : 1. ^/(4a;)+v/(42;-7) = 7. 2. V{x-\-14.)+V{x-U)=lL 3. x/(a; + ll)4-\/(a;-9) = 10. 4. v/(9a;+4) + v/(9a;~l)=3. 5. \/(a;+4aZ>)=2a— \/:r. 6. v/(a;-a)+v/(2;-Z>)=x/(a~^>). 7. a+a;~N/(2a:?; + ir')=&. 8. a-{-x^-y/{a^ + 'bx + x^)=b. XXX. Quadratic Equations. 223. Quadratic equations are those in which the square of the unknown quantity is found, but no higher power of it. Of these there are two species : 1. Pure Quadratics, in which the square only of the un- known is found without the first power, as a;'-9=0; ic'-a'=Z^^ &c. 2. Affected Quadratics, where the first power enters as well as the square, as a;'— 3a; + 2=:0; ax^-\-hx^c\ &c. 224. Quadratic equations are (Art. 1G6) also called Equa- tions of tlie second degree. The two species also are distin- Quadratic equations ; two species of, and two ways of designating them. QUADEATIC EQUATI0:N'S. 201 guished as, 1st, Incomplete equations of the second degree ; and 2d, Complete equations of the second degree. We shall, however, generally use the notation of Art. 223.* 225, To solve a Pure Quadratic Equation : Find tlie value of x^ ly the ride for solving simple equa- tions ; then take the square root of both sides of this result, we thus find the value of x, to which we must prefix the douUe sign =b (144). Such equations will therefore have two equal roots with contrary signs. Ex. 1. x^-^=^. Here a:^=9, and x=^^. If we had put db:?;=:±3, we should still have had only these two different values of a;, viz., a; = +3, a;=— 3; since —x—^'d gives x=—d, and —x=—^ gives x= +3. * Ex. 2.' ^(3:?;= + 5)-i(a;'' + 21) = 39-5a;'. Reducing, 121a;'^=:1089; .*. x^=^, and ^=±3. a;^ + 2 9 Ex. 3. a;^-2~ 7' To examples like this the principle of fractions, (Art. 134, vi), may be applied with advantage when the unknown quantity does not enter in both sides of it. * The term affected was introduced by Vieta, about the year 1600. It is used to distinguish equations which involve or are affected with different powers of the unknown quantity from those which contain one power only. (Lund.). To Boh'e a pure quadratic. 202 ELEMEJS^TARY ALGEBRA. {x'-i-2)-lx'-2)~9-7' that is, 2^_16 T'~~2'' Ex. 4. - — =4. ir"— 16, and x—±4c. (134, Yi) V4: + X' 5 X 3' squaring, 4: + x' 25 x' -9' again, (134, iy and i). x' 9 ,9 , , -_-; r,x--; a:-±f. Examples — 48. 1. ix'=U-3x\ 2. x' + 5=^x'--16. 3. (a;+2)^=4^ + 5. '■ it+i-.-»- 3 17 4.x' 6x'~3' a; 7 dx' 15a;'' 4- 8 ^'4 6 ~ 2x' , ^ x^ a;' -10 ^ 50+a? ^- ^- 5 15 =^ 26 . 3:2;*-27 90+4^;^ ^ Ar^^' + S 2.^^-5 7ii;'^-25 ^' _2 , O + 2 . A ~7. lU. 11. 12. a;''^+3 ^ aj'-h9 '10 15 ~ 20 10:r^ + 17 _ 12^N-2 _ 5a;'' -4 18 ■~lla;''-8~ 9~"' 140;" + 16 2a;'' + 8 ___ 2^ 21 ~8a;''-ll~'3"' QUADKATIC EQUATjg^is^S. 203 2 2 ^16 + af-'^25^^_l_ 226. An affected quadratic, or comiDlete equation of tlie second degree^, may alivays he reduced to the form x^ + px + q = 0, ivliere the coefficient of x^ is +1, and p and q represent known numbers, luliole or fractional, positive or negative. For, let all the terms be brought to one side, and, if neces- sary, change the signs of all the terms, so that the coeflScient of ^ may be a positive number ; then divide every term by this coefficient, and the equation takes the assigned form, X' -^px-{-q—^. jN'ow in this equation we have x^ ^px——q', and adding (kvf ^^ ^^^^ side, we get '^^ ^-px'\-\p' =\p^ —q\ by this step the first side becomes a complete square (Art. 153); and taking the square root of each side, prefixing, as before, the double sign to that of the second side, we have x-V\p^±^W^\ which expression gives us, according as we take the upper or lower sign, two roots of the quadratic. 227. From the preceding we derive the following rule for the solution of equations containing an afiected quadratic : By reduction and transposition arrange the equation so that the terms involving x^ and x ai^e alone on one side, and the coefficient of x" is + 1 ; then add to each side the square of half the coefficient of x, and talce the square root of each side, prefixing the double sign to the second. To what form may an affected quadratic always be reduced ? Rule for the solution of equations containing an affected quadratic ? 204 ELEMENTARY ALGEBRA. We thus obtain a simple equation from which x is readily found. Ex. 1. x'-6x==7. Here x'-ex-}- 9=^7 + 9 = 16; whence ^—3 = ±4, and a;=3+4=7, ori^=3-4=-l; so that 7 and —1 are the two roots of the equation. Ex. 2. x''-i-Ux=95. Here a;' + 14a; + 49=: 95 + 49 ==144; whence a; + 7= ±12, and ir= — 7±12=5, or —19. Examples — 49. Solve 1. x''-2x=8. 2. a:' + 10:?;=-9. 3. x^'-Ux^VZO. 4. a;''-12a;=-35. 5. a;' + 322;=320. 6. i^;' + 100a; =1100. 228. If the coefiicient of x be odd, its half will be a frac- tion. Its square may be indicated on the first side bv using brackets. Ex.1. • x'-6x=z-6. Here x''-5x+{iy= -6 + ^=1; whence a;— 1=±|, and a;=| + |=f ==3, or x=^—^=^=2, Ex. 2. x'-'X=l. Here x'-x+{iy=i+i=l; whence cc— ^=±1, and a;=i+l=l|; ora;=J— 1 = — ^. quadratic equations s. 205 Examples — 50. iSolye 1. ^' + 7a;=8. 2. ^j'^-IS^j^GS. 3. x' + 2dx=-100. 4. i?;' + 13ic=-12. 5. a;' + 190:= 20. 6. cc' + 1112:= 3400. 229. If the coefficient of a; be a fraction, its half will, of course, be found by halving the numerator, if possible; if not, by doubling the denominator. Ex.1. Solve x' + y^x=19. Here x' + \^x+{iy=19-i-^^=^^; whence a;+|=±^, and a;= — 1 + ^=3, or x=—^—^='-6^, Ex.2. Solve x'+^x=U, Here x'+h'-x+my=U-hm='^'^y whence x+\^=±^^y and x=-{i+U=n, or ^=_^— 1-|=-10. ♦ Examples— 51. Solve 1. x'-ix=d4.. 2. x'-ix=27, 3. x'+^xz=86. 4. x'-^x=lU. 5. x'-{-j\x=U6, 6. ^'^ -f|:z;=147. 230. In the following examples the equations will first re- quire reduction ; and since the rule requires that the coeffi- cient of x^ shall be + 1, if it have any other coefficient we must first divide each term of the equation by it ; and if its coefficient be negative we must change the signs of all the ' terms. 206 ELEMENTARY ALGEBRA. Ex. 1. Solve -3x' + 20x + 6 = 0. Here 3:^;'- 20:^-5 = 0, and x^—^^-x=^; therefore, x'—^^-x+^^=^^; /, o^^KlOiv'iis)^ the roots being here surd quantities. Ex.2. Solve —. T\+-2 — T=-r- 2(a;-l) x^—1 4 Here we first clear of fractions by multiplying by ^x^^—l), which is the least common multiple of the denominators. Thus 2(a; + l)+12rr:cc'^-l. By transposition, a:'^— 22;=15; adding 1', x'-2x +1 = 16 + 1 = 16; extracting the square root, a;—! = ±4; therefore, a;=ldb4=5, or —3. Ex.3. Solve _+^-^^-=_^_. Multiplying by 570, which is the least common multiple of 15 and 190, ^ 10 + i?; ^ ^ whence 190(^x-50)^^^^_^^ 10 + a; ' and 190(3^-50)=:(210~40^)(10 + ir); that is, 570a;-9500=2100-190^-40a;'; therefore, 40a;'' + 7602:= 11600 ; or a;' + 19a;=290; addmg (^— J , a:'' + 19^+ (^yj =290+-^-=--^ ; QUADRATIC EQUATIOJS^S, 20T 19 39 extracting the square root, x +—= dz— ; whence a;=— — dz— =10, or —29. ^ , ^ , a;+3 i?:— 3 2:?;— 3 Ex.4. Solve -— ^4- ^ = r- x+2 x—2 x—1 Clearing of fractions, (x + d) {x-2) (^-l) + (^-3) {x + 2) {x-1) = (2x-d) {x + 2) {x-2) ; that is, x'-'7x + 6+x'-2x'-6x + Q=2x'~dx'Sx + 12; or 2a;'-2a;'-122;4-12=2a;'-3:z;'^-8a;+12; therefore, x^—4:X=0; adding 2^ * x'-4.x + 2':=4:; extracting the square root, a;— 2 = ±2, whence a;=2±2 = 4 or 0. Remark. — We have given the last three lines in order to complete the solution of the equation in the same manner as in the former examples ; but the results may be obtained more simply. For the equation x^—Ax = ^ may be written (a?— 4)a?=0; and in this form it is sufficiently obvious that we must have either oj— 4=0, or x—0, that is, a^ = 4 or 0. The student will observe that in this example 2a;^ is found on both sides of the equation, after we have cleared of fractions ; accordingly, it Ci,n be removed by subtraction, and so the equation remains a quad- ratic equation. Examples — 52. 6 1. ■ X 3. ^x''-^x=^^{llx + lS). 4. lla;^-9ri;-lli. 5. i{x'-3):=i{x-3), 6. 2r^Hl-:ll(.r+2). S:08 ELEMEJS'TARY ALGEBRA. X 1 ^ a;+33 4 9a;-6 .. a;+3 4-3; _ 4 ^x-Q a; ' ~ 2 " 4 32 12 _JL__?L 12 ^ a;4-l_13 ' 5-a;'^4-a;"a; + 2' a;+l ri: ~ 6 ' 231. Sometimes, on completing the square, the second side of the equation becomes 0. For example, take the equation ic^ — 14:^j = — 49. This giyes a;^-14a; + 49=0 ; .-. {x-iy=0'y .\ x=7. In this case we say the quadratic equation has two equal roots. 232. Solve x'-^x + lZ^O, By transposition, x^ —Qx— — !^ ; adding 3', a;'-6:z:+9 = -13 + 9 = -4. If we try to extract the square root, we have cr-3=:±Vir4. In this case the quadratic equation has no real root, and this is expressed by saying that the roots, are imaginary or impossiUe, 233. An equation of the form ax^-{-'bx-\-c=0, or ax^ ^hx = — c (where a, d, and c are any quantities whatever), may be solved by what is called the Hindoo method,* as follows, without diyiding by the coefficient of x^. Multiply every term hy 4a, that is, 4 times the coefficient of x^ a7id add V, that is, the square of the coefficient of x, to both sides : the first side will be a " complete square." Thus, 4:aV + 4.adx + b'=I?'- 4.ac. * This method is given in the Bija Ganita, a Hindoo treatise on Algebra. QUADRATIC EQUATIONS. 200 Extracting the square root, .'.x=^{^d±>/¥^^^c). (1) Ex. 1. If dx^ + 2:r=85, find x. Multiplying by 4x3, or 12, 3 6a;' +240; =1020; adding 2\ or 4, 36ic*+24a; + 4=1024; extracting root, 6a; + 2=±32; B:r=:±32-2=:30, or -34; /. x=z5, or — 5f. Ex. 2. If 6x' -dx^ 2i= 0, find x. Transposing, 6x^ — 9x= — 2 J ; multiplying by 4x5, or 20, 100:r'— 180^;= -45 ; adding 9', or 81, 100cc'-180a:+81 = 81-45=36; extracting root, lOx— 9 = dz6; whence, 10a;= 9 ± 6 = 15, or 3 ; 15 3 .•.:.=^,or-; = 4, or-. The student will find it well to apply at once (by memory) the formula (1) above obtained for x, Ex. 3. (3:i;-2) {l-x)=4:, or dx'-5x-{-6 = 0. Here a:=:i(5±^^25-72)=l(5±^^^^^), the roots being impossible. Or, since it appears that the equation ax^ + I)x-= — c is re- ducible to the simple equation 2ax-\-b=:dt:^b''—4:ac, the What two abbreviated forms of Bolution are suggested ? 14 ^]0 ELEMENTAEY ALGEBRA. student may readily acquire the habit of writing down the simple equation from any proposed equation without the aid of any intermediate steps. Ex.4. Solve 3x'+6x=^2, 62; + 5 = ±V25 + 12X42; that is, 6a; + 5 = ±23 ; ,\x= 3, or — 4|. EXAMPLES- _f;? ^ 2x 2x-6 _. _ 2x + 9 4.X-3 _ 3x-16 5a; 3x—2 _^ 4a;+7 5— a;_4a; • ^T4~2a;-3~"' ' 19 "^3T^~"9'' x—1 ^-2_2^+13 ^+1 ^+^_2^+13 • x-\-l'^x + 2~ x + 1^' * a;-l'^a;-2~ ic+1 ' 2a;-l 3^-l_5^--ll 14a:-9 a;^-3 a; + l "^ a; + 2 ~ a;-! * * ^~ 8a;-3 "ic + l* 9. a'a;^-2a'i?;+a'-l = 0. 10. 4a'a;=(a^-^^ + a;)^ ^^ X a x h ^^1111 11- r+-=r+-. 12. r+i^^=-+: a X h X ' X x + h a a+b' 234. If r, r' represent the two roots of x^+px+q=0, then —p = r + r\ Sind q=rXr'. YoTr=-lp-hViip'-q), r'^-ip-VUp'-q) ; .-. r + r'=—2y, and rXr'==iy~(iy— g) = ^. Hence, luhen any quadratic is reduced to the form x^' + px + q^O, the coefficient of 2d term, ivith sig?i changed, = su7n of roots, and the 3d term = product of roots. To what are the sum of the roots and the product of the roots of a quadratic equa- tion, respectively, equal ? QUADEATIC EQUATIOXS. 211 Thus, in (Ex. 1, 227) the equation, when expressed in this form, is x^ — 6x—7 = 0, and the roots are there found, 7 and —1; and here -{■6 = '7-\-{ — l) =^ sum of roots, and —7 = 7X (— 1) = product of roots. So, also, ax'^-\-hx + c=0, expressed in this form, becomes x^-\ — x^ — = 0; .-. = sum of roots, — = product a a a a "- 235. If T, r' he the roots of x^ ■\-px-\-q — ^, then x^-\-px-\-q—(x—r^ (x—r'). For, (236) x^ -\rpx-\-q—x^ —(r ^r') x-^r r\ =x^ —rx--r'x+r r' = {x—r) {x—r'). So, also, if r, r' be the roots of ax'^+bx-{-c=0, b c that is, of x''-\ — X-] — =0, a a we have ax'' + bx + c~a (x^-i — x + —]—a{x—r) {x—r'). 236. Hence we may form a quadratic equation with any two given roots. Thus with roots 2 and 3 w^e shall have (^-2) {x-^)=x''-bx-{-^=0. With roots —2 and J, we have {x + 2) (x—D—x' ■\-lx—i =0; or clearing it of fractions, 4iz;^ + 7r?^— 2 = 0. If one of the roots be 0, the corresponding factor will be x—Oy or x. Thus with roots. and 4, we have x{x~4,) — 0, or x^—A:X = 0. (Compare Ex. 4, Art. 230.) In such a case, then, x will occur in every terr)i of the equation, and may be struck out of each; but we must notice, always, that when we thus strike out x from every term of an equation, x—^ satisfies the equation, and is therefore one of the roots. A quadratic equation may be formed from any two given roots. a;=0, a root. 212 elementary algebra. Examples — 54. Form equations with the following roots : 1. 7 and -3. 2. f and -|. 3. -6 and -5. 4. 2| and 0. 5. 10 and -10. 6. a-f— and a . 7. — 1 + n/2 and — 1— \/2. a a m XXXI. Equations which may be solyed like Quadratics. 237. There are many equations which are not strictly quadratics, but which may be solved by the method of com- pleting the square. We will give some examples. 238. Ex. 1. Solve x'-W=^, Adding (|)^ x'-W + {iy=^^-^=^i^; extracting the square root, a;^— |==b|; whence, a;'=|d=|=:8 or —1 ; extracting the cube root, a; ==2 or —1. This method applies, evidently, in all cases where the lowest of the two exponents of the unknown quantity is one- half of the highest exppnent. Ex. 2. a; + 4a;2=21; required x, a;+4a;^+4=r21+4=25; iz;i + 2=:±5; a;*=:±5-2=3 or -7; :,x=^ or 49. What other equations may be solved like quadratics ? QUADRATIC EQUATIONS. 213 Ex.3, a;"' + ^""^=6; required ic. X ^z= — - — =2 or —6; .-. x=l or f 239. Equations may be proposed containing quadratic iTirds, from which, by performing the operations of trans- posing and squaring, once or oftener (Art. 223), we obtain afiected quadratic equations. Ex. 1. x+V6x-{-10 = S ; to find x. By transposition, 'v/52;+10=:8— :?:; squaring, 5a; + 10 =64— 16:?;+ a;'; x'-21x=-54c; 441 441 225 ^«__21a;+— =— -54=— ; 21 ^15 21±15 ^_ .. a;= — ^r — =18 or 3. We have thus found two values of x ; but on trial we find that 18 does not satisfy the equation if we suppose that 1/0:2; + 10 represents the positive square root. The value 18 satisfies the equation x— y 5.^ + 10=8. Ex. 2. Solve 2x-V{x'-3x-3) = d, Transposing, 2x—9 = V{x^—3x—3); squaring, 4:?;^--36a; + 81=cc^— 3.^—3; transposing, 3^;'— 33.^+84=0; dividing by 3, ' x'- llo; + 28=0. 214 ELEMEKTARY ALGEBRA. By solving this quadratic we shall obtain x^=^^l or 4. The value 7 satisfies the original equation ; the value 4 belongs strictly to the equation 2a; + \/(a;^ — 3a;— 3) = 9. Ex. 3. Solve N/(aj + 4) + v'(2:?; + 6)3=N/{8^+9). Squaring, aj4-4 + 2a; + 6 + 2v/(:2; + 4)v/(2a;4-6) = 8a; + 9 ; transposing, 2\/(:?; + 4)\/(2a; + 6) = 52:— 1; squaring, 4(2; + 4) (2a;+6) = 25^^— 10::i; + l; that is, 8a;' + 56a; + 96 = 252;'— 10a; + l ; transposing, 17:?;^— 66:?;— 95 = 0. By solving this quadratic we shall obtain cc=:o, or — f^ The value 5 satisfies the original equation ; the value — ^ belongs strictly to the equation v/(2^ + 6)-v/(:?; + 4)=:x/(8:2; + 9). 240. The student will see from the preceding examples that in cases in which we have to square in order to reduce an equation to the ordinary form, we cannot be certain, with- out trial, that the values finally obtained for the unknown quantity belong strictly to the original equation. 241. Solve x^ ■\-Zx-\-?