m MEMU Edward Brig. it \ (J rmmy^-^- •Elements of Algebra-. G. A. WENTWORTH, A.M., PROFESSOR OF MATHEMATICS IN PHILLIPS EXETER ACADEIIY \-- BOSTON, U.S.A.: PUBLISHED BY GINN & COMPANY. 1889. r / WENTWORTH'S SERIES OF MATHEMATICS. First Steps in Number. Primary Arithmetic. Grammar School Arithmetic High School Arithmetic. Exercises in Arithmetic. Shorter Course in Algebra. Elements of Algebra. Complete Algebra. College Algebra. Exercises in Algebra. Plane Geometry. Plane and Solid Geometry. Exercises in Geometry. PI. and Sol. Geometry and PI. Trigonometry. Plane Trigonometry and Tables. Plane and Spherical Trigonom#ry. Surveying. PI. and Sph. Trigonometry, Surveying, and Tables. Trigonometry, Surveying, and Navigation. Trigonometry Formulas. Logarithmic and Trigonometric Tables (Seven\ Log. and Trig. Tables (Complete Edition). Analytic Geometry. Special Terms and Circular on Application, COPYRIGHT, 1881, BY G. A. WENTWORTH. Ttpography by J. 8. Gushing & Co., Boston, Prbsswork by Ginn & Co., Boston. PREFACE. THE single aim in writing this volume has been to make an Algebra which the beginner would read with increasing in- terest, intelligence, and power. The fact has been kept constantly in mind that, to accomplish this object, the several parts must be presented so distinctly that the pupil will be led to feel that he is mastering the subject. Originality in a text-book of this kind is not to be expected or desired, and any claim to usefulness must be based upon the method of treatment and upon the number and character of the examples. About four thousand examples have been selected, arranged, and tested in the recitation-room, and any found too difficult have been excluded from the book. The idea has been to furnish a great number of examples for prac- tice, but to exclude complicated problems, that consume time and energy to little or no purpose. In expressing the definitions, particular regard has been paid to brevity and perspicuity. The rules have been deduced from pro- cesses immediately preceding, and have been written, not to be committed to memory, but to furnish aids to the student in fram- ing for himself intelligent statements of his methods. Each principle has been fully illustrated, and a sufficient number of problems has been given to fix it firmly in the pupil's mind before he proceeds to another. Many examples have been worked out, in order to exhibit the best methods of dealing with different classes of prob- lems and the best arrangement of the work ; and such aid has been given in the statement of problems as experience has shown 797971 IV PREFACE. to be necessary for the attainment of the best results. General demonstrations have been avoided whenever a particular illustra- tion would serve the purpose, and the application of the principle to similar cases was obvious. The reason for this course is, that the pupil must become familiar with the separate steps from par- ticular examples, before he is able to follow them in a general demonstration, and to understand their logical connection. It is presumed that pupils will have a fair acquaintance with Arithmetic before beginning the study of Algebra ; and that suffi- cient time will be afforded to learn the language of Algebra, and to settle the principles on which the ordinary processes of Algebra are conducted, before attacking the harder parts of the book. "Make haste slowly" should be the watchword for the early chapters. It has been found by actual trial that a class can accomplish the whole work of this Algebra in a school year, with one reci- tation a day ; and that the student will not find it so difficult as to discourage him, nor yet so easy as to deprive him of the re- wards of patient and successful labor. At least one-fourth of the year is required to reach the chapter on Fractions ; but, if the first hundred pages are thoroughly mastered, rapid and satisfactory progress will be made in the rest of the book. Particular attention should be paid to the chapter on Factoring; for a thorough knowledge of this subject is requisite to success in common algebraic work. Attention is called to the method of presenting Choice and Chance. The accomplished mathematician may miss the elegance of the gen- eral method usually adopted in Algebras; but it is believed that this mode of treatment will furnish to the average student the only way by which he can arrive at an understanding of the principles underlying these difficult subjects. In the preparation of these chapters the author has had the assistance and cooperation of G. A. Hill, A.M., of Cambridge, Mass., to whom he gratefully acknowl- edges his obligation. PREFACE. The chapter on the General Theory of Equations has been con- tributed by Professor H. A. Howe of Denver University, Colorado. The materials for this Algebra have been obtained from English, German, and French sources. To avoid trespassing upon the works of recent American authors, no American text-book has been con- sulted. The author yeturns his sincere thanks for assistance to Rev. Dr. Thomas Hill ; to Professors Samuel Hart of Hartford, Ct. ; C. H. Judson of Greenville, S.C. ; 0. S. Westcott of Racine, Wis. ; G. B. Halsted of Princeton, N.J. ; M. W. Humphreys of Nashville, Tenn. ; W. LeConte Stevens of New York, N.Y. ; G. W. Bailey of New York, N.Y. ; Robert A. Benton of Concord, N.H. ; and to Dr. D. F. Wells of Exeter. He has also the pleasure of expressing his obli- gations to Messrs. J. S. Gushing and F. E. Bartley, to whose superior taste and judgment the typographical excellence of this book is due. There will be two editions of the Algebra: one of 350 pages, designed for high schools and academies, will contain an ample amount to meet the requirements for admission to any college ; the other will consist of the Elementary part and about 150 pages more, and will include the subjects usually taught in colleges. Answers to the problems are bound separately in paper covers, and will be furnished free to pupils when teachers apply to the publishers for them. An}^ corrections or suggestions relating to the work will be thank- fully received. G. A. WENTWORTH. Phillips Exeter Academy, May, 1881. CONTENTS, CHAPTER I. Definitions: Quantity and number, 1 ; numbers, 2 ; algebraic numbers, 4 ; factors and powers, 7 ; algebraic symbols, 9 ; algebraic expres- sions, 10; axioms, 11; exercise in algebraic notation, 13; simple problems, 14. CHAPTER II. Addition and Subteaction : Addition of algebraic numbers, 16 ; addition of monomials, 17; addition of polynomials, 19 ; subtraction of algebraic numbers, 20; subtraction of monomials, 21 ; subtraction of polynomials, 22 : parentheses, 25 ; simplifying algebraic expressions by removing parentheses, 26 ; the introduction of parentheses, 27. CHAPTER III. Multiplication: Multiplication of algebraic numbers, 28 ; law of signs in mul- tiplication, 29 ; multiplication of monomials by monomials, 30 ; the product of two or more powers of a number, 30 ; multiplica- tion of polynomials by monomials, 31 ; multiplication of polyno- mials by polynomials, 32; special cases of multiplication ; square of the sum of two numbers ; the square of the difference of two numbers; the product of the sum and difference of two numbers, 37; square of a trinomial, 39 ; the product of two binomials of the form x + a and x + b, 40 ; the powers of binomials of the form a ±6, 42. CHAPTER IV. Division : Division of algebraic numbers, 44 ; division of monomials by monomials, 46 ; division of powers of a number, 46 ; division of polynomials by monomials, 48 ; division of polynomials by poly- nomials, 49; use of parentheses in division, 53; special cases of division ; the difference of two equal odd powers of two numbers divisible by the difference of the numbers, 54 ; the sum of two ALGEBRA. Vll equal odd powers of two numbers divisible by the sum of the num- bers, 55 ; the difference of two equal even powers of two numbers divisible by the difference and by the sum of the numbers, 56 ; the sum of two equal even powers of two t ^mbers when each exponent is the product of an odd and an even factor divisible by the sum of the powers expressed by the even factor, 57 ; general definitions of addition, subtraction, multiplication, and division, 58. CHAPTER V. Simple Equations : Definitions, 59 ; transposition of terms, 60 ; solution of simple equations, 60 ; problems in simple equations, 62. CHAPTER VI. Factors : Case in which all the terms of an expression have a common simple factor, 68 ; case in which the terms of an expression can be so arranged as to show a common compound factor, 69 ; resolution into binomial factors of trinomials of the form of x^-^ {a + b)x-\- ah, 70 ; of the form oi x'^ — {a + h)x + ah, 71 ; of the form of a;^ + (a— J) x — ah, 72; of the form of x'^ — {a — h)x — ah, 73; of the form of x^ + 2ax ^-a^, 74 ; of the form of x'^ — 2ax-\- c?, 75 ; resolution of expressions of the form of two squares with the negative sign between them, 76; resolution of the difference of two equal odd powers, 78 ; resolution of the sum of two equal odd powers, 78 ; resolution of the sum of two equal even powers, when possible, 79; resolution of trinomials of the form of x^ + x^y^+y*, 80; of the form of 2a;'^ + 5aa; + 2a^ 81 ; resolution of polynomials which are perfect powers, 82 ; resolution of polynomials composed of two trinomial factors, 83 ; resolution of polynomials when a compound factor of the first three terras is also a factor of the remaining terms, 84. CHAPTER VII. Common Factors and Multiples : Highest common factor, 88 ; method of finding the highest common factor by inspection, 88 ; method of finding the highest common factor by division, 90; principles upon which this method depends, 90 ; this method of use only to determine the compound factor of the highest common factor, 92 ; modifications of this method required, 93 ; lowest common multiple, 98 ; method of finding the lowest common multiple by inspection, 98 ; method of finding the lowest common multiple by division, 100. VIU CONTENTS. CHAPTER VIII. Fractions: Reduction of fractions to their lowest terms, 103; reduction of fractions to integral or mixed expressions, 106 ; reduction of mixed expressions to the form of fractions, 107 ; reduction of frac- tions to the lowest common denominator, 110 ; addition and sub- traction of fractions, 112; multiplication of fractions, 120; division of fractions, 122 ; complex fractions, 124. CHAPTER IX. Fractional Equations: Reduction of fractional equations, 130; reduction of literal equations, 134. CHAPTER X. Problems Producing Fractional Equations, 137. . CHAPTER XI. Simultaneous Equations of the First Degree, 151 ; elimination by addition or subtraction, 152 ; elimination by- substitution, 154 ; elimination by comparison, 155 ; literal simul- taneous equations, 159. CHAPTER XII. Problems Producing Simultaneous Equations, 166. CHAPTER XIII. Involution and Evolution: Powers of simple expressions, 181 ; law of exponents, 181 ; powers of a binomial when the terms of the binomial have coeffi- cients or exponents, 182 ; powers of polynomials by the binomial method, 182 ; roots of simple expressions, 184 ; imaginary roots, 184 ; square roots of compound expressions, 186 ; square roots of arithmetical numbers, 188; cube roots of compound expressions, 190; cube roots of arithmetical numbers, 193. CHAPTER XIV. Quadratic Equations: Pure quadratic equations, 196 ; affected quadratic equations, 198 ; literal quadratic equations, 203 ; resolution of quadratic equations by inspection, 206 ; number of roots of an equation, 208 ; formation of equations when the roots are known, 209 ; determination of the character of the roots of an equation by in- spection, 209 ; determination of the maximum or minimum value ALGEBRA. IX of a quadratic expression, 211 ; higher equations which can be solved by completing the square, 212 ; problems involving quad- ratics, 214. CHAPTER XV. Simultaneous Quadkatic Equations: Solution when the value of one of the unknown quantities can be found in terms of the other, 219 ; when each of the equations is homogeneous and of the second degree, 222 ; when the equa- tions are symmetrical with respect to the unknown quantities, 223 ; problems producing simultaneous quadratics, 226. CHAPTER XVI. Simple Indeterminate Equations: The values of the unknown quantities dependent upon each other, 228 ; method of solving an indeterminate equation in posi- tive integers, 228. CHAPTER XVII. Inequalities : Fundamental proposition, 234 ; an inequality reversed by changing the signs, 234. CHAPTER XVIII. Theory of Exponents: Fractional and negative exponents, 236 ; the meaning of a fractional exponent, 237 ; the meaning of a negative expone .it, 237 ; laws which apply to positive integral exponents apply also to fractional and negative exponents, 238 ; exercise with mono- mials having fractional and negative exponents, 239 ; exercises with polynomials having fractional and negative exponents, 240 ; radical expressions, 242 ; reduction of surds to their simplest forms, 243 ; comparing surds of the same order, 244 ; comparing surds of different orders, 245 ; addition and subtraction of surds, 247 ; expansion of binomials when the terms are radical expres- sions, 249 ; rationalization of the denominator of a radical expres- sion, 249 ; imaginary expressions, 251 ; square root of a binomial surd, 252 ; equations containing radicals, 255 ; solution of an equation with respect to an expression, 257 ; reciprocal equations, 258. CHAPTER XIX. Logarithms: Common system of logarithms, 261 ; the characteristic of a logarithm, 263 ; the mantissa of a logarithm, 264 ; logarithm of CONTENTS. a product, 264 ; logarithm of a power and of a root, 265 ; loga- rithm of a quotient, 266 ; a table of four-place logarithms, 270 ; general proofs of the laws of logarithms, 276 ; solution of an expo- nential equation by logarithms, 277. CHAPTER XX. Ratio, Proportion, and Variation: Ratio, 278 ; commensurable and incommensurable ratios, 279 ; theorems of ratio, 281 ; proportion, 284 ; theorems of proportion, 284; variation, 292 ; direct variation, 293 ; inverse variation, 293, CHAPTER XXI. Series : Infinite series, 299 ; finite series, 299 ; converging series, 300 -, arithmetical series, 301 ; geometrical series, 308 ; limit of the sum of an infinite geometrical series, 313 ; harmonical series, 314. CHAPTER XXII. Choice. Binomial Theorem: Fundamental principle, 317; arrangements or permutations, 320 ; number of arrangements of n different elements taken all at a time, 320 ; number of arrangements of n different elements taken r at a time, 321 ; number of arrangements of n elements of which p are alike, q are alike, etc., 324 ; number of arrangements of n different elements when repetitions are allowed, 325 ; selectioa? or combinations, 326 ; number of selections of r elements from n dif- ferent elements, 326 ; the number of ways in which p elements can be selected from p+r different elements the same as the num- ber of ways in which r elements can be selected, 327 ; value of r for which the number of selections of n different elements taken r at a time is the greatest, 330 ; the number of ways of selecting r elements from n different elements when repetitions are allowed, 336 ; the number of ways in which a selection can be made from n different elements, 337 ; the number of ways in which a selec- tion can be made from p-\-q-]-r elements of which p are alike, q are alike, r are alike, 338 ; proof of the binomial theorem when the exponent is positive and integral, 342; general formula for the expansion of (a + .-c)**, 343; general term of the expansion of a binomial, 344 ; greatest term of the expansion of a binomial, 345 ; proof of the binomial theorem when the exponent is fractional, 346 ; when negative, 348 ; applications, 349. ELEMENTS OF ALGEBEA. CHAPTER I. Quantity and Number, /^l. Whatever may be regarded as being made up of <^ parts like the whole is called a Quantity. 2. To measure a quantity of any kind is to find how many times it contains another known quantity of the same kind. 3. A known quantity which is adopted as a standard for measuring quantities of the same kind is called a Unit. Thus, the foot, the pound, the dollar, the day, are units for measuring distance, weight, money, time. 4. A Number arises from the repetitions of the unit of / measure, and shows how many times the unit is contained ^"^Sw^the quantity measured. 5. "When a quantity is measured, the result obtained is expressed by prefixing to the name of the unit the number which shows how many times the unit is contained in the quantity measured ; and the two combined denote a quan- tity expressed in units. Thus, 7 feet, 8 pounds, 9 dollars, 10 days, are quantities expressed in their respective units. 