>sf(x^ ^Zx-%):=^^. Subtracting 2 from both sides, a;'-f3:?;-2 + 3v/(^' + 3a;— 2)=4. Thus on the left-hand side we have two expressions, namely, \/(a^ -\-Zx—%), and x'^ + ^x—2, and the latter is the square of the former ; we can now complete the square. Adding (-|)^ i2;^ + 3^_2 + 3x/(^' + 3a;-2) + (|y=4 + |=^; extracting the square root, N/(a;'' + 3a;-2)+|=zb|; therefore, v/(:r' + 3a;-2) = -f ±f =1 or -4. QUADRATIC EQUATIONS. 215 First siii:)pos8 \/{x^ + 3x—2) = 1 ; squaring both sides, 2;'^ 4-32'— 2 = 1. This is an ordinary quadratic equation ; by solving it we T. n T4- • ^ -3±^/21 shall obtain x= . z l^ext, suppose V{x^-{-3x—2) = —4z. Squaring both sides, r?;^+3:?;— 2 = 16. This is an ordinary quadratic equation ; by solving it we shall obtain a; =3 or —6. Thus on the whole we have four values for x, namely, -3±n/21 3 or —6, or 2 o -4-/9-1 But we shall find, on trial, that only the values ■ 2 will satisfy the given equation x'+3x + 3V{x' + 3x-2) = 6, but the values 3 or — 6 satisfy the equation x'' + dX'-3V{x'-{-dx-2)z=6. 242. The method pursued in the example in the last article applies whenever an expression may be formed which, con- taining all the unknown terms outside of the surd, is the same as the surd expressio?i, or is a rmilUple of it, 2IB. Equations of the third degree are sometimes proposed in which it is intended to find one of the roots by inspection or trial, and the two remaining roots by solving a quadratic equation. x-{-A: x—4: d + x 9—x Ex. 1. Solve X-\r4: 9 — x 9-hx' Bring the fractions on each side of the equation to a com- mon denominator. Thus: What method of solution is explained iu Art. 243? .216 ELEMEKTARY ALGEBRA. 16x d6x that IS, -16""81-cz;' Here it is obyious that x=0 is a root (Art. 236). To find the other roots we begin by dividing both sides of the equa- tion by ^x. Thus : 4__ 9_^ ic^ - 16 ~ 81 -:?;'/ therefore, 4(81~:r^)=:9(a;'-16) ; /. 132;'=324 + 144=468; .-. x' = 36; .-. x=do6. Thus there are three roots of the proposed equation, namely, 0, 6, —6. Ex. 2. Solve x'-7xa' + 6a'=0. Here it is obvious that a;=fl^ is a root. We may write the equation, x^—a'' = '7a\x—a); and to find the other roots we begin by dividing hj x—cu Thus, x''-]-ax+a'=7a^. By solving this quadratic we obtain x=2a, or —3a. Thus there are three roots of the proposed equation, namely a, 2a 3a. Examples — 55. 1. a;'-13.^'4-36=0. 2. x-5Vx-U=0. 3. x-^V{x+^)=::Z 4. x'+y/{x'' + 9)=21. 5. 2v/(a;''-2a:+l)+:?;'=23 + 2a;. 6. x'-2x'+x' = 36. 7. 9V{x'-9x-{-2S) + 9x=x'i-S(>, QUADEATIC EQUATIONS. 217 9. x'-4.x''--2V{x'-4.x' + 4:)=31. 10. a; + 2v/(:?;' + 5:2; + 2) = 10. 11. ^x+^/{x''h'7x + 6) = 19, 12. v/(a; + 9) = 2v/.T-3. 13. 6 s/{l -x')+6x=7. 14. v/(3:c-3)+ N/(5a;~19) = v/(2aj + 8). :?;+ \/(12<^''— ^) a + l 15 X— Vil^a^—x) a- ..> 1 1 1 1 ^ 16. ^ + -H - + ^=0. x + 7 ^— 1 cc + l x—1 x+V{2-x')^x- V{2-x-') ^ ' 19 x-^a x—a b + x h—x x—a x + a h—x b + x' 20. x' + 3ax''=4:a\ XXXII. Problems which lead to Quadratic Equa- tions CONTAINING OnE UNKNOWN QUANTITY. 214. In the solution of problems depending on quadratic and higher equations there may be two or more values of the- root, and these values may be real quantities, or impossible. In the former case, we must consider if any of the roots are excluded by the nature of the question, which may altogether reject fractional, or negative^ or surd answers; in the latter case, we conclude that the solution of the proposed question IS arithmetically impossible. Prob. 1. Find two numbers, such that their sum is 13, and their product is 42: What is said of the different roots of quadratic equations ? 10 21H ELEMEJS^TARY ALGEBllA. Let X be one of the numbers, tben Id—x will be the other ; and then, x{lS-x)=i2; x=7 or 6. .-. 13-a;=:6 or 7. Thus the two numbers are 7 and 6. Here, although the quadratic equation gives two values of i», yet there is really only one solution of the problem. Pkob. 2. What number, when added to 30, will be less than its square by 12 ? Let X = the number ; then whence « = 7or— 6. And here the latter root would be excluded if we required only posi- tive numbers. Pkob. 3. A person bought a number of oxen for $600 : if he had bought 3 more for the same money, he would have paid $10 less for each. How many did he buy ? Let X be the number bought ; then the price actually given for each was — , and therefore, 600 600 ^^ x + S~ X ' whence a^=12 or —15, which latter root is rejected by the nature of the problem. Prob. 4. There are four consecutive numbers, of which, if the first two be taken for the digits of a number, that number is the product of the other two. Let X, x + 1, x + 2, x + S, be the four numbers required; then 10^ + (aj + 1) = the number whose digits are x, and x + 1. Therefore, by the question, (x •^2){x + S) = IO.2; -\-{x + l); or x'' + 5x + e=zllx + l; whence ic = 5 or 1. QUADKATIC EQUATIO^^S. 219 Hence the numbers required are 5, 6, 7, 8, or 1, 2, 3, 4, both of which results satisfy the conditions of the problem. Prob. 5. Find two numbers whose difference is 10 and whose product is one-third of the square of their sum. Let X = the smaller, and a; + 10 = the greater ; then, whence 2J=— 5±5\/^, which values are impossihle. And the solution of the question is arithmetically impossible, as may easily be shown, since it calls for two numbers whose product is equal to the sum of their squares. 245. The reason why results are sometimes obtained, as in Prob. 3, which do not apply to the problem proposed, seems to be that the algebraic language is more general than the ordinary language in which the problem is stated ; and thus the equation which expresses the conditions of the problem will also apply to other conditions. It will be a profitable exercise for the student, when it is possible, by suitable changes in the statement of the problem, to form a new problem, corresponding to the result which was inapplicable to the original problem. Thus in Prob, 3 it will be found that " 15 " oxen is the answer of the following problem : Find the number of oxen bought for $600, when, if the person had bought '^ fewer oxen, he would have paid $10 more per head. Examples — 56. 1. Find the three consecutive numbers whose sum is equal to the product of the first two. 2. The sum of two numbers is 60, and the sum of their squares is 1872 : find the numbers. Why are some of the results ohtainefl inapplicable? 220 ELEMENTARY ALGEBIIA. 3. The difference of two numbers is 6, and their product is 720 : find the numbers. 4. Find three numbers, such that the second shall be two-thirds of the first, and the third one-half of the first, and that the sum of the squares of the numbers shall be 549. 5. Find the number which added to its square root will make 210. 6. There are two numbers, one of which is f of the other, and the difference of their squares is 81 : find them. 7. A and B together can perform a piece of work in 14| days, and A alone can perform it in 12 days less than B alone : find the time in which ^. alone can perform it. 8. In a certain court there are two square grass-plots, a side of one of which is 10 yards longer than a side of the other, and the area of the latter is -^^j of that of the former : what are the lengths of the sides ? 9. A detachment of troops was arranged in a column with 5 more men in depth than in front; the arrangement was changed so as to increase the front by 845 men ; this left the column 5 men deep : find the number of men in the detachment. 10. There is a rectangular field, whose length exceeds its breadth by 16 yards, and it contains 960 square yards: find its dimensions. 11. A person bought a certain number of oxen for $1200, and after losing 3, sold the rest for $40 a head more than they cost him, thus gaining $295 by the bargain: what number did he buy ? 12. The fore-wheel of a carriage makes 6 revolutions more than the hind-wheel in going 120 yards ; but if the circum- ference of each were increased by 3 feet, the fore-whc^el would SIMULTANEOUS EQUATIONS. 221 make only 4 revolutions more than the hind one in the same space : what is the circumference of each ? 13. By selling a horse for £24, I lose as much per cent, as it cost me : what was the prime cost of it ? 14. Find the price of eggs per dozen, when two less in 24 cents' worth raises the price 2 cents per dozen. 15. There are three equal vessels, A, B, and C ; the first contains water, the second brandy, and the third brandy and water. If the contents of B and C be put together, it is found that the mixture is nine times as strong as if the contents of A and (7 had been treated in like manner: find the proportion of brandy to water in the vessel C, XXXIII. Simultaneous Equations involving QUADEATICS. 246. We shall now give some examples of simultaneous equations which may be solved by means of quadratics. There are three cases in which general rules can be given for' the solution of these simultaneous equations of two un- known quantities. 217. I. When one of the equations is of the first degree, and the other is of the second degree : Eule. — From tlie equation of the first degree find the ex- pression for either of the unhnoivn quantities in terms of the other, and substitute this expression in the equation of the second degree. This will give a quadratic equation from which the value' of one unknown is found. Example. Given, 3:r4-4?/=18 ] , ,, -, . ^ ^ ^ ^ r to hnd x and u, hx^ — Zxy~ 2 ) -^ Solution of Simultaneous Equations involving quadratics. Case I. 222 ELEME2!^TAIIY ALGEBEA. From the first equation, y= — ; substituting this value in the second equation, . whence, 20x''—54:X + 9x''=8; that is, 29x''-6^x=8. From this quadratic we shall find that x=2,ov-±; and tlien by substituting these in the first equation we find that 2/=3, or --. 218. II. When the two equations are of the second de- gree, and all those terms which contain x and y are homo- geneous, with respect to these quantities : EuLE. — Put J=YX in both equations; obtain by division an equation in wJiich v is tJie only unknown; v being deter- mined, X and J may then be found. Example. Solve 2x''—xy=z6Q \ 2xy—y^=4.8 ) Putting 't^ — vx and substituting for y, x\2-v)=^Q, and ^^(2^;--^;')=48; whence, by division, 2v-v\ 48 _ 6 ^ 2~v ~56~ 7 ' f n w^hence, v——,or,v=2. The latter value is inapplicable; the first gives, ^=±7, ^— ± 6. This method is also applicable to Case I. Example. x^ + xy-\-y^=l 2x-\ Case n. SIMULTANEOUS EQUATIOIs^S. 223 V Here puttiug vx for y, ' x\l + v + v'')=7 (1); x{2 + dv)=S (2). Therefore, by dividing (1) by the square of (2) x" disappears, and we *^^^'^' (2 + 3^~64' whence, ^=2, or 18; and, from (2), ^i^-^'^)=^^ that is, i ^=^' and i 2/=^^=^ ; or aj(2 + 54)=8; ( or a,'=t; ( or y=vx=2^. 249. III. When each of the two equations is symmetrical with respect to x and y, put u+v for x, and u— y for j, [Definition. — An expression is said to be symmetrical with respect to X and y when these quantities are similarly involved in it. Thus, each of the expressions, x^-{-x^y'^ + y^, 4xy + 6x + 5y—ly 2x'^—3x^y—dxy'^+2y*t is symmetrical with respect to x and y.] Example. x^+y^=18xy\ (1) x + y=12 [ (2). Put u+v for X, and u—v for y ; • then (1) becomes {u+vy+{u—vy = lS{u + v) (u—v), or, u'-{-3uv'=9(u'-v') (3); and (2) becomes {u + v)-{-{u—v)=12; whence, u=6. Putting this for u in (3), • 21Q-{-lSv' = 9{d6-v'); whence, t; = =h 2 ; .-. x—u-{-v=6±:2=8oy4:; and, ^='Z^— -^=6=^2=4 or 8. Case III ;— Symmetrical Equations. 224 ELEMEKTAllY ALGEBRA. 250. The preceding are general methods for the solution of equations of the kind referred to^ and will sometimes suc- ceed also in other equations ; yet in many of these cases a little ingenuity and experience will often suggest steps by which the roots may be found more simply. Ex. 1. Solve 3x'-2xy=16 ) (1) 2x + 3^=^:12 ) (2) Multiplying (1) by 3, 9x''-6xy=4t6, (2) by 2x, 4a;' + 6xy=2^; ', adding, 13cc' = 45-f 24^;, or 13.^^—240; =45, whence x=d or ~ly3^. Equation (2) gives y=^^ (12— 2:?;) =2 or 4if. Ex.2. Solve x'+y'=26) (1) 2:?;^ =24 ) (2) Here adding, x^ + 2x2/ ^y"^ =4:9, whence a; + ^=±7; subtracting, x^ — 2xy-\-y''=: 1, whence iz;—^=:±l. If x+y^ + 'l) or, if x + y^+'l) and x—y=-\-l ) and x—y= — l ) then 22^=8, and x=4:, then 22;= 6, and x=3, also 2y=6, and ^=3; also 2y=8, and ^=4. Similarly, by combining the equation x-{-y= — 7 with each of the two cc— ^=±1, we should get the other two pairs of roots, 0;=— 4, y=—dy and x=—3, y=—4, Ex. 3. Solve x+y = b\ {1) x' + y'^^b) (2) This may be solved by the method of Art. 249, and alstj as follows : Bv division, — -^ =-^ ; x+y 5 that is, x" -xy + ?/ =13. (3) SIMULTANEOUS EQUATIOIs^S. 225 From this equation, combined witli x+y=5, we can f:nd X and 2/ by the first case, or we may complete the soluticn thus : x + y=5; (4) squaring, x^ + 2xy + 7/^=26; also, (3) x''-xy+if=13; therefore, by subtraction, 3:?;?/= 12; or, xy=:4:; and, 4zxy—16, (5) Subtracting (5) from (4); x''—2xt/ + y^=9; extracting the square root, x—y=:iz3. We haye now to find x and y from the simple equations x + y=:6, x—y=^±:3. These give x=l or 4, y=^ or 1. Ex.4. Solve x'+xy+tf = 19, x'+xy + y*=U3, By division, __^:.__; that is, x'^—xy-\^y^=7. We have now to solve the equations x'^ + xy + y'^=zl9, x''—xy-hy^=7. By addition and subtraction we obtain successively a;'+/=z:13, xy=6. Then proceeding as in Ex. 2, we shall find x=dzd or d=2, y=±2 or =t:3. Examples — 57. 1. ^^{3x + 6y)+i{^x-dy) = 6U\ 2. x' + y'=2bl 3.T^-h2/ = 179) x-\-y= 1 ) 22Q ELEMEKTARY ALGEBRA. 3. x'-\-y'=25) 4. 2{x-y) = ll\ 5. x'+xy=^m\ 4^+3^=24) xy=20) x''-y'=.ll) 6. x-ij=2l- '7. x'=\^-y'-4:xyl 16{x'-y'):=:^16xy) x-y=2 ) 8. xy^{x-^) (y + i) I 9, x +y = 6) xY^(x' + d) (^^-4) ) x'-Vf^T2 ) 10. 3^^ + 2a;+^=485 ) 11. x -y — \\ Zx=2y \ x^-f^l9^) 12. a:'+^' = 189) 13. x-^y^a\ 14. xy=a^ \^ x^y^xy''=.V^^) x'+y' = b') x-^y=b 3 15. Vx + Vy=Sl 16. x'-[-:.:i/=a') x+y=9) y'' + xy=lf) 17. l^ + 9{x + y)=2{X'^y)\ ^-{x-y) = {x-y)\ 18. x^—xy=a{x^-l)-\-h-^l, xy—y'^—ay^-l. cc' Z> xy x^ If' ' xy 21. x^=^ax-\-'by, y'^—ay-\-'bx. 251. We shall now giye some problems, to be solved by equations of the second degree, with more than one unknown quantity. Ex. 1. The sum of the squares of the digits of a number of two places is 25, and the product of the digits is 12 : find the number. Let X, y be the digits, so that the number will be \^x^-y\ then 2:^+^^=25, and xy—\2, from which equations we get a; =3, y — ^^ or a: =4, ^=3, and the number will be 34 or 43. In this case both the roots give solutions. Ex. 2. Find two numbers, such that their sum, their product, and the difference of their squares may be all equal. SIMULTAISTEOUS EQUATIONS. 22? Here assume x+y and x—yioT the two numbers; [this step should be noticed, as it simplifies much the solution ol' problems of this kind :] then their sum = 2x, their product —x^^y"^, and the difference of their squares =42;^; .-. (1) 2x^4.xy, (2) 2x=x''~y''] from (1), y=i-; from (2), 2x=x^—^; whence, 2;= |-(2=fc v/5) ; and, .\ x + y=i {3±V6), x—y=i{l±:V6), the numbers required. Ex. 3. A man sets out from the foot of a mountain to walk to its summit. His rate of walking during the second hal£ of the distance is half a mile per hour less than his rate during the first half, and he reaches the summit in 5^ hours. He descends in 3| hours, by walking at a uniform rate, which is one mile per hour more than his rate during the first half of the ascent: find the distance to the summit, and his rates of walking. Let 2x denote the number of miles to the summit, and suppose that during the first half of the ascent the man X walked y miles per hour. Then he took — hours for the first X half of the ascent and — — hours for the second. ^-2 Therefore, ^-|—L-=5i (1). Similarly, ^==3f (2). From (2), 2^=^(f/ + l); 15 therefore, x=—{y-\-l). 4 ~ 8' 228 ELEMEJS'TAllY ALGEBEA. From (1), x(2y-j^=~tj(^y-jy Therefore, by substitution, whence, 16{y + l) {4:y--l)—Uy{2y—l); and, 28^'— 8% + 15 = 0. 5 From this quadratic equation we obtain y—3, or -^, /Co 5 The value -^ is inapplicable, because, by supposition, y is 1 15 greater than — . Therefore, y = 3; and then x=:-^, so that the whole distance to the summit is 15 miles. Examples — 58. 1. The sum of the squares of two numbers is 170, and the difference of their squares is 72 : find the numbers. 2. The product of two numbers is 192, and the sum of their squares is 640 : find the numbers. 3. The product of two numbers is 60 times their differ- ence, and the sum of their squares is 244 : find the numbers. 4. Find two numbers, such that twice the first, w^ith three times the second, may make 60, and twice the square of the first, with, three times the square of the second, may make 840. 5. Find two numbers, such that their difference multiplied into the difference of their squares shall make 32, and their sum multiplied by the sum of their squares shall make 272. 6. Find two numbers, such that their difference added to the difference of their squares may make 14, and their sum added to the sum of their squares may make 26, EXAMPLES. 229 7. Find two numbers, such that their product is equal to their sum, and their sum added to tlje sum of their squares equal to 12. 8. The difference of two numbers is 3, and the difference of their cubes is 279 : find the numbers. 9. A man has to trayel a certain distance, and when he has travelled 40 miles he increases his speed 2 miles per Hour. If he had trayelled with his increased speed during the whole of his journey, he w^ould have arrived at his des tination 40 minutes earlier, but if he had continued at his original speed he would have arrived 20 minutes later : find the whole distance he had to travel. 10. A number consisting of two digits has one decimal place ; the difference of the squares of the digits is 20, and if the digits be reversed, the sum of the two numbers is 11 * find the number. 11. A person buys a quantity of wheat, which he sells so as to gain 5 per cent, on his outlay, and thus clears £16. If he had sold it at a gain of 5 shillings per quarter, he would have cleared as many pounds as each quarter cost' him shil- lings: find how many quarters he bought, and what each quarter cost. 12. Two trains start at the same time from tw^o towns, and each proceeds at a uniform rate toward the other town. When they meet it is found that one train has run 108 miles more than the other, and that if they continue to run at the same rate they will finish the journey in 9 and 16 hours re- spectively : find the distance between the towns, and the rates of the trains. 13. A and B take shares in a concern to the amount, alto- gether, of $2500; they sell out at par — A at the end of 2 years, B at the end of 8 years — and each receives, in capital and profit, $1485: how much did each embark? 230 ELEMENTARY ALGEBRA. 14. Find three numbers, such that if the first be multi- plied by the sum of tha second and third, the second by the sum of the first and third, and the third by the sum of the first and second, the products shall be 26, 50, and 56. XXXIV. Eatio. 252. Ratio is the relation which one quantity bears to an- other with respect to magnitude, the comparison being made by considering what multiple, part or parts, the first is of the second ; or, in other words, what fraction the first is of the second. Thus, if one quantity be two-thirds of another quantity, the former is said to be to the latter in the ratio of 2 to 3, for if both be divided into respectively equal parts, the former will contain two, and the latter three of these equal parts. And thus the ratio of 2 to 3 and. the fraction f, express the same idea ; for f indicates that unity has been divided into 3 equal parts, and two of them are taken. 253. The ratio, then, of one quantity to another, is repre- sented by the fraction obtained by dividing the former by the latter. Thus, the ratio of 6 to 3 is |, or 2 ; that oi aioh is 7-; that of 15 to 40 is 77:, or -; that of 4a to 6Z> is ^, or 2a ^y. Of course the two quantities compared must be of the same kind, or one could not be a fraction of the other. (See Venable's Arithmetic, Art. 171.) 254. The ratio oi aio h is expressed, either by two points placed between the quantities, as a : 5, or for shortness, by its measure, -r-. The first of the quantities, a:!), is called the antecedent term of the ratio, and the latter the consequent. Ratio. Antecedent and Coneequent. EATIO. J>31 255. A ratio is said to be of greater or less ijiequaliiy ac- cording as the antecedent is greater or less than the conse- quent. 25G. If the antecedents of any ratios are multiplied to- gether, and also the consequents, a new ratio is obtained, which is said to be compounded of the former ratios. Thus, the ratio, ac : hd^ is compounded of the two ratios, a : 1), and c : d. When the ratio a : Z> is compounded with itself, the re- sulting ratio is a^ '.h^\ this ratio is called the duplicate ratio of (2 : Z> ; and the ratio a^ : V^ is called the triplicate ratio of a:h, 257. Problems upon ratios are solved by representing them by their corresponding fractions, and then treating these fractions by the ordinary rules. Thus, If the terms of a ratio he riiultiplied or divided hy the same quantity, the ratio is not altered, ^ a ma ' h mo Thus ratios are com/pared with one another by reducing the fractions which measure these ratios to common denomina- tors, and comparing the numerators; and they are com- pounded by multiplying together the fractions which meas- ure them. Thus, also, a ratio may be reduced to its lowest terms by dividing the numerator and denominator of its fraction by their g.c.d. Ex. 1. Compare the ratios 5 : 7 and 4 : 9. ■^iis. 14, H; whence 5 : 7 > 4 : 9. Ex. 2. Find the ratio of | : f . Ans. 4 -^ |-=| X |=:||. Ex. 3. What is the ratio compounded of 2 : 3, 6:7, 14 : 15 ? • • Ans. f X f X ii=TV or 8 : 15. Ratio of greater or less inequality. Componnd, duplicate, and triplicate ra« t7X)s. Solution of problems upon ratios. 232 ELEME^^TARY ALGEBRA. Ex. 4. Reduce to its lowest terms, a''—x^ : a^ ■{■2ax-\'X^. . (a—x) (a + x) a—x Ans. 7 — 7—xr~-. — \= — ; — ? ^^ a—x:a+x, {a + x) [a+x) a+x 258. If to botli terms of the ratio, a : b, the quantity x bo added, that ratio will be increas'ed or diminished according as a is less or greater than b. For, a . ^ ^ a-\-x b + x' if ab + ax . b{b^x) ^ ab-hbx "' < b{b+xy that is. if ab-\-ax > or < ab+bx; that is, if ax > or < bx; or if a > or < b; which shows the truth of the proposition. Examples — 59. 1. Compare the ratios 3 : 4 and 4:5; 13:14 and 23 : 24 ; 3:7, 7 : 11, and 11 : 15. 2. Of a-\-b:a — b and a"^ + b'^ :a^—b^, which is greater, supposing a>b? 3. What is the ratio b inches to c yards ? 4. Find the ratio compounded of 3 : 5, 10 : 21, and 14 : 15 ; of 7 : 9, 102 : 105, and 15 : 17. a ~\~ ax -f- x^ 5. Find the ratio compounded of -^ ^ and . a —ax + ax^—x a^—ax-\-x^ a+x 6. Compound x'-9x+20 :x^-6x and x'-13x+A2 :x^~5x. RATIO. 233 7. Compound the ratios a-\-h : a—d, a^-{-b'^ : {a-hby, {a'-by:a'-I)\ 8. "What is the ratio compounded of the duplicate ratio of a+b : a—b, and the difference of the duplicate ratios of a : a and a : b, supposing a>b? 9. What quantity must be added to each term of the ratio a : b, that it may become equal to the ratio c:d? 10. Show that a—b:a + b^a^—b^:a^ + b'^, according as a: b is Si ratio of less or greater inequality. 11. Find two numbers in the ratio of 3 to 4=, such that their sum has to the sum of their squares the ratio of 7 to 50. 12. Find two numbers in the ratio of 5 to 6, such that their sum has to the difference of their squares the ratio of 1 to 7. 13. Find x so that the ratio x : 1 may be the duplicate of the ratio Six, 14. Find x so that the ratio a —x :b--x may be the dupli- cate of the ratio a : b. XXXY. Proportion. 259. When two ratios are equal, the four quantities com- posing them are said to be proportional to one another ; thus, (t c a\b~c\d\ that is, \i ——~, then a, b, c, d, are proportion- als. Thus, four quantities are proportional when the first is the same multiple, part or parts of the second, as the third is of the fourth. This is expressed by saying a is jfo b as q is to d, and denoted thus, a\b\\c\d\ or thus, a\b = c:d\ or thus, — — —, b d Proportion. 234 ELEMENTARY ALGEBRA. The first and last terms in a proportion are called the ex- tremes, the other two the means. Problems on proportions, like those on ratios, are solved by the use of fractions. 260. (1.) When four qua7iUties are proportionals, the Xyroduct of the extremes is equal to the product of the means, a c For if — = — , then ad—dc, h d (2.) Hence, if three terms of a proportion are given, we can find the other from the equation ad—lc. Thus Ic .ad ad ^ he "=!' ^=T' 'S^ ^=a- (3.) If a : 1=^1 : d, we have ad—I'' ; that is, if the first be to the second as the second is to the third, the product of the extremes is equal to the square of the mean. In this case a, l, d, are said to be in continued proportion, 261. If the product of two quantities he equal to that of tivo others, the* four are proportionals, the factors of either product heing the extremes, and of the other the means. For, let ad— he, dividing by M, , y=^7^ or a\l—c\d, 262. \i a\'b—c\d, and c\d=e:f, then a\h—e:f, _, a c ^ c e For _^_and^=-^; therefore, -7-=-^, or a: J=e: /l ' 263. li a:'b=c:d, and e:f—g:h, then ae:hf=cg:dh. Extremes. Means. Demonstrate Art. 260 (1), (2), and (3). Demonstrate Art. 261; Art. 262; Art. 263. PROPOitTiois". 235 1-1 a c ^ e a For _=_and-^=^; ^ ae eg t.^ ^j-l This is called comjpounding the two proportions. And so we may compound any number of such proportions. Thus, if a\l — c : d, a" : If—c" : d% &c. 2M. If four quantities le proportionals, they are propor- tionals when talcen inversely. That is, if a:h=c:d, then 'b:a=d\c. For (Art. 134, i), if | = |, l-^f=l--|; that is, — = — , or Z>: a=:<^: c. a c 265. If four quantities le proportionals, they are propor- tionals when talcen alternately. That is, if a:d=c'.d, then a : c=h: d. For (Art. 134, u), since -r=-j, -yX — ^-yX — ; d c d G that is, ' — =-7» or a : c=d : d. c a 266. If four quantities are proportionals, the first together loitli the second is to the second, as the third together with the fourth is to the fourth, therefore (Art. 134, iii), —j—— , or a-\-'b : d=c + d:c, c 267. Also, the excess of the first above the second is to the second as the excess of the third above the fourth is to the fourth. Art. 264; Ai't. 265; Art. 266; Art, 267. 236 ELEMENTARY ALGEBRA. (X c For -7- — -7; therefore (Art. 134, iv), by subtracting 1 from each of these equals, a—1) c—d 77 -, , —7 — = — 7-, OY a—o: o~c—d\d. d 2G8. We have also (134, v), a^l) d c:^d d adtzh c^d hade a c or a±Z> : a—G::^d : c, which, by inversion (264), gives a'.a-±^l)=c\c±d. 269. When four quantities are proportionals, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference. For (Arts. 266 and 267), ct-\-l) c-\-d -, a—h c—d therefore, that is, d ^ d ' l ~ d a+d ^ a—h_ci-d c—d^ a—h~c—d'^ or a-\-'h\a'-'b^c-^d\c—d. 270. If four quantities form a proportion, we may der.ye from them many other proportions*, all equally true. Thus, if — =— , then — 7=-^ or ma : m'h~c\ d, d mo d Similarly, ma : h — mc \ d, a\mJ)=c:md, a:d — mc\ md ; and in like manner ad , h d ^ — : — =c:dy a: — =c: - , &c. mm mm Art. 268; Art. 269 ; Art. 270, PK0P0RTI02s\ 237 That is^ either the Jirst or fourth terms of any proi)ortion may be multiplied or divided by any quantity, provided that either the second or tliird be multiplied or divided by the same. Ct G 271. As'ain ; if a:I?=c:d, then — = — ; ° d m a m c ma mc and — X-7 = — X-T? or —.=—.; n n d no nd or ma : nh — mc : na. Then, by the preceding Articles, or by Art. 134, ma±7ib mc±;?f/^ 7)ia '~ mc ^ madcnl) fnc^hnd whence, also, = ; a c or ma dtznb: a = mc±nd: c, A2:am(Art. 269); 7= •. ma— no m 0—714 or ma-{-nI):ma—nd=mc + nd:mc—nd, 272. (1) In like manner, if a:b=c: d—e :/, &c., by which a c 6 it is meant a : h=c : d, or a : h=^e : f, &c., so that —=1—=--:, -^ h d f &c. Then r^ : Z>=a + c+6 : l-^-d^-f, &c. For, let — =ri?^=— =— ; i\\Qn a—hx, c—dx, e=fx\ h d f ^ J ^ therefore, a-^-c-^-e—lx-^dx -\-fx —{h + d 4-/ ) x, .-. a; or — =^-— -— - , or a: o=a-\-c+e:d + d-}-f, o-i-d-\-f That is, if there be a7iy 7iumber of quantities in projjortioii, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the co7iseqttents. Art. 271; Art. 272 (1), (2), (3). 238 ELEMEKTARY ALGEBRA. (I C Again (2) ; the equations above deriyed from — =a;=-y =-^5 &c., give ma^=^mbx^ nc=7idx, pe=pfx, &c. /. 7na-\-nc +2^c — (in h-\-nd -\-pf ^x\ ., „ a ma + nc+pe therefore, . x ot -:r-=^ 7-^; • mo-\-na+pf a _c So also (3), since -.=-^-=:-, &c., r^=- = -; Since d=- (Art. 260), this is ^=i. , T r, a _a -\-c -{-e _ ma + nc -^pe merelore, -_---— --^_^^^^_^^^^^^^ ; and so on for any number of terms and any like powers. Ex. 1. Find a fourth proportional to \, J, and \. a ^ " ' I- Ex. 2. Find a mean proportional to 2 and 8. Since Vz=iaG (Art. 260), this is n/(2x8)= v/16 = 4. Ex. 3. \i a\'b—c\d^ express (a^d) — {b-\-c) in terms of a, Z>, c only. Here {a^-d)-{l^c) = (a^^-^--{M-c) ^a^—ab—ac-\-'bc_{a—b) {a—c) ~^ a "' a ' Examples— 61. 1. Find a fourth proportional to 3, 5, 6 ; to 12, 5, 10 ' to h h h 2. Find a third proportional to 4, 6 ; to 2, 3 ; to |, \, PROPOETIO^^ 2o9 3. Find a mean proportional to 4, 9 ; to 4, |-| ; to 1|, l-^^g-. 4. 1^ a '.!):'. h'.c, then, a^ -^If \a + c:\ a^—W : a — c. 5. If | = |, show that (a^l) (0 + (^)r=^((; + ^)'^r:r|^(a + ^)% 6. If a : Z> : : c : ^, and m:n\\p:q, then ma-\-7id : ma—nb : :pc + qd :pc—qd, 7. If a\h\\l)\c, then a^— Z^'^ : «^ : : Z^"^ — c'^ : c. 8. li a\l)\:c\d:\e:f, then a— e : Z>— /: : c : d 9. If ^ : Z> : : Z> : (?, then mo^—'nF \ma—nc\ ipa^-i-qH^ :pa-{-qc. Solve the equations, 10. -Vx^ y/h'^ \/X'-\/'b=a:'b, 11. 2; + a: 2.?;— ^:=3a; + ^ :4a;— fl^. 12. a;+y + l::r+^+2=:6:7 y + 2:^; : ;z/-2a;=12a; + 62/— 3 : 62/'-12a;-l. 13. xi^'l—y.^—^-.x—y, 14. What number is that to which if 1, 5, and 13 be sev- erally added, the first sum shall be to the second as the sec- ond to the third ? 15. Find two numbers in the ratio of 2^ : 2, such that, when diminished each by 5, they shall be in that of l-J- : 1. 16. A railway passenger observes that a train passes him, moving in the opposite direction, in 2'^, whereas, if it had been moving in the same direction with him, it would have passed him in 30" : compare the rates of the two trains. 17. A quantity of milk is increased by watering in the ratio of 4 : 5, and then three gallons are sold ; the rest being mixed with three quarts of water, is increased in the ratio of 6:7: how many gallons of milk were there at first ? 240 ELEMEA^TAEY ALGEBRA. XXXVI. Arithmetical Progression. 273. Quantities are said to form a Series when they ^7-0- ceed by a laio, i. e., when any one quantity may be obtained from those which precede it by a rule, which is the Imv of the series. 