2 ALGEBRA. When a question about a quantity includes the unit, the answer is a number ; when it does not include the unit, the answer is a quantity. Thus, if a man who has fifteen bushels of wheat be asked fiow many bushels of wheat he has, the answer is the number, fifteen ; if he be asked how much wheat he has, the answer is the quantity, fifteen bushels. A number answers the question, How many ? a quantity, the question, How much ? Numbers. 6. The symbols which Aritlunetio employs to represent numbers are the figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The natural series of numbers begins with 0; each succeeding number is obtained by adding one to the preceding number, and the series is infinite. 7. Besides figures, the chief symbols used in Arithmetic are: -f (read, plus), the sign of addition. — (read, minus), the sign of subtraction. X (read, multiplied by), the sign of multiplication. -7- (read, divided by), the sign of division. = (read, is equal to), the sign of equality. Exercise. — Read : 7 + 12-19. 8 + 3-5= 20-15 + 1. 9- 4= 5. 24 + 6= 10 X 3. 6x 4 = 24. 14-7 + 5= 6x 2. 48-3 = 16. 9x5 = 180-- 4. 8. Any figure, or combination of figures, as 7, 28, 346, has one, and only one, value. That is, figures represent NUMBERS. • 3 particular numbers. But numbers possess many general properties, which are true, not only of a particular number, but of all numbers. Thus, the sum of 12 and 8 is 20, and the difference be- tween 12 and 8 is 4. Their sum added to their difference is 24, which is twice the greater number. Their differ- ence" taken from their sum is 16, which is twice the smaller number. 9. As this is true of any two numbers, we have this gen- eral property : The sum of two numbers added to their differ- ence is twice the greater number ; the difference of two numbers taken from their sum is twice the smaller number. Or, 1. (greater number -f smaller number) + (greater number — smaller number) = twice greater number. 2. (greater number -\- smaller number) — (greater number — smaller number) = twice smaller number. But these statements may be very much shortened ; for, as greater number and smaller number may mean any two numbers, two letters, as a and b, may be used to represent them; and 2a may represent twice the greater, and 25 twice the smaller. Then these statements become : /I. (a + 5) + (a-^>) = 2(Z. 2. {a-\-b)-{a-b) = 2b. In studying the general properties of numbers, letters may represent any numerical values consistent with the conditions of the problem. 10. It is also convenient to use letters to denote numbers which are unknown, and which are to be found from certain given relations to other known numbers. ALGEBRA. Thus, the solution of the problem, " Find two numbers such that, when the greater is divided by the less, the quo- tient is 4, and the remainder 3 ; and when the sum of the two numbers is increased by 38, and the result divided by the greater of the two numbers, the quotient is 2 and the remainder 2," is much simplified by the use of letters to represent the unknown numbers. 11. The science which employs letters in reasoning about numbers, either to discover their general properties, or to find the value of an unhnown number from its relations to known numbers, is called Algebra. Algebraic Numbers. 12. There are quantities which stand to each other in such opposite relations that, when we combine them, they cancel each other entirely or in part. Thus, six dollars gain and six dollars loss just cancel each other ; but ten dollars gain and six dollars loss cancel each other only in part. For the six dollars loss will cancel six dollars of the gain and will leave four dollars. An opposition of this kind exists in assets and debts, in income and outlay, in motion forwards and backwards, in motion to the right and to the left, in time before and after a fixed date, in the degrees above and below zero on a thermometer. From this relation of quantities a question often arises which is not considered in Arithmetic ; namely, the sub- tracting of a greater number from a smaller. This cannot be done in Arithmetic, for the real nature of subtraction consists in counting backwards, along the natural series of numbers. If we wish to subtract four from six, we start at six in the natural series, count four units backwards, and ALGEBRAIC NUMBERS. arrive at two, the difference sought. If we subtract six from six, we start at six in the natural series, count six units backwards, and arrive at zero. If we try to subtract nine from six, we cannot do it, because, when we have counted backwards as far as zero, the natural series of nuinhers comes to an end. 13. In order to subtract a greater number from a smaller it is necessary to assume a new series of numbers, beginning at zero and extending to the left of zero. The series to the left of zero must ascend from zero by the repetitions of the unit, precisely like the natural series to the right of zero ; and the opposition between the right-hand series and the left-hand series must be clearly marked. This opposition is indicated by calling every number in the right-hand series a positive number, and prefixing to it, when written, the sign + ; ^^^ by calling every number in the left-hand series a negative number, and prefixing to it the sign — . The two series of numbers will be written thus : 2, -1, 0, +1, +2, +3, +4, J I I \ \ I L_ If, now, we wish to subtract 9 from 6, we begin at 6 in the positive series, count nine units in the negative direction (to the left), and arrive at — 3 in the negative series. That is, 6 - 9 = — 3. The result obtained by subtracting a greater number from a less, when both are positive, is always a negative number. If a and b represent any two numbers of the positive series, the expression a — b will denote a positive number when a is greater than b ; will be equal to zero when a is equal to b ; will denote a negative number when a is less than b. If we wish to add 9 to — 6, we begin at — 6, in the ALGEBRA. negative series, count nine units in the positive direction (to the right), and arrive at -f 3, in the positive series. We may illustrate the use of positive and negative num- bers as follows : 5 8 • 20 \ \ 1 DA C Suppose a person starting f^t A walks 20 feet to the right of A^ and then returns 12 feet, where will he be ? Answer : at (7, a point 8 feet to the right of A. That is, 20 feet — 12 feet = 8 feet ; or, 20 - 12 = 8. Again, suppose he walks from A to the right 20 feet, and then returns 20 feet, where will he be ? Answer: at A, the point from which he started. That is, 20 — 20 = 0. Again, suppose he walks from A to the right 20 feet, and then returns 25 feet, where will he now be ? Answer : at D, a point 5 feet to the left of A, That is, if we consider distance measured in feet to the left of A as forming a negative series of numbers, beginning at A, 20 — 25 = — 5. Hence, the phrase, 5 feet to the left of A, is now expressed by the negative number — 5. 14. Numbers provided with the sign -{- ov — are called algebraic numbers. They are unknown in Arithmetic, but play a very important part in Algebra. In contradistinc- tion, numbers not affected by the signs + or — are termed absolute numbers. 15. Every algebraic number, as -f- 4 or — 4, consists of a sign + or — and the absolute value of the number; in this case 4. The sign shows whether the number belongs ' to the positive or negative series of numbers ; the absolute [ value shows what place the number has in the positive or ' negative series. > FACTORS AND POWERS. / 16. When no sign stands before a number, the sign + is always understood ; thus, 4 means the same as + 4, a means l ^he same as + a. But the sign — is never omitted. f 17. Two numbers which have one the sign + and the f other the sign — , are said to have unlike signs. 18. Two numbers which have the same absolute values, but unlike signs, always cancel each other when combined ; thus + 4-4 = 0, +a-a=0. 19. The use of the signs + and — , to indicate addition and subtraction, must be carefully distinguished from their use to indicate in which series, the positive or the negative, . a given number belongs. In the first sense, they are signs of operations, and are common to both Arithmetic and Al- gebra. In the second sense, they are signs of opposition, and are employed in Algebra alone. Factors and Powers. 20. When a number consists of the product of two or more numbers, each of these numbers is called a factor of the product. When these numbers are denoted by letters, the sign X is omitted ; thus, instead of axb, we write ab ; instead of a X b X c, we write abc. The expression abc must not be confounded with a~\-b-\-c', the first is a product, the second is a sum. If a = 2, 5 = 3, c = 4, then aJc? = 2 X 3 X 4 = 24 ; a4-5 + c = 2 + 3 + 4= 9. 21. Factors expressed by letters are called literal factors ; factors expressed by figures are called numerical factors. 8 ALGEBRA. 22. A known factor of a product which is prefixed to an- other factor, to show how many times that factor is taken, is called a coefficient. Thus, in 1c, 7 is the coefficient of c ; in lax, 7 is the coefficient of ax, or, if a be known, 7a is the coefficient of x. When no numerical coefficient occurs in a product, 1 is always understood. Thus, ax means the same as lax. 23. A product consisting of two or more equal factors is called a power of that factor. 24. The index or exponent of a power is a small figure or letter placed at the right of a number, to show how many times the number is taken as a factor. Thus, a\ or simply a, denotes that a is taken once as a factor ; a^ denotes that a is taken twice as a factor ; a^ denotes that a is taken three times as a factor ; and a** denotes that a is taken n times as a factor. These are read : the first power of a ; the second power of a ; the third power of a ; the «th power of a. a^ is written instead of aaa. a" is written instead of aaa, etc., repeated n times. The meaning of coefficient and exponent must be care- fully distinguished. Thus, 4a ^^ a -\- a -\- a -\- a ; a^ = aXaXaXa. If a -- 3, 4a •-= 3 + 3 + 3 + 3 = 12. a4^3x 3x3x3 = 81. 25. The second power of a number is generally called the square of that number ; thus, a^ is called the square of a, because if a denote the number of units of length in the side of a square, a^ denotes the number of units of surface in the square. ALGEBRAIC SYMBOLS. The third power of a number is generally called the cube of that number ; thus, o? is called the cube of a, because if a denote the number of units of length in the edge of a cube, a^ denotes the number of units of volume in the cube. Algebraic Symbols. 26. Known numbers in Algebra are denoted by figures and by the first letters of some alphabet ; as, a, b, c, etc. ; a\ b\ d, read a prime, b prime, c prime, etc.; ai, b^, Ci, read a one, b one, c one. Unknown numbers are generally denoted by the last let- ters of some alphabet ; as, x, y, z, x' , ?/, /, etc. 27. The symbols of operations are the same in Algebra as in Arithmetic. One point of difference, however, must be carefully observed. When a symbol of operation is omit- ted in the notation of Arithmetic, it is always the symbol of addition ; but when a symbol of operation is omitted in the notation of Algebra, it is always the symbol of mul- tiplication. Thus, 456 means 400 4-50 + 6, but 4 a.b means 4 X a X 5 ; 4| means 4 + f , but 4^ means 4X7- b 28. The symbols of relation are =, >, <, which stand for the words, " is equal to," " is greater than," and " is lesd than," respectively. 29. The symbols of aggregation are the bar, | ; the vin- culum, ; the parenthesis, ( ) ; the bracket, [ ] ; and X y. the brace, \ \. Thus, each of the expressions, , {po-\-y),[x-\-y\ \x-\-y\, signifies that rr + y is to be treated as a single number. 10 ALGEBRA. 30. The symbols of continuation are dots, , or dashes, , and are read, " and so on." 31. The symbol of deduction is .*., and is read, "hence," or " therefore." Algebraic Expressions. 32. An algebraic expression is any number written in algebraic symbols. Thus, 8 c is the algebraic expression for 8 times the number denoted by c. 1 a^ — 2>ab is the algebraic expression for 7 times the square of the number denoted by a, diminished by 3 times the product of the numbers denoted by a and h. (33. A term is an algebraic expression the parts of which are not separated by the sign of addition or subtraction. Thus, 3a5, 5^y, 2>ah -^bxy are terms. 34. A monomial or simple expression is an expression which contains only one term. 35. A polynomial or compound expression is an expression which contains two or more terms. A binomial is a poly- nomial of two terms. A trinomial is a polynomial of three terms. 36. Like terms are terms which have the same letters, and the corresponding letters affected by the same expo- nents. Thus, la^cx^ and —ba^ca^ are like terms; but 1 d^ca^ and — bac^x^ are unlike terms. 37. The dimensions of a term are its literal factors. 38. The degree of a term is equal to the number of its dimensions, and is found by taking the sum of the expo- nents of its literal factors. Thus, Sxy is of the second degree, and hx^y^ is of the sixth degree. AXIOMS. 11 39. A polynomial is said to be homogeneous when all its terms are of the same degree. Thus, 1 x^ — b x^y -\- xyz \^ homogeneous, for each term is of the third degree. 40. A polynomial is said to be arranged according to the powers of some letter when the exponents of that letter either descend or ascend in order of magnitude. Thus, ?>ax^ — 4:bx^ — 6 a:r -j- 8 ^ is arranged according to the descend- ing powers of x, and 8b — 6 ax — Abx^ + 3a:r' is arranged according to the ascending powers of x. 41. The numerical value of an algebraic expression is the number obtained by giving a particular value to each letter, and then performing the operations indicated. 42. Two numbers are reciprocals of each other when their product is equal to unity. Thus, a and - are reciprocals. r"^ Axioms. 43. 1. Things which are equal to the same thing are equal to each other. 2. If equal numbers be added to equal numbers, the sums will be equal. 3. If equal numbers be subtracted from equal numbers, the remainders will be equal. 4. If equal numbers be multiplied into equal numbers, the products will be equal. ^,>^. If equal numbers be divided by equal numbers, the quotients will be equal. 6. If the same number be both added to and subtracted from another, the value of the latter will not be altered. 7. If a number be both multiplied and divided by an- other, the value of the former will not be altered. 12 ALGEBRA. Exercise I. / If a = l, 5 = 2, c = 3, c^ = 4, e = 5, /= 0, find the nu- merical values of the f6llowing expressions : -. n toz,.o o/ . 4:ac , 8bc 6cd 1. 9 a + 2 + 3 c — 2/. 4. — — |- . ode 2. 4:e~Sa-Sb + 5c. ,^1 ^e + bcd-^^. zac 3. 8abc-bcd+9cde-def. 6. aJc^ + Jcc?^ _ ^^^2 _|_y3^ 7. eH6e2Z)' + Z>^-4e^5-4eZ>^ '^* a'b' ' c'~b' 9. c^" I ,, b'-i-d' In // ' V + d^-bd 10. <+i'" 12. ^-'^' c y e'j^ed+d'' In simplifying cohipound expressions, eacli term must be reduced to its simplest form before the operations of addi- tion and subtraction are performed. Simplify the following expressions : 13. 100 + 80 --4. 15. 25 + 5x4- 10 --5. 14. 75-25x2. 16. 24-5x4-10 + 3. % 17. (24 - 5) X (4 --10 + 3). Find the numerical value of the following expressions, in which a = 2, ^> = 10, .r = 3, y = 5 : 18. .^^ + 4^X2. 20. 3:ir+7?/-f-7 + aX?/. 19. xy — lbb-^b. 21. Q>b — 8y -^2y Xb — 2b. ALGEBRAIC NOTATION. 13 22. (6 5-8y)--2yX h-{-2h. ^. (6 5-8y)--(2yx5) + 25. 24. U-(d>y-^2y)xh-2b. ^ i 25. 6^^(6—2/) — 3a; + 5:cy^ 10a. [y'"" ^^ Algebraic Notation. 26. Express the sum of a and b. 27. Express the double of x. 28. By how much is a greater than 5 ? 29. If a; be a whole number, what is the next number above it? 30. Write five numbers in order of magnitude, so that x shall be the middle number. 3|^3Vliat is the sum oi x-^-x-^-x + written a times? 32. If the product be xy and the multiplier x, what is the multiplicand ? 33. A man who has a dollars spends b dollars ; how many dollars has he left ? 34. A regiment of men can be drawm up in a ranks of b men each, and there are c men over ; of how many men does the regiment consist ? 35. Write, the sum of x and y divided by c is equal to the product of a, b, and m, diminished by six times c, and increased by the quotient of a divided by the sum of X and y. 36. Write, six times the square of n, divided by m minus a, increased by five b into the expression c plus d minus a. 37. Write, four times the fourth power of a, diminished by five times the square of a into the square of b, and increased by three times the fourth power of b, \ 14 ALGEBRA. Exercise II. That the beginner may see how Algebra is employed in the solution of problems, the following simple exercises are introduced : 1. John and James together have $B. James has twice as much as John. How much has each ? Let X denote the numher of dollars John has. Then 2x = numher of dollars James has, and x-\- 2x= number of dollars both have. But 6 = number of dollars both have .