274. Quantities are said to form an Aritlimetical Series, or to be in Arithmetical Progression, when they proceed by a common difference. Thus, the following series are in a.p. : 1, 3, 5, 7, 9, 12, 8, 4, 0, -4, ... . a, a-\-d, a + 2d, a + 3d, .... In the first and third the quantities increase as the series proceeds ; in the second, they decrease ; the common differ- ences being 2, —4, and d, respectively, which are found by subtracti7ig any term from tlie term folloiving ; therefor.?, when the progression is a decreasing one, the common differ- ence is negative. 275. Given a, the first term, and d, the common difference of an Arithmetical Series, to find 1 the n*^ term, and S the sum of n ter^ns. Since a is the first term, and d the common difference, the second term is a + d', the third term is a-\-2d', the fourth term is a + 3d; and so on, where the coefiicient of d is less by one than the number of the term. So in the 71^^ term we shall have {7i—l)d; therefore, the 71"' term l=a+{n-l)d. (1) Again, the sum of the terms, S=a'h{a+d) + {ai' 2d) + &c., -f {I- 2d) + {l-d)+l; Series. Arithmetical Series, or Progression. Common difference, how found; when ne^tive. How to find the last term, and the sum of the scricB. AKITHMETICAL PROGKESSIOK^. 241 and by writing the series in the reverse order, we have also 8=l-{-{l-d)-{-{l-M)-\-kQ. + {a+M) + {a^cl)-\-a. Therefore, by addition, 2/S'=(« + Z) + (^+?) + (a+Z) &c., to n terms; .-. ^8^ {a + 1)71', and since l=a-\'{n—l)d, we have also, S=\^a-^{n—l)cT\—. (3) The equation (2) furnishes the following rule : The sum of any nmnber of terms in A. p. ^5 equal to the product of the numher of terms into half the sum of the first and last terms. Ex. 1. Find the sum of 20 terms of the series 1, 2, 3, 4. Here a=l, d—l, ^^=20; using formula (3), ;S^=r[2 + (20-l)lp/, or =(2 + 19)-2/=21XlO=210. Ex. 2. Find the 9th term, and the sum of 9 terms of 7, 5i, 4, &c. -^ Here ^=7, ^=— |, 7^— 9; .-. Z==7+(9-l)x-|=7-8x|=~5; and ^=1(7-5)^9. Ex. 3. Find the 13th term of the series -48, -44, —40,. &c. Here a= -48, ^==4, n=ld', .-. Z==-48 + (13-1)4=0. Ex. 4. Find the sum of seven terms of i+i + i, &c. Formulas (1), (2), (3). 11 242 ELEMEiifTARY ALGEBRA. Here a — ^, d=—^, 71=7; here, as in Ex. 1, we are not required to find I; ,\ using formula (3), ^zz.(l + 6X-i)l-(l-l)J=0. ' The series continued to seven terms is i, i, ^, 0, —^, —i, Examples — 62. Find the last term and the sum of 1. 2+4 + 6 + &C. to 16 terms. 2. 1 + 3 + 54-&C. to 20 terms. 3. 3 + 9 + 15 + &c. to 11 terms. 4. 1 + 8+15 + &c. to 100 terms. 5. — 5— 3 — 1 — &c. to 8 terms. 6. 14-I + -I+&C. to 15 terms. Find the sum of 7. f +t\ + tt + &c. to 21 terms. 8. 4-3-10~&c. to 10 terms. 9. i+f + 1 + &C. to 10 terms. 10. |_|_i^i_&c. to 13 terms. 11. l + 2|- + 4i + &c. to 20 terms. 12. |-|4— |i— &c. to 10 terms. 276. By means of the equations, (1) l=a+{n~iyd, (2) S = {a + l)j, and (3) S={2a+{n-l)d}^, when any three of the quantities a, d, I, n, S, are given, we may find the others. We may also use these equations to solve many problems in Arithmetical Progression. ARITHMETICAL PROGRESSIOI^r. 243 • Ex. 1. The sum of 15 terms of an A. p. is 600, and the common difference is 5 : find the first term. Since S =600, n=15, and d=o, we have by (3), 600=(26^ + 14x5)-V-; .-. 600=(a + 35)15; .*. a + 35 = 40; .-. a=5. Ex. 2. What number of terms of the series 10, 8, 6, &c. must be taken to make 30 ? ^^=30, ^=10, d=:-2; /. by (3), 30 = [20-2{n-l)]-; .-. {22-27z)^r=:30; that is, 7^' — 11^=— 30, and the roots of this quadratic are 5 and 6, either of which satisfies the question, since the 6th term is 0. Ex. 3. How many terms of the series 3, 5, 7, &c. make up 24? Here ;S'=24, a=3, ^=2; therefore, by (3), 24=r[6+2(^-l)]4^; whence,, n =4,, or —6, of which the first only is admissible by the conditions of the question. 277. Ex. 4. Find the Arithmetical Mean between two quantities a and b. Let X denote this mean ; then since a, x, and b are in A. P. x—a=h—x; 1 a + h \v hence, x=:———; that is, the arithmetical mean between two quantities la half the sum of the quantities. Ex. 5. Insert five arithmetical means between 11 and 23. 244 ELEMENTARY ALGEBRA. Here we have to obtain an A. p. consisting of seven terms, beginning with 11, and ending with 23. Thus, a=ll, Z=23, ^=7; therefore, by (1), Art. 276, 23 = 11 + 6^; .-. d=2. Thus the whole series is 11, 13, 15, 17, 19, 21, 23. 278. Ex. 6. The sum of three numbers in a. p. is 21, and the sum of their squares is 155 : find the numbers. Let X = the middle number, and y the common dif- ference; then x—y, x,x^-y, represent the three numbers; then {x—y)-\'X-\-{x^-y)= 21) and {x-^Jy^Vx'-\-(x■\■yy=zlbb ) or reducing, ^Xz=z 21 ) 3:?;H2?/^=155 f whence, 2;=7, ^=±2; and the numbers are 5, 7, 9. Examples — 63. 1. The first term in an A. p. is 2, the common difference 7, and the last term 79 : find the number of terms. 2. The first term of an A. p. is 13 j-^, the common differ- ence — f, and the last term |: find the number of terms. 3. The first and last of 40 numbers in a. p. are 1^ and If : find the other terms, and the sum of the series. 4. Insert 3 arithmetical means between 12 and 20. 5. Insert 5 arithmetical means between 14 and 16. 6. Insert 7 arithmetical means between 8 and —4. ARITHMETICAL PROGRESSIO:^'. 245 7. Insert 8 arithmetical means between —1 and 5. 8. The first term of an arithmetical progression is 13, the second term is 11, the sum is 40 : find the number of terms. 9. The first term of an arithmetical progression is 5, and the fifth term is 11 : find the sum of 8 terms. 10. The sum of four terms in arithmetical progression is 44, and the last term is 17: find the terms. 11. The sum of fiye numbers in arithmetical progression is 15, and the sum of their squares is 55 : find the numbers. 12. The seventh term of an arithmetical progression is 12, and the twelfth term is 7 ; the sum of the series is 171 : find the number of terms. 13. A traveller has a journey of 140 miles to perform. He goes 26 miles the first day, 24 the second, 22 the third, and so on: in how many days does he perform the journey? 14. A sets out from a place and travels 2i miles an hour. B sets out 3 hours after A, and travels in the same direction, 3 miles the first hour, 3i miles the second, 4 miles the third, and so on : in how many hours will B overtake A, XXXYII. Geometrical Progression. 279. Quantities are said to be in Geometrical Progression when they proceed ly a comvion factor ; that is, when each is equal to the product of the preceding by a common factor. This common factor is called the common ratio, or simply the ratio. Thus the following series are in geometrical progression : Geometrical Progression. 246 ELEMEKTARY ALGEBRA. 1, 3, 9, 27, 81 ... . ^> ^} h iQy it __JL __4 ___l 6 a, ar, ar"^, &c. The commo7i ratios being 3, ^, — f, and r, respectively. The common ratio is found dy dividing any term hy the term ivJiicli immediately precedes it; therefore, if the quantities are alternately + and — , the ratio is negative, 280. Given a tJie first term and r the common ratio of a geometrical series, to find 1 the nth term, and S the sum of n terms. Here, since a is the first term and r the common ratio, the second term is ar, the third term is ar"^, the fourth term is ar^, and so on; where the index of r in any term is less hy one than the number of the term. Thus then the ?zth term l—ar^~'^. (1) Again, 8—a^-ar^ar'^-\-kQ,., +ar"-^; and TS=.ar-{-ar''-\-ar^-\-kQ,,, +ar"-Har"; therefore, by subtraction, rS—S^ar'^—a, the other terms disappearing. Hence, 8— —z=a.^ ^ (2); ox B— (3), smce rl=iar\ Ex. 1. Find the 6th term, and the sum of 6 terms of 1, 2, 4, &c. Here a—\, r=2, n—^\ .•.?=1X 2^=32; and /S'^^^-^ 63. /C — 1 Ex. 2. Find the 8th term, and the sum of 8 terms of 81, -27, 9, &c. The common ratio, how found; when negative. To find the /ith term, and the sum of n terms. GEOMETRICAL PROGRESSJOIn^ 247' Here a— SI, r= — l, n-. therefore^ )re, ?=81x(^|-)'-3*X-^ = -|-3=:-^; and ;S^=:Ii ^ = 60-2-0- — 3— 1 Ex. 3. Find the sum of 8 terms of the series, 4, 2, 1, ^, (&c. • Here a—^, r—^, n—S; therefore, without finding I, a ^V2^"V _ ^V^~2V _255 2_255 i-1 1-i 64^1"" 32* Ex. 4. Find the sum of 1— f +-^— &c. to 4 terms. Here a=l, r=— |, 71 = 4; ,(-iy- 4* . 4^-3* -1 o-i-1 . ^-IX.^— ^^ -»^I — j^- 3 256-81 __175___25 - 7.3' ~ 27' Ex. 5. Find the sum of 2^— 1+|— &c. to 5 terms. Here ^=f? ^=-"1? n=6; (-1)- ••^ -^- _|^i 2- -1-1 2- I __5^ ^ 32 + 3125 _3157__ ""2* 7* 5' ~14.5^~ ^* Examples — 64. Find the last term and the sum of 1. 1+4 + 16 + &C. to4terms. 2. 5 + 20+ 80 +&c. to 5 terms. Formulas (1), (2), (3). 'ZiS ELEMEIs^TARY ALGEBRA. 3. 3 + 6 + 12 + &c.to6terms. 4. 2— 4 + 8- &c. to 8 terms. 5. l-4+16-&c.to7terms. 6. 1-2+2'- &c. to 10 terms. Find the sum of 7. i + i + tV + &c. to 8 terms. 8. i + i + f + &c. to 6 -terms. 9. 3 +i_|.2 +&C. to 6 terms. 10. 3-i + -^j—&c. to 5 terms. 281. If the terms of a geometrical progression decrease numerically as the series proceeds, then the common ratio r is a proper fraction ; that is, r is less than 1. Therefore tho powers r^, r^ r*, .... r'^, are still less than 1, and ar^ less than a. Both the numerator and denominator of the fraction S= z— are then negative, and we may write it ^ a—af a ar^ 1— r "1— r 1— r ISTow, the greater we take the number of terms n, the less will be the value of ar"" ; and therefore, by taking n suffi- ciently great, we may make as small as we please. Hence by making n sufficiently great, we render the value of S as near as we please. 1—r ^ This result we enunciate thus : In a Geometrical Progres- sion in loliich the common ratio is a iwojper fraction, hy talcing a sufficient number of terms the sum of the series can be made to differ as little as ive please from> . is said then to be the Limit of the sum of the series 1—r a, ar, ar"^, &c., when r<l ; or we say, for shortness (but not correctly), S=z is the sum of the series to infinity. Using the same language, these series are called infinite Geometrical Progressions. ' When the common ratio is a proper fraction. The limit of the sum oi a series, when r<l. Infinite Geometrical Progressions, GEOMETRICAL PROGEESSIOK". 249 It is common to denote the Limit of such a sum by 2. Ex. 1. Find the Limit of the sum of the series 1 + i + i + &c. Here a=l, r=i; .*. :s=- — -=— = 2; that is, the more 1— -2- Y terms we take of tliis series, the more nearly will their sum = 2, but will neyer actually reach it. Ex. 2. Fmd the sum of 2i—^ + ^—&G. ad infinitum. Here a=n, r^-^, :. ^=i3|z^= 1^^=1=2 A- 282. Eecurring Decimals are examples of infinite Geo- metrical Progressions. Thus, for example, .3333 .... denotes iV + yfo + 1A-0+&C:, a G. p. of which the first term <^=yV ^^^ ^^ ^^2,^0 r=^. Hence we may say that the Limit of this decimal is Again; .3242424 .... denotes 33_+^|a_ + .^^_2a__+&c. Here the terms after -^-^ form a G. p. of which the first term = y 0^0-0^ ^^^ ^^^ common ratio is y^-g-. Hence the Limit of this series is iQOQ - — ^2^_. Therefore the limiting ^ Too" value of the recurring decimal is 3_ 24 _ 3x99 + 24 3 (100-1) + 24 _ 324-3 ^ 10"^ 990 ~ 990 "" 990 ~ 990 ' and this value accords with the rule in Arithmetic. (See Venable's Arithmetic, Eecurring Decimals.) Examples — 65. Find the Limits of the sums of the following series : L 4+2 + 1+&C. 2. | + ^ + | + &c. 3. i— iV+A + <^^'- Recurring Decimals. 11* 250 eleme:s'taky algebra. 4. I-H-I-&C. 5. l-i + i-&c. 6. l-| + ^-&c. Find the limiting values of the following recurring decimals : 10. .151515 11. .123123123 12. .4282828 283. By means of the equations of Geometrical Progres- , „ 1 rv Ir—a af — a a Bion, yiz., l—ar''-^. 8— -= -, :2=- , we may solve many problems respecting series of this kind. It is not, how^ever, generally easy to find 7i from the other quantities, because it is an exponent. The method of logarithms will serve to find it in all cases. Ex. 1. Find a geometrical series, whose 1st term is 2 and 7th term -^j. Here a=2, 1=-^, n=7; ,-, -^=2r% and r°=-g^, whence r=dt:^, and the series is 2, ±1, i, ±J, &c. Ex. 2. Given 6 the second term of a geometrical series and 54 the fourth, to find the first term. 54 dT^ Here 6=ar, 54=ar^: -'- -yr— — , or 9=r^: 6 ar n hence r=:dt3, a= — = ±:2, r Ex. 3. Insert three geometrical means between 2 and 32. Here we have to obtain a geometrical progression con- sisting of five terms, beginning wath 2 and ending with 32. Thus, a=2, 1 = 32, n=5; therefore, 32=2r^; .-. r'=16; .-. r=2. Thus the series is 2, 4, 8, 16, 32. Ex. 4. How many terms of the series 2, —6, 18, &c., must be taken to make —40 ? GEOMETRICAL PROGRESSIO:^'. 251 Here a=:2; r=-3; S=-4:0; therefore, -3-1 ' hence 2(— 3)"=162; /. (_3)^==:81. But we know that 81=3*; therefore, n=4:. Examples — 66, • 1. Insert 3 geometrical means between 1 and 256. 2. Insert 4 geometrical means between 5^ and 40-|-. 3. Insert 4 geometrical means between 3 and —729. 4. The sum of three terms in geometrical progression is 63, and the difference of the first and the third term is 45 : find the terms. 5. The sum of the first four terms of a geometrical pro- gression is 40, and the sum of the first eight terms is 3280 : find the progression. 6. The population of a country increases annually in G. p., and in four years was raised from 10,000 to 14,641 souls : by what part of itself was it annually increased ? 7. The sum of an infinite geometrical series is 3, and the sum of its first two terms is 2f : find the series. 8. The sum of an infinite geometrical series is 2, and the second term is — | : find the series. 9. A body moves through 20 miles in the first one-mil- lionth part of a second, 18 miles in the second millionth part of a second, and 16| miles in the third millionth part of a second, and so on forever: what is the limit of the dis- tance from its point of departure, which it can attain ? 252 ELEMENTAEY ALGEBRA. XXXVIII. Harmonical Progression. 284. Quantities are said to be in Har7nonical Progression when their reciprocals are in A. p. Thus, since 1, 3, 5, &c., ^, —i, — f, &c., are in aiithmetical progression, their recip- rocals, 1, ^, ^, &c., 2, —2, — f, &c., are in Harmonical Pro- gression. The term Harmonical is apj)lied to series of this character from the fact that musical strings of equal thickness and tension will produce harmony when sounded together, if their lengths be as the reciprocals of the arithmetical series of the natural numbers 1, 2, 3, 4, &c. 285. If three quantities, A, B, and C, are i7i Harmonical Progression, tlien 2uiU A : : : A—B : B— C, For, by definition, -j, ^ , and -^ are in arithmetical pro gression; therefore, 1 2_A i. B^A^G'S' multiplying by ^^(7, AG-BG=AB-AO; that is, G {A-B)=A (B-G) ; therefore. A: G::A—B:B—G, wliicli loas to le ^proved. This property is sometimes given as the definition of Har- monical Progression, and the property gi^^en as the defini- tion in Art. 284 deduced from it. 286. We cannot find a convenient expression for the sum of any number of terms of a harmonical series ; but many problems with regard to such series may be solved by invert- ing the terms, and then treating these reciprocals as in arith- metical progression. Harmonical Progression. Demonstrate Art. 285. To fin(^ the sum of the terms of a Harmonical Progression. HAEMOKICAL PROGRESSION. 