-. x-\-2x=Q., or 3a; =6. and x=2. Therefore, John has $2, and James has $4. 2. A stick of timber 40 feet long is sawed in two, so that one part is two-thirds as long as the other. Eequired the length of each part. Let 3 X denote the numher of feet in the longer part. Then 2x = numher of feet in the shorter part, and 3a; + 2a; = number of feet in both together. But 40 = number of feet in both together ; .-. 3a; + 2a; = 40, or 5a; = 40, and a; = 8. Therefore, the longer part, or 3^, is 24 feet long ; and the shorter, or 2rr, is IG feet. Note. The unit of the quantity sought is always given, and only the number of such units is required. Therefore, x must never be put for money, length, time, weight, etc., but always for the required number of specified -j^nits of money, length, time, weight, etc. The beginner should give particular attention to this caution. PROBLEMS. 15 3. The greater of two numbers is six times the smaller, and their sum is 35. Required the numbers. 4. Thomas had 75 cents. After spending a part of his money, he found he had twice as much left as he had spent. How much had he spent? 5. A tree 75 feet high was broken, so that the part broken off was four times the length of the part left standing. Required the length of each part. 6. Four times the smaller of two numbers is three times the greater, and their sum is 63. Required the num- bers. 7. A farmer sold a sheep, a cow, and a horse, for $216. He sold the cow for seven times as much as the sheep, and the horse for four times as much as the cow. How much did he get for each ? 8. George bought some apples, pears, and oranges, for 91 cents. He paid twice as much for the pears as for the apples, and twice as much for the oranges as for the pears. How much money did he spend for each ? 9. A man bought a horse, wagon, and harness, for $350. He paid for the horse four times as much as for the harness, and for the wagon one-half as much as for the horse. What did he pay for each ? 10. Distribute $3 among Thomas, Richard, and Henry, so that Thomas and Richard shall each have twice as much as Henry. 11. Three men. A, B, and 0, pay $1000 taxes, B pays 4 times as much as A, and an amount equal to the sum of what the other two pay. How much does each pay ? CHAPTER 11. Addition and Subtraction. 44. An algebraic number which is to be added or sub- tracted is often inclosed in a parenthesis, in order that the signs -f and — which are used to distinguish positive and negative numbers may not be confounded with the + and — signs that denote the operations of addition and subtrac- tion. Thus, + 4 + (— 3) expresses the sum, and -f- 4 — (— 3) expresses the difference, of the numbers + 4 and — 3. 45. In order to add two algebraic numbers, we begin at the place in the series which the first number occupies, and count, in the direction indicated hy the sign of the second number, as many units as are equal to the absolute value of the second number. Thus, the sum of -}- 4 + (+ 3) is found by counting from + 4 three units in the positive direction, and is, therefore, + 7 ; the sum of -f 4 + (— 3) is found by counting from + 4 three units in the negative direction, and is, therefore, + 1- In like manner, the sum of — 4 -f- (+ 3) is — 1, and the sum of - 4 + (— 3) is - 7. That is, (1) +4 + (+3)=:7; (3) -4-f(+3) = -l; (2) +4 + (-3)-l; (4) _4 + (-3)=-7. I. Therefore, to add two numbers with like signs, ^?ic? the sum of their absolute values, and prefix the common sign to the sum. II. To add two numbers with unlike signs, j?nc? the differ- ence of their absolute values, and prefix the sign of the greater number to the difference. ADDITION. 17 Exercise III. I. +16 + (-ll)= 3. +68 + (-79) = ^3. -15 + (-25)= 4. _7 + (+4) = , 5. +33 + (+18) = 6. + 378 + (+ 709) + (- 592) = 7. A man lias §5242 and owes $2758. How much is he worth ? 8. The First Punic War began B.C. 264, and lasted 23 years. When did it end ? 9. Augustus Caesar was born B.C. 63, and lived 77 years. When did he die ? 10. A man goes 65 steps forwards, then 37 steps backwards, then again 48 steps forwards. How many steps did he take in all ? How many steps is he from where he started ? Addition of Monomials. 46. If a and Z) denote the absolute values of any two numbers, 1, 2, 3, 4 (§ 45) become : (1) +a + (+5) = a + J; (3) -a + (+Z') = -a + 5'; (2) +a + (-5) = a-5; (4) -a + {~b) = -a-b f Therefore, to add two terms, write them one after the other J \with unchanged signs. it should be noticed that the order of the terms is im- material. Thus, + a — Z> = — 5 + a. Ifa = 8 and b = 12, the result in either case is — 4. ^ 47. 3a + 5a + 2a + 6a + a = 17a. — 2c-3c-c-4c-8c = -18c. Therefore, to add several like terms which have the same 18 ALGEBRA. sign, add the coefficients, prefix the common sign, and annex the common symbols. 48. 7a-Qa+lla-{-a-5a-2a = l9a — lSa = 6a. -^a - 15a- 7 a-j-Ua — 2a = 14:a -27 a = -lSa. Therefore, to add several like terms which have not all the same sign, find the difiference between the sum of the positive coeffi£ie7its and the sum of the negative coefficients, prefix the sign of the greater sum, and annex the common symbols. 49. 5a - 25 + 3a = 8a -23. — 3aa:4-8?/ + 9aa; — 4(? = 6aa; + 8?/ — 4c. Therefore, to add terms which are not all like terms, combine the lih terms, and winte down the other terms, each preceded by its proper sign. A Exercise IV. 1. 5a5 + (-5aZ>)= 6. 7ab-^{-bab) = 2. Si1ix-\-{—2mx)= 7. 120my + (— 95wy) = 3. —\2>mng-\-{—7mng)^ 9^. - ?>2, ab"" -[- {II a^) == 4. -5^^ + (+8x'^)= 9. -7bxy-{-{-\-20xy)--= 5. 25m?/'^ + (-18my^)= 10. -\- lb a^ x"" ^ {- a" x") = 11. -5^m^ + (+7Z'"'m^) = 12. 5a + (-35) + (+4a) + (-75)- /7I3. 4aV + (- lOxyz) + (+ 6a' c) + (- 9xyz) + (-ll«'^) + (+20a;yz) = 14. 3a;'i/ + (-4aJ) + (-2ww)+(+5a;V) ■\-{r-x'y)-\-{-4.x'y) = ADDITION. 19 Addition of Polynomials. 50. Two or more polynomials are added by adding their separate terms. It is convenient to arrange the terms in columns, so that like terms shall stand in the same column. Thus, (l)2a'-Ba'b + 4:ab'i- b' {2)-2x'y +6y^-l a? + 4:0?b-lab''-2b^ -^x'y^lxy' +5 -3a'+ a'b-3ab'-4:b^ 6x'y +2 2a^ + 2a''b + 6ab'-SP x^y - f 2a'i-4:a'b -86' -2x'y -5 Add: Exercise V. /9 1. 5a + 35 + c, Sa-^3b + Sc, a + 35 + 5c?. 2. 7a-4:b-{-c, 6a + 35-5(?, -12a + 4:C. 's.a + b — c, b-\-c — a, c-j-a—b, a + b — c. 4. a-i-2b-i-Sc, 2a-b — 2c, b—a-c, c — a-b. 5. a — 2b4-Sc-4:d, Sb -4:c-{- bd-2a, "2^ \^ 6c-6d-\-Sa~4:b, 7c?- 4a + 56 -4c?. 6 x'-4:x'+bx-'6, 2xf-7x'~7x'~Ux + 6, '' -x^^x'^-x-^-S. 'l. x'-2a^-\-Zx\ s^-\-x''-{-x, 4:x' + bx\ ""^ 4-3:?r-4, oa:^ — 2a; — 5. 8. a^-\-Zab-'-Zo?b-b\ 2a'i-ba'b~Gab'-7b\ a'-ah'-\-2b\ ^^2a6 -2>ax'' + 2a'a;, I2ab - 6a'a; + 10aa;^ ' " ax^ — ^ab - hd'x. 20 ALGEBRA. 11. Ssc'^ — xy~{-xz — Sy'^-\-4:yz — z^, —bx'^—xy — xz-{-byz, 6a;2_6y-62, ^z-byz-{-^z^, -4;r^ + y^ + 3yz + 32l 12. m*^ — 3m*w — 6m^w^, -{-Tn^n^ -\-m^n^ — bm^n, J 7 m"^ 7^■^ + 4 w^n^ — 3 m^*, — 2 m^n^ — 3 ww* + 4 n^ 2 mw* + 2 n^ + 3 m^ — ri^ -]- 2 ??i^ + 7 ??2^n. Subtraction. 51. In order to find the difference between two algebraic numbers, we begin at the place in the series which the minu- end occupies, and count in the direction opposite to that indi- cated hy the sign of the subtrahend as many units as are equal to the absolute value of the subtrahend. Thus, the difference between +4 and + 3 is found by counting from +4 three units in the negative direction, and is, therefore, +1; the difference between +4 and — 3 is found by counting from -[-4 three units in ih.e positive direc- tion, and is, therefore, -\-l. In like manner, the difference between —4 and +3 is — 7 ; the difference between —4 and —3 is —1. Compare these results with results obtained in addition : (1)4-4- (+3) = = 1 + 4 + (-3) = = 1. (2)+4-(-3) = = 7 + 4 + (+3) = = 7. (3)-4-(+8) = = -7 -4 + (-3) = = -7 (4)_4-(-3) = = -1 ~4 + (+3) = = -1 Or, (1) + 4 - -(+3) = + 4 + (-3). (2) +4-^ - (- 3) = + 4 + (+ 3). (3) -4- -(+3) = -4 + (-3). (4)-4- -(-3). = -4 + (+3). SUBTRACTION. 21 52. From (1) and (3), it is evident that subtracting a positive number is equivalent to adding an equal negative number. From (^) and (4), it is evident that subtracting a nega- tive number is equivalent to adding an equal positive number. To subtract, therefore, one algebraic number from another, change the sign of the subtrahend, and then add the subtra- hend to the minuend. Exercise VI. / 1. +25-(-f-16)= 3. - 31-(+58) = 2. -50 -(-25)= 4. +107 -(-93) = 5. Rome was ruled by emperors from B.C. 30 to its fall, A.D. 476. How long did the empire last ? 6. The continent of Europe lies between 36° and 71° north latitude, and between 12° west and 63° east longi- tude (from Paris). How many degrees does it extend in latitude, and how many in longitude ? Subtraction of Monomials. If a and b denote the absolute values of any two num- bers, 1, 2, 3, and 4 (§ 51) become : (1) 4- a -(+5) = a- 5. (3) - a- {-\-b) -= ~ a- b. (2) J^a-{-b) = a + b. (4) - a- {-b) = - a-^b. To subtract, therefore, one term from another, change the sign of the term to be subtracted, and write the terms one after the other. -^ -9- 22 ALGEBRA. Exercise VII. 1. 5x-(-4:x)= 6. 11aa^-(-24:aa^) = 2. -Sab-(+6ab)= 7. 5a^x -(-Sa^x) = 3. Sab^-(+10ah^)= 8. -4:X7/-(-5xy) = 4. 15mV — (-7mV)= 9. 8aa; — (-3ay) = 5. -7a7/ — (-Sai/)= 10. 2aPy — (+ahj) = 11. 9.'>;2_|_(5^)_(_|.g^^j^ 12. 5a;2y-(-18^y) + (-10.^V) = ^^13. 17aa;«-(-aa;«)-(+24aa;^)- w 14. — 3 aJ + (2m:r) — (— Amx) ~ ^^15. 3a-(+25)-(-4c) = Subtraction of Polynomials. 53. When one polynomial is to be subtracted from an- other, place its terms under the like terms of the other, change the signs of the subtrahend, and add. From Aa^ — ^x^ij— xy^^^'f take 2ci?-~ o^y-\'bx'f — ?>if Change the signs of the subtrahend and add : 4:0? — ?>o;^y— x7/-\-2'i^ -2:^3+ x'y~bxif-\-Zf ^a^-^x'y-QiXif + bf From a^a?-\-2 a?oi? — 4 ax^ take a'^ + 4a^x^-?>a?a?--4ax^ SUBTRACTIOX. 23 In the last example we have conceived the signs to be changed without actually changing them. The beginner should do the examples by both methods until he has ac- quired sufficient practice, when he should use the second method only. Exercise VIII. 1. -Frnm f\n. - 9.h - n i^]^(^ 9^-^7)_-^,3^ r '•a^ + lcc-O. rz 2. YYomSa — 2h-i-3ct2i\ie2a — 1h-c-b. 3. From1x^-8x-ltsike5x^-6x + 3. 4. From4^*-3^-2a.-2-7x + 9 6. From :r^— 3^y — y^ + ?/2; — 2r take x^ + 2.ry + dxz- dif — 2r. 7. 'From a^-2>ci?h-{-^aI?-h^ """^ take - o? -{-ZaH - Zal? + V. 8. Yxomx^ — hxij-\-xz — if^-1yz-\-2z^ ^~ ^ -^'^ - ■ take^-a:y-x^ + 2?/2; + 3^2. 9. Yrom2ax^+Sahx~4:h'-x + 12b^ take ax^ — 4 ahx + hx^ — 5 h^x — x^. 10. From (jx^-7x^'i/ + 4:xf-2f — 5x^-{-xi/-4:y^ + 2 take3a?2-- W?r+ x^^^r-f-^ "^xP- - .ry 4^ From a^ — h^ take 4 d from t] ^a?h + (ja''h^-\-4:ah^-2h\ , 1^. From x^i/ -?>x?f -\- 4:xy* - f i^kQ - x^ + 2 x^y - 4 xy' ^•^^ — 4y^. Add the same two expressions, and subtract ^ the former result from the latter. 13. Froma^^^ a'bc - 8 aPc - aV + ahe'-6 1\^ 2a}hc. ^oh^c + 2ah(f~f)}f(f, 21 ALGEBRA. 14. From 12a-\-Sb -5c-2d take lOa-b -{-4:C-Sd, and show that the result is numerically correct when a = Q, 5 = 4, c = l, d=b. 15. What number must be added to a to make h ; and what number must be taken from 2 a^ — 6 a^ 5 -f G ab^ — 21^ to leave a^-7a25-3539 16. From 2x^-f-2xy^z^ take oc^ -y^ '^2xy - z\ 17. From 12ac-\-^cd—^ take - 7 ac — 9cc^ + 8. 18. From - Qa? + 2ab -Zc^ take ^a^-\-^ab - 4^1 19. From ^xy — ^x — Zy-^-l take 82:y — 2:r + 3y + 6. 20. From — a^bc — aW c + ci^c^ — aS(? take a^bc + a5^c — abc^ + aSc. ''^From 7a;2-2a; + 4 take 2a^ + Sx--l. 22. From 3 a:^ + 2 rry — y^ take —x^~oxy-{-Sy^, and from the remainder take Sit^-j-Axy — 5y^. 23. From a:r^ — by^ take (?a;^ — dy^. \ 24. From a:r + J:?; + Z)y + cy take aa: — 5:r — Z»y + cy. 25rTrom 5ar^ + Ax ~ Ay + SyHsike bx^—Sx + Sy-{-7/^ 26. From a^js,,^ X2a5(?- 9aa;2 |-g^]jg Aab^ — Qacx + Sa^x. 27. From a^ _ 2 aJ + ^ - 3 5^ ^^i^g 2a^ — 2ab + 3P. 28. From the sum of the first four of the following expres- sions, a'-\-b' + c'+ d\ ^ J^M-\-^^, a'-c^ + P- d-\ a^ - b^ + c-2 + d\ b^-\-c'+d^- a\ take the sum of the last four. 29. From 2o?-2y''~z^ take Zy'^ -\-2x? -z", and from the remainder take 3s^ — 2?/^ — x^. 30. From a^ — 2 a^ (? + 3 ac^ take the sum oi c^c — 2d?-^2 ac^ and a^ — a(f--a^c. ' PARENTHESES. 25 Parentheses. I 64. From (§ 52), it appears that (1) a + (+b) = a + b. ^^. (2) a-\-(—h) = a — h. (3) a — (+h) = a — b. (4) a — (—h) = a-}-b. The same laws respecting the removal of parentheses hold true whether one or more terms are inclosed. /Hence, when an expression Avithin a parenthesis is preced(^d by a plus sign, the parenthesis may he removed. / "W^ien an expression within a parenijhesis is preceded by a minus sign, the parenthesis may be removed if the sinn '^every term within the 'parciiiheds be changed. Thus : i' (1) a + (^ — c) = a + 5 — e. (2) a — {]) — c) = a — b-\-c. ■ 55. Expressions may occur with more than one paren- thesis. These parentheses may be removed in succession by removing yirs/f, the innermost jmrcnthesis ; next, the inner- most of all that remain, and so on. Thus : (1) a~\b-{c-d)\ = a — lb-- c-\-d\, ^= a — b -\- c ~ d. (2) a-[b-\c + {d-7^)\-\ = a-[b-\c-\-{d-e+f)\l = a — \b — c — d-{' e — /], = a — b'\-c^d—e +/. 26 ALGEBRA. Exercise IX. Simplify the following expressions by removing the paren- theses and combining like terms. 1. (af Z») + (^ + c)-(a + c'). \ 2. {2a-h-c)-{a--2h-\-c). 3. {2x-y)-{2y-z)~{:2z-x). 5. {2x - y + Zz) -^ {- X - y ~ 4:z) - {?>x -2y - z). 6. (3a - ^> + 7c) - (2a + 3Z>) - (5Z» - 4c) + (3c -- a). 7. 1 - (1 - a) + ( 1 - a + a^) - ( 1 - a + a^ - a'). 8. a~\2h~(Sc + 2b)~al 9. 2a-\b~(a-2b)l 10. 3a-\b + (2a~-h)-(a-b)l U.- 7a-[3a-;4a-(5a--2a)n. 12. 2:r + ^-32|^-|^3:r-22Af-4+5^-(4y-^30). 13. (^3a-2Z^) + (4c-aj||^ j^/^ (2/> 4- 3aj) -- cj L4. a-r2air!(3a--4a)|--5a-S6a-k'^a + 8a^^ 15? 2a -<35 -f^c) -^5Z> - (Gc |^ gE^ + 5c^ 16/ a - |2Z> -f |3c - 3a - (cv^^^ + }2a - |z> -f 6|||]. 17. 16 - :r - [7^- - 58:r - (9:^; - 3a; - 6x) \ ]. 18. 2a-[3Z) + (2Z>--c)-4c + 52a-(3Z'-J'=^)S]. 19. a - [2b + S3c - 3a - (a + Z^)J + 2a - (b + 3c)]. 20. a-[5<^-Ja-(3c-3Z^)4-2c-(a-2^-c)i]. PARENTHESES. 27 56. The rules for introducing parentheses follow directly from the rules for removing them : 1. Any number of terms of an expression may be put within a parenthesis, and the sign plus placed before the .; whole. P 2. Any number of terms of an expression may be put within a parenthesis, and the sign minus placed before the whole ; provided the sign of every term within the paren- thesis he changed. It is usual to prefix to the parenthesis the sign of the first term that is to be inclosed within it. Exercise X. Express in binomials, and also in trinomials : 1. 1a-Zh-^.c^d^Ze-y. 2. a-2:r + 4y-30-2^> + c. 3. a« + 3a^-2a3-4a2 4-a-l. 4. -3a-2Z» + 2c-5(Z-e-2/. (^ ax — hy — cz — hx ■\- cy -\- az. 6. 2a^~?,x^y^4:X^f-^x^f-\-x7/-2y\ 7. Express each of the above in trinomials, having the last two terms inclosed by inner parentheses. Collect in parentheses the coefficients of x, y, z in 8. 2ax ~ Q>ay -\- ^hz ~ ^bx — 2cx — Z cy, 9. ax — hx-\-2ay-\-%y-\'4:az — ?>hz — 2z. 10. ax ~ 2hy -{- b cz — 4:hx — ?> cy -\- az — 2cx — ay -\- ^bz. 11. l2aX'^V2ay-^4:by-l2bz-lbcx-\-Q>cy-\-^cz. 12. 2otit — 2>by-^ltcz^^p^^k^-{-2cx^ CHAPTER III. Multiplication of Algebraic Numbers. 57. The operation of finding the sum of 3 numbers, each equal to 5, is symbolized by the expression, 3 X 5 = 15, read, " three times five is equal to fifteen " ; or, by the expression 5 X 3 = 15, read, " five multiplied by three is equal to fifteen." 58. With reference to this operation, this sum is called the product ; one of the equal numbers is called the multi- plicand ; and the number which shows how many times the multiplicand is to be taken is called the multiplier. 59. The multiplier means so many times. The multipli- cand can be a positive or a negative number ; but the mul- tiplier can only mean that the multiplicand is taken so many times to he added, or so many times to he subtracted. 60. If we have to multiply 867 by 98, we may put the multiplier in the form 100 — 2. The 100 will mean that the multiplicand is taken 100 times to be added; the — 2 will mean that the multiplicand is taken twice to be sub- tracted. In general, a multiplier with -f before it, expressed or understood, means that the multiplicand is taken so many times to he added; and a multiplier with — before it means that the multiplicand is taken so many times to be sub- tracted. Thus, MULTIPLICATION. 29 (1) + 3 X (+ 5) = (+ 5) + ( + 5) + (+ 5), or (+ 15). (2) + 3 X (- 5) = (- 5) + (- 5) + (- 5), or (- 15). (3) -3 X (+ 5) = - (+ 5) - (+ 5) - (+ 5), or (- 15). (4) - 3 X (- 5) - - (- 5) - (- 5) - (- 5), or (+ 15). From these four cases it follows, that, in finding the product of two numbers, 61. Like signs produce plus ; unlike signs, minus, ■ y Exercise XI. 1.-17x8= 4. -]8x-5 = 2. - 12.8 X 25 = ^. 43 X - 6 = 3. 3.29 X 5.49 = ^^457 X 100 = 7^\(-358-417)X-79--- y (7.512 -J- 2.8940 X (- 6.037 +S 13.9630 = 62. The product of more than two factors, each preceded by — , will be positive or negative, according as the number of such factors is even or odd. Thus, -2x-3x-4= 4-6x-4= -24. -2x-3x-4x-5 = -24x-5==+ 120. . 9. 13x8x-7=- 10. - 38 X 9 X - 6 = ai. - 20.9 X- 1.1 X8 = ^^- 78.3 X - 0.57 X + 1.38 x - 27.9 = 3\ V- 2.906 X - 2.076 x - 1.49 x 0.89 = 30 ALGEBRA. Multiplication of Monomials. 63. The product of numerical factors is a new number in whicli no trace of the original factors is found. Thus, 4 X 9 = 36. But the product of literal factors can only be expressed by writing them one after the other. Thus, the product of a and h is expressed by ah ; the product of ab and cd is expressed by ahcd. 64. If we have to multiply 5 a by —4 J, the factors will give the same result in whatever order they are taken. Thus, ^aX-4:h = bX-4:XaXh = -20Xab = ~mab. 65. Hence, to find the product of monomials, annsx the literal factors to the product of the numerical factors. 66. €? X a^ = aaX aaa = aaaaa = a^. a^ X a^ X a* = aaX aaa X aaaa = aaaaaaaaa ^= a^. It is evident that the exponent of the product is equal to the sum of the exponents of the factors. Hence, 67. The product of two or more powers of any nurnber is that number with am exponent equal to the suvi of the expo- nents of the factors. Exercise XII. 1. + a X + 5 = -h ab. 6. - ^p X 8m — -24pr:t. 2. +aX-b = -ab. 7. 3a'X— a' = -3al 3. — aX+5 = — aS. 8. —oax2a^ = -~6a\ 4:. —aX — b^-\-ab. 9. QaX — 2a^ 5. 7aX5J = 35a5. 10. 6mnX9m = MULTIPLICATION. 31 15. 5a*" X —2a" = 16. 3a' ^'X 7a'x'^ 17. 7aX— 45 X — 8c 18.^8a6'X 3acX-4c' = 11. SaxX—4:by = 12. — ScmX dn = 13. —lab X 2ac = 14. ^TTi^x X 3'ma;' = 19. 27a5 X — 39mp X 18a^ = 20. 6a5'3/^X 25VX — 5a'y = 21. 7m'xXSmx^X — 2mq = 22. — 3^2'' xQp^qX 8p' q^ = 23. 2a'm^a:* X 3am^:r' X 4 a^mx'' = 24. 6x''7/z'X--9x''fz'X-Sx*7/z = 25. 3aa:X 2a7?2 X — 4ma:X 5'-= 26. 7am' X 35' w' X — 4:ab X a'bn X — 2b^nX — mn^ = Of Polynomials by Monomials. 68. If we have to multiply a + b by n, that is, to take {a-\-b)n times to be added, we have, X w = (a + 5) + (a + 5) + (a + 5) . . . .n times, — a-\-a-\-a...n times + b-{-b-\~b n times, = aXn + 5xn, / ■ = an-\- on. As it is immaterial in what order the factors are taken, n X {a -\- b) = an -\~ bn. In like manner, {a -\- b -\- c) X n = an -\- bn -{- en, or, n {a -{- b -]- c) = an -\- bn -\- en. V 32 ALGEBRA. Hence, to multiply a polynomial by a monomial, 69. Multiply each term of the 'polynomial by the monomial, and add the partial products. Exercise XIII. 1. (6a-55)x3c=-18ac-15^c. 2. (2 + 3a-4a'-5a=^)6a' = 12a^ + 18a-^-24a*-30a^ 3. 5a(3& + 4c-c?) = 15a5 + 20ac-5ac/. 4. — Zax (— hy — 2 C2; + 5) = 3 abxy + 6 acxz — 15 ax. 5y^4a^-3^)x3a5 = ;8a2-9aZ>)x3a' = ;3a:'-42/^ + 52^)x2:r2y = 'd^x — ba^x"^ + ax^ -\- 2x*) X ax'^y = 9a' -i-Sa'b' - 4:a'P -b')x- 3ab' = 'Sx' — 2x'y -Ixy'i- y') X-bx'y = 4 xy' + bx'y + 8:r') X - Sx''y = 5 + 2ab + a'b')x-a' = [—z — 2xz^ + bx'^yz'^ — ^x^y"^ + ^x^y^^z) X — 2>x^yz = Of Polynomials by Polynomials. 70. If we have a + 5 + c to be multiplied by m + ^ + J», we may represent the multiplicand a-\-b ^ chj 3L Then, M(m-^n^p) = Mx m+ Mxn^ Mxp. If now we substitute for J[f its value, (a + 5 + c) (m + ?2 -\-p) = (a + ^ + ^) X m + (a + ^ -f c) X w ■\-\a-\-b-\-c)Xp\ MULTIPLICATION. 33 or, (a -{- b -\- c) (m -{- n +^) = am + bm + cm -\- an -\~bn -{-en -\- op -{-bp -\- cp. That is, to find the product of two polynomials, 71. Multiply the multiplicand by each term of the tnultiplier and add the partial products ; or, multiply each term of one factor by each term of the other and add the partial products. 72. In multiplying polynomials, it is a convenient ar- rangement to write the multiplier under the multiplicand, and place like terms of the partial products in columns. Thus: (1) ba -^ b 3« - 4 5 Iba^-l^ab -20ab + 2W 15a^-38ai + 24^^ (2) Multiply 4:r + 3 + So;'' — 6:r^ by 4 — 6:r' - 5rr. Arrange both multiplicand and multiplier according to the ascending powers of x. 3+ 4:x+ bx""- Qx^ 4- bx- Qx'' 12 + 16:i; + 20a;'-24a;^ - Ibx — 20x^ - 25a;' + 30:r* - 18 .r^ - 24 a.-' - 30^^ + 36 .t^ 12+ x~\Sx^-l?>x^ +36a,-^ (3) Multiply 1 + 2^ + a:* — 3^' by a;' - 2 - 2:r. Arrange according to the descending powers of x. x'—^x^-{'2x +1 a^-2x -2 x'- -^x'-'r^x' -2x' -2x + 6 x' X'-A:X''- ~2x -4.T-! x' - ~bx' + 7 x'-\-2x'~ -^x-\ 84 ALGEBRA. (4) Multiply a^ -\- b^ ^ c^ — ah ~ be — ac hj a -\- h -\- c. Arrange according to descending powers of a. a^ — ab — aG-^-h^— bc-\- & a -j- b-{- c a^ — d^b — a^c + ab^ — abc + ac^ + a}b -ab^- abc ~\-b^ -b^c-\-bc^ -{-a^c — abc — ac^ -f b'^c — bc^ -f c^ a^ -Zabc +¥ -\- c' The student should observe that, with a view to bringing like terms of the partial products in columns, the terms of the multiplicand and multiplier are arranged in the same order. In order to test the accuracy of the work, interchange the multiplicand and multiplier. The result should be the same in both operations. iCISE AiV. (^^' a' + aV + x' by a' — x\ Exercise XIV. Multiply : 1. x^ — 4: by x"^ -\-b. /^2. y — 6 by 2/ + 13. 4. x"^ -\- xy -\- y"^ hj x — y . 5. 2x~-y by x-\-'2>y. 6. 2a;^ + 4:t^' + 8;r+16by 3;2;-6. 7. x^ -\~ x"^ -\- X — Ihj X — 1. 8. x^ — 3 ax by a; + 3 a. 9. 'Ib^-^^ab-a^hj -bb-\~1a. 10. 2a + Z>bya + 25. 12. a^ — ab -\- b"" hj a -\- b . 11. a^-\-ab + b''hy a—b. 13. 2a^ - 55^ by 3a'-4a^». 14. - a^ + 2a'b -b'hy Aa'-}- Sab. 15. a' + aZ) + b' by a' - ab + b\ MULTIPLICATION. 35 16. a'-^a'b-{-^ah^-b'hYa''-'lab + h\ 17. a; + 2y-32by a:-2y + 32. 19. x^ -\- xy -{- 'if hj x^ -\- xz -\- z^ . 20. o? -\- h^ -\- c^ — ab — ac — bchYa-\-b-\-c. 21. x^ — xy^y"^ -{-X -\-y -\-\hj X -\-y —1. Arrange the multiplicand and multiplier according to the descending powers of a common letter, and multiply : 22. bx + ix''^x^-2^hY x'' + ll'-^x. 23. rHll:r-4:r'-24by a:' + 5 + 4:r. 24. a;* + a;'-4:r-ll + 2.r^by :i-'-2a;+3. 25. ~bx'~x''-~x-\- x' + 13a;^ by x" - 2 - 2x. 26. 3 2' + a;^ - 2x' - 4 by 2^; + 4:rH 3:r^ + 1. 27. 5a* + 2a^b'' + aZ>' - 3a'^ by baJ'b - 2a5H 3a'6^ + b\ 28. 4 t/7y - 32 ay' - 8 a'y' + 16 a'y' by a'y' + 4 a'y* + 4 a'/. 29. 3m' + 3w' + 9?/27i' + 9m'n by Gw^n^^ - 2 w?i* I — 6 w'' n'^ + 2 ??z* 72 . 30. ^a'b + 3a^6* - 2ab'> + 6« by 4a* - 2a5-^ - 36*. Find the products of: 31. X — ?>, X ~\, X -\- 1, and x -\- 3. 32. x^ —X -{- 1, o;^ + a; + 1, and x' — x^ -\- 1. 33. o?-{-ab^ b\ a^ - ab -\- b\ and a* - a'5*' + 6*. 34. /^a?^^a^b-\-ab\ 4a' + 3aJ + 5^ and2a'6 + 6l 35. x-^ a, x-\-2a, x — ?>a, x~4:a, 2.ndi X -\-ba. 36. 9a^ + 5^ 27a'-6\27aH^', and 81a*- 9a'6' + 6^ 36 ALGEBRA. 37. From the product of y^ — 2yz— z^ and y'^~\-2yz— ^ take the product oiy^ — yz~2z^ and y^ + yz — 2z^. 38. Find the dividend when the divisor = '6a^ ~ ab ~Sb^, the quotient =a^b — 2b^, the remainder =~-2ab^ -6¥. The multiplication of polynomials may be indicated by inclosing each in a parenthesis and writing them one after the other. When the operations indicated are actually per- formed, the expression is said to be simplified. Simplify : 39. (a-\-b-cXa-i-c~b)(b-{-c~d)(ai-b-}-c). 40. (a -{-b)(b + c)~ (c + ri) (d -}- a) - (a + c) (b - d). 41. (a + b + c + df+ia-b-c + dy -i- la~b + c~ dy + (a-h b ~ c - df. 42. (a + b-{-cy-a(b + c~a)-b(a + c-b)~c(a + b~c). 43. (a~b)x--{b-c)a~^(b~x)(b-a)~(b-cXb + c)l 44. (m-\'n)m — \h^}^^f)'^ — (n — m)n}. 45. (a-5 + cf 4P^-« -'^)-[^(a + ^+^) — c(a — b — c)]l. 46. (p^-{-q^)r-(p + q)(p\r-q\~q\r-2?l). 47. (da^f-^y^) {x^ ~ f) ~ \Zxy - 2f\ \Zx{x^ + f) Wf -2y(f-{-Sxy-^x')ly. 48. a'~\2ab~[~(a+lb-clXa-\b-cl) + 2ab] ' ~4:bcl~(b-{-cy. 49. lac-(a-b)(b-}-c)l~b\b-(a-c)l 50. 5 K« - h)^ -^G2/l - 2\a(x--y)~ bx\ 51. {x - l)(ar --2) - 3a;(a; + 3) ^2\{x + 2)(:r + 1) - 3| . MULTIPLICATION. 37 52. \i2a + hf^(a-2bf\x\(:?>a-2bf-{2a-Uf\. 53. ^{a-?>b)(a + U)-2{a-&bf-2{cv' + U^). 54. x" {x" + ?/)2 - 2 :^^ 2/- i^x -^7j){x-y)-{x'~ ff. 55. IG (r/ + Z/-^)(a2 - Z^^) - (2 a - 3)(2 a + 3)(4 a? + 9) + (26-3)(2Z> + 3X46' + 9). 73. There are some examples in multiplication which occur so often in algebraical operations that they should be carefully noticed and remembered. The three which follow are of great importance: (1) a + b (2) a — b (3) a -\- b a + b a~ b a — b a2+ ab €?— ab a^+ab ab-\-W - ab + b^ ^ab- -b' ct'-\-2ab-\-b^ a^~2ab-^b' a' -b'' From (1) we have (a + bf = a^ + 2 ab + b^. That is, 74. The square of the sum of tivo numbers is equal to the sum of their squares + twice their product. From (2) we have {a - bf =^ a^~2ab + b\ That is, 75. The square of the difference of two numbers is equal to the sum of their squares — twice their product. From (3) we have (a + b){a -b) = a^- b\ That is, 76. TJie product of the su7n and difference of two num- bers is equal to the difference of their squares. '' ^ -^^- ssed by symbols is called a 38 ALGEBRA. 78. By using the double sign rlz, read plus or minus, we may represent (1) and (2) by a single formula ; thus, in which expression the upper signs correspond with one another, and the lower with one another. By remembering these formulas the square of any bino- mial, or the product of the sum and difference of any two numbers, may be written by inspection ; thus : Exercise XV. 1. (127/ - (123)2 = (127 + 123)(127 - 123) -- 250 X 4 = 1000. 2. (29)2 =(30 -1)2 = 900 -60 + 1 = 841. 3. (53)2 = (50 + 3)2 = 2500 + 300 + 9 = 2809. 5. {^oJ'x-bT'yf^^^a^x' ~20a^x^y-\-2bx^if. (3. {^aI/c-\-2a?c%^al/c- ■2a'c^) = ^aH'c^-4.a'c\ 7. {x + yf = 15. (ab-\-cdf^ 8. (y-.)2 = 16. (3m7i-4)2 = 9. (2a; + 1)2 = 17. (12 + 5^)2 = 10. (2a + 55)2 = 18. {^xf-yz'f = 11. (1-^2)2 = 19. {^ahG-hcdf = 12. {Zax-^.x'f^ 20. {^a?-xy''f = 13. (l-7a)2 = 21. {x-\-y){x-y) = 14. (5.-ry + 2)2 = 22. (2a + 5)(2a-5)=; MULTIPLICATION. 39 23. {S-x)(S + x) = 24. {Sab-\-2b'){Sab-2b') = 25. (4:x'-3f)(iAx' + Sf) = 26. (aV-5y)(«'^ + %*) = 27. (6x7/-by')(i6xi/+bf) = 28. (4a;^-l)(4a;5 + l) = 29. (1 + Sab')(l-Sab') = 30. (aa; + 5y)(aa7-5y)(aV+%') = 79. Also the square of a trinomial should be carefully noticed. a + b + c '^ -^ ^' -' 'j ■ a + b + G a' + ab + ac ah + 5^+ he ac -{- bc-\- & 2*^^. a' + 2a6 + 2ac + S'^ + 25c + c^ = a'^ + S'^ + c' + 2a6 + 2ac + 25c. It is evident that this result is composed of two sets of numbers : I. The squares of «, 5, and c ; II. Twice the products of a, b, and c taken two and two. Again, a - - b~ c a - - b- c a^- - ab- ac - ab + 5^ + bc — ac + bc + & a^- -2ab- 2ac- + 5^ + 2hc-\- c' a^-\-V-\-c^~ 2ab ~2ac^ 2bc, 40 ALGEBRA. The law of formation is the same as before : I. The .squares of a, b, and c ; II. Twice the products of a, b, and c taken two and two. The sign of each double product is + or — according as the signs of the factors composing it are like or unlike. The same law holds good for the square of expressions containing more than three terms, and may be stated thus : 80. To the sum of the squares of the several terms add twice t) the product of each term by each of the terms that follow it. By remembering this formula, the square of any polyno- mial may be written by inspection ; thus : Exercise XVI. 1. (a; + y + z)2= 9. {a^ -\- b^ -^ (?f = 2. (x-y-\-zf= 10. {:t?--f-^f = 3. (m+n-p — qf= 11. {x-\-2y -^zf -= 4. (:r2^2a;-3)2= 12. (0^-2^-^-^^)^=- 5. {x'~Q>x-\-1f=^ 13. {x' + ^x-^f^ 6. {2a?-1x + ^y= 14. {a^-bx-\-1f = 7. {x^-{-f-zy= 15. (2:r2-3a;-4)*- 8. {x^-4:x'y''-\-yy= 16. (x + 2y -\- Z z)^ = 81. Likewise, the product of two binomials of the form r + «, x-\-b should be carefully noticed and remembered. X +5 (2) x-b x+^ X -3 x' + bx a^-bx 3a; 4-15 -3a:+15 3^-\-Sx+lb a^-^x+lb MULTIPLICATION. 41 (3) a; + 5 (4) :r - 5 X -3 ■ X +3 a;^ + 5a7 x^ — bx • -3a;-15 +3a;-15 a;2 + 2a;-16 x^~2x-lb It will be observed that : I. In all the results the first term is a? and the last term is the product of 5 and 3. II. From (1) and (2), when the second terms of the bino- mials have liJce signs, the product has the last term positive ; the coefficient of the middle term = the sum of 3 and 5 ; the sign of the middle term is the same as that of the 3 and 5. III. From (3) and (4), when the second terms of the bi- nomials have unlike signs, the product has the last term negative ; the coefficient of the middle term = the difference of 3 and 5; the sign of the middle term is that of the greater of the two numbers. 82. These results may be deduced from the general formula, {x^a){x + b) = x'-{-{a + b)x-\-ab, by supposing for (1) a and b both positive ; (2) a and b both negative ; (3) a positive, b negative, and a>b; (4) a negative, b positive, and a>b. By remembering this formula the product of two bino mials may be written by inspection ; thus : 42 ALGEBRA. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Exercise XVII. rc + 2)(a: + 3)= 11. {x - c){x - d) ■= :r + l)(a; + 5)= 12. (:?;- 4y)(a: + y) -= x-^){x-&)= 13. {a-2b){a~bh)-= x-^){x-l)= 14. (:r2 + 2y2)(^4-y2^)^ x-^){x+l)= 15. {a^~?>xy){x'-\-xy)- x-2){x-\-b)^ 16. (a:r - 9)(a:r + 6) == :i;-3)(a:+7)-= 17. (a: + a)(^-5) = x-2){x-4.)^ 18. (:r-ll)(a; + 4) = a;+l)(a7+ll)= 19. (:r+12)(a;-ll) = x-2a){x + 2>a)= 20. (a; - 10)(a: - 6) = 83. The second, third, and fourth powers oi a-{-h are found in the following manner: a + h g + h ah + h'' {a^hy = d' + 2ab-\-b'' a +1 a' + 2a''6+ aJ)" d'b + 2ab^ + b ^ (a + by = a' + Sa'b + Sab' + 6' a +b a* + 3a^6 + 3a'^5^+ «^' (a + ^)* = a* + ^d'b + 6a^6'^ + 4a5H 6* MULTIPLICATION. 43 From these results it will be observed that : I. The number of terms is greater by one than the ex- ponent of the power to which the binomial is raised. II. In the first term, the exponent of a is the same as the exponent of the power to which the binomial is raised ; and it decreases by one in each succeeding term. III. h appears in the second term with 1 for an expo- nent, and its exponent increases by one in each succeeding term. IV. The coefficient of the first term is 1. V. The coefficient of the second term is the same as the exponent of the power to which the binomial is raised. VI. The coefficient of each succeeding term is found from the next preceding term by multiplying its coefficient by the exponent of a, and dividing the product by a num- ber greater by one than the exponent of b. 84. If h be negative, the terms in which the odd powers of h occur are negative. Thus : (a - hy = a* - 4a'6 -i- 6 a'b^ - 4a6» + h\ Exercise XVIII. Write by inspection the results : 1. {x-{-af--=- 5. (a;-fa)*= 9. {x-{-yf = 2. (x — ay= 6. (x — ay= 10. {x — yf^ 3. {x-^Vf=^ 7. (a; + iy= 11. (:r + l)' = 4. {x-lf= • 8. {x-iy= 12. {x-Vf = CHAPTER IV. Division. 85. Division is the operation by which, when a product and one of its factors are given, the other factor is deter- mined. 86. With reference to this operation the product is called the dividend ; the given factor the divisor ; and the required factor the quotient. 87. The operation of division is indicated by the sign h- ; oy the colon : , or by writing the dividend over the divisor 12 %vith a line drawn between them. Thus, 12 -f- 4, 12 : 4, —-, sach means that 12 is to be divided by 4. 88. + 12 divided by + 4 gives the quotient -|- 3 ; since only a positive number, + 3, when multiplied by -f- 4, can give the positive product, + 12. § 61. + 12 divided by — 4 gives the quotient — 3 ; since only a negative number, — 3, when multiplied by — 4, can give the positive product, + 12. § 61. — 12 divided by -f 4 gives the quotient — 3 ; since only a negative number, — 3, when multiplied by + 4, can give the negative product, — 12. § 61. — 12 divided by — 4 gives the quotient -f 3; since only a positive number, -f 3, when multiplied by ~ 4, can give the negative product, — 12. * § 61. DIVISION. 45 I (1) ^ = + 3. (3) =f = -3. (2) Z^ = -Z. (4) ^ = + 3. From (1) and (4) it follows that 89. The quotient is positive when the dividend and divisor have like signs. From (2) and (3) it follows that The quotient is negative when the dividend and divisor have unlike signs. 90. The absolute value of the quotient is equal to the quotient of the absolute values of the dividend and divisor. Exercise XIX. , +264_ ^ +3840_ ^ 106.33 1, : o. — — O. : -30 ^ - 2568 4. + 4 -4648 -8 7. -264 + 24 8. -3670 -85 9. + 6.8503 + 12 10. 11. -4.9 6 42.435 + 34.5 -7.1560 + 324 -1 -3.14159 - .31831 - 61 - 31.4159 46 ALGEBRA. Division of Monomials. 91. If we have to divide abc by he, aahx by ahy, 12 abc by — 4:ab, we write them as follows : abc aahx ax 12 abc r, DC aby y —±ab Hence, to divide one monomial by another, 92. Write the dividend over the divisor with a line between tlienn ; if the expressio7is have common factors, remove the common factors. If we have to divide a^ by a^, a^ by a*, a* by a, we write them as follows : a' aaaaa , — a'- aa 9l- aaaaaa o = = aa^ a^ a' aaaa a^_ aaaa ^3 = = aaa = a . 93. That is, if a power of a number be divided by a lower power of the same number, the quotient is that power of the number whose exponent is equal to the exponent of the dividend — that of the divisor. Again, a^ aa 1 1 a^ aaaaa aaa a' a^ aaa 1 1 '5' a^ aaaaa aa a* ' aaaa 1 1 a' aaaaaaaa aaaa a' 3k DIVISION. 