253 Ex. 1. Continue to three terms each way the H. p., 2, 3, 6. Here, since -J-, i, ^ are in a. p. with common difference — I", the arithmetical series continued each way is 1. h h h h h ^-h -i therefore the harmonical series is 1,1,1,2,3, 6, infinity, -6 -3. Ex. 2. Insert five harmonical means between f and -fj. Here we have to insert five arithmetical means between | and -1^^-. Hence by equation (1) Art. 275, -i/ =1 + 6(7; there- fore, 6^— f, and d=-^-^''y hence the A. p. is t 2.A 2_6 2 7 J.8. 2JL JL5_ • ^i 165 16, iti 165 165 8 ? and therefore the H. p. is 287. The geometrical mean G between two quantities a and l is the geometrical mean between their arithmetical mean A and their harmonical mean H, Eor, the arithmetical mean between a and l is (Art. 277), A="^. (1) h P The geometrical mean 6^ gives --^=—; .*. G'^a'h and G—^/ab, To find the harmonical mean ZTwe have ———^=-- ; h H H a '%ah therefore, aE—al—al — lH. or H— — -7: (2) a-\-o multiplying (1) and (2), we get AH———X ^ — al—G\ Therefore G—s/ AH, or G is the geometrical mean be- tween A and H, Demonstrate Art. 287. 254 ELEMEKTAKY ALGEBRA. Examples— 67. 1. Continue the Harmonical Progression 6, 3, 2 for three terms. 2. Continue the Harmonical Progression 8, 2, l^- for three terms. 3. Insert 2 harmonical means between 4 and 2. 4. Insert 3 harmonical means between — and — . o /vL 5. The arithmetical mean of two numbers is 9, and the harmonical mean is 8 : find the numbers. 6. The geometrical mean of two numbers is 48, and the harmonical mean is 46^2-g-: find the numbers. 7. Find two numbers, such that the sum of their arith- metical, geometrical, and harmonical means /j 9|, and the product of these means is 27. 8. Find two numbers, such that the product of their arith- metical and harmonical means is 27, and the excess of the arithmetical mean above the harmonical mean is 1|-. XXXIX. Pekmutations and Combinations. 288. The Permutatio7is of any number of things are the different arrangements which can be made of them by placing them in different orders, taking either all the things, or a certain number of them at a time, together. Thus the 'permutations of a, Z*, c, taken all together, are ale, achy hca, cia, cah, lac ; taken two together, are ac, ca, al^ la, Ic, cl. 289. Note. — Some writers on Algebra restrict the word permufa- tions to the case where the things are taken all together, and call tlie PERMUTATIOIsrS A^^D COMBII^TATIONS. 255 sets in all otiier cases Variations^ or Arrangements. But this distinction is not always observed, and we shall use the word permutations in all cases. 290. The number of permutations of n tilings, tciken two together, is n (n— 1) ; taken three together, ^5 n (u — 1) (n— 2). Let there be n different things, a, i, c, d, &c. Remove one of them, a-, there will be ^—1 things, h, c, d, &c., left; now place a before each of these n—1 things; we thus get n—1 permutations, n things taken tiuo together, in which a stands first. JSText remove d from the n things ; there will remain oi—l things, a, c, d, &c. ; and placing b before each of these we get 71—1 permutations of n things tahen two together, in which b stands first. Similarly placing c before each one of the other letters, we find n—1 permutations, in w^hich c stands first; and so on for the rest. Therefore, on the whole, there are n{7i—l) permutations of n things talcen two together, or ttuo and ttoo, as is the usual phrase. Therefore there are also {n—1) {^^— 2) permutations of n—1 things taken tioo together. Let now a, one of the n things, be again removed ; the re- •maining n—1 things, by what w^e have just proved, gives {n — 1) {n—2) permutations when taken tivo together; put a before each of these permutations ; we thus get {7i — l){7i—2) permutations, each composed of three things, in w^hich a stands first. Similarly, there are {7i—l) (^^— 2) permuta- tions each, of three things in which b stands first ; similarly, there are as many in which c stands first, and so on for the rest. Therefore there are 72 {n — 1) {71— 2) per77iu tat ions of 7i thiiigs talcen three together, 291. We observe at once that the second term of the -usi} factor of the product which expresses the number of permu- tations in each case of the preceding article, is numerically less by one than the 7iumber of things taken together. From Permutations. Demonstrate Art. 290. 256 ELEMENTARY ALGEBRA. these cases we miglit infer by induction that this is a general law, and that the number of permutations of 7i letters taken r together is n{;n—l) {n—2) .... (?^— r+l), and this we can now demonstrate. For, suppose this law to hold for the number of permuta- tions of n things, a, l, c, d, &c., taken r—1 together, which would therefore be 71(71-1) (n-'^) .... (^^-(r-l)+l). ]^ow leaye out a\ there will be 7^— 1 things, h, c, d, &c., and the permutations of these, taken r—1 together, will be, by the preceding result, {71-1) {n-2) {7i-l-[r-l)+l); that is, {71-1) {71-2) {7i-r+l). Set a before each of these permutations; we get thus (^—1) (n— 2) {7i—r-\-l) permutations taken r to- gether, in wiiich a stands first. Similarly, we have as many in which b stands first ; as many in wiiich c stands first, and so of the rest ; therefore on the whole there would be, n{7i-l) {71-2) {7i-r+l) permutations of 7i things taken r together.^ If then the formula holds when the 7i things are taken r—1 together, it will hold when they are taken r together; but it has been proved true when they are taken 3 together ; it holds, therefore, when they are taken 4 together ; and therefore, when taken 5 together, and so on ; that is, 7i{7i — l) {71—2) {7i—r+l) represents the Ttwmher of per7nutations of ti tilings taken r togetlie7% for all values of r (these values being limited only by the definition). General law for the number of Permutations of n letters taken r together. PERMUTATIONS AND COMBUST ATIONS. 257 292. Hence, denoting by P^, P^, P^, &c., P^, the number of permutations of n things taken 1, 2, 3, &c. r, together, we have from the preceding formula, P^^n, P^==n(n-1), P,=n{n-l){n-2), &c. P,,=n{n—1) (^— r + 1). 29B. If T=7i, that is, if all the quantities are taken to- gether, then the number of permutations (P) of n things is n{n—l) (71—2) (n—n + l); that is, n{n—l) {n—2) .' . . 1; or reversing the order of the factors, we have, P=1X2X3 . . . . Xn. This result we may enunciate thus : The number of Permutations of n things, taken all together, is equal to the product of the natural numbers from 1 to n, inchisive. Thus the number of permutations of 8 letters, taken all together, is 1x2x3x4x5x6x7x8. 294. For the sake of shortness, the continued product, 1.2.3.4 . . . . n, is often denoted by [^; thus [^ denotes the product of the natural numbers, from 1 to ^ inclusive. The symbol \n may be read factorial n. Ex. [8, (read factorial eighty, denotes the product 1X2X3X4X5X6X7X8. 295. To find the number of permutations of n things, which are not all different, taken n, i, e, all together. Express the formula by the use of the symhols Pi, P21 Pz^ ^tc. The nuii> ber of Permutations where r=:-n. The symbol \n. Demonstrate Art. 295 258 ELEMEis^TARY ALGEBRA. Let there be n letters ; and suppose jp of them to be a's, q of them to be ^'s, r of them to be c's, &c. ; the number of per- mutations of them, taken all together, will be, 1.2.3 . . . . ^ 1.2.3 .... ^Xl.2.3 .... ^X&c* , . For let N be the number of such Permutations. Suppose now that in any one of them we change the ^ a's into dif- ferent letters; then these letters might be arranged in 1.2.3....^ different ways, and so instead of this one permutation, in which p letters would have been «'s, we shall now haye 1.2.3 . . . . p different permutations. The same would be true for each of the iV^ permutations ; hence, if the p a's were all changed to different letters, we should have all together 1 . 2 . 3 .... j^X indifferent permutations of n letters, whereof still q are 5's, r are c's, &c. So, if in these the q b's were changed to different letters, we should have 1.2.3 .... gXl.2.3 . . . .^XiV" differ- ent permutations of n things, whereof still r would be c's ; and so we may go on, until all the n letters are different. But when this is the case we know that their whole number of permutations ==1.2.3 . . . . n. Hence, 1.2.3 ... .i?Xl. 2.3 ... . gX&c.XiV'=1.2.3 . . . n, 1.2.3 . , . . n and -^-12,3 i?Xl.2.3 .... gX&e.' This value of iV^ may be written by the notation of Art. 294; thus, \n Ex. 1. How many changes can be rung with 5 bells out of 8 ? How many with the whole peal ? Here P, =38.7.6.5.4=6720; P=8.7.6.5.4.3. 2 1 = 40320. PERMUTATIOJ^^S Al^D COMBIJsTATIOKS. 259 Ex. 2. How many differ tnt words may be made with all the letters of the expression a^Jfc ? 12 3 4 5 6 Here are 6 letters, 3 a's, and 2 Z>'s ; .-. N= ' ' ' ' ' = 60. JL./V.d X 1"V Examples — 68. 1. How many changes may be rung with 5 bells out of 6, and how many with the whole peal ? 2. In how many different orders may 7 persons seat themselves at table ? 3. How many different words may be made of all the let- ters of the word laccalaureus ? 4. How many different words may be made of all the letters of the word Mississippi? 5. How many different words may be made of all the letters of the word Alabama? 6. Of what number of things are the permutations 720 ? 7. There are 7 letters, of which a certain number are a^% ; and 210 different words can be made of them : how many a's are there ? 8. If the number of permutations of n things taken 4 together is equal to twelve times the number of permu- tations of n things taken 2 together, find n, 296. The Comlinations of any things are the different col- lections or sets that can be made of them, without regarding the order in which the things are placed. Thus the com- binations of a, b, c, taken two together, are ab, ac, bc\ of a, b, c, d, three together, are abc, abd, acd, bed. 297. It is readily seen that each combination which con- tains r things will furnish 1.2.3 r permutations of Combinations. Demonstrate Formula (Art. 298). 260 ELEMENTAKY ALGEBIcA. • these things taken all together. For we haye seen, Art. 291, that^z things give 1.2.3 , . , , n permutations. Thus the combination abc supplies 1.2.3, or 6 permuta- tions, abc, act, lac, lea, cal, da, 298. The numler of comlinations of n different tilings, talcen r together, is n (^—1) (^—2) .... (?^— r+1) 1.2.S r * For, each combination of r things will supply 1.2.3 . . . . r permutations of r things ; hence, if Cr denotes the number of combinations of n things, r together, we have 1.2.3 .... rxC^=number of permutations of n things, r together, = y^—n {n—\) (^--2) (n—r + l) ; ^ _n{n—l) {n—2) .... (n—r + l) .-. G,- 1.2.3 . . . . r • Therefore, (7i=— , C^= ^ ^^' , 0,=^ — ^23 ' ®^-' where Ci, Co, C^, &c., express the number of combinations of n letters taken one and one, two together, three together, &c. «Aa rn\. ' n n{n-l){n- 2) (^-r+ 1) 299. The expression Cr — -^ \ 9,^ ^ ~^^ may be put in a yery conyenient form ; for, by multiplying the numerator and denominator of the aboye fraction by 1.2.3 .... {n—r), it becomes n jn-l) {n-2) (n-r + l) X (n-r) 3.2.1 1.2.3 r X 1.2.3.... (^-r) 1.2.3 n lH: "1.2.3 r X 1.2.3 (?z-r)~" [r | PERMUTATIONS AND COMBINATlOjSfS. 201 ■'■ <^ -7-^- (1) \r \7i — r 300. The number of comUnations of n things taken r to- gether is the. same as the ^lumber of combinations ofn things taken n— r together. For, to find the number of combinations of n things taken n—r together,. we simply write n—r for r in formula (1). We get thus \n \n '^''~ \n—r\7i—{ n—r)^\n—r\)r_^ which is equal to Cr , which was to be proved. The truth of this proposition is also evident from a very simple consideration, viz., that when we take r things from ^^ things, n—r things will be left; and for every different collection containing r things there will be a different col- lection left containing n—r things; therefore the number of the former collections must be equal to that of the latter. Ex. 1. Eequired the number of combinations of 20 things taken 18 together. Here the number of combinations of 20 things taken 18 together is equal to the number of combinations of 20 things taken 2 together, that is, (7is=C,-=^f^=:10xl9 = 190. Ex. 2. Eind the number of combinations of 10 things, 3 and 6 together. ^ ^ 10.9.8 ^__ Ann 10.9.8.7 ^,^ Here C,=^-^j^=120, and C,= C= ^ 2.3.4 ^ Ex. 3. How many words of 6 letters might be made out of the first 10 letters of the alphabet, with two vowels in each word ? state th** r^Tinciple explained in Art. 300. 262 ELEMEKTARY ALGEBRA. In these 10 letters tliere are 7 consonants and 3 vowels; and in each, of the required words there are to be 4 conso- nants and 2 vowels : now the 7 consonants can he combined four together in 35 ways, and the 3 vowels, two together, in 3 ways; hence there can be formed 35x3 = 105 different sets of 6 letters, of which 4 are consonants and 2 vowels : but each of these sets of 6 letters may be permuted 6.5.4.3.2.1 = 720 ways, each of these forming a different word, though the whole 720 are composed of the same 6 letters ; hence the number required=105X 720=75600. Examples — 69. 1. How many combinations can be made of 9 things, 4 together ? how many, 6 together ? how many, 7 together ? 2. How many combinations can be made of 11 things, 4 together ? how many, 7 together ? how many, 10 together ? 3. A person having 15 friends, on how many days might he invite a different p'arty of 10 ? or of 12 ? 4. Fintl the number of combinations of 100 things, taken 98 together. 5. Four persons are chosen by lot out of 10 : in how many ways can this be done? on how many of these occasions would any given man be taken ? 6. The number of combinations of ^+1 things, 4 to- gether, is 9 times the number of combinations of n things, 2 together : find n. 7. How often may a different guard be posted, of 6 men out of 60 ? on how many of these occasions would any given man be taken ? 8. How many words may be formed, each consisting of three consonants and a vowel, out of 19 consonants and 5 vowels. Bi:ts"OMIAL TIIEOKE^.!. ^ 263 XL. Binomial Theorem. 301. The Bi7iomial Theorem is the name given to a rule discovered by Sir Isaac JSTewton, by means of which any bi- nomial may be raised to any given power much more expe- ditiously than by the process of repeated multiplication given in Involution. 302. To prove the Binomial Theorem when the index of the poiver is a positive ivhole ntimher, (Bobillier's Proof.) By actual multiplication the successive powers of the bi- nomial a-\-x are found to be as follows : {a-\-xy=a-{-xi {a-\-xy=:a^-\-2ax-{-x^ ; {a+xy=a^ + da'x-\-dax'+x'' ; (a + a;) * = a' + 4a'a; + 6aV + 4:ax^ + x* ; which, by dividing the first by 1, the second by 1.2, the third by 1.2.3, the fourth by 1.2.3.4, and using the factorial nota- tion of the preceding chapter to denote the continued pro- ducts 1.1, 1.2, 1.2.3, &c., may be written thus : (a + xy _a' x^ [1— |i + li' (a + xy _ a^ g' eg' x^ {a^xy a^ a^ x" a' x^ x^ [3 ""]3 ^ l^'^"[ili^]3 ' {a + xy a' a^ x' ^^ a' x"" o^ in which a laio of formation is easily perceived in relation to the exponent of the power of the binomial. The same Binomial Theorem. Proof of the Binomial Theorem, when the index of the power is a positive whole number. 