47 ^ 94, That is, if any power of a number be divided by a ^ higher power of the same number, the quotient is expressed \ by 1 divided by the number with an exponent equal to the \ exponent of the divisor — that of the dividend. ^ ^ Exercise XX. + a 7. \Oab_ 2bc 13. — 3 bmx _ — a 8. x' 14. ab'c'__ abc -ah_ J 9. -12ai^' l: 15. nv'p^x*' + a -¥ ' mp^x^ — a 10. 35 abed _ 5bd 16. -5labdy\_ ?>bdy ^mx 2x 11. abx 5aby 17. 225 m^y _ 25 my'' 12a* _ -3a 12. 21 a' -3a^ 18. 30 ab — 12ac)-^4:a = 2b — Sc. 2. (16am - lObm + 20cm) -~- 5m = -^a + 2b — 4:0. 3. {lSamy—21bny + 3Qcpy)-^ — 9y = 4. (21ax~18bx + 15cx)-~ — Sx = 5. (12a^~Sa^-\-4:x)~4:X = 6. (Sa^-6a/' + 9x^~12x^)~Sx^ = 7. (3bm^y + 28m^f-14:mf)-~-7my = a (4:a*b-6a^b^+12a^b^)-~2aH = 9. (12a^f-15x*7/-24:a^y)-r--3ar'y=^ 10. (I2a^y*-24:x^y' + d>6a^7/-12a^y')^12x'y'=-^ 11. (3a*~2a'b-aH')-^a' = 12. (Sa^yz" + 6^V^ " l^^fz^ + ISx^z) ~ - Sx'yz =- 13. (~lQa^bH^ + Sa'b^c'-12a'b^c^)-i--4:a^b^c^'^ DIVISION. 49 Of Polynomials by Polynomials. 97. If the divisor (one factor) = a^h-\- c, and the quotient (other factor) = n-\-p-^ q, {an-{-bn-{- en -\- ap -{- bp -\- cp -i-aq + bq-i- cq. The first term of the dividend is an; that is, the product of a, the first term of the divisor, by n, the first term of the quotient. The first term n of the quotient is therefore found by dividing an, the first term of the dividend, by a, the first term of the divisor. If the partial product formed by multiplying the entire divisor by n be subtracted from the dividend, the first term of the remainder ap is the product of a, the first term of the divisor, by p, the second term of the quotient. That is, the second term of the quotient is obtained by dividing the first term of the remainder by the first term of the divisor. In like manner, the third term of the quotient is obtained by dividing the first term of the new remainder by the first term of the divisor, and so on. Therefore, to divide one polynomial by another, 98. Divide the first term of the dividend by the first term of the divisor. Wnte the result as the first term of the quotient. Multiply all the terms of the divisor by the first term of the quotient. Subtract the product from the dividend. If there be a remainder, consider it as a new dividend and proceed as before. 50 ALGEBRA. 99. It is of great importance to arrange both dividend and divisor according to the ascending or descending pow- ers of some common letter, and to keep this order throughout the op)eration. Exercise XXIL Divide (1) a'' + 2ab-i-b' by a + b; (2) a' - b' hj a -f b ; a''-}-2ab^b'\a + b d' - b'\a + b g^ -f- Cib a-\- b a' + cib a — b ab -\-b^ ■ — ab — b'^ ahj-b^ -ab-b' (3) a'~2ab-^b' hj a-b; ar-2ab + b'\ a-b a^ — ab a — b - ab + b' - ab-\-b' (4) 4 a V - 4 a V -j-x'-a' by x' x^ — 4 d^x* + 4 a V — a^' \ x"^ — -3aV + 4aV- -a« -3aV + 3aV aV- -a' aV- A (5) 22a^5H155*-f3a*-10a^^-22aJ^by a^ + 3&^-2aZ>: 3 a* - 10 a'6 + 22 a^b"" - 22 a5^ + 15 b' | a'^ - 2r<^> + 3^)'^ 3a^- 6a.^&+ 9a^^>' 3a' ~ 4a6 + 5^^" - 4a^^<^ + 13a''^^-22a6^ - 4a35+ 8a^^^-12a5^ 5a^6^-10a^'^ + 156* ba^b-'-lOab'+lW DIVISION. 51 Divide 6. 7. 10. f* 13. T 22. 23. 24. 25. 26. 27. 28. 29. x^-nx^V2. by :r-3. :r^ + a: - 72 by a; + 9. 2a;'-a;' + 3a; — 9 by 2:r — 3. 6a:'+14:f2-4:r + 24 by 2x + 6. 3:r' + a: + 9r'-l by 3a; -1. 7:r=' + 58:r- 24:^2 -21 by 7a; -3. x^ — 1 by a; — 1. ""^^-^^^^ a^-lab'^h^ by a-h?^- ^* - 8iy_by^j;J^ ^ — it ^y ^ "~ v- a^ + 32^^ by a + 25. 2a*+27a5-'-815' by a + 36.' ~ a;'' + 11 x^ - 12a; - bx' + 6 by 3 + x' - 3a;. ^* -_ 9a;' + .^' -- 16.2; - 4 by a;' + 4 + 4a;. 36 + :r*-13a;' by 6 + a;' + 5a;. a;* + 64 by a;^ + 4a; + 8. ^i _|_ :^3 _|_ 57 _ 35^ _ 24.2;=' by a;^ - 3 + 2a;. l-x~^x'-x' by l + 2a; + a;l .^6 _ 2^3+1 by .T'-2a;+l. a' + 2a'b' + 9b' by a'-2a6 + 36l 4^5_^3_|_4^ by 2 + 2a;' + 3a;. a' - 243 by a - 3. 18a;* + 82a;'^ + 40-67a;-45a;' by 3a;'' + 5-4a;. a;* - 6a;?/ - 9a:' - y' by a^-' + ?; + 3a;. 52 ALGEBRA. 30. x^-^^a?'i^-^x^y-^y'^\)jo^-Zxy^2f. 31. x^ -\- x^ y^ -^ y^ \ij x^ — xy -\- y^ . 32. 3?-\-o?-\-x^y-\-y^ — ^x^ — :i^y^\>Y0^-{-x — y. 33. 2x^ — Z'f-^xy — xz — \yz~z^\)^'^x-\-Zy-\-z. 34. 12 + 82^:2 +106:i;^-70ar'- 112:^2 ~38:r by3-5:r+7x2. 35. a;'' + y* by ^* — s^y + ^r^y^ — :ry^ + y*. 36. '2.x^-\-'l3?f-'lxf-n:x?y-y''hj'lx'-\-'f-xy. 37. 16a;* + 45^y2_|_^^4]^y4^_2^y_^yj. 38. 32a'^6 + 8a3^«-a5*-4a2^*-56a452 by Z;^ - 4 a2Z> + 6 aZ^l 39. l + 5a;«-6a;nyl-a; + 3.r2. 40. l-52a*3<-51a35ny4a252 + 3a5-l. 41. x^y — x]/ hj oi^y-{-2x'i^ — 2oi?'jf — y*. 42. a;«+15a;V'+15^y* + /-6:r*y-6:r3/'-20:r3/ by:r3-3:r2y + 3:ry2_2/3^ 43. a^ + 2a3Z>^-2a*3«-2a«Z>-6a25^-3a5« bya3-2a2Z>-c^Z/l 44. 81a;«y + 185^/-54a;«y2_i3^^4_i3^^6_9y by3a;'' + a;22/2 + y^ 45. a* + 2a85 + 8a2 52 4- 8 ah^ -\-l^¥hj a" + 4:b\ 46. 83/«-a:« + 21a:33/3_24^y5by3a;y-a;2_^^ 47. 16aH9^' + 8a262by4a2+3Z»2_4a5. 4& a« + 53 + c3-3a5cbya + 5 + d?. 49. «»+ 8^3 + c»- ^aho by a*+ 45^ + ^-^ - ac- 2a6 -2.be. 50. a^ + ^Ht'^ + Sa^i + Sa^Hya + ^. + c. DIVISION. 53 100. The operation of division may be shortened in some cases by the use of parentheses. Thus : ^ ^(^a + h + c)ar^ +(ah + ac + he) x + ahc\x_jj-b ^+( +^ )^ a^ + {a + c)x-{-ac (a + c) a:^ + (ab -{■ac-\- be) x (a +e)x^-\-(ab -\-hc)x acx + abc acx +ahc Exercise XXIII. Divide 1. a^ (J) + e) -^ W {a - e) + (^ {a ~b)-\- abc hy a + h^ c. 2. a? — (a-{-h ■{■ c) a^ -{- {ah -\- ac -]-be) X — ahe \)joi^—{a-\-b)x-\- ah. 3. a?-2aoi^-\-(c? + ah - h^x-c?b + aV^ by a; - a + J. 4. x^ — {a^ — h — e) x^— (b — e) ax -{- he hj x^ — ax + e. 5. ?/^ — (m + n + ^) 2/^ + {inn + 7np-\- np) y — mnp by y ~p, 6. a;H(5 + a)^-(4-5a + 5)a;2_(4^_^55)^^4j \>j x^ -\- b X — 4i. 7. x^-{a+h-\-e-\- d)a^^{ah-\-ae+ad-\-he-\-hd^cd)c(? — {ahe -\-abd-\-aed-\- bed) x-{- abed by a^—{a-{-c)x-\-ac. 8. x^ — {m — e)x*-{-{n — em -{- d) a^ -{- {r -\- en — dm) x^ -\- {er + dn) x-\-dr by :c^ — moc^ -\-n^-{-r. 9. a;* — m.x* + no(? — n^x? + m.x — 1 by a; — 1. 10. {x^y)^^Z{x-\-yJz-\-Z{x^y)z^-\-Z^ 'oy{x-\-yJ-\-^{x-\-y)z-\-^. 54 ALGEBRA. 101. There are some cases in Division which occur so often in algebraic operations that they should be carefully noticed and remembered. Case I. The student may easily verify the following results : (1) ^!zi^ = a2 + a5 + Jl a — b (2) 2W^i8Z^_9^2_^6^j_f_4j2^ 3a — 26 (3) -5). 6. (a^-l)-(a-l). 3. (a«-216)--(a-6). 7. (1 - 8^.^) - (1 - 2:r). 4. (.r^ - 343) - (.r - 7). 8. (a^ ~S2b') -i- {x-2b). 9. (8a^x^-l)-^(2ax-l). 10. (l-27a^f)^(l-Sxy). 11. (64a3i3-27a;^)-f-(4a5-3a;). 12. (243a^-l)--(3a-l). 13. (32a^-243^>^)--(2a-3Z>). Case II. (1) ^^^a'-ab + b^ (2) 2J^ + y ^9.t^-6;n/ + 4y. da; +^y (3) ^' + ^' = a'-a'b + a^^^^ - a&^ + 6". a-{-b (4) 243.2;^ + 32 ^y^ _ g^ ^, _ ^^ ^^ _^ gg ^^ - 24 rr?/^ + IGy". 3:i; + 2?/ From these results it may be assumed that : 103. The sum of two equal odd powers of two numbers is dmsible by the sum of the numbers. The quotient may be found as in Case I., but the signs are alternately plus and minus. 66 ALGEBRA. Exercise XXV. Write by inspection the results in the following examples : 1. {a^ + f)^{x-{-y). 5. (8a«r^+l)--(2a:r + l). 2. {z^ + ^)^{x^y). 6. {:^J^21f)~{x + ?>y). 3. (l + 8a«)--(l + 2a). 7. (a^ + 325^) - (a + 25). 4. (27a« + 53)--(3a + Z^). 8. (512:r32/« + 2^) -(8a;y + 2). 9. (729a3 + 216 5«)--(9a + 6Z»). 10. (64 a^ + 1000 h^) - (4 a + 10 5). 11. {<6^o?¥+21x^)-r-{4:ah + 2>x). 12. (2:« + 343)--(:r+7). 13. {21s^if + %^)---{^xy^2z). 14. (1024a^ + 2435^)--(4a + 35). Case III. (1) ^^/! = :r + y. (2) ■2-:Z.:^' = ^-3_|_^2 _^ 2_^.^5^ a; — y x~y x"- (3) r^2^=^^__y. (4) ? ^ ^ ^^ ._._ ^y _!_ ^^^2 _. ^^ ^ + y -^ ^' ^^4-y From these results it may be assumed that : 104. The difference of tivo equal even powers of two num.- hers is divisible by the difference and also by the sum of the numbers. When the divisor is the difference of the numbers, the quotient is found as in Case I. When the divisor is the sum of the numbers, the quotient is found as in Case II. DIVISION. 57 EXEECISE XXVI. Write by inspection the results in the following examples : 1. {x^-y^)-^{x-y). 8. (16 a;'*- 1) -- (2:r + 1). 2. {x'^-y^)-^{x^y). 9. (81a^a;^-l)--(3ax-l). 3. (ct^-x^)-^{a-x). 10. (fi\a^x^-l)-^(^ax-\-V). 4. {a^--x^)---{a + x). 11. {fo^a^ -h^) -^ (2a-b). 5. {x^~^\y^)-~{x-?>tj). 12. (64a^-5«)^(2a + Z»). 6. (2-4-81?/*)^(.r + 3y). 13. (:r«-729?/«)-:-(^-3?/). 7. (16:f^-l)^(2:r-l). 14. (:r« - 729 y«) -- (a; + 3 ?/) . 15. (81a^-16(7^)--(3a-24 16. (81a''-16c^)--(3a + 26). 17. (256 a^ - 10,000) -- (4 a - 10). 18. (256 a^ - 10,000) H- (4 a +10). 19. (625 a,'' -1)^ (5 a; -1). Case IV. It may be easily verified that : 105. The sum of two equal even poivei'S of two nurtiberb, is not divisible by either the sum or the difference of the number's. But when the exponent of each of the two equal powers is composed of an odd and an even factor, the sum of the given powers is divisible by the sum of the powers expressed by the even factor. Thus, x^ + y^ is not divisible by x-^y or by x — y, but is divisible by a;^ + ?/". The quotient may be found as in Case 11. 68 ALGEBRA. Exercise XXVII. Write by inspection the results in the following examples : 1. {x'-\-y')-^{x' + f). 6. (:r^^ + l)-(^' + l). 2. (a«+l)-^(a^ + l). 7. (64:s« + y«)-^(4^^ + 2/')- 3. {a}'-\-y'')-^{o:' + y'). 8. (64 + a«) -^ (4 + a^). 4. (5^'' + 1)-^ (5^+1). 9. (729a« + J«)^(9a2 + 5'^). 5. (a^M-^'')-(a' + ^')- 10. (729c*' + l)-^(9c2+I). Note. The introduction of negative numbers requires an exten- sion of the meaning of some terms common to arithmetic and algebra. But every such extension of meaning must be consistent with the sense previously attached to the term and with general laws already established. Addition in algebra does not necessarily imply augmentation, as it does in arithmetic. Thus, 7 + (— 5)= 2. The word sum, however, is used to denote the result. Such a result is called the algebraic sum, when it is necessary to distinguish it from the arithmetical sum,, which would be obtained by adding the absolute values of the numbers. The general definition of Addition is, the operation of uniting two or more numbers in a single expression written in its simplest form. The general definition of Subtraction is, the operation of finding from two given numbers, called minuend and subtrahend, a third number, called difference, which added to the subtrahend will give the minuend. The general definition of Multiplication is, the operation of find- ing from two given numbers, called multiplicand and multiplier, a third number, called product, which may be formed from the mul- tiplicand as the multiplier is formed from unity. The general definition of Division is, the operation of finding the other factor when the product of two factors and one factor are given. CHAPTER V. Simple Equations. 106. An equation is a statement that two expressions are equal. Thus, 4:r-12 = 8.' 107. Every equation consists of two parts, called the first and second sides, or members, of the equation. 108. An identical equation is one in which the two sides are equal, whatever numbers the letters stand for. Thus, {x -f- h) {x~b) = x'~ h\ 109. An equation of condition is one which is true only when the letters stand for particular values. Thus, ^ + 5 = 8 is true only when a; = 3. 110. A letter to which a particular value must be given in order that the statement contained in an equation may be true is called an unknown quantity. 111. The value of the unknown quantity is the number which substituted for it will satisfy the equation, and is called a root of the equation. 112. To solve an equation is to find the value of the unknown quantity. 113. A simple equation is one which contains only the first power of the unknown quantity, and is also called an equation of the first degree. 60 ALGEBRA. 114. If equal changes he made in both sides of an equa- tion, the results will he equal. § 43. (1) To find the value oi x m x -\-h^=a. x-\- h = a; Subtract h from each side, x -{- b—b — a — h; Cancel + & — &, x=a—b. (2) To find the value of x in x — h = a. x — b = a; ■ Subtract — b from each side, x—b + 6 = a + 6 ; Cancel — 6 + 6, x=a-\-b. The result in each case is the same as if h were trans- posed to the other side of the equation with its sign changed. Therefore, 115. Any term may he transposed from one side of an equation to the other provided its sign he changed. For, in this transposition, the same number is subtracted from each side of the equation. 116. The signs of all the terms on each side of an equa- tion may be changed ; for, this is in effect transposing every term. 117. When the known and unknown quantities of an equation are connected by the sign + or — , they may be separated by transposing the known quantities to one side and the unknown to the other. 118. Hence, to solve an equation with one unknown quantity, Transpose all the terms involving the unknown quantity to the left side, and all the other terms to the right side: SIMPLE EQUATIONS. 61 combine the like terTns, and divide both sides hy the coefficient of the unknown quantity. 119. To verify the result, substitute the value of the unknown quantity in the original equation. Exercise XXVIII. Find the value of x in 1. 5a;- 1 = 19. 8. 16a;- 11 = 7rr+ 70. 2. 3a; + 6=12. 9. 24a; - 49 = 19a;- 14. 3. 24a;=7a; + 34. 10. 3a; + 23 = 78 -2a;. 4. 8a; -29 = 36 -3a;. 11. 26 -8a; = 80- 14a;. 5. 12 -5a; = 19 -12a;. 12. 13 - 3a; = 5a;- 3. 6. 3a; + 6-2a;=7a;. 13. 3ar- 22 = 7a; + 6. 7. 5a; + 50 = 4a; + 56. 14. 8 + 4a; = 12a;- 16. 15. 5a; -(3a; -7) = 4a; -(6a; -35). 16. 6a;- 2 (9 -4a;) + 3 (5a; -7) = 10a;- (4 + 16a; + 35). 17. 9a;-3(5a;-6) + 30^0. 18. a; -7 (4a; -11) = 14 (a;- 5) -19 (8 -a;) -61. 19. (a;+7)(a;-3) = (a;-5)(a;-15). 20. (a;-8)(a; + 12) = (a; + l)(a;-6). 21. {x - 2) (7 - a;) + (a; - 5) (a: + 3) - 2 (a; - 1) + 12 = 0. 22. (2a; -7) (a; + 5) = (9 -2a;) (4 -a;) + 229. 23. 14-a;-5(a;-3)(a; + 2) + (5-a;)(4-5a;)=45a;-76. 24. (a.- + 5)2-(4-a;)2 = 21a;. 25. 5(a;-2)2+7(a;-3)2=(3a;-7)(4a;-19) + 42. 62 ALGEBRA. Exercise XXIX. PROBLEMS. 1. Find a number sucb. that when 12 is added to its double the sum shall be 28. Let X = the number. Then 2 a; = its double, and 2 a; + 12 = double the number increased by 12. But 28 = double the number increased by 12. .•.2a;+12=28. 2a; -^28 -12, 2 a; = 16. 2. A farmer had two flocks of sheep, each containing the same number. He sold 21 sheep from one flock and 70 from the other, and then found that he had left in one flock twice as many as in the other. How many had he in each? Let X — number of sheep in each flock. Then a; — 21 = number of sheep left in one flock, and a; — 70 — number of sheep left in the other. . • . a; - 21 = 2 (a; - 70), a; -21 = 2a; -140. a; - 2a; =. - 140 + 21, ~- a;:::. -119, X = 119. 3. A and B had equal sums of money ; B gave A $5, and then 3 times A's money was equal to 11 times B's money. What had each at first ? Let X = number of dollars each had. Then x-\- 5 = number of dollars A had after receiving |5 from B, and a; — 5 = number of dollars B had after giving A |5, SIMPLE EQUATIONS. 63 .•.3(a; + 5)-ll(x-5); 3 a; + 15 = 11 a; -55; 3a; -11 a; = -55 -15; -8a; = -70; a; = 8|. Therefore, each had $8.75. 4. Find a number whose treble exceeds 50 by as much as its double falls short of 40. Let X = the number. Then 3 a; = its treble, and 3 a; — 50 = the excess of its treble over 50 ; also, 40 — 2 a; = the number its double lacks of 40. .•.3a; -50 = 40 -2a;; 3a; + 2a; = 40 + 50; 5a; = 90; a; = 18. 5. What two numbers are those whose difference is 14, and whose sum is 48 ? Let X = the larger number. Then 48 — a; = the smaller number, and X — (48 — x) = the difference of the numbers. But 14 = the difference of the numbers. .-. a; - (48 - a;) - 14 a; — 48 + a; = 14 2a; = 62: a; = 31. Therefore, the two numbers are 31 and 17. 6. To the double of a certain number I add 14, and obtain as a result 154. What is the number ? 7. To four times a certain number I add 16, and obtain as a result 188. What is the number ? 8. By adding 46 to a certain number, I obtain as a result a number three times as large as the original number. Find the original number. 64 ALGEBRA. 9. One number is three times as large as another. If I take the smaller from 16 and the greater from 30, the remainders are equal. What are the numbers ? 10. Divide the number 92 into four parts, such that the first exceeds the second by 10, the third by 18, and the fourth by 24. 11. The sum of two numbers is 20 ; and if three times the smaller number be added to five times the greater, the sum is 84. What are the numbers ? 12. The joint ages of a father and son are 80 years. If the age of the son were doubled, he would be 10 years older than his father. What is the age of each ? 13. A man has 6 sons, each 4 years older than the next younger. The eldest is three times as old as the youngest. What is the age of each ? 14. Add $24 to a certain sum and the amount will be as much above $80 as the sum is below $80. What is the sum? 15. Thirty yards of cloth and 40 yards of silk together cost $ 330 ; and the silk cost twice as much a yard as the cloth. How much does each cost a yard ? 