264 ELEMENTARY ALGEBRA. law of formation of the terms of the expansion is found to hold for {a + xy, (a-\-xy, &c. Now the introduction of the new factor « + a; in order to convert {a + xY~^ into (a + ic)% inyolves precisely the same processes as the introduction of the same factor {a + x) to conyert {a + xY into {a-\-xy. It is reasonable then to assume that if the law is true for {a-\-xy'~^, it is true for [a + xY] now we know by actual multiplication it is true for {a-\-xy\ hence it is true for {a-{-x)\ and hence for {a-{-xy, &c. Therefore the law holds generally — viz., for any positive whole number exponent we have {x-\-aY _ cc** a x""-' a^ x"^"" a^ x""-^ a" ^ \n ~~\n'^\l |/^-"l"^]2~ j^^^ "^ |3 \n-3 "^"j^ ' which may be written, \n \n I. ix + aY=--x'' + TT-T^— ;^^''~' +- [1 \n-l [2 1 ^-2 \n \n + ,^ , ^ 6^V-^ + + — J==— a'x^^. . . ,'\-a\ [3 1^—3 [r \n — r Or by cancelling the like factors in the coefficients, II. {x + aY=x''-{- nax""-' + — ^ — ^ a^x'"-' + —^ —^ ^ [± li [r 303. In these expressions I. and II. the corresponding terms \n Ir In— aV-' (1) and nin-l){n-2) (n-r+l)^^^^^ are the same, and they express the term which has r terms before it; that is, the (r + 1)*^ term. This term is called the BINOMIAL THEOREM. 265 General Term; and both forms of it, (1) and (2), should be carefully noted and remembered. 304. If the binomial is written {a-\-xy, the expansion I. would be \n \n li r^~^ 1^ |y^— 2 ^ \n the general term being , ^ ~_ — a^-V, the exponents of a and X being interchanged. Similarly, II. becomes {a-{-xy=a'' + na'^''x-\' ^\~ ^ a^-V+ + — ^^ — -7 ^ — -^ a^-^a;^+ x\ And if «^=1, this last giyes IIL(l + ^)-l+n^+^A'+ ^^^-^)^^-! )^'.... If l£. w (w-1) (w-2) .... (n-r+l) ^, ^„ 305. I. The law of the exponents of the terms in the ex- pansion of the binomial formula is, that the exponent of the leading letter of the binomial is in the first term n, and that of the second letter is o ; and the former decreases by unity and the latter increases by unity, in each successiye term to the last term, in which the exponent of the leading letter is 0, and that of the second letter is n; and the sum of the ex- pone7its of the tiuo letters in any term is always n, the ex- ponent of the Mnomial. Moreover, the exponent of the second letter in any term expresses the number of terms which precede that term, and the exponent of the leading letter expresses the number of terms which follow it. Thus The law of the exponents of the terms. 12 260 ELEMEKTARY ALGEBRA. we can easily write the exponents of the letters in any required term. II. The numerical coefficients of the first and last terms are 1 ; the coefficient of the second term is 7i, or the number of combinations of 7i things taken singly ; the coefiicient of the third term is the number of combinations of 7i things taken 2 and 2, &c., &c.; the coeflS.cient of the (r + 1)*^ term is the number of combinations of n things taken r together. And since the coefficient of the term which has r terms before it is -] — , , and the coeflBcient of the term which has r vr \n—T \n terms after it or n—r terms before it is -i — =^— , it follows m—r [r that the numerical coefficients of any tivo terms equidistant from the heginning and end are the same, 306. From the above it will be seen that to find the coeffi- cient of any term we may use either of the following rules : KuLE I. — The coefficient of any term is the exponent of the hijiomial, taken factorially, divided hy the product of the ex- ponents of the ttvo letters in that term, taken also factori- ally, i. e., divided hy the product of the number of terms tvhich precede it and the number of terms lohich follow it, taken factorially. Examining Expansion II., Art. 302, we have for finding the coefficient of any term from the preceding term, KuLE II. — Multiply the coefficient of the preceding term hy the exponent of the leading letter in that term, and divide the product hy the numher of terms ivhich precede the required term, NoTE.-^The first rule is used always when we wish to find any terra without finding the preceding terms. Since from Art. 305 all the coefficients after the middle term, or (first middle term when there are two), repeat To lind the coeflScient of any term,— Rule I. and Rale 11. BIK03<IIAL THEOREM. 267 tliemselves, after having found all tlie terms as far as the middle, we may for the remaining terms simply write down the coefficients already found, in an inverted order, as in the following examples. Ex. 1. {a + xy =:««+ Jl_ a^a;+_[|_ aV+Ji_ aV + Jl__ aV + «&c.: 1^ m l\i liii or, / , N8 8,8, 8.7 ,,, 8.7.6 . , 8.7.6.5 , . , = a' + Sa'x + 28aV + 5 6aV + 70a*x' + 5 6aV +28a'x' + Sax''+x\ Ex. 2. {a+xy = a''+^ a'x+1^1 aV + ^l^^a*x'+&c., =a'-{' la'x+^laV + d6a'x'+36a'x'+21aV + 7ax' + x\ 307. If the second term of the binomial is negative, the second term of the expansion is negative, and every alternate term also negative, as is evident by the rule of signs of powers. Thus, {a-xy = a'-la'x^2la'x^-3^aV-\-3ba'x'-2la^x'-\-lax'-x\ 7.6 , 7.6^ = l--'ix-\-Ux'-3^x'-hdbx'-2lx' + W-x\ Ex.4. {3x-\yy 6 .. ..,, ^ . 6.5,. ,_, ,, 6.5.4, Ex.3. (l-«jy = l-4-:c + f^rc^--|:^a;^ + &c. =(3^)^- Y (3^)^(1^)+^ (3a:)* {^yy-^^xy a^)'+&c. = 729a;''-6x243a;^Xi^+15x81:z:*Xiy'-20x27a;'Xi^y' + 15x92:^XTV^*-6x3a;X^^^* + A/ = 729a;' - 729a;> 4- ^-^^x'f - ^\^x^f + W^>* " -h^f + A^'- Ex. 5. Find the 8th term in the expansion (x.-{-ay^ 268 ELEMEIS-TARY ALGEBRA. The exponents of 8tli term give x^a\ |11 Hence the term is -r^ — rr- x'a^ ; or, cancelling out like factors, Ex. 6. Find the middle term of {a—by\ The middle term is the 7th. Hence it is 112 |6 |6 Examples — 70. 1. {l + x)\ 2. (a-dxy. 3. {1-xy. 4. (a-xy, 5. (1+^)'^ 6. (1-2^)^^ 7. {a-3xy. 8. (2x+ay. 9. (2«-3^)^ 10. {l-ixy\ 11. {i-ixy\ 12. Find the 8th term (independently of the rest) in (a--xy. 13. Find the 98th term in {a-hy'\ 14. Find the 5th term in {a'-by\ 15. Find the middle term of {a-{-xy\ 16. {x' + xyy. 17. {a'-x'y, is. {a' + ¥y. 308. The Binomial Theorem is true not only for n a pos- itive integer, bnt for n negative or fractional. But the dis- cussion in this case is not sufficiently elementary for this book. SCALES OF JS"OTATIOK. 269 XLI. Scales of Notation. 309. In the common system of Arithmetic numbers are expressed by the use of 9 figures called digits, and one ci- pher. This is effected, we know, by giving to each digit a local, as well as its intrinsic, value. The local values of the figures increase in a tenfold proportion in going from right to left ; in other words, the local values of the digits pro- ceed according to the poiuers of 10 froin right to left. Thus, 4296 may be expressed by 4000 + 200 + 90 + 6, or 4XlO^ + 2XlO'^ + 9XlO + 6. A system of notation is called a scale. In the common system or scale, the number 10 is called the radix or base of the scale. 310. It is purely conventional that 10 should be the radix ; and therefore there may be any number of different scales, each of which has its own radix. Notation is then the method of expressing numbers by means of a series of powers of some one fixed number, which is said to be the dase of the scale in which the numbers are expressed. (We use the word number here in the sense of whole number,) If the digits of a number JSf, of n digits (including among the digits for convenience), be a^, a^, a^ , . . a^^^, reckoning from right to left, and r be the radix, N may be expressed by the formula, Obs. 1. — It will be noted that since the units figure does not contain r, the highest power of r will be one less than the number of figures in the number expressed. Obs. 2.— In any scale of notation every digit is necessarily less than r, therefore r-1 is the greatest digit, and r-1 expresses the number of digits, and the number of figures used in any scale including is equal to r. Arithmetical Notation. Scale. Radix, or Base. The general formula for ex- pressing numbers. ^70 ELEMElsTTARY ALGEBKA. 311. If r=2 the scale is called the Binary; r=3 Ternary ; r=4: Quaternary ; T=6 Quinary ; r=6 Senary ; &c., &c. T= 10 Denary ; ^=11 Undenary ; r=zl2 Duodenary, The digits, including the cipher, in the Binary scale are 1, ; Ternary ^,2,0; Quinary 1, 2, 3, 4, 0; &c., &c. Nonary 1, 2, 3, 4, 5, 6, 7, 8, 0; Denary 1, 2, 3, 4, 5, 6, 7, 8, 9, 0; but in the duodenary scale we must have two additional characters to express ten and eleven ; we therefore put t for ten, and e for eleven. .•. Duodenary digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, t, e,0; also undenary 1, 2, 3, 4, 5, 6, 7, 8, 9, t, 0. All numbers are supposed to be expressed in the common, or denary scale, unless otherwise stated. 312. To express a given number in any proposed scale. Let N be the number, and r the radix of the proposed scale. Then if «„, a^, a^, &c. be the unknown digits, iV=:an-i^"~' + fl^^2^"~'+ . . ^-ay + ay + a^r+a^. If now N be divided by r, the remainder is a^. If the quotient be divided by r, the remainder is a^. Names of different scales. To express a given number in any given scale. SCALES OF 2s"0TATI0i^. 271 303... ...2 .•. 1st remainder ^^=2 50.... ...3 2d " a=^ 8 ...2 3d " «,=:2 1 ...2 4tli " a, =2 If the second quotient be divided by r, the remainder is a^ ; and so on, until there is no further quotient. Hence the repeated divisions of the given number N, by the radix of the proposed scale, give as remainders the re- quired digits of the number in the proposed scale. Ex. 1. Express 1820 of the common or denary scale, in a scale whose base is 6. 6 1820 6 6 6 6 .-. the number required is 12232. This is easily verified, for 1X6'+2X6'+2X6' + 3X 6 + 2 = 1820. This verification gives the method of transforming a number from any other scale to the denary. By the method of division given above, a number may be transformed from miy given scale to any other of which the radix is given. It is only necessary to bear in mind through- out the process that the radix of the scale of the given num- bers is not 10, but some other number. Or the same thing may be done by first expressing the number (as by verifica- tion above) in the denary scale, and then proceeding as in Ex. 1. Ex. 2. Transform 12232 from the senary scale to the quaternary. To transform a number from one scale to another. 4 12232 4 -2035.. ..0 4 305.. ..3 4 44.. ..1 4 11.. ..0 1.. ..3 272 ELEMENTARY ALGEBRA. (Observe, that in dividing 13 by 4, 12 does not mean twelm, but 1 x6 + 2=:8 ; so also, 23 is fifteen^ 32 is twenty^ and so on; i. e. we must convert each partial dividend to the denary scale as we proceed.) , 1st remainder ^^^^O 2d " a,=:3 3d " a,=l 4tli " ^3=0 5th " a,=0 .'. the number required is 130130. This number transformed to the denary scale is, 1x4^ + 3x4*4-0x4^1x4^ + 3x4 + 0=1820. Ex. 3. Transform 3256 from a scale whose radix is 7, to the duodenary scale. twelve 3256 twelve 166 4 .*. 1st remainder aj,=: 4; 11 1 .-. 2d " a=l] .'. the number required is 814. (Observe in this division that 33 is twenty-three, and the remainder, eleven, is multiplied by 7 and added to the next figure, 5, giving eighty- two for the next partial dividend, &c.) Examples — 71. t. Express the common number 300 in the scales of 2, 3, 4, 5, 6. 2. Express 10000 in the scales of 7, 8, 9, 11, 12. 3. Express a million in the duodenary scale. SCALES OF IS'OTATION". 273 4. Transform 27z^ and 7007 from the undenary to the oc- tenary scale. 5. If the number 803 is expressed by 30203, show that the new scale is the quaternary. 6. The number 95 is expressed in a different scale by 137 : find the base of this scale. 313. The common processes of Arithmetic are all carried on with numbers expressed in any one of these scales as with ordinary numbers, observing that when we have to find what numbers we are to carry in Addition, &c., we must not divide by ten, but by the base of the scale in which the numbers are expressed. Ex.1. Addition, r^A: r=zl 32123 65432 21003 54321 33012 43210 22033 1444 31102 65001 332011 326041 Subtraction, r=3 r=12 7^^348 he^t^ 201210 102221 21212 1^864 Ex. 2. Multiply the numbers 1049 and 1^5 together in the duodenary scale. 1049 lg5 51^9 e443 1049 202329 -duodenary. .*. the product is 202329=2X12^ + 2X12' + 3x12^+2x12 + 9 = 501585~denary. 12^ 274 ELEMEKTAKY ALGEBRA. Ex. 3. Divide 234431 by 414 (quinary), and extract the square root of 122112 (senary). 234431 41 414)234431(310 ' 2302. 122112(252 4 122112 44 234340 423 414 45)421 401 122024 41 542)2012 1524 • 44 814. To find the greatest and least numders expressed hy a give7i number of figures in arty proposed scale. Let r be the base of the scale, and n the number of digits ; then the number will be greatest when every digit is as great as it can be, that is, =r— 1. Thus the number will be (r-l)r"-^+(r-l)r"-2+ .... +(r-l)r' + (r~l)r+r-l; or, (r-1) (r^Hr"-'+ .... ^r^^r-{-l). But the quantity in the second parenthesis is the sum of the terms of a geometrical progression, of which the first term is r"~^, the ratio r, and the last term 1. This is equal to 7-^1 — - . We have then for our greatest number, (r—1) -: or, r"— 1. Again, the number will be least when the digit on the left IS 1, and all the other figures 0, in which case it will be equal to r^\ Ex. 1. In the denary scale the greatest number of 3 fig- ures is 10'- 1:^999; and the least is 10', or 100. To find the greatest and least numbers expressed by a given number of figures in any proposed scale. LOGAKITHMS. 275 Ex. 2. In the senary scale the greatest number of 3 fig- ures is 555 = 6^ — 1 = 215, denary; and the least number of 3 figures is 100 = 6^=36, denary. Examples— 72. 1. Extract the square root of 33224 in the scale of six. 2. Show that 144 is a perfect square, in any scale whose radix is greater than four. 3. Show that 12345654321 is divisible by 12321 in any scale greater than six. 4. Multiply the common numbers 64 and 33 in the binaiy and quaternary scales, and transform each result to the other scale. 5. Divide 51117344 by 675 (octenary), 37542627 by 42?f (undenary), and 29^96580 by "2U^ (duodenary). 6. Extract the square roots of 25400544 (senary), 47610370 (nonary), and 32^75721 (duodenary). 7. Express in common numbers the greatest and least that can be formed with four figures in the scales of 6, 7, and 8. 8. Show that 1331 is a perfect cube in any scale of nota- tion whose radix is greater than three. XLII. LOGAEITHMS. 315. A geometrical progression whose first term is 1 and ratio any number, as a, may be written a\ a', a", a\ a\ a', a', a\ a^ &c., a^ ; and the indices form the arithmetical progression 0, 1, 2, 3, 4, 5, 6, 7, 8 n. 27G elemein^tary algebra. In this A. p. each term measures the order of the ratio of the corresponding term in the geometrical progression to 1.* Hence these indices are called the measures of the ratios of the numbers in G. P. to 1, or the Logarithms of these numbers. 316. From the rules established in the earlier chapters, we know that to multiply or divide any two terms in the first series we have only to add or subtract their indices — i, e,, the corresponding terms in the second series; also, to raise a number of the first series to a given power, we multiply its index or corresponding term in the second series by the in- dex of the power ; also, to extract a given root of any number in the G. p., we divide its index or corresponding term in the A. p. by the index of the root. 317. It is evident from the above that if a geometrical progression can be formed which shall represent with a suffi- ciently close approximation all numbers from 1 to 10,000.. and the terms of the arithmetical progression corresponding to this G. p., in the same manner as in Art. 315, be calculated, and both series be recorded in a table, that much trouble may be saved in arithmetical computation by operating solely on the terms of the A. p., and finding from the table the numbers of the g. p. corresponding to the results. 318. Such tables have been calculated, and are called TaUes of Logarithms, To see how this may be efiected, let a=10 m. the system (Art. 315) ; we have then lo^ 10^ lo^ lo^ lo^ lo^ lo^ lo^ lo^ lo^ &c., (i) and 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, (2) are the logarithms of the corresponding terms of the first series ; that is, in a system of logarithms whose base is 10, * The ratio of a? -.1 is the duplicate ratio of a: 1. The ratio of a^ : 1 is the triplicate ratio oi a\l. And the ratio of a" : 1 is called n times the ratio of a ; 1. Thus the indices Tneasure tJie ratios. LOGAEITHMS. 277 O=log. 10" or log. 1; l=log. 10^ or log. 10; 2=log. 10' orlog. 100; 3= log. 10' or log. 1000; 4=log. 10* or log. 10,000 ; 5=log. 10' or log. 100,000; &c., &c., &c. It is manifest now that the arithmetical mean between any two terms of the series (2) will be the logarithm of the geometrical mean between the two corresponding terms of the series (1). The arithmetical mean of and 1 is —^r— = .5. The geometrical mean between 1 and 10 is ^/fxlO ==3.16227 + ; and therefore .5 = the logarithm of 3.16227. The arithmetical mean of .5 and 1 is .75. The geometrical mean of 10 and 3.16227 is Vl0x3.162"27= 5.62341 + ; whence .75 = the logarithm of 5.62341. The arithmetical mean of 1 and 2 is 1.5. The geometrical mean of 10 and 100 is 31.62277+; whence 1.5 = the logarithm of 31.62277. And by repeating this process, with immense labor, the inventor of logarithms, N"apier (a. d. 1618), and his suc- cessor in these calculations, Briggs (a. d. 1624), calculated tables of logarithms of natural numbers from 1 to 100,000, But (he labor of calculating logarithms is much diminished 278 ELEMEl^TARY ALGEBEA. by the use of series which cannot find place in an ele- mentary work like the present ; besides, as will be seen, the chief labor is with the prime numbers. 319. These logarithms in the common table of which 10 is the base, are the indices (entire or fractional) of the powers to which 10 is to be raised to obtain all natural numbers ap- proximately. Thus, .30103 is the logarithm of 2, means that 2Q.3oio3__2. And this may be yerified by developing 10-^*^^°^ .30103 ~ (1 + 9)ioooob by the binomial formula. 820. 10 is the most convenient base, but any positive number except 1 may be taken as the base. E"apier took 2.71828 as his base. In general, by taking any positive number (except unity) for a base, we may express any posi- tive number as some power of it. And thus logarithms may be defined to be the indices of the powers {entire or frac- tional) to which ice raise a fixed number, called the hase, to oUain the series of natural numhers. Each logarithm is the representative of its corresponding natural number. 321. In the common system, of which 10 is the base, it is clear that the logarithm of every number between 1 and 10 is a decimal fraction ; that of everv number between 10 and 100 is 1 with a decimal fraction annexed; that of any number between 100 and 1000 will be 2 with a decimal frac- tion annexed, &c. The integral part of a logarithm is called the characteristic of the logarithm ; and the decimal part is called the Mantissa, or " handful.'^ Thus is the characteristic of the logarithms of numbers between 1 and 10 ; 1 is the characteristic of the logarithms of all numbers between 10 and 100 ; 2 that of the logarithms of all numbers between 100 and 1000, &c. And in general, the character- istic of the logarithm of any number is alioays less ly unity than the number of figures in the given number. 322. Tables of logarithms, arranged in convenient form, are usually given in books on Trigonometry, and with them LOGARITHMS. 279 explanations of the mode of finding in the table the loga- rithms corresponding to a given number, or the number corresponding to a given logarithm. The table below is a portion of such a table of logarithms. Logaritlims^ to hose 10, of all Prime Numbers f yo7n 1 to 100. No. Logarithms. No. Logarithms. No. Logarithms. No. Logarithms. 2 0.3010300 19 1.2787536 43 1.6334685 71 1.8512583 3 0.4771213 23 1.3617278 47 1.6720979 73 1.8633229 7 0.8450980 29 1.4623980 53 1.7242759 79 1.8976271 11 1.0413927 31 1.4913617 59 1.7708520 83 1.9190781 13 1.1139434 37 1.5682017 61 1.7853298 89 1.9493900 17 1.2304489 41 1.6127839 67 1.8260748 97 1.9867717 323. We will now show more fully the properties of loga- rithms, which render them so useful in diminishing the labor of arithmetical calculations. 324. In the same system, the sum of the logarithms of two numbers is the logarithm of their product ; and the difference of the logarithms of two numlers is the logarithm of their quotient. Let m and n be the two numbers ; lei x = log. m, and y = log. n; let a be the base of the system; then a''=7n, on and a^—n\ hence a'^'^—mn, and a'^^——\ or x-^y is the m log. mn, and x—y is log. — ; that is, log. m + log. ^ =:log. n 7)1/ mn ; and log. m — log. n — log — . n Ex. 1. Log. 6=log. 2+log. 3 = .3010300 + 4771213 =.7781513. Ex. 2. Log. mnp=log. mn + log. p=log. m + log. n-\-log,p. Ex. 3. Log. 5=log. 10-log. 2=:l-log. 2=.6989700. Ex. 4. Log. |=:log. 7-log. 5=.1461280. 280 ELEMENTARY ALGEBRA. Ex. 5. Log. .07=log. T^=log. 7-log. 100:=:. 8450980 -2, which is written thus, 2.8450980; it being understood that in this position of the negative sign it belongs only to the characteristic 2, and not to the mantissa, which is still positive. Ex. 6. Log. ^\=.4771213-1.9867717 = -1.5096504, which maybe written thus: -2 + (1 -.5096504) =2.4903496. S25. If the logarithn of a number be multiplied by m, the product is the logarithm of that number raised to the m.th fower. Let iV^be the number whose logarithm is x\ then a^=N\ therefore a'^'^=N'^\ that is, mx is the log. of N''^\ or log. N'^—mx^m log. N, Ex. 1. Log. (13)^=5xlog. 13 = 5X1.1139434=5.5697170. Ex. 2. Log. b^—%j log. b. Ex. 3. Log. 4= log. 2=^ = 2 log. 2 = .6020600. Ex. 4. Log. (a''-xy=^2 log. (a + a;)+2 log. {^a-x), Ex. 5. Log. (a"'Z>"c^. . .)=m log. a^-n log. b-Vp log. c-f-. . . 326. If the logarithm of a number be divided by m, the quotient is the logarithm of the mth root of that numMr Let a;=log. iV, or a'^—N^ X log. N then arfi — N^^ or log. iV^»n= m m ^ ^ r K^ %-^ .6989700 ,^,^,^^ Ex.1. Log. 5*=—^= J- — =.1747425. Ex. 2. Log. y -|-=— log. a-~ log. b, Ex. 3. Log. Va^—x^=i log. {a + x)-{-i log. (a—x). LOGAEITHMS 281 Ex. 4. Given log. 128=2.1072100 to extract the 7th root of 128. Log. VlM=\ log. 128=1 (2.1072100) =.3010300=log. 2. .•.V'l28=2. Ex. 5. Log. -VTt^^ log. 1-1 log. 71= -I log. 71 =4-X ~L8512583=| (-2+(l-.8512583))=4- (2.1487417). ]^ow in order to diyide this log. by 7, we place it under the form 7 + 5.1487147, so that the negative characteristic may become a multiple of 7 ; then 4- ( 7 + 5.1487147) = f.7355306. 327. We can now see how the work of computing a table of logarithms is facilitated by the application of the above properties of logarithms. For the logarithms of the com- posite numbers are all found by adding together the loga- rithms of their prime factors. 328. While we are at liberty to take any number except 1 as the base of a system of logarithms, we can now under- stand the great advantages of the system which has 10 as a base ; that is, the advantages of having the base of the scale of notation the same as the base of the system of logarithms. For, 1st, the characteristic of the logarithm of any whole number is always one less than the number of figures in the given number. Hence when the number of figures is given we know the characteristic ; and when the characteristic of the logarithm is given, we know the number of figures in the required number. 2d. By every multiplication or division of a number by 10, the characteristic of its logarithm is increased or dimin- ished by unity. For log. 1280= log. (128X10) =log. 128 + log. 10=log. 128 + 1, 282 ELEMENTARY ALGEBRA. log. 128u0=log. 128 + log. 100=:log. 128+2, log. 128000= log. 128+log. 1000=log. 128 + 3, log. 12.8=.log. (128^10)=log. 128-log. 10=log. 128-1, log. 1.28=log. (128 -f- 100)= log. 128-log. 100=:log. 128-2, log. .128=log. 128-log. 1000=log. 128-3, &c. From the tables of logarithms, log. 128=2.1072100. Therefore, log. 1280 =3.1072100, log. 12800 =4.1072100, log. 128000 =5.1072100, log. 1280000=6.1072100, log. 12.8 =1.1072100, log. 1.28 =0.1072100, log. .128 =1.1072100, log. .0128 =2.1072100. We* observe that the logarithms of all numbers which contain the same significant figures, arranged in the same manner, have the same mantissa, or decimal parts ; that is, the mantissa of the logarithm remains the same however we may change the corresponding number, by annexing ciphers or by inserting a decimal point in it, changing the position of its decimal point to the right or left. Moreover, the logarithm of a decimal fraction has a nega- tive characteristic greater by unity than the number of O's between the decimal point and the first significant figure of the number. 329. To find a fourth proportional to three given numbers, using logarithms. Let the numbers be a, b, and c; let x = required fourth be proportional. Then a : b=c:x; .*. x=—. LOGARITHMS. 283 Therefore log. x=log, J + log. c— log. a. Hence tlie rule : From the sum of the logarithms of the second and third terms subtract the logarithm of the first term : the remainder will be the logarithm of the fourth proportional The fourth proportional may then be found from the tables. Note. — In the following examples use the table of logarithms given in Art. 322. Examples — 73. 1. Find the logarithms of 8, 9, 12, 20, 25, 60. 2. Find the logarithms of |, i, f, .03, ■^\, .0033. 3. Eequired the logarithms of 168, 1.04, and 3690. 4. Given the logarithms of 3 and 7 : find the logarithm of 14700. 5. Find the logarithm of 83349, from the logarithms of 3 and .21. 6. Determine the logarithms of V^-^% and \/l.625, by means of those of 2, 3, 5, and 13. 7. Find a fourth proportional to the quantities 1.3, .0104, and 2.375, by logarithms. 8. Find by means of logarithms the number of figures in the results of the involutions of 2^° and 3'^ 9. Find the logarithm of V^J" X V^f" X Vf. ANSWERS TO EXAMPLES. Examples— 1. I. 22, 2. 26. 3. 89. 4. 564. 5. 274. 6. 10. 7. 6. 8. 6. 9. 34. 10. 39. 11. 6. 12. 5. 13. 9. 14. 5. 16. x + a + b, 17. x-}-x-^a + x + a + b. 18. x—a. 19. aS'-^. 20. x~a + b, 21. 2: + a:+2+2:-f2+3. 22. 4xlO + 5+yV 23. 100a;+10^ + ^, 100;^ + 10y + .'?;. 24. X7i + bn. 25. lla;+5. 26. ~. 27. — . "^ ^* 'a; a;4-3' iV— r 28. — y^. Examples — 2. 1. 55. 2. 81. 3. 94. 4. 8. 5. 27. 6. 81 7. 13. 8. 11. 9. 31. 10. 15. •11. 10. 12. 3 3. 3. U. 127. answers to examples. 285 Examples — 4. 1. 5. 2. 16. 3. 9. 4. 224. 5. 459. 6. 7. 7. 74. 8. 12. 9. 8. 10. 238. 11. 420. 12. 144. 13. 43. 14. 15. Examples — 5. 1. 2a+2b. 2. 2a, 3. 2a-2b. 4. 2a, 5. 2a+2^. 6. 2+m + n. 7. 7m— 1. 8. 4:xy-h4:X. 9, p—q-^-S, 10. eah-bc, 11. 15a-9^. 12. dx'-df, 13. 9« + 95+9c. 14. 4:z;+2y + 4^. 15. a—b. 16. 3a;-3a-2Z>. 17. 2a+2b. 18. a + ^ + c. 19. --2« + 2J + 2^. 20. 2a;' - 2x' - 8x + 10. 21. 5^* + 4a;' + 3^' + 2^- 9. 22. 4:a' + 2a'h-4.a¥+b'-n\ 23. a'x + da\ 24. 6aJ-9a'a; + 7aa;' + ^x^ 25. 5a;^ 26. 10a;^ + 83/'4-12a;+12. Examples — 6. 1. 3^ + 45. 2. 4:a+2c, 3. a+5b + 4:C + d. 4. 2.T^-2a;--4. 5. 3a;* -a;' -14a; +18. 6. x'-ax + 2a\ 7. -5a;y-5a;^4-2i/' + 7/^. 8. 3a;' + 13a:y-16a;2;-y'-13y;2. 9. 2a'-6a'b + 6ab'-2b\ 10. 3a;'' 4- 4a; + 16, a;H8a;'. 286 eleme^'tary algebra. Examples— 7. 1. 4.a-4:X. 2. 4:a'-4.a'c. 3. x'-3y'-3z\ 4. 2ax'' + 2by'+2cz\ 5. a'-dd'+3c\ 6. 2ab-i-U\ 7. 0. 8. -3x-y-h4.z. 9. 8a;-8. 10. -4:C + 4:d. Examples — 8. 1. {a-h+c)x'-{I)-c-{-d)x^-{c + d+e)x, 2. 2{ax-dy). 3. {a-\-d)x^—{a—5b)xy-}-{a—c)y\ 4. 2{ax-\-cy), 2b{x+y). 5. — («— 5J):r+(2(^+3Z> + c)y, (aj— 45 — c)^+(«^— 35— 2^)?/, (5— c):c+(3a— c)y. 6. {6a—b)x—{2a-db'-6c)y, —{a-^-c)x+{a—b+2c)y, (4.a-b-c)x-{a-2b-7c)y. Examples — 9. 1. abx^y% —mnx^, 2a'cx'^y, ab'^c^, d^bc^^ -~^y« 2. x^—x'y^-xy'', —a^x-\-d^x^—aQ^^ —abx^-\-(^bx^—ab'^x^ x'y-3x^y^-\'3xY-xy\ 3. 20" ■\-1ab-\-W,2ac-bc-Ud+3bd. 4. ^x' + lZxy + Qy^ Mb'-ab'-Ub*. 5. x'+Qx'-rlx-^e, x'-ex'-^Ux-G. 6. a*+a'-2a'+3a-l, a*-a'-8a'4-a+l. ^ VNIYER" AKS^RS TO ;gXAMPLES. 287 SAUfQ : 7. Six' -if. 8. a''\-S2b\ 9. x'-ia'x+da'. 10. 27a' + ¥ + S-lSab, 11. x'-y'+z' + Sxyz 12. a'-l. 13. a'-8Z>'-27c'-18a^c. 14. a'-h2aW-{-b\ 15. a;'— (a+c)cc'' + («^c + ^)^ — be; x*—(a''—b+c)x' + a{b + c)x—bc 16. 1 — (a— l)a;— (a— d + l)^' + (fl^ + Z>-c)^' — (^>+c)2;* + c:?;'. Examples— 10. 1. a'-2ax+x\ l-^ix^+4:x\ 4a* + 12a' + 9> 9a;''-24a;y^-lG?/^ 2. 9 + 12:^: + 4a;^ 4:i;^-12:ry + 9/, a*-6a'c?;+9aV, b'x*'-2bcx'y + c'xy. 3. 4^^^-!, 9aV-Z>^ a;*-l. 4. x'+4.x + d, x'+3x'-4:, a'b'-qb-Q, 4.aV-Sabx+db\ 5. x'-5aV + 4:a\ 6. mV-13mV2;y + 36^y. 7. 4^1 8. x'+4.y\ 4.a*-6a'b'-}-b\ 9. a' + 2r/^»+^>^-c^ «'^^>''+2^c + c^ a'-b'-2bc—c\ 10. «'-2aJ + ^>''-c^ -a^ + 2«5-^>•'+c^ -^'+Z>'-2Z^c + c'. 11. 4ta'-b' + 6bc-9c', -4:a' + 12ac-hb'-9c\ 12. 4a^-Z>''-6^^c-9c^ -4a'-{-4.ab-b' + dc\ 13. a=+2ac+c'-J'-2^>^-^^ a'' + 2ac?+6Z'-^'-2^^c-c', 288 ELEMENTARY ALGEBRA. + ^ab - ib% a' + 6ac + 9c'- W + Ud- d\ . Examples — 11. I. 5a;'. 2. -3a^ 3. ^xy. 4. -MhW 5. ^a'Wy\ 6. x^-'^x^L 7. -aM-4a-5. 8. x^-Zxy^A.f, 9. 5«^P+a^-4. 10. 15a'^'-12«JH9aJc'-5c\ 11. x-L 12. a;- 8. 13. a;^ + a; + 3. 14. 3ii;'^-2a; + 4. 15. 3:z;'' + 2:z; + l. 16. x^-Zx^l. 17. 2;^ + a;* + 2;' + ^'+i?;+l. 18. a^-\-ab-l\ 19. a;^ + 3^>+9^^''+272/'. 20. x^-x'y + xy\ 21. i?;*+ii^> + i^y + :2^^'+y'. 22. a*-2a^Z> + 46^^Z>^-8t^Z>' + 16^\ 23. 2a'-Mb + l^aV-mh\ 24. a;''+^y+^'. 25. a;' + 2:r2/ + 3/. 26. x^^^x-{-2. 27. a;'-3:c-l. 28. ic'-5a;+6. 29. :z:'-4a; + 8. 30. a;'' + 5:i;+6. 31. x-c, 32. a;'-j9a; + 5'. 33. y'-{m-l)y^-{m-n-l)y''-{m-l)y + l. 34. a' + Z>' + c' + fl^&-flJc + Z>c, a' + ^'+c' + a^ + «c-2><:. 35. a—ax-{-ax^ — ax^-{- -^ — , l + 5a:+15a:' + 45a;' + - — ^ l-{-x l—6x 36. l + 2a: + 3a;^ + 4a;H , ^ , . 1— 2a;4-ar AKSWlrKS TO EXAMPLES. 289 Examples — 12. 1. v.—x,a^-\-a^x^ a^x^ + ax^ + a?^ w'-a^x + a^x^-d^x^ + ax*' — x^, 2. 3rc + l, 5:c-l, 2ic-3. 3. Zmn-h, hn^-n^, 4. l-2z + 4^', 92:' + 3^ + l, l--2a;+4^'-8a;'. 5. ^' + 3a;>+9:?;2/^ + 27^', a*-2^^Z> + 4a'^Z>'-8a^^^ + 16^>*, x''-x'Y + xhf-xY + x'y^-y'\ 6. ia'-iaZ^^-^^ x\f-xYz-\-xyz'-z\ 7. «+Z> + c, a+^— ^. 8. (o^ + y)''— (a; + ^)2; + 2;^=a;'' + 2a:?/+^''— ir^;— y^ + 2;^ Examples— 13. 1. (l-2:r) (l+2.r), (a-32:) (<^+3a;), (3m-27^) (37?i+2^), a;X5a-2) (5a + 2), xy{4:X-6ij) (4a; + 5?/). 2. (^+^) (a;^_:r^ + 2/^), (^-7/) (x'+xtj + f), (l + xy) (l-xy+xY), (^-1) (^+1) (^' + 1). ^y((^y-^") {cty^-x"), 2«Z''c(<^-2c) («+2c). 3. x\bx-a) {bx-\-a), a\a-3I)') {a + SI?'), (2x-S) (4^^ + 6.'?;+ 9), (a-2h) (a' + 2ad + 4.b'), x'y{a + dy) {a''-3ay + 9y"), 4. {x+2){x'-2x'-\-4.af-Sx+16), x'{a + dx) {a' -3ax+9x'), {2x'+f) {4.x'-2xY+/), {ab'+c') {ab'-c") {a'h'^-c^ abc(a-\-cy. 290 ELEMEl!fTARY ALGEBKA. 