16. Find the number whose double increased by 24 exceeds 80 by as much as the number itself is less than 100. 17. The sum of $500 is divided among A, B, 0, and D. A and B have together $280, A and $260, and A and D $220. How much does each receive? 18. In a company of 266 persons composed of men, women, and children, there are twice as many men as women, and twice as many women as children. How many are there of each ? SIMPLE EQUATIONS. 65 19. Find two numbers differing by 8, such tbat four times the less may exceed twice the greater by 10. 20. A is 58 years older than B, and A's age is as much above 60 as B's age is below 50. Find the age of each. 21. A man leaves his property, amounting to $ 7500, to be divided among his wife, his two sons, and three daugh- ters, as follows : a son is to have twice as much as a daughter, and the wife $500 more than all the chil- dren together. How much was the share of each ? 22. A vessel containing some water was filled by pouring in 42 gallons, and there was then in the vessel seven times as much as at first. How much did the vessel hold? 23. A has $ 72 and B has $ 52. B gives A a certain sum ; then A has three times as much as B. How much did A receive from B? 24. Divide 90 into two such parts that four times one part may be equal to five times the other. 25. Divide 60 into two such parts that one part exceeds the other by 24. 26. Divide 84 into two such parts that one part may be less than the other by 36. Note I. When we have to compare the ages of two persons at a given time, and also a number of years after or before the given time, we must remember that both persons will be so many years older or younger. Thus, if a; represent A's age, and 2x B's age, at the present time, A's age five years ago will be represented by x — b; and B's by 2 a; — 5. A's age five years hence will be represented by a; + 5 ; and B's age by 2 a; + 5. QQ ALGEBRA. 27. A is twice as old as B, and 22 years ago he was three times as old as B. What is A's age ? 28. A father is 30 and his son 6 years old. In how many years will the father be just twice as old as the son ? 29. A is twice as old as B, and 20 years since he was three times as old. What is B's age ? 30. A is three times as old as B, and 19 years hence he will be only twice as old as B. What is the age of each ? 31. A man has three nephews ; his age is 50, and the joint ages of the nephews is 42. How long will it be be- fore the joint ages of the nephews will be equal to that of the uncle ? Note II. In problems involving quantities of the same kind expressed in different units, we must be careful to reduce all the quan- tities to the same unit. Thus, if X denote a number of inches, all the quantities of the same kind involved in the problem must be reduced to inches. 32. A sum of money consists of dollars and twenty-five-cent pieces, and amounts to $20. The number of coins is 50. How many are there of each sort ? 33. A person bought 30 pounds of sugar of two different kinds, and paid for the whole $2.94. The better kind cost 10 cents a pound and the poorer kind 7 cents a pound. How many pounds were there of each kind ? 34. A workman was hired for 40 days, at $ 1 for every day he worked, but with the condition that for every day he did not work he was to pay 45 cents for his board. At the end of the time he received $22.60. How many days did he work ? » SIMPLE EQUATIONS. 67 35. A wine merchant has two kinds of wine ; one worth 50 cents a quart, and the other 75 cents a quart. From these he wishes to make a mixture of 100 gallons, worth $2.40 a gallon. How many gallons must he take of each kind. 36. A gentleman gave some children 10 cents each, and had a dollar left. He found that he would have re- quired one dollar more to enable him to give them 15 cents each. How many children were there ? 37. Two casks contain equal quantities of vinegar; from the first cask 34 quarts are drawn, from the second, 20 gallons ; the quantity remaining in one vessel is now twice that in the other. How much did each cask contain at first ? 38. A gentleman hired a man for 12 months, at the wages of $ 90 and a suit of clothes. At the end of 7 months the man quits his service and receives $33.75 and the suit of clothes. What was the price of the suit of clothes ? 39. A man has three times as many quarters as half-dollars, four times as many dimes as quarters, and twice as many half-dimes as dimes. The whole sum is $7.30. How many coins has he altogether ? 40. A person paid a bill of $15.25 with quarters and half- dollars, and gave 51 pieces of money altogether. How many of each kind were there ? 41. A bill of 100 pounds was paid with guineas (21 shil- lings) and half-crowns (21 shillings), and 48 more half-crowns than guineas were used. How many of each were paid ? CHAPTER VI. Factors. 120. In multiplication we determine the product of two given factors ; it is often important to determine the /actors of a given product. 121. Case I. The simplest case is that in which all the terms of an expression have one common factor. Thus, (1) o[^']-xy = x{x + y). (2) 6a3 + 4a2 + 8a-=2a(3a2 + 2a + 4). (3) 18 a^h - 27^262 + 36 a^ = 9 a5 (2 a^ - 3 a^ + 4). Exercise XXX. Resolve into factors : 1. 5^2 -15a. 4. ^xSj -12x^1/'' ^^xif. 2. 6a^ + 18a2-12a. 5. y^ - aif -^hif + cij. 3. 49a;2_2i^_|_i4. e. 6 a^^^- 21 a^Z^2_|_ 27,///. 7. 54 ^/ + 108 :iV - 243 x''y\ 8. 45 xhf - 90 x^y' - 360 x^. 9. 70 ay - 140 ahf + 210 aif. 10. 32a^Z'« + 96a«6«-128a«i«. FACTORS. 69 122. Case II. Frequently the terms of an expression can be so arranged as to show a common factor. Thus, (1) a?-\-ax-\- bx -\- ah — {pi^ + ax) + (bx + ah), ^= X (x -\- a) -\- h {x -\- a), = (x + h){x-{-a). (2) ac — ad—hc-\-hd={ac — ad) — {hc — hd), = a{c — d) — h{c ~ d), = (a-h)(c- dy Exercise XXXI. Resolve into factors : 1. 0!^ — ax — hx-\-ah. 6. ahx — ahy -\-pqx — pqy. 2. ah-\~ay — hy — y'^. 7. cdx^ + adxy — hcxy — ahif. 3. hc-{-hx— ex — 01^. 8. ahcy — ¥dy — acdx + hd^x. 4. mx + mn + cix + an. 9. ax — ay ~bx -{- hy. 5. cda^ — cxy -\- dxy — ?/-. 10. (?(i/ — cyz + c??/2; - if. 123. The square root of a number is one of the two equal factors of that number. Thus, the square root of 25 is 5 ; for, 25 = 5 X 5. The square root of a* is o? ; for, a^ = a^ X c?. The square root of c^h'^(? is ahe ; for, a^&V = ahc X ahe. In general, the square root of a power of a number is ex- pressed by writing the number with an exponent equal to one-half the exponent of the power. The square root of a product may be found by taking the square root of each factor, and finding the product of the roots. 70 ALGEBRA. The square root of a positive number may be either posi- tive or negative ; for, a^ = a X a, or, a^ == — (2 X — a ; but throughout this chapter only the positive value of the square root will be taken. 124. Case III. From § 73 it is seen that a trinomial is often the product of two binomials. Conversely, a trino- mial may, in certain cases, be resolved into two binomial factors. Thus, To find the factors of The first term of each binomial factor will obviously be x. The second terms of the two binomial factors must be two numbers whose product is 12, and whose sum is 7. The only two numbers whose product is 12 and whose sum is 7 are 4 and 3. .-. ^+7^ + 12 = (:r + 4)(a; + 3). Again, to find the factors of ^ ~\- 5x7/ -\- 67/. The first term of each binomial factor will obviously be x. The second terms of the two binomial factors must be two numbers whose product is 6 7/, and whose sum is 5y. The only two numbers whose product is 6?/^ and whose sum is 5y are 3y and 2y. ... ^ _|_ 5^3/ + Qf = (^ + 33/) (x + 2y). FACTORS. 71 Exercise XXXII. Find the factors of : 1. a;' + 11a; + 24. 2. ^^+ll:6- + 30. 3. ?/^ + 17y + 60. 4. z^ + 132+12. 5. a;' + 21^7+ 110. 6. y^ + 35y + 300. 7. ^^ + 23^> + 102. 8. x'+^x^^. 9. a;' +7^7 + 6. 10. a' + 9a5 + 8^»l 11. a;' + 13aa7 + 36a'. 12. y' + 19j9y + 48^1 13. 0^4-29^2 + 100 g^ 14. a* + 5a' + 6. 15. 2« + 4z' + 3. 16. a^52+18a5 + 32. 17. a;y + 7a7y + 12. 18. z^«+10z^ + 16. 19. a2+9a5 + 205l 20. 07^ + 9:^;' + 20. 21. aV+14a^:r + 335l 24. 5V + 18aJc + 65al 22. aV + 7a^^ + 10:r^ 25. ? V + 23 rs2 + 90 2l «3. xy0' + 192'y2 + 48. 26. mV4-20mV^^ + 5iy^'. 125. Case IV. To find the factors of a;^-9:r + 20. The second terms of the two binomial factors must be two numbers whose product is 20, and whose sum is — 9. The only two numbers whose product is 20 and whose sum is —9 are —5 and —4. .-. ^^ - 9:r + 20 = (:r - 5) {x - 4). 72 ALGEBRA. Exercise XXXIII. Resolve into factors : 1. x'-lx + lO. 13. a'b'c'-lSabci- 22. 2. :r2-29^ + 190. 14. a;' — 15a; + 50. 3. a'^- 23a + 132. 15. a;^ - 20a: + 100. 4. 5^ — 306 + 200. 16. aV — 21aa; + 54. 5. z^~432 + 460. 17. aV-16a&:r + 395'^ 6. x'-7x-{-6. 18. aV-24acz + 14321 7. ^*-4aV + 3a*. 19. :c^-20a; + 91. 8. a;2-8;r+12. 20. a;^ — 23:r + 120. 9. 2^-572 + 56. 21. 2*^-532 + 360. 10. /-7/+12. 22. x''-(a + c)x-i-ac. 11. xY -21xy^ 26. 23. yV - 2d>ahyz + 187 a''^»^ 12. a*5«-lla'^6^ + 30. 24. c^d' -2>0ahcd-\- 221 a'b\ 126. Case V. To find the factors of x''-\-2x-^. The second terms of the two binomial factors must be two numbers whose product is — 3, and whose sum is + 2. The only two numbers whose product is — 3 and whose sum is + 2 are + 3 and — 1. .■.x}-\-2x-2> = {x + ^){x-V). FACTORS. 73 k Exercise XXXIV. Resolve into factors : 1. a^^^x-l. 8. aM-25a-150. 2. x' + bx-M. 9. h^ + U^-4:. 3. y'+Ty-eO. 10. 5V + 3^(7 -154. 4. f+12y~^b. 11. c-i^ + lSc-^-lOO. 5. z'-\-llz-12. 12. 6-2+17^-390. 6. 22_|_i32_140. 13. a2 + a-132. 7. a2_^13a-300. 14. a,y 2^ ^ 9 a;y2 - 22. 127. Case VI. To find the factors of ■ a? — bx—QQ. The second terms of the two binomial factors must be two numbers i j 4 • a a whose product is — bo, and whose sum is — 5. The only two numbers whose product is— 66 and whose sum is — 5 are — 11 and + 6. .\x'-bx~66 = (x-ll)(x + 6). Exercise XXXV. Resolve into factors : 1. x'-3x-2S. 6. a'-lba-lOO. 2. y2_7y_i8. 7. t.io_9^;>_io. 3. ^-9^7-36. 8. :r2-8a;-20. 4. z^-llz-60. 9. f-5mj~b0a\ 5. :^-lSz-U. 10. aW-Sab-4:. 74 ALGEBEA. 11. aV-3a:r-54. 14. fz''-by''z^-%^.. 12. c2c^2_24ec^_i80. 15. a^^^- 16aZ> - 36. 13. aV-a^c?-2. 16. :^- (a — 5)^-a5. We now proceed to the consideration of trinomials wliicli are perfect squares. These are only particular forms of Cases III. and IV., but from their importance demand special attention. 128. Case VII. To find the factors of a;2+ 18^ + 81. The second terms of the two binomial factors must be two numbers ^j^^^^ ^^^^^^^ j^ g^_ and whose sum is 18. The only two numbers whose product is 81 and whose sum is 18 are 9 and 9. .-. 0,-2 + 18:r + 81 = (a; + 9) (a; + 9) = (^ + 9)2. Exercise XXXVI. Eesolve into factoMt : 1. ;r2+12:zr + 36. 8. y''-\-\<6fz^-\-^^z\ 2. ^ + 28:r+196. 9. y« + 24y«+144. 3. a:2 + 34:r + 289. 10. a;V+ 162 a;2 + 6561. 4. 22 + 22 + 1. 11. 4a2+12a^2_|_9j4_ 5. 2/2 + 200y + 10,000. 12. 9:^2^^ + 30^^ + 2522. 6. z''+ 1422 + 49. 13. 9:?;2_^i2:ry + 4y2. 7. a?2 + 36a;y + 324y2. 14. 4 aV + 20 a Vy + 25 o^y . FACTORS. 75 129. Case VIII. To find the factors of a;'' -18 a; + 81. The second terms of the two trinomials must be two numbers whose product is 81, and whose sum is — 18. The only two numbers whose product is 81 and whose sum is — 18 are — 9 and ~ 9. .'.a^-18x + 81 = (x-'9)(x-9) = (x-df. Exercise XXXVII. 1. a'^- 8a 4- 16. 10. ^xY -20xyz + 2by'z\ 2. a^- 30a + 225. 11. 16a;y- 8a;yV + 3/V. 3. a;^- 38a; + 361. 12. da'b'c'-Qab'c'd+bVd'. 4. a;' — 40a; + 400. 13. 16 x^ - 8 xY -\- x^y\ 5. y^ - lOOy + 2500. 14. aV - 2 a'bx'y' + by. 6. 3/* -20/ +100. 15. 36a;y-60a;y' + 252/*. 7. y^ - 50y2 + 625 2^ 16. 1 - 6 a6^ + 9 a'b\ 8. a;*-32a;y+256y*. 17. 9mV-24mw+16. 9. 2^-342^ + 289. 18. Wx^-Uhs^y + Qxy. 19. 49a^-112a5 + 64^>^ 20. 64a;y - 160a;y2 + 100a;V. 21. 49a=^^>V-28a^>ca; + 4a;l 22. 121a;*-286a;'V + 169y^ 23. 289a;y2' - 102a;yVc^ + 9yYd\ 24. 361 a;y22 - 76 abcxT/z + 4 a'b'c\ 76 ALGEBRA. 130. Case IX. An expression in the form of two squares, with the negative sign between them, is the product of two factors which may be determined as follows : Take the square root of the first number, and the square root of the second number. The sum, of these roots will form the first factor ; The difference of these roots will form the second factor. Thus: (1) a:'-h'' = {a + h){a-h). (2) a?-{h-cf=^\a + {h-c)\\a-(h-c)], = [a-j-h — cl{a — b-{-c]. (3) (a~by-(c~dy=^{(a-b)+(c-d)]{(^a~h)-~(c-d)], = la — b-{-c—dl\a~b — c-{-dl. 131. The terms of an expression may often be arranged so as to form two squares with the negative sign between them, and the expression can then be resolved into factors. Thus: a' + b' -c' - d' + 2ab + 2cd, =^a' + 2ab + b'-c^ + 2cd-d\ = (a' + 2ab + b') -(c'-2cd+ d'), ^(a + by-(c-d)\ = l(a + b) + (c-d)U{a + b)~(o-d)\, = {a + b + c-dllai-b-c + dl 132. An expression may often be resolved into three or more factors. Thus : (1) x''-7/'' = (x' + i/')(x'-f) ^ = (x' + f) (x' + y*) {x' + f) (x' - f) = {x' + f) {x' + y') (x' + y^) (x + 2/)(x- y\ FACTORS. 77 (2) 4 {ab + cdj - (c^ + b' -c'- dy, = I2(ab + cd) + (a' + b'- c'-d')] \2(ab + cd) - (a' -]-b'-c'- d')l = \2ab + 2cd + a' -j-b'-c'- d'} \2ab -\-2cd - a^ -V -\- c^ ^ d^\, = \a' + 2a5 + b^) - {*. 17. x'-2yz-y''-z\ 5. a*-l. 18. x''-2xy-^y''-z\ 6. a«-5l 19. a' + 12^)c-45'^-9cl 7. a'-l. 20. a' — 2ay + y'^ — a;'-22r2;-2l 8. 36r^;'-49y^ 21. 2a:y-a;^-y2 + z^ 9. 100a:y-121a'6l 22. x^-^y'' -z^-d''-2xy-2dz. 10. l-49a;^ 23. x" -y"" -\-z^ - a^ -2xz^2ay. 11. a*- 2551 24. 2a5 + a'^ + 5'-cl 12. {a-by-c\ 25. 2:^3/- ^2_ y2^^2_|_52_2a5. 13. x'-ia- h)\ 26. («a; + 5y)^-l. 78 ALGEBRA. 27. \-x'-f-\-2xy. 31. {x^Vf-{y-Xf. 28. (5 a -2)2 -(a -4)2. 32. d^ ~ a? -\- ^xy - ^f . 29. a^-2ah-\-h''-:^. 33. a^- 5^ - 2^c - c^. 30. (:^+l)2-(y+l)l 34. 4a;*-9a;2^5^_]^^ 133. Case X. a? — y'' Since — — — = .r2 + rry + y^ and • _^_^4_|_^^_^^y2_^^^_|_^4^ and so on, it follows that the difference between two equal odd powers of two numbers is divisible by the difference between the numbers. Exercise XXXIX. Eesolve into factors : 1. a'^-W. 6. 8:^3 -272/^. 2. a?-^. 7. 64y«-1000zS. 3. a?-Z^Z. 8. 729:^3 _5i2y3. 4. 2/3 _ 125. 9. 27a3-1728. 5. 2/3 _ 216. 10. 1000 ««- 133153. 134. Case XI. Since T =^^— Q^^ + Q^% and ^^±^ = .'r^-;r3y + a;2^- 0:3/3 + y*, and so on, it follows that the sum of two equal odd powers of two numbers is divisible by the sum of the numbers. FACTORS. 79 Exercise XL. Resolve into factors : 1. c^^f. 6. 216a3 + 512G3. 2. ^ + 8. 7. 729^:3+1728^. 3. :r3 + 216. 8. 2i^-\-'if. 4. 2/^ + 64z8. 9. x^-\-'if. 5. 64 ^'3+ 125^3. 10. 32^^ + 243 c*. 135. Case XII. The sum of any two p&wers of two numbers, whose exponents contain the same odd factor, is divisible by the sum of the powers obtained by dividing the exponents of the given powers by this odd factor. Thus, x^ A- ij^ i^--'-<^f+f- In like manner, x^''-^'^2f, which is equal to ^^°-f(23/)^ . is divisible by x'-\-2y', but x^ + y\ whose exponents do not •' contain an odd factor, and :r^ + y^°, whose exponents do not contain the same odd factor, cannot be resolved into factors. t Exercise XLI. Resolve into factors : ^ 1. a^-{-b\ 3. a;^ + 2/^. 5. rr^ + l. 7. ^^a^ + a^. 2. ai« + §^. 4. 6« + 64c«. 6. a^+l. 8. 729 + ^6. 80 ALGEBEA. 136. Case XIII. For a trinomial to be a perfect square, the middle term must be twice the product of the square roots of the first and last terms. The expression a;* + x^y^ + y* will become a perfect square if ^y be added to the middle term. And if the subtraction of o^y from the expression thus obtained be indicated, the result will be the difference of two squares. Thus: x' + xY + / = {x' + 2xY + /) - xY, = {x' + yy-xY, = {x" + 2/' + xy) {x^ + y' - xy), or, {x" -{-xy-\- y"") {x^ - xy -^ y'). Exercise XLII. Resolve into factors : 1. a^ + a'b'+b'. 8. 49m*+ llOmV -f 81w*. 2. dx' + dxy + 4:y\ 9. 9a* + 21aV + 25c*. 3. 16x'-11xy + y\ 10. ^9a'-lba''b^+121b\ 4. 81a' + 2Sa'b'i-16b\ 11. 64: x' + 128 xy + 81 y\ 5. 81a'-28a'b'i-ie>b\ 12Mx' -^1 xy + 9y\ 6. 9^* + 38:ry + 49/. 13. 25x' -4:lxy + 16y'. 7. 2ba'-9a'b' + 16b\ 14. 81x' --d4.xy i-y'. *If, in Example 12, 9 y* = (- 3 2/^)2^ then 25a;y sheuld be added to 4a;4 _ yjx^y^ -v 83/*, in order to make the expression a perfect square. That is, we should have : (4a;* - 12a;y + 92/*) - 25a;V, = (2x2-32/'^)2-25icy, = (2 a;2 _ 3 ^2 ^ 5 a._y) (2 aj2 _ 3 2/2 _ 5 a^^/), or, (2a;2 ^hxy- 2,y^){^x'' - bxy - Sy^). FACTORS. 81 137. Case XIV. To find the factors of 6x' + x-12. It is evident that the first terms of the two factors might be 6:r and r^, or 2 a; and 3 a;, since the product of either of these pairs is 6^^ Likewise, the last terms of the two factors might be 12 and 1, 6 and 2, or 4 and 3 (if we disregard the signs). From these it is necessary to select such as will produce the middle term of the trinomial. And they are found by trial to be 3 a; and 2x, and —4 and -|-3. .-. 