5. {3x-l) (3a; + l) (dx' + l), {x^2) (x+2) {x' + 2x + 4:) {x'-2x + 4.), x\x-'b)\ x\x-ay (x+a)\ 6. (4a;— 5) (2a; + 1), {a+^h) {a-b), 7{x-y){x+v). 7. {x-yy{x+7/)% {c + a-I)){c-a + I?), Sal?, 8. (x+yY, mn{m--7i), 61){a—b). 9. 2{x+y){4.x-y), 2{x-y){4.y-x), ^y(x-\-y), 10. {a-V'b){a' + ah^l)''), (a-by, 0. Examples— 14. 1. (x + l) {x + 6), {x + 4:) {x + 5), {x-2) {x-d), {x-3) (x-5), (a;+l)(a;+7), {x^l){x-9), 2. {x + 3) {x-2), {x-3) {x + 2), {x-d) {x+1), {x+5) {x-3), {x+S) {x-1), {x-9) {x+1). 3. {2x+3){2x + l), (4a; + l)(a; + 3), (4a;-l)(a;+3), (2a;-3) (2a;+l), (3a;-2) (a;+2), (3a; + 4) (2a;-l). 4. (4a; + l)(3a;-2), 2(6a;~l)(a;-l), (4a; + l)(3a;-l), (a; + 4) {x-3), (3a;-5)(a;+l). 5. a\x—a) {x—2a), a{a—Sx) (a + 2a;), db{Za—2li) (a+5), (2«+a;)(2a-a;)(3a'+a;'). 6. xy{2x'\-y) {x-\-2y), 3y\3x-+2y) {x-y), a\3ax-l) {2ax-\-l), x\2b-dx){db+x). ANSWERS TO EXAMPLES. 291 Examples — 15. 1. 2x'{a + xy. 2. x\a+x)\ . 3. ab{a-l))\ I. 2(^-1). 5. x\x^l), 6. 2(a;+a), 7. a\x-\-l), 8. 3(6?:?; + 2). 9. x-l, 10. iz; + 5. 11. a;-10. 12. x'-x+l, 13. a; + 3y. Examples— 16. 1. 3x-2. 2. 2a; + 3. 3. dx+6, 4. 8a;'' + 14a;- 15. 5. 4a;-5. 6. x' + 2x-d. Examples — 17. 1. 3:r~2. 2. dx-2, 3. 2(:rH2:r+l). 4. y-2. 5. :^;-2«. 6. x+d. 7. 3{:?;-|-3). 8 x' i-y'. 9.'a{a-{-h). 10. a{a'-V). 11. a;^~2:?;^-i'y'. 12. a;'+4a; + 4 Examples — 18. 1. 12aWc, S6x^y% ax^y—axy\ ab^—ad^. 2. 120a*b% 10a'b\ ISOOaV. ?92 ELEMENTARY ALGEBRA. 3. 6{a'-b'), 12a{a'-l), 120xy{x'-y'), 4. 24.a'b\a'-I)'), d6xf{x'-y'). 5. (^ + l)(^H-3)(2;-4). 6. {x + 2){x + 4:){x' + 3x + l), 7. x{2x + l){dx-l){4.x + 3), 8. {x'-5x+6){x-l){x-4:). 9. (^' + 3a; + 2)(:?:-3)(:r+5). 10. {x'+x + l){x'+l){x + l){x-l), 11. 36a^Z^V. 12. 120{a + by{a-hy, 13. 24(a-^)(a'+Z>^). ^ 14. 106ab'{a+I)){a-'b), 15. ^'-1. 16. 0:^-1. 17. x''-l. 18. (.T + 1) (:?:+2)(^ + 3). Examples — 19. 1. 3:;^f -^. 2. 4ac+7r. 3. 2aH- — . 4. 2x--f- 3. ^4. _A^. 6. 2a; ^. 7. a;^ + 3aa; + 3a^+- ^^^' a; + 3* * x—3' * ' x—2a 8. ^-1 — , \ . 9. x'+x'i-x-hl-^-^. x—x + l x—\ ANSWEES TO EXAMPLES. 293 a-^b ' '• 3{a + by x' + l Examples — 20. ^ 2a^x ^^ a-i-b „ a+b . 2ax "%"• ~2b~' ~^:-b' H^-^'y"" h[a—b) ' a—b ' ' x + 6' c cc + T rt CC4-3 ^^ i?;+5 ^^ x—b CK— 5 ic— 7 x-hc x + G iA 3:r— 4 -„ x-\-a—b—c ^. x + 3 IZ. —. ^. Id. — — 1 • 14. 4^;— 3' ' x-\-b—a—c ' x^ — 2x-\-b X~'~0 ^ g^ X -IT u ^ jy X •O- '_ m-Ul- .-r:--! x^ + a' a'+g'y + y a:'- 5a: a-5 "a "' z' ' a^ + J' ' x + b ' «+&• "■ 12a;" 12a;" 12a;" 4(a;-l) 3(a;- l) ix ^'' 40«"-l)' 4(a;^-l)' 4(a;''-l)* a(a; + «) — a;(a;+«) a;' — aa; .294 ELEMENTARY ALGEBRA. fa(a+d) l{a-h) ab V \ _a{^h)(^y^ ^^' {x-iy{x+if {x-^iy{x+iy' {x-iy{x+iy' 4:{x-iy 6{x-l){x+l) {x-iy{x-{-iy' {x-iy{x+iy' ^^ a{x^ + ax+a^) a^—x^ ax an x^+ax-{-a^ x^—ax+a^ a^ 1 x' + a'x' + a'' x' + a'x' + a'' x' + a'x' + a*' Examples — 21. a'4-2>' da''—ab+2¥ 26a -20b 2 • 2{a-{-b)b' 6{a-b)b ' 12 ab a'+b' a^±b^ g'-gh + b^ g-y g-^r a'-b'' a'-b' • ANSWERS TO EXAMPLES. 295 4. -^, 0. ' ^ 6 " "• 4a'- ■r 2a; a-hhx ^ l + x^ + x"" b + ax ' x^{x'^ + iy' ' x+y x^ — y^ ' * a'{a + x)' X —y a\x—a) x-\-y x-^x^-\-Zx^ l-\-2x-\'':^x^ Examples — 22. 1 '^ 5^ x{a-{-b) — ab {x—a){x—by («— a)(«— 5)* 3 ^ 4 ^-^-^ d* v/'* 6 ^ 7. 1. c(c—a) {c—b) Examples — 23. 1. 1^. 2. 1. 4. 1 296 ELEMENTARY ALGEBRA. 5. x—a. ®- ab • 8. ax 9 i^+yy x^^-f a'-xr 11. X abc x-y 14. x" a" y' A. 15. 1. y 7. x^c X-\-l) ^« x'' — ((jr''^oJ'x—a^ Id. ~"T"3~^ 1. 4. d{a-iy b{a+b) 7. a + x cc+y 10. 1 13. 5.^-1 ifi aj'-Ga" 2. Examples — 24. 9cV ^ 1 16a V x+y ^ x{a-^2x) 2x a x—y o. • "• a;— a' * c^-a—V ^-1\' 19 /-^' :ca 2/ 6^^+a^ + l .. fa^ + a^)(^Ha ') 17« . lo. • Examples— 25. 1. -. 2. 1. 3. -^. 4. x^l. X x^\ ANSWERS TO EXAMPLES. 297 5. 1. 6. ^-f. x—5 7. A. 8. 0. 9. i. 10. 2f. 11. 0. 12. 0. 13. a. Examples — 26. 1. w,-27«w^||^, -^f. 2. a;' + 6a;' + 12^ + 8. 3. :?;'-8^' + 24:c'*-32a' + 16. 4. iz;' + lore' + 90:?;' + 270:?;' + 405a; + 243. 5. l + 10a; + 40a;' + 80a;' + 80a;' + 32a;'. 6. Sm'-12m' + 6m-l. 7. 81a;* + 108a;' + 54a;' + 1 2a; +1. 8. 16a;*-32^a;' + 24aV-8a'a;+a\ 9. 243a;' + 810^a;* + 1080a'a;' + 720a'a;' + 24.0a'x + 32a\ 10. 64^' -1446^'^* + 108^5' ~27Z>'. n. aV-3a'xy + daxtf-y\ 12. aV + 4f^V + 6aV + 4aa;'+a;^ 13. 32a'm'-80a'm'+80a'm'-40a'm'+10am'-m'^ U a'-da'b+3a'c-h3ad'-6a'bc + dac'-b'+db'C'-ddc'+c\ 15. l-3a; + 6a;'-7.^'' + 6a;*-3a;'^ + a;^ 298 elementabt algebea. Examples — 27 1. 4:{aO + ad-\-U-^cd). 2. 2(a' + 2«c + c'+Z>' + 2M+6Z'). 3. l+2ir+3^'+2^' + a;\ 4. l-2aj + 3a;'-2a;N-a;*. 5. l + 'Zx-x'-^x' + x'. 6. l + 6^+13.T' + 12r^;' + 4:?;*. 7. l-6a; + 152;'-18^' + 9a:\ 8. 2(4+25^^+16^*). 9. l-^x + Zx^-x' + ^x^-^-x'. 10. l-\-^-{-10x'-\-2Qx'-\-^bx'-^'il4.x' + l%x\ Examples — 28. 1. ±:2a¥c\ dtiWy\ ±10a*Z>V. .3^^ ^-j^ +^^y ^- 5^ ^ 8a ^ 4a&^' aV^ _^' 4^' Ga^g*^ "^^ 2 ^ 3a;^^ 5a* ^ 7 * Examples — 29. 1. x'+x+l. 2. l-a; + 2a;^ 3. a;'+3.T + 8. 4. a;'*-2a;-2. 5. l-2a;+3aJ^ 6. 2a;*-i?;'-2. 7. x'-ax + 2a\ 8. ic''-a2;+S^ 9. a;' - 6a;' + 12a;- 8. 13 ANSWEKS TO EXAMPLES. 299 10. x'-\'2ax^-Wx-a\ 11. l-x + x^-x^-^-x'. 12 |5_^^|^. 13. 1-x, a-2. 14. 2a-35. 15. x''—xi/ + y\ Examples — 30. 1. 421, 347, 69.4, 737, 1046, 4321. 2. 2082, 20.92, 1011, 20.22, 129.63. 3. 1.5811, 44.721, .54772, .17320, 10.535, .03331, .06324, .07071. Examples — 31. 1. x + 2y. 2. a-S, 3. a; + 4. 4. 2^-35. 5. a+Sb, 6. 2x—7y. 7. m—4:nx, 8. ao?— 5Z'2;, 9. a' + 2a + l, 10. a;'-4:z; + 2. 11. a'-ab + b'. 12. a-5+c;. Examples — 32. 1. 21, 23, 25, 32, 4.7, 48, 64, 9.6. 2. 114, 11.7, 125, 108, 1.41, 192. 8. 1.357, .5848, .2154, L587. 300 ELilMENTABY ALGEBRA. Examples— 33. 1. 5. 2. 2. 3. 3. 4. 4. 5. -h .. d—a to. . 7. 3. 8. 1. 9. 4. 10. —\a. 1. -4. 12. |. 13. -|. 14. ^'. 15. x=5. Examples — 34 1. 42. 2. 12. 3. 12. 4. 5. 5. 7. 6. 4. 7. 5. 8. f. 9. 7. 10. ^(25a-18^). 11. 7. 12. If 13. 11. 14. 5. 15. 2^. 16. 3. 17. 2. 18. 4. 19. 2. Examples — 35. 1. 10. 2. 8. 3. 12. 4. 6. 5. -7. 6. 16. 7. 5. 8. 31 9. -6. 10. 5. 11. 8. 12. J. 13. 3. 14. 2. 15. 7. 16. If 17. i. 18. 1. 19. 17. 20. 2. 21. 4. 22. 2. 23. 18. 24. 8. 25. x=2. 26. a;=-||. 27. a;:^-7. 28. :r=4. 29. a:=-L 30. 20. 31. 3. 32. 5. 33. a- 5. ANSWERS TO EXAMPLES. 301 34. b—a, 35. a-\-b 37. ah 38. ^«*,. a + b a+h—c 40. a^l-Vc^d 41. c. 43. i{a + b + 3). Examples — 36. 39. 42. a-hb ' a-hb b—a 1. 12. 2. 9. 3. 120. 4. $1.75. 5. 35, 13. 6. 513, 466. 7. 15. 8. 31, 18. 9. 15. 10. 90, 60. 11. IsToYember 20tli. 12. 16. 13. 37, 30, 20. 14. 20. 15. 41. 16. 88. 17. $36, $12, $16. 18. 5. 19. £45, £57, £63, £65. 20. 15, 5. 21. 98f miles from B ; lOf hours. 22, 10, 14, 18, 22, 26, 30. 23. 28, 14. 24. 88, 44. 25. 5, 6. 26. 22, 7, 12 gallons. 27. 3000. 28. 18, 3, 3. 29. 24000. 30. £140. Examples — 37. 1. 45 gallons. 2. 2450, 196, 98. 3. 84. 4. 15 feet by 11 feet. 5. 20 lbs., 15 lbs., 15 lbs. 302 ELEMENTAKY ALGEBEA. 6. $240. 7. 3i days. 8. 75. 9. 1504. 10. 1540, 880, 616. 11. 10 lbs. 12. 18, lOf, 6i days. 13. $1.05, $1.17. 14. 6f oz. 15. 654. 16. 76,30. 17. 21-3^ hrs., lOJ^lirs. 18. 12,16. 19. 10, 15, 3, 60. 20. 240, 180, 144 days. 21. 12. 22. 20, 80. 23. 5^^. 24. 240. 25. 24. 26. 60. 27. 25. 28. 7 hours, 5-^', 6 hours, 16^'. 29. 40 minutes past eleven. 30. $100000000, 31. 7, 15, 48. 32. 189. Examples — 38. ^ mna « m{nb—a) n{a^ml) m+n * n—m ^ n—m ma na 4. m-\-n m + n mpa npa nqa mp+np + nq^ mp-i-np + nq^ mp-hn^J-^nq ^ ml—na ^ dbc „ d 5. . 6. -1 TT-. 7. n—m ' ab + ac + bc ' b + c be b + c' ANSWEES TO EXAMPLES. 303 EXAMPLES-~39. 1. 10; 7. 2. 17; 19. 3. 2; 13. 4. 4; 1. 5. 5; 5. 6. 21; 12. 7. 19; 2. 8. 38i; 70. 9. 6; 12. 10. fif; IM- "• 5; 7. 12. 2^; 1. 13. x=l, y=ri. II. ic=10, y=24 15. a;=144, 2/=:216. 16. .2; .2. 17. 10; 8. 18. 12; 3. 19. 3; 2. 20. a; J. 21. a\ I. 22. -^; -^,. 23. ^; .. 24. ^gr; -gj. 25. -^; -^. 26. -^; 0. 27. «; I, a + b^ a + b a+b Examples — 40. 1. x=l, y=2, z=d. 2. x=7, y=10, z=9. 3. x=6, y=6) z=:7, 4. x=4:, y=z—6, z=6. 5. a;=3-5, y=6, z=-2, 6. a;=l|, i/=2f, ^=:-12. 7. x=:2, y=-dy z=:L 8. a:=12, y=12, z=l2. 9.x=6,y=:7,z=-3. 10. |; f; f. 11. a;=i(J + c--a), &c. 12. x=%{a + b + c)-'a, &c. 13. a:=K^ + c),&c. 14. x=y=z=:- ^^^ ^ab + bc+ca 304 ELEMENTARY ALGEBRA. Examples — 41. 1. ^5g. 2. 48. 3. 108 sq. ft. 4. 4 liours, 6 hours. 5. 20, 30, 60. 6. 24, 72. 7. 49; 21. 8. 45; 63. 9. |. 10. (24-1)20. 11. 1 ; 2. 12. 50 yards; rates 4 and 5 yards per minute. 13. 11, and 5, gallons. 14. A. D. 1752. 15. 50; 75. 16. 90; 72; 60. 17. 4; 2. 18. 8; 5. 19. 4 miles walking, 3 miles rowing, at first. 20. 30 ; 50 miles per hour. 21. 60 miles; passenger train 30 miles per hour. 22. 150; 120; 90. 23. rr=40, ^=160, ^^=480. a-\-b a—b 24. 25. 771C — all + am . {n—h) ^ i7i—nc + dn{a—7n) rob— an ^ mh—an l + a* a^—h 2n 2n 2a ' 2a ' ' m—V m + 1* Examples — 42. 1. x^-\-x^, +2:^, +a:3; ah"^ + ah^ + ah"" -^ a^. 2. ah^-\-a^-Va'b^+ah^) ah'''\-al)'+ah' + ah\ ANSWERS TO EXAMPLES. 305 a'b-' + 3a'd-' + 6ab-'+4.a-'b + 2a-^b^ ; 1 1 A _i_ _A_ a'^ b''^ c''^ a-'b^ ab-^' 1 3 5 4 2 a-'b''^ a-'b^ a-'b'^ a'b-^^ a'b-'' 1 4 2 1 ^^ 3a-^Z>^c^ ^ a'b c-' '^ ab-'c~' ^ 3abc ' 1 2 3 5 '^ 3.-, o o ' . S.7I.2+ ,4 5. v/^+2V«'^+3Va'+4V6x + Va', Va V{a:'b) 2V(ac') V{Fc') Vjbc ') Vb''^ 2v/c "^ 3Vb' "^ 4Va "^5V^'' ^' a'^ b^'^abc'^ aW ' Va''^ Vb' "^ ^fa' ^Vb'' a 6> be a vb vb Va Examples — 43. 1. i. 2.1 3.^. 4.100. 5.^. 6. ^-^ 7. a^ 8. ^-^ 9. a-\ 10. a^"^- 11. .7:^-^1. 12. a-b. 13. .T^+2.'z;^ + a:-4 S06 ELEMENTARY ALGEBRA. 14. x'+l+x-\ 15. a-'-l. 18. a'-3a^ + da-^-a--\ 17. a'' + 2ahi + ab-x^yK 18. x^+x^y^ + x^y^ + y^. 19. a^ + t«^^^+^^ 20. lex-^-nx-^y-^^+dy-^' 21. a;+2/. 22. a^-ah^+hK 23. a*+2>^-c*- 24. a;* + 2iz;M+3a;*a+2A^-}-^'. 25. x^-2x"K 26. ^r-2-ar\ 27. aJ"* + 1 + a-^Z». Examples — i4. 1. 64^ 81* {i)K {i)\ {i)K 8i 2. 25^, (V)*, (K)^ (K)i, {i(^^+2a^+J^)}i; 125^ (1^)4, (^a^)^, ftW)^ {i(a^+3a^^+3«&»+J')}i 6561-^ (n^3^)-% (a«)-^ (^)"*. 4. v/125, v/3, v/12, v/|, x/i v/320. 5. V54, V256, V2048, V3, V|, V^. 6. v/(4a), v/(98a^a;), j/^. 7. v/(2a5), v/(6a':.), |/||;, ^^, .^{^'-^r'). ANSWERS TO EXAMPLES. 307 8. 3v/5, 5v/5, 36v/3, 3V5, 18V2, iv/G, V12, V54, 6. 9. 4V2, 8V2, 6V48, fv/2, ^v/2, fV2, ■|v/21, |V150, V375. 10. 2v/3, 15v/3, |v/3, ^n/3, iv/3, ^VB. Examples — 45. 1. v/108, v/112; V81, V80; V120, V128, V135. 2. 7v/2. 3. 9V4. 4. |v/3. 5. '^. 9. 2+|n/6. 10. iCv/S+v/S+v/S), iv/6+iV33+iV120. 11. ^(2y2+v/3), v/5 + 1, v/5-v/2, 4+v/2, i(7+3v/5). Examples — 46. 1. v/3+1. 2. 3+v/2. 3. v/5-v/3. 4. 2V6-3V2, 5. 4v/2-3. 6. iV6-l. 7. 2-^|v/3. 308 ELEMENTARY ALGEBRA, Examples — i7. r 4 2. 50. 3. 25. 4. H. 5. {a--h)\ (,. a. 7. -^^. 8. -3^3^. Examples — 48. 1. ±3. 2. ±3. 3. ±1. 4. ±4. 5. ±i. 6. ±2^. 7. ±|. 8. ±5. 9. ±3. 10. ±5. U. ±2. 12. ±2. 13. ±%/3. 14. £B=d=3. Examples — 49. 1. 4, -2. 2. -1, -9. 3. 20, -6. 4. 7, 5. 5. 8, -40. Examples— 50. 6. 10, -110. 1. 1, -8. 2. 17, -4. 3. —5, -20. 4. -1, -12. 5. 1, -20. 6. 25, -136. answers to examples. 309 Examples — 51. I. 0, -51. 2. 6, -ii. 3. 8|, -10. 4. 14, -lOf. 5. 12, -12tJj. 6. 13, -11^ Examples— 52, 1. 10, 2. 2. 3, -1. 3. 2, -I 4. U, -i|. 5. If, -n. 6. 7, -li. 7. 3, i- 8. i(-9±3v/3). 9. 3, if. 10. 3, -f 11. |(37±V57). 12. 2, -3. Examples — 53. 1. G, 3^V 2- 6, -4|. 3. 1, lOf. 4. 3, -8,'^. 5. 5, -l^-V 6. 5, li. 7. 5, -li. 8. 31, 0. • 9. a±-. 10. (a±by. a 310 ELEMENTARY ALGEBRA. Examples — 54. 1. x'-'ke-21= :0. 2. 6a;''+5a;- -6= 0. 3. cc^'+llx+SO: =0. 4. Sx^-Sx-- =0. 5. a;'-100=0. 6. a;'-3aa;+fl'- -4=0. a 7. a;'' + 2a;-l=0. EXAMPLES — 55. • 1. ±2, ±3. 2. 49. 3. 4 4. ±4, 5. 5, -3. C. 3, -2. 7. 12, -3 8. 9, -12, 9. ±3. 10. 2. 11. 4. 12. 16. 13. 1, |. 14. 4. 15. 3a'. 16. 0, ±5. 17. 0, ±V2. 18. 2, ±1. 19. 0, ±v/(a5). 20. a, -2flr, —2a. EXAMTLES— 56. 1. 3, 4, 5. 2. 36, 24. 3. 30, 24. 4. 18, 12, 9. 5. 196. 6. ±12, ±15. 7. 24. 8, 15 yards, 25 yards. 9. 4550. 10. 40 yds. by 24 ANSWERS TO EXAMPLES. 311 11. 16. 12. 4 yards, 5 yards. 13. £60, or £40. 14. 8d. 15. Equal. Examples — 57. 1. a;=7, ?/=±4. 2. x=4:, y=-dj a;=-3, y=4. ^ S. a;=4, ^=3,) 4. cc=:8, ^/^^^^ 5. x=:6, y=5, ) 6. a:=5, y=^,) 7. z=:5, y=3j 8. cc=3, y=4:J 9. a;=4, 2/=^J 10- ^=10, y=15,^ x=2, y=4.5 a;=-.10|, y=-16i.\ 11. a;=3, ^=2,^ 12. a:=5, 2/=4,) t;=4, y=5.$ 13. a;=i{aiFv/(25'-a'')},i 14. x=i{±V{W + b') + I>},-i U. x=8,y=l,-i ,„ ^ «' =1, y=.8.i ^^' ''=^T777;rn?i' y=^- x=l, i/=8. $ v'(a'' + 5^)' ^~ ^{a' + d'Y 312 ELEMENTARY ALGEBRA. 18. fl^ + Z> + l, --7—; I?,- a-hl ' ' a + 1 19. ±-|; ±3Z>. 20. ±1-; ±2^. d 4: 21. 0, a-Vh, h{a-l))^hV {{a-Vdh){a-l))}. Examples — 58. 1. 11; 7. 2. 8; 24. 3. 10; 12. 4. 18- 8: 6; 16, 5. 5; 3. 6. 4; 2. 7. 2; 2. 8. ?; 4. 9. 60. 10. 6, .4. 11. 160; £2. 12. 756; 36; 27. 13. £275, £225. 14. 2, 5, 8. Examples — 59. *• ia? io ^ irf^ TFt 5 mz> ihh, AW 2. «+J. *'-"363- ^- 4 .4 - »'— lla; + 28 4. A, |. 7. 1. ANSWERS TO EXAMPLES. 313 8. -('iti)" 9. i^l^. IJ. c. 8. o{a — o) o—d 12. 35, 42. 13. 4. 14. — ^. a-\-o Examples — 61. 1. 10, 4|, 2tV 2. 9, ^, If. 3. 6, 1|, If. 10. ^(^1) ' ^^ a + h,OY h{a-h). 12. .^' = l, y-=i\ 13. rr:=±9, //=±3. 14. 3. 15. 25, 20. 16. 8:7. 17. 6. Examples — 62. 1. 32, 272. 2. 39, 400. 3. 63, 363. 4. 694, 34750. 5. 9, IG. 6.-1, 0. 7. -28. 8. -275. 9. 16i. 10. -m. 11. 336f. 12. -84. Examples— 63. 1 12. 2. 20. 3. I^Vt^ 1-^V^ &c., &c. ; 6 = 60. 4. 14, 16, 18. 5. 141 14|,... 6. 6J, 5,... 314 ELEMENTARY ALGEBRA. 7. -h h'- 8. 10, 4. 9. 82. 10. 5, 9, 13, 17. 11. 1, 2, 3, 4, 5, 12. 18, 19. 13. 7. 14. 5. Examples — 64. 1. 64, 85. 2. 1280, 1705. 3. 96, 189. 4. __256, -170. 5. 4096, 3277. 6. -512, -341. T. tV^. 8. 1H|. 9. 4,VV 10. 2f-||. Examples — 65. 1. 8. 2. H. 3. i 4. ^. 5. I 6. 4. 7. tV 8. 1. 9. 1/,. 10. h. 11. iV^. 12. lU. Examples — 66. I. 4, 16, 64. 2. 8, 12, 18, 27. 3. -9, 27, -81, 243. 4. 3, 12, 48; or 81, -54, 36. 5. 1, 3, 9, . . 14 ANSWERS TO EXAMPLES. S15 6. ^^. 7. 2 + | + f + &c. ; or 4-| + |-&c 8. 3_|4.|_&c. 9. 200 miles. Examples — 67. 1. iil. 2. f, A, 2. 3. 3,i^. 4. A. tV. a. 5. 6, 12. 6. 36, 64 7. 1, 0. 8. 3, 9. Examples— 68. 1. 720, 720. 2. 5040. 3. 19958400. 4. 34650. 5. 210. 6. 6. 7. 4. 8. 6, Examples— 69. I. 126, 84, 36. 2. 330, 330, 11. 3. 3003, 455. I. 4950. 5. 210, 84. 6. IL 7. 50063860, 5006386. a 116280. Examples — 70. 1. l + Gx+lox' + 20x' -^iDx' -\-6x' -hx', 2. a'-loa*x + dOaV'-270a'x''^4:06ax*-24Saf. 316 ELEMENTARY ALGEBRA. 4. a'-9a'a; + 36aV~84aV + i2GaV-126aV-f 84aV -36aV + 9ax''-x^ 5. 1 + 12x-{-66x'-{-220x' + ^9dx' + 792x + 924a;" + 792a;' '\'4.mx'+220x' + 6Gx'' -\-12x'' +x'\ 6. l-20a;-fl80a;^-9G0a;^4-33G0x*-8064a;^ + 13440a;^ -15360a;' + 11520a;''-5120a;' + 1024a;^°. 7. ^•-18r/'a; + 135c'^V-540aV + 1215aV-14o8aa;* + 729a;". 8. 25Ga;^ + 1024r/a;' + 1792aV 4- 1792aV + 1120aV + 448a'^a;' + 112aV + lea'x-^-a', a 128a'-1344a«a; + 6048r^V-15120aV + 22680«.V --20412r^V + 10206aa;''-2187a;^ 10. l-^6x■i■\^-x''--16x' + ^^x'-\^x'' + ^%^-x''-^x'' + ^,d' U. l-i^x + %^x^^^i-x' + ^\^x'-ii^x'^iUx'-¥A^' 12. 36aV. 13. -l^^XAVy. 110 14. 4t05a"b'. 13. t^=^V, [5 |o_ answers to examples. 317 Examples — 71. 1. 100101100, 102010, 10230, 2200, 1220. 2. 41104, 23420, 14641, 7571, 5954. 3. 402854. 1 511, 22154. 6. a Examples— 72. 1. 152. 4. 100001000000 (binary) == 201000 (quat.). 5. 57264, 95494, eltS. 6. 4112, 6543, 62/^. 7. 1295, 216; 2400, 343; 4095, 512. Examples — 73. 1. .9030900, .9542426, 1.0791813, 1.3010300, 1.3979400, 1.7781513. 2. r.5228787, 1.3979400, 1.6020600, 2.4771213, 2^5228787, 375185140. 3. 2.2253093, .0170334, 3.5670265. 318 ELEMENTARY ALGEBRA. 4. 2 + log. 3 + 2 log. 7. 5. Gh-2 log. 3 + 3 log. .21. 6. 2 log. 2-t log. 3 + f log. 5-1, and I log. 13 -| log. 2 7. .019. 8. 4 and 6. 9. 1.8035700. ' 1 _</x ^. ) A* ^- P"^^ V J V .^^ x./ f rn ^ If U. C. BERKELEY LIBRARIES ^ %^ iiiiiiiiii nil nil nil III II nil mil mil III III! ^.v^ ^. ^\. s %,