6a;^ + a; - 12 = (3a; - 4) (2a; + 3). Exercise XLIII. Resolve into factors : 1. I2a;2-5a; — 2. 13. 6aV + aa;-l. 2. 12a;2-7a;+l. 14. Qb'-7bx-3x\ 3. Ux' — x-l. 15. 4:X^-i-8x + S. 4. 3a;''-2a;-5. 16. a''~ax-6x\ 5. 3a;2 + 4a;-4. 17. 80" + Uab -15b\ 6. 6a;^ + 5a;-4. 18. Go" -19ac + 10c\ 7. 4a;2 + 13a; + 3. 19. Sx' + S4:X7/ + 21y\ 8. 4a;' + 11 a; -3. 20. 8a;' - 22 a;y - 21 3/^ 9. 4a;'- 4a;- 3. 21. 6a;' + 19a;y — 7y'. 10. x'-Sax + 2a\ 22. 11a' - 23a& + 2^'. 11. 12a* + a'a;'-a;^ 23. 2c' -13cd-i- Qd\ 12. 2a;' + 5a;y + 2y'. 24. 6y' + 73/2 - 3^'. 82 ALGEBRA. 138. Case XV. The factors, if any exist, of a polyno- mial of more than three terms can often be found by the a]3plication of principles already explained. Thus it is seen at a glance that the expression a^-Sa^^ + Sa^^-Js fulfils, both in respect to exponents and coefficients, the laws stated in § 83 for writing the power of a binomial ; and it is known at once that c^-^a^b + Sab'~b^ = (a- bf. Again, it is seen that the expression a^-2xi/ + f + 2xz~~2yz-}-2? consists of three squares and three double products, and from § 79, is the square of a trinomial which has for terms X, y, z. It is also seen from the double product — 2 a;y, that x and 3/ have unlike signs ; and from the double product 2xz, that x and z have like signs. Hence, a^ — 2xy-i-y^ + 2xz — 2yz ■\-^^{x — y-\- zf. Exercise XLIV. Resolve into factors : 1. a8 + 3a2^-f3a52 + 53. 4. a:^+4a;3y+6:rV+4:2.y_^^4 2. a« + 3«2 + 3a + l. 5. x" - \3? -\-^o? -^x-\-\. 3. a^-Za^-\-^a-\. 6. o^-^a^c-^<6a^(?-^a(?-^G^. 7. x^-\-2xy-\-'i^-^2xz-\-2yz^^. 8. x^~2xy^y''-2xz-\-2yz-\-z^. 9. a^ J^W J^ c" J^2ab ~2ac~2hc. \ FACTORS. 83 139. Case XVI. Multiply 2a:-y + 3bya; + 2y-3. 2x - y + 3 X + 2y-3 2a:^— 'xy -\-K>x 4:xy-2y^ +6y -6a; + 3y-9 2a:2 + 3^y _ 2y2. _ 3^^ _|_ 9y _ 9 It is to be observed that 2a?+3xy — 23/'*, of the product, is obtained from (2ar— y) X (a; + 2y) ; that — 9 is obtained from 3 x — 3 ; that — 3 X is the sum of 2a; X — 3 and a; X 3 ; that 9?/ is the sum of 2y X 3 and — 3/ X — 3. From this result may be deduced a method of resolving into its factors a polynomial which is composed of two tri- nomial factors. Thus : Find the factors of ^3^-lxy-2>f-'^x-\-^Q\y~ 27. The factors of the first three terms are (by Case XIV.) 3 a; + ?/ and 2 a; — 3 2/ . Now —27 must be resolved into two factors such that the sum of the products obtained by multiplying one of these factors by 3 a; and the other by 2 a; shall be —9 a;. These two factors evidently are — 9 and + 3. That is, (6a;2_7^^_3^_9^_|_30y_27) = (3:r + y-9)(2^-3y + 3). 140. The following method is often most convenient for separating a polynomial into its factors : Find the factors of 23(^-bxy-^2f-\- Ixz - byz + 32^. 1. Reject the terms that contain z. 2. Reject the terms that contain y. 3. Reject the terms that contain x. 84 ALGEBRA. Factor the expression that remains in each case. 1. 2x'-bxy + 2y'={x-2y){2x-y). 2. 2a;2 + 7a;2 + 322=(a; + 32)(2a; + 2). 3. 22/2-53/2 + 322 = (_2y + 3z)(-2/+z). Arrange these three pairs of factors in two rows of three factors each, so that any two factors of each row may have a common Unri. Thus : x — 2y, a? + 32, —2y-\-^z; 2j; — y, 2x + 2, — 3/ + 2. From the first row, select the terms common to two factors for one trinomial factor : x — 2y-\-?>z. From the second row, select the terms common to two factors for the other trinomial factor : 2x — y-\-z. Then, 2x'-bxy-^2y^+lxz-byz + 2>z^ = (:x-2y-\-Zz){2x-y + z). 141. When a factor obtained from the first three terms is also a factor of the remaining terms, the expression is easily resolved. Thus : (3) x^-?>xy-\-2'f-2>x-\-^y, = {x-2y){x-y)-Z{x~2y), = (x-2y)(x-y-S). EXEECISE XLV. Resolve into factors : 1. 2:^- 5:^^ + 22/2-17^ + 13^ + 21. 2. ex'-Slxyi-Qi^-bx-by-l. 3. 6x^ — 5xy — 6y^ — x — 5y — l. 4. 5x^ — 8xy + Sf-\-1x-by + 2. 5. 2x^~xy-Sf-Sx + 7y + e>. FACTORS. 85 6. a;2_25y2_10a;-20y + 21. 7. ^x^ — ^xy-^-^'if' — xz — yz — '^. 8. ^c^-\-xy-f-Zxz-\-^yz-^z^. 9. ^y^-nxy-^f-^Z^xz-byz-^^, 10. ^oc^-%xy-\-Z^-Zxz-\-yz-'l^. 11. 2^^ - xy -Zf -^yz-1z^. 12. ^x'-\Zxy-\-^y''-\-Vlxz-\Zyz^^^. 13. :r2-2a:y + 2/2 + 5a;-52/. 14. 2a;2_|_5^^_3^_4^2_|.2y2. Exercise XL VI. MISCELLANEOUS EXAMPLES. The following expressions are to be resolved into factors by the principles already explained. The student should first carefully remove all monomial factors from the ex- pressions. 1. 5a;2_i5^_20. 9. a^ + a^+i. 2. 2r^-16a;* + 24a;«. 10. x" -f-xz^yz. 3. Za%^-^ah-\2. 11. ah - ac -W -\-hc. 4. a^+2aa;4-a:^+4a+4:r. 12. ?):^ — Zxz — xy ^yz. 5. c?-1ah-\-h''-(?. 13. a^-x'-ah-hx. 6. :z;2_2:ry+y2_c2+2cc?-c^2 ^4^ ^2 __ 2 aa; + a.-^ + a - :c. 7. ^-x'-'l:^-x\ 15. 3:r2-3y2-2:r + 2y. 8. a^-W-a-h. 16. a;* + a;^ + a:^ _|_ ^,_ 17. aV-aV-aV+1. 18. Z:^-23^y-'^nxf^\^^^. 86 ALGEBRA. 19. 4:x'~x'-{-2x-l. 28. 4a2_4a5 + 5l 20. x' — f. 29. 16x2 -80a:y+ 100 ?/2. 21. x' + f. 30. 36aVy2_255Vt/2. 22. 729 — :r«. 31. 9 ri;^^^ - 30 xy^^ + 25 2^. 23. a;"y + y". 32. 16 x^ — .2^. 24. aV-c^. 33. c(r^-2x7/-2xz-{-f+27/z + z 25. x^ + 4:x~21. 34. a2-aZ>-6S2_4^^12^. 26. 3a^-21a5 + 3052. 35. a;2 + 2a;y + 3/2_^ _^ _ g 27. 2x^-4.0^^-6x^1/'. 36. (a + &)^-c^ 37. x2_^y_6y2_4^_^12y. 39. Sa^ -llx2/ + 6f. 38. l-a: + a:2__^ 4^^ :r2 + 20:r + 91. 41. (:r-y)(rc2-22>^-(:r-0)(;r2_^) 42. a;2-5;r-24. 50. f- 4:7/ -117. 43. (:r2-2/2_^2^)2_4^^2 5^^ x^-^-Qx-lSb. 44. 5a;3^ + 5a;2^2;-602'22. 52. 4a2- 12a5 + 95^- 4^^. 45. 3:r«-:r2 + 3a:-l. 53. (a -f- 3 5/ - 9 (^ - c)2. 54. 9x^-4,f-{-4:i/z-z^. 55. 65V-7J:^;3_3^4_ 56. a«-^>3_3a5(a_i). 57. x^-}-y^-{-Sx7/(xi-y). 58. a«-53-a(a2_52)-f Z,(a-^.)2. 59. 9ar'y'-Sxf-67/\ 60. 6:i;2^i3^^_|. g^^ 61. 6a2^2 -a53- 12 5^ -62. a^ + 2ad+d^-4P-}-12bc-9c'. 63. :?;3-2a;2^ 4- 4x^2-8?/^. 64. 4 aV - 8ia5:r + 3 ^.^ 46 x^ — 2 mx + w^ — n^. 47. 4:a'b'-(a' + b'-c') 48. a^ + cd^. 49. l-14A + 49aV. FACTORS. 87 65. 18.^2 -24;ry + 8y2 + 9a; -6y. 74. 16a^x-2x\ 66. 2a^ + 2.T7/~12y^-^6xz-{-lS7/z. 75. S2ba^-4:bf. 67. (x + 7/y-l-x2/(x + y+l). 76. x-21x\ 68. x^-y^-z'-^2yz + x + i/-z. 77. x^^-2/^\ 69. 2:i;2 + 4a:y + 22/2-j-2aa7 + 2ay. 78. 49' 121 70. 71. 72. 73. 79. 16-81y^ 80. 122'' -2^-6. 81. a^ — x^ + x — l. 82. :r2 + 2a; + l-?/*. 16 a'bi-S2abc + 1260^. m^p — m^q — r?p + r^q. 12 a:r^ — 14 axy — 6 ay^. 2x^-\-^:x?-n^x. 83. 49(a-^)2-64(m-n)2. 84. X-(^l±^l^ \ 2ab )' 85. ^2- 53^ + 360. 86. x^-2x^y-^x^-4:X-{-Sy-^. 87. 2a5-25c^-ae + ce + 2^<2_5e. 88. 125r^ + 350:i:y+245:ryl 89. a« + r/^ + a''^Ha'^'+«'^' + «^'. 90. 2 a*.r — 2 a^cx + 2 ac":r — 2 c%. 91. 6^-5^y-6y2 + 3^2 + 15y2-92«. 92. ^x'-9xy-\-2y^-?>xz-yz-z\ 93. 3a2-7a5 + 2Z>2 + 5a(?-5/^c4-2c2. 94. x^-2k?-{-x^-^x-\-S. 95. 5a;2-8a;y + 32/2-5a; + 3y. 96. a2-2ac^+c^2_4j2_^12^>c-9c2. 97. {f-x-6){c(?-x-20). CHAPTER VII. Common Factors and Multiples. 142. A common factor of two or more expressions is an expression which is contained in each of them without a remainder. Thus, 5 a is a common factor of 20 a and 25 a ; 2>3!^if is a common factor of 12:ry and Ibx^i^. 143. Two expressions which have no common factor ex- cept 1, are said to he prime to each other. 144. The Highest Common Pactor of two or more expres- sions is the product of all the factors common to the expressions. Thus, 3 a^ is the highest common factor of 3 a^, 6 a^, and 12 a^ 6x^y^ is the highest common factor of lOx^i/ and 16x^7/^. For brevity, H. 0. F. will be used for Highest Common Factor. (1) Find the H. C. F. of 42 a%'x and 21 a^^V. ^2a%^^ =2xSx7Xa^Xb^Xx; 21 a^^ V ^SxIXa^Xb^Xx"". .-. the B..G.F. = 3x1Xa^Xb^Xx, = 21 a^b^x. (2) Find the H. C. F. of 2^ + 2 ax^ and 3 abxy -f 3 ba^y, 2a^x -{- 2ax^ = 2 ax (a -\~ x) ; 3 abxT/ -f 3 bx^i/ = 3 bxi/ (a -f x). .'. the B..C.¥.=x(a + x'). COMMON FACTORS AND MULTIPLES. 89 (3) Find the H. 0. F. of 8aV - 24:a^x + 16a^ and 12aa^2j - 12ax7/ - 24ay. 8a'x'-24:a'x+16a^ = 8a'(x'-Sx + 2), = 2'a'(x~l)(x~2); I2ax^y - \2axy - 24ay = \2ay {a?-x - 2), = 22x3ay(a; + l)(:r-2). .-. theH.0.F. = 22«(a;-2), • =^a{x-2). Hence, to find the H. 0. F. of two or more expressions : Resolve each expression iyito its lowest factors. Select from these the lowest power of each comrnon factor, \' and find the product of these powers. I Exercise XL VII. FindtheH.C.F. of: ^ 1. ISa^Vc^and 2>^a'bcd\ 2. 17;?^^ 34/^, and b\p^x^-^x + 2>, ^x^ -\-^x- 12, and 12a;^ - 12. 20. 6 (a - by, 8 (a' - bj, and 10 (a* - b'). 21. x'-y\(x-{-yy,Sindx^ + Sxy + 2y\ 22. x^ — 3/^ a;^ — y^ and :r'^ — 7^?/ + 6yl # 23. x^-l,x^-l,SiZidx' + x--2. 145. "When it is required to find the H. 0. F. of two or more expressions which cannot readily be resolved into their factors, the method to be employed is similar to that of the corresponding case in arithmetic. And as that meth-d consists in obtaining pairs of continually decreasing numbers which contain as a factor the H. C. F. requir d ; so in alge- bra, pairs of expressions of continually decreasing degrees are obtained, which contain as a factor the H. 0. F. re- quired. The method depends upon two principles : 1. Any factor of an expression is a factor also of any multiple of that expression. Thus, if F represent a factor of an expression A, so that A = nF, then mA = mnF. That is, mA contains the factor F. 2. Any common factor of two expressions is a factor of the sum or difference of any multiples of the expressions. Thus, if F represent a common factor of the expressions A and B so that A = mF,2Jidi B = nF; ' then pA =pmF, and qB = qnF. Hence, pA ± qB =pmF± qnF, = (pm + qn)F. | That is, pA ± qB contains the factor F. \ COMMON FACTORS AND MULTIPLES. 91 p> 146. The general proof of this method as applied to numbers is as follows : Let a and h be two numbers, of which a is the greater. The operation may be represented by : b)a{p 42)154(3 nF)mF{p ph 126 pnF ~V) h (q ~28) 42 (1 ~^ nF(q qc 28 qcF ~d)c(r 14)28(2 ~F)cF(c rd ^ cF p, q, and r represent the several quotients, c and d represent the remainders, and d is supposed to be contained exactly in c. The numbers represented are all integral. Then c = rd, h = qc + c? == qrd + d = {qr + 1) d, a = pb -{■ c ^ pqrd + pd -\- rd, = {pqr +p + r)d. .•. c? is a common factor of a and b. f It remains to show that d is the highest common factor of a and b. m Let/ represent the highest common factor of a and b. Now c = a —pb, and / is a common factor of a and 6. .-. by (2) /is a factor of c. Also, d=b — qc, and / is a common factor of b and e. .■. by (2) /is a factor of d. That is, d contains the highest common factor of a and b. But it has been shown that c? is a common factor of a and b. .'. d is the highest common factor of a and b. Note. The second operation represents the application of the method to a particular case. The third operation is intended to rep- resent clearly that every remainder in the course of the operation contains as a factor the H. C. F. sought, and that this is the highest factor common to that remainder and the preceding divisor. 92 ALGEBRA. 147. By the same method, find the H. C. F. of 2a^ + x-S)4:C(^ + 8:t'- x-Q(2x + ^ 4:^-i-2:^-6x Qx' + bx-e 2a; + 3)2a;H x-S{x-l 2 x^ + ?>x -2x-Z :. the H. 0. Y. = 2x + 3. -2:i:-3 The given expressions are arranged according to the descending powers of x. The expression whose first term is of the lower degree is taken for the divisor ; and each division is continued until the first term of the remainder is of lower degree than that of the divisor. 148. This method is of use only to determine the com- pound factor of the H. 0. F. Simple factors of the given expressions must first be separated from them, and the highest common factor of these must be reserved to be multiplied into the compound factor obtained. Find the H. 0. F. of 12a;^ + 30a,-3 - nx^ and 32.^^ + Mx^ ~ 176x. 12x' + SOa^~na^ --^ Qx' (2x^ -^ 5x -12). S2a^ + 84.^^ - 176 x - 4:x(8x^ + 21a.- - 44). Qct^ and Ax have 2x coiiwion. 2^-2 + 5a;- 12)8.^2 + 21 a;-44(4 Sx^ + 20x-A8 X+ A)2x^-\-bx~12{2x~^ 2x'' + d>x -3.T-12 .-. the H. C. F. = 2x {x + 4). - 3a;- 12 COMMON FACTORS AND MULTIPLES. 93 149. Modifications of this metliod are sometimes needed. (1) FindtheH.C.F. of4a;2_g^_5g^ndl2^2-4a;-65. 4:0^ -8x- 5) 120^- 4:^-65(3 2037-60 The first division ends here, for 20a: is of lower degree than ix^. But if 20a; — 50 be made the divisor, 4ar^ will not contain 20a; an in- tegral number of times. Now, it is to be remembered that the H. C. F. sought is contained in the remainder 20 a; — 50, and that it is a compound factor. Hence if the simple factor 10 be removed, the H. C. F. must still be con- tained in 2 a; — 5, and therefore the process may be continued with 2 a; — 5 for a divisor. 2x-6)4:a^- 8:r-5(2a; + l 4:X^~10X 2x-6 2x-5 .-. theH.0.F. = 2a:-5. C2) Find the H. 0. F. of 210.-3 - 4:x^ - 15a; - 2 and 21:^3 - 32^2 - 54a; - 7. 21a^~4:x''-lbx--2)21x'-S2x'-b^x~'J(l 2\7?- 4a;^-15a;-2 -2%x'-'^^x-6 The difficulty here cannot be obviated by removing a simple factor from the remainder, for — 28 a:^ _ 39^ _ 5 j^g^g ^^ simple factor. In this case, the expression 21ar' — 4ar^— 15a; — 2 must be multiplied by the simple factor 4 to make its first term divisible by — 28 a,^. The introduction of such a factor can in no way affect the H. C. F. sought ; for the H. C. F. contains only factors common to the remain- der and the last divisor, and 4 is not a factor of the remainder. The signs of all the terms of the remainder may be changed; for if an expression A is divisible by — F, it is divisible by + F. 94 ALGEBRA. The process then is continued by changing the signs of the remain- der and multiplying the divisor by 4. 28x^-{-S9x + 5)84:a^- 16x'-~ e>Ox~ 8(Sx -ISSx"- Ibx- 8 Multiply by - 4, —4 532^H^00^+32{19 6S2x'+Ulx-j-95 Divide by - 63, — 63 ) -441 a; -63 7x-^ i 7x-\-l)28x' + S9x + 5{4:x + 5 28:r^+- 4:x 35^ + 5 .-. tlieH.0.F. = 7a; + l. 35;r + 5 (3) Find the H. C. F. of 8a^-\-2x-^ and 6^(^ + 5x^-2. 6x^+ 5x'- 2 4 8x^-\-2x-^)2^x^ + 20x'- 8 {?>x+l 24:0^+ Qx"- 9x Ux^+ 9x- 8 Multiply by 4, 4 56:^2 + 86:1; -32 56a;^+14a;-21 Divide by 11, ll )22a;-ll 2x~ l)8x'-{-2x-~S{4:x-{-S 8x'-4:x 6x — S .-. the H. 0. Y. = 2x- 1. (jx-S In this case it is necessary to multiply by 4 the given expression 6a^ + 5ar^ — 2 to make its first term divisible by 8ar^, 4 being obvi- ously not a common factor. COMMON FACTORS AND MULTIPLES. 95 The following arrangement of the work will be found most convenient : 8:^ + 2:r-3 8r'-4:x 6a^+ 5a^~ 2 4 - 8 6x-S 6x-S 24:^ + 20x^- 8 24^+ ex"- 9x Sx Ux'-i- 9x~ 4 b6a^+S6x- 56:^2+14:^- ll)22:i;- 2x- -32 -21 -11 - 1 + 7 4:r + 3 150i From the foregoing examples it will be seen that, in the algebraic process of finding the highest common factor, the following steps, in the order here given, must be carefully observed : I. Simple factors of the given expressions are to be re- moved from them, and the highest common factor of these is to be reserved as a factor of the H. 0. F. sought. II. The resulting compound expressions are to be ar- ranged according to the descending powers of a common letter ; and that expression which is of the lower degree is to be taken for the divisor ; or, if both are of the same degree, that whose first term has the smaller coefiicient. III. Each division is to be continued until the remainder is of lower degree than the divisor. IV. If the final remainder of any division is found to contain a factor that is not a common factor of the given expressions, this factor is to he rermoved; and the resulting expression is to be used as the next divisor. V. A dividend whose first term is not exactly divisible by the first term of the divisor, is to be multiplied by such an expression as will make it thus divisible. ALGEBRA. Exercise XL VIII. FindtheH.C.F. of: 1. 5:t'-\-4:X-l, 20a;2^21a?-5. 3. 6a* + 25a3-21a2 + 4a, 24a* + 112a'- 94^^ + 18a. 4. 9a;3_|_9^_4^_4^ 45a:« + 54:^2- 20^- 24. 5. ^Ix^-Sx^ + Qc^^-Scc", 162a;« + 48^-3 _ 13^2 _|_5^_ 6. 20a;«-60ri;2_j_5o^_20, 32:r<- 92a;« + 68a;2_24^. 7. 4a;2_8^_5^ 12:^_4:r-65. 8. 3a3-5a2^-2a:^, 9 a' - 8 a^^r - 20 a:^^. 9. 10^ + x'-9x + 24:, 20 a;* -17a:2 + 48:^-3. 10. 8rc«-4^-32a7-182, 36^ - 84 ^r^- Ilia; - 126. 11. bx'(12a^+4:x'+11x-d), 10x{24:x^-b2x'-\-14:x~l). 12. 9x^^-x'f-20xy\ ISa^y -ISxY -2xf ~S2/\ 13. 6x^-x-15, 9a;2_3^_20. 14. 12a^-9x'-\-bx + 2, 245^+10:^ + 1. 15. 6a^+15x^-6x + d, 9a^ + 6x^-51x + SQ. 16. ^a^-x^y-xf~5f, 7a^+ 4:0^^ i^ 4:X7/ -Sf. 17. 2a3-2a2-3a-2, 3a3- a^- 2a- 16. 18. 122/8 + 2y2_94y_ 60, 482/3 -24y2 -3482/ + 30. 19. 9x(2x*-6:^-:^+15x-10), 6x'(Ax^-{-Ga^-ix'-15x-15'). 20. Wx*-^2a^-15x'+5x-^2, Sbx'+a^-llbx'+SOx + l. 21. 2lx*-4:a^-15x'-2x, 2la^ -^2x' -~b4.x-1. 22. 9a;V-22^2/3-3V+102/'. 9r^y-6aV+^y -25:r/. COMMON FACTORS AND MULTIPLES. 97 23. 6a^-^x*-ll^-S^-Sx-l, 24. x*-as^-a^a^-a^x-2a\ ^x^ -laci^ + Sa'x -2a^ 151. The H. C. F. of three expressions will be obtained by finding the H. 0. F. of two of them, and then of that and the third expression. For, if ^, B, and Care three expressions, and D the highest common factor of A and B, and U the highest common factor of B and C, Then B contains everj factor common to A and B, and B contains every factor common to B and C. .'. E contains every factor common to A, B, and O. Exercise XLIX. FindtheH.C.F. of: 1. 2^ + a;-l, x'-[-bx-\-4:, a?-\-l. 2. f-f-y-^l, 32/2-2y-l, f-f-{-y-l. 3. af'-4:cc^+^x-l0, o^-{-2a^-2>x+20, ci^-{-bx'-^x-\-2>b. 4. x«- 7^ +16:^-12, 3:r3-14:r2+16:r, 5. f-bif-\-lly-lb, 2^_2^+3y + 5, 2f~1f+lQy-lb. 6. 2a^-{-?>x-b, ^x'-x~2, 2x' + x-^. 7. a^-1, o?-~x'-x-2, 2x^-x'~x-Z. 8. x^'~Zx-2, 2j?^?>j?-\ x?^\. 9. \2{x'-y'). 10 (:.-^-/), 8(xV + ^y')- 10. x^-^x\f, :i^y-\-y\ x^-\-x^if^y\ 11. 2{fy — x'f), 2>{p[^y — x'f), 4:{x^y~xy^), b{x?y — x'f). 98 ALGEBRA. Lowest Common Multiple. 152. A common multiple of two or more expressions is an expression which is exactly divisible by each of them. 153. The Lowest Common Multiple of two or more ex- pressions is the product of all the factors of the expressions, each factor being written with its highest exponent. 154. The lowest common multiple of two expressions which hdve no common factor will be their product. For brevity L. C. M. will be used for Lowest Common Multiple. (1) Find the L. 0. M. of 12 a\ Ubc", 36a5l Ubc'^^ Xlbc", 36ab' = 2^xS^ab\ .'. the L. C. M. = 22 X 32 X Ta^^V = 252 a'b'c". (2) Find the L. 0. M. of 2a2 + 2aa:,6a2-6:r2, 3a2_6a:r + 3:r2. 2a^-j-2ax =2a(a-\-x), Qa'-6x' =2x3(a + :r)(«-.r), Sa^-6ax-]-3x^ = S(a- xf. .-. the L. C. M. = 6 a (a + ^) (« - :^-)'- Exercise L. ^ind the L. C. M. of: 1. 4 A, 6aV, 2ax'. 4. a^-1, x^-x. 2. 18 ax^, nay\ \2xy. 5. cv'-b'^, a^^ab. 3. x^, ax + x^. 6. 2:r-l, 4^:2- L COMMON FACTORS AND MULTIPLES. 99 7. a + h, a^ + h\ 9. x'-x, ^-1, a?+l. 8. x^-l, ^+1, x^~l. 10. x^-l, x'-x, a^-l. 11. 2a + l, 4a2-l, Sa^+l. 12. {a + hf, a^-h\ 13. 4(1 + ^), 4(1-^), "li^-o?). 14. a;-l, ^2_|_^_|_i^ a;3_i_ 15. :^-f, {x-\-y)\ {x-y)\ 16. :^-y\ ^{x-y)\ \K^-\-f). 17. 6(:^;2_|_^^>)^ 8(2:y-y2), I0(a;2_^>)^ 18. a.^ + 5:2; + 6, a,-2 + 6:r + 8. 19. a^-a-%), a^-\-a~\% 20. a:2_^ 11^ + 30, ^+12a; + 35. 21. ^^2-9:^-22, :c2_i3^_|_22. 22. ^ah{c?-Zab-\-'2.h% 6a\a' + ab ~ 6b'). 23. 20(^-1), 24(a;2_^_2), 16(a^ + x-2). 24. 12:i7(2;2-2/2), 2a;2(^^^)2^ Zf{x~y)\ 25. {a — b){b — c), (b — c)(c — a), (c — a)(a — b). 26. (a — 6)(a — c), (b — a)(b — c), (c — a)(c — by 27. :^-Ax' + Sx, x^ + a^-Ux", a^ + Sx^-4:X^. 28. xhj — xrf, Zx{x — yf, ^y(x — yf. 29. (a+5/-(^+c^)^ (a + e)2-(5 + ^)^ (a+^)2-(5 + ^)'. 30. (2:r-4)(32:-6), (a; -3) (4 a; -8), (2a;-6)(5:r-10). 155. When the expressions cannot be readily resolved into their factors, the expressions may be resolved by find- ing their H. 0. F. 100 ALGEBRA. I. Find the L. C. M. of Sf. 6a^ 6^ — 4^X2/^ 9a^ 2 22 V - ^f 18:^ 18 ri:^ lly )33:r^.y-44V-22.v^ 3a;2 2a; 4:xy — 2y Hence, 6ar»-lla;2^ + 22/8 = (2a7-2/) (3a;2_4^y_2^2>)^ and 9o?-22xf-^f = {^xA^^y)\zo?~^xy-2y'). :. theL.0.M.=-(2a;-y)(3a;-f 4y)(3ri-2-4^y-2y2). In this example we find the H. 0. F. of the given expres- sion, and divide each of them by the H. 0. F. 156. It will be observed that the product of the H. 0. F. and the L. 0. M. of two expressions is equal to the product of the given expressions. For, Let A and B denote the two expressions, and D their H.C.F. Suppose A = aD, and B = bl) ; Since I) consists of all the factors common to A and B, a and b have no common factor. .'. L. 0. M. of a and b is ab. Hence, the L. 0. M. of al) and bl) is abl). Now, A = aD, and B=bD] .'. AB = abD X D. . AB D =^abD=^ the lowest common multiple. That is, The L. 0. M. of two expressions can be found by dividing their product by their H. 0. F. Or, by dividing one of the expressions by the H. 0. F., and 'inultiplying the result by the other expression. COMMON FACTORS AND MULTIPLES. 101 157. To find the L. C. M. of three expressions, A, B, C. Find M, the L. C. M. of ^ and ^ ; then the L. 0. M. of M and (7 is the L. 0. M. required. Exercise Li. Find the L. CM. of: 1. Qx'-x-^, 21^2_i7^_l_2^ 14a,-2 + 5^-l. 3. :?^-27, x'-l^x + SQ, s^-Sx^~2x+6. 4. 5:^2+19:^-4, 10a;2_|_2^3^_3_ 5. Ux' + xy-Qf, 18a;2 + 18:ry-20?/2. 6. x^-2a^ + x, 2x*-2a^-2x-2. 7. 12x' + 2x-4:, 12:^2 -42a; -24, 12a;2_ 28^-24. 8. a^-Qx^+llx-e, x^-9x'-{-2Qx-24., a^-Sx'+ldx-li 9. a^-4:a^ c(^+2ax^ + 4:a^x + Sa^, a?-2ax^^^a^x-^9,a?, 10. o?^2y?y-xy^-2f, :i? -2a?y -xf ^2f. 11. l+i?+/, 1-p+i^^ \-\-f-\-f. 12. (1-a), (1-a)^ (l-a)«. 13. {a-\-cJ-h\ (a + hf-c', {h-{-cf-o?. 14. ^(?-^c'y-\-cf-f, 4:c^-(^y-?>cf. 15. m3-8m + 3, m« + 3m' + m + 3. 16. 20n^ + 7i2-l, 25n^ + 5^3-71-1. 17. 5^-2^3 _^52-85 + 8, 453-1252 + 95-1. 18. 2r^-8r*+12r3-8r2 + 2r, Sr^-Gr^ + Sn CHAPTER VIII. Feactions. 158. The expression -• is employed to indicate that a units are divided into h equal parts, and that one of these parts is taken ; or, that one unit is divided into h equal parts, and that a of these parts are taken. 159. The expression ~ is called a fraction, a is the nu- merator, and h the denominator. 160. The numerator and denominator are called the terms of the fraction. 161. The denominator shows into how many equal parts the unit is divided, and therefore names the part ; and the numerator shows how many of these parts are taken. It will be observed that a letter written above the line in a fraction serves a very different purpose from that of a letter written below the line. A letter written above the line denotes number ; A letter written below the line denotes name. 162. Every whole number may be written in the form of a fraction with unity for its denominator ; thus, a = -• FRACTIONS. 103 To Reduce a Fraction to its Lowest Terms. 163. Let the line AB be divided into 5 equal parts, at the points C, D, E, F. ^ 1 I I I I I I 1 I I I I I I I U C D E F Then^i^is-fof J.^. (1) Now let each of the parts be subdivided into 3 equal parts. Then AB contains 15 of these subdivisions, and ^.i^^ con- tains 12 of these subdivisions. .-. J.i^is||of ^^. (2) Comparing (1) and (2), it is evident that |- = xf- In general: If we suppose ^^ to be divided into h equal parts, and that AF contains a of these parts, Then^i^is^of^^. (3) Now, if we suppose each of the parts to be subdivided into c equal parts. Then AB contains he of these subdivisions, and ^i^ con- tains ac of these subdivisions. .-. ^i^is^of^^. (4) be Comparing (3) and (4), it is evident that a _ae h ~ he Since — is obtained by multiplying by c both terms of the fraction -, and, conversely, - is obtained by dividing by e both terms of the fraction ^, it follows that be 104 ALGEBRA. I. If the numerator and denominator of a fraction be multiplied by the same number, the value of the fraction is not altered. II. If the numerator and denominator be divided by the same number, the value of the fraction is not altered. Hence, to reduce a fraction to lower terms. Divide the numerator and denominator hy any common factor. 164. A fraction is expressed in its lowest terms when both numerator and denominator are divided by their H. 0. F. Eeduce the following fractions to their lowest terms : (1) (2) (3) (4) 0^ _ {a — x) jo? ■\-ax-^QiF) _ a? ■\- ax -\- o? c? — 0^ {a — x^{a'\-x) a-\-x a^+7a+10 __ (a + 5)(a + 2) ^ ft + 5 a2 + 5a + 6 (a + 3)(a + 2) a + 3' 6r^-5a;-6 ^ (2a; - 3) (3 a; + 2) ^ 3a;-f 2 8a;2_2a;-15 (2a;- 3) (4a; + 5) 4a; + 5* a^-7a^+16a-12 3a3-14a2+l6a * Since in Ex. (4) no common factor can be determined by inspection, it is necessary to find the H. 0. F, of the numerator and denominator by the method of division. Suppress the factor a of the denominator and proceed to divide ; FRACTIONS. 105 7aF+16a- 12 3 3a« -21a2 + 48a- -14^2+ 16a 36 - 7a2 + 32a- 3 36 -21a^ + 9Qa-~ ~-21a'+98a- 108 112 -2j- 2a + 4 a~ 2 •. the H. G.F. = a 3a?-14a + 16 3^2 6a - 8a+16 ~ 8a + 16 a-7 3a-8 2. Now, if a» — 7a2 + 16 a - 12 be divided by a - 2, tbe re- sult is a^ — 5 a + 6 ; and if 3 a^ — 14 a^ + 16 a be divided by a — 2, tbe result is 3 a^ — 8 a. . a^-7a^ + 16a-12 _^ a^-5a+-6 *' 3a3-14a2+16a 3a2-8a ' 165. When common factors cannot be determined by in- spection, the H. 0. F. must be found by the method of division. Exercise LIT. Reduce to lowest terms : 2. ar^-9x + 20 :i;2-7a; + 12' c(^-2x~S :r2-10:r + 21' x' + x+l' a^ + 2a^7/^ + y^ x^ — v^ a^ + l 9. 10. a« + 2a2 + 2a+l a^-a-20 a2 + a-12* :^ -4a;^ + 9^-10 a^ + 2x'~Sx + 26' a;^-5a;^+ll^-15 x'-ar'+^x + b ' X* -f- x^2/ -f- X7/^ — y ^ x^ — a^y — x]^ — y^ 106 ALGEBRA. :r3 + a;2_^_l- a^^ab-2b^' ' a^-a^-2x-{-2' '4:(ia'b-abJ' 4:ar'-12ax + 9a' a^ + 2ah + P-(r' 8x^-21a^ ' ' a' + ab-ac ' ' 9a?-\-^ab~2b^' ' ^x'y-bxy^-Qf' o?-b^-2bc-c' a^-(5 + g + ^)' ' a^ + 2ab + ¥-(^' ' {a-bf~{c + df' x* — a?-2x-\-2 ^o(^ — bx — Q ' 2a?-x~\ ' ' 8:^:2-2:^-15' ^g x^-Qx' + Ux-G 28 ^* + ^y' + 2/* x^--2x^ — x-\-2 {^ — y) K.^ — if') ^g ^x^~2?>x'-\-l^x-^ 29 ^' + ^' Qa^-l1a^+llx-2 x^-x'y^-\-y'' x'~x^-x + l 3Q {a?-\-b'){a' + ab + W) ' x^-2x^-x'-2x-\-X * {00^-1^)10? -ab-\-b'') To Eeduce a Fraction to an Integral or Mixed Expression. ^ I 1 Change ' to a mixed expression. X — 1 {x^-\-\)-^{x-V) = x^-\-x-\'\-\—^' Hence, 166. If the degree of the numerator of a fraction equals or exceeds that of the denominator, the fraction may be changed to the form of a mixed or integral expression by dividing the numerator by the denominator. FRACTIONS. 107 The quotient will be the integral expression, the remainder (if any) will be the numerator, and the divisor the denom- inator, of the fractional expression. Exercise LIII. Change to integral or mixed expressions : 1. . 6. . x—1 ba — x 2. ^ 3;r2 + 6^ + 5 Q 2x^~bx~2 0„ • . O. . X-\-4l X — 4: c^ — ax-\-x^ Q o?-\-h ^ a-YX a — o To Reduce a Mixed Expression to the Form of a Fraction. 167. In arithmetic 5f means 5 4- f . But in algebra the fraction connected with the integral expression, as well as the integral expression, may be posi- tive or negative ; so that a mixed expression may occur in any one of the following forms : .a a , a a 108 ALGEBRA. Change n + — to a fractional form. Since there are b hths in 1, in n there will be n times h bths, that is, nb bths, which, with the additional a bths, make nb-{-a bths. In like manner : .a nb-{-a a 7ib — a ^ 2. ZL ' , a — nb-{-a -"+5= b' 1 a —nb — a TT and — n — - = ; . Hence, 168. To reduce a mixed expression to a fractional form, Multiply/ the integral expression by the denominator^ to the product annex the numerator, and under the result write the denominator. 169. It will be seen that the sign before the fraction is transferred to the numerator when the mixed expression is reduced to the fractional form, for the denominator shows only what part of the numerator is to be added or subtracted. The dividing line has the force of a vinculum or paren- thesis affecting the numerator ; therefore if a Tninus sign precede the dividing line, and this line be removed, the sign of every term of the numerator Tnust be changed. Thus, a—-b cn—-(a-~b) en — a + 5 FEACTIONS. 109 x — 1 (1) Change to fractional form x-~l-{ . I ^ .-1+^ X _a^ — x-{-(x — l) = ■» X __o?~x-\-x— 1 = > X 0?~1 X x-\ (2) Change to fractional form x — 1 x-\ X -1- X x^ — x-~{x--'\) X x'- ~x-x-\-\ X x"- -2a; + l Exercise LIV. Change to fractional form : x + y 1. 1-^-2/ K f;._9A_?«^4^' ^ - , x—y 2. 1-L ^ 3. ?>x x-\-y 1 + 2:^2 X 4. a — X'\ ■ a — x 5a — 65 6. a^-l «^ + f. a + 5 7. ..^ 2-3a + 4a2 ^^ 5-6a • a 3^ 5a^-3 2a 110 ALGEBRA. 9. — '—z + 1. 15,2a — o — -. a~o a-\-b 10. ^-1. 16. 3^-10 • ^1 a-\-h :r + 4* 11. -^-(^ + 3/). 17. :r2 + :r + l+ ^ ^ + y a;— 1 12. §^Zli2^ + 6a + 3^. 18. :.3_ 3^_ 3^ (3 -^) 13. a_l + -J_. 19. a2_2aa; + 4:r2- ^^ a-\-l a-i-2x 14. 07 + 5 — . 20. a; — a + vH £_ LA. x — 6 ^ x-\-a Lowest Common Denominator. 170. To reduce fractions to equivalent fractions having the lowest common denominator : Zx 2v 5 Reduce 7—^, — ^, and --— ^ to equivalent fractions hav- 4a^ 3 a Qar ing the lowest common denominator. The L. CM. of4a^ 3a, and Qa^-=l2a^. If both terms of — ^ be multiplied by 3 a, the value of the fraction will not be altered, but the form will be changed to —~ ; if both terms of ^ be multiplied by 4 a^, iZd O Ob Q 2 the equivalent fraction -^-^ is obtained ; and, if both terms 12 a** 5 . 10 of — ^ be multiplied by 2, the equivalent fraction - — - is ba 12a obtained. rSAOTIONS. Ill Hence, ^„ ^, A areequalto ^^, |g, ^3, respectively. The multipliers 3 a, 4a^, and 2, are obtained by dividing 12 a^ the L. C. M. of the denominators, by the respective denominators of the given fractions. 171. Therefore, to reduce fractions to equivalent frac- tions having the lowest common denominator, Mnd the L. 0. M. of the denominators. Divide the h. CM., hy the denominator of each fraction. Multiply the first numerator hy the first quotient, the sec- ond hy the second quotient, and so on. The products will he the numerators of the equivalent fractions. The L. 0. M. of the given denominators will he the denom- inator of each of the equivalent fractions. Exercise LV. Reduce to equivalent fractions with the lowest common denominator : 1. , — r- — . 5. 2. 3. 6 ' 18 ■ ' {a-h){h-cy {a-h){a-cy 2x~4:y Sx-~Sy „ 4:0^ xy b^ ' 10a; ' ' S(a + hy 6{a^-h^)' 4:a-~bc 3a-2c ^ 8:?; + 2 2:?;-l Sx + 2 bac ' 12a^c 5 6 f. a — hm -, c — hn z. -, -1 ^- o. , i, . l — xl — or mx nx 112 ALGEBRA. Addition and Subtraction op Fractions. 172. To add fractions : Reduce the fractions to equivalent fractions having the lowest common denominator. Add the num^erators of the equivalent fractions. Write the result over the lowest common denominator. 173. To subtract one fraction from another : Reduce the fractions to equivalent fractions having the lowest common denom,inator. Subtract the numerator of the subtrahend from the numer- ator of the minuend. Write the result over the lowest common denominator. (1) Simplify, 4^+§^. The lowest common denominator (L. 0. D.) = 15. The multiphers are 3 and 1 respectively. 12 a: + 21 = 1st numerator, 3 a: — 4 = 2d numerator, 15 a: + 17 = sum of numerators. . 4a: + 7 , 3a:-4 _^ 15a:+17 •* 5 15 15 * ^ON a- r-P 3a — 45 2a — b-\-c , 13a — 4(? (2) Simplify, 3~ + i2 ' The L. 0. D. = 84. The multipliers are 12, 28, and 7 respectively. 36 a — 48 Z> = 1st numerator, — 56 a + 28 5 — 28 (? = 2d numerator, 91a — 28 (? = 3d numerator. 71a — 20 5 — 56hy x-y {x-yf 2xy{x — y) 2xy{x-^y) (1) Simplify -^- — — -. a — b a + b a^ — b^ TheL.G.D. = {a-b){a + b). The multipliers are a + b, a — h, and 1, respectively. FRACTIONS. - 115 20^ -j- S ab -\- b^ = 1st numerator, — 2a^ -\- Sab — ^'^ = 2d numerator, — Qab =^ 3d numerator. = sum of numerators. . 2a-{-b 2a — b 6ab _q " a — b a-\-b o^—h^ (2) Simplify -^ ^^^ + 1 + -1^^ • or — 'If x-\-y x^ -\- 'If The L. C. D. = (:r + y) {x - y) {x' ^f). The multipliers are rr^ + y^, (a; — y) (a;^ + y^), {x + y){x — y), {x^ + 2/2), (x +y){x — y), respectively. ^y + y* "= 1^*^' numerator, — x^-\- 2x^y — 2x^y'^ + 2xif' — 3/* = 2d numerator, X*' — y* = 3d numerator, 2x^y — 2xy^ = 4th numerator. 4 o^y — o^y''' — y* =^ sum of numerators. Sum of fractions = ^—, — A — —- x'-y' Exercise LVIII. Simplify : 1,1, 2a « X x^ X \-\-a \ — a \ — G^ \~x \ — x \-^x^ L 2 -^ 1 I 2^ 4 ^4--JL_ + _^' \—x \-\-x \-\- o? y x-\-y x^-\- xy x—\ , X — 2 , X — S X -2 x-S x-4. ^ 3 , 4a 5a^ b. j- X —a {x — ay {x — of 116 ALGEBEA. 1 x + 2 {x-\-l){x-\-2) .a — h , h — c . G — a x — a . x — h {a — hf y. -j- X— b x — a {x — a)(x — h) 10. ^ + 3/ 2^ J ^y-^ y x-\-y yix'-f) - a-\-'b . 5-f-g I g + g {h ~c){c — a) {c — a) {a — b) {a — b)(b — c) o? — bc , W — ac , c^ -\- ab ' (a + b)(a + c)~^ (b + a)(b + c)~^ (c + b)(c + a) a X c?-\-^ a — x a-\-'2,x (a — x){a-{-2x)' 13. 14. , ± ^ - ,-^ ,+ 6 (a — b){b — c) {a — b){a — c) {a — c)(b — c) ^^ x — 2y _ 2x + y 2x 16. x{x — y) yix-\-y) a^-f a — b a — b (a — b) (x -\- y) x{a-\-b) y(a-{-b) xy{a-\-b) Zx x-\-2y Zy 18. {x^ryf ^-f {^~yf a — c a — b {a + bf-(^ {a + cf-b''' 19. ^ + ^ _ ^ — ^ , a5 {x — y) ax -{-by ax — by a V — Wy^ FRACTIONS. 117 174. Since -^ — a, and h ' -6 it is evident that if tlie signs of both numerator and de- nominator be changed, the value of the fraction is not altered. Again, «:^^ -(«-&) ^-«+5^&:z^_ c — d —-{c — d) ~c-\-d d — G Therefore, if the numerator or denominator be a com- pound expression, or if both be compound expressions, the sign of every term in the denominator may be changed, provided the sign of every term in the numerator be also changed. Since the change of ih.Q sign before the fraction is equiva- lent to the change of the sign before every term of the numerator of the fraction, the sign before every term of the denominator may he changed, provided the sign before the fraction be changed. Since, also, the product of + a multiplied by + J is ab, and the product of — a multiplied by — 5 is ab, the signs of two factors, or of any even number of factors, of the de- nominator of a fraction may be changed without altering the value of the fraction. By the application of these principles, fractions may often be changed to a more simple form for addition or subtrac- tion. (1) Simplify 2 _ 8^^!^ X 2a:— 1 1— 4ar Change the signs before the terms of the denominator of the third fraction, and change the sign before the fraction. The result is, 2_ 3 2x~2, ' X 2a:- 1 ^x'-V in which the several denominators are written in symmetrical form. 118 ALGEBRA. The L. C. D. = ^ (2:z; - 1) (2 :^; + 1). Scc^ — 2 = 1st numerator, — 6^ — Sx = 2d numerator, — 2a^-{-Sx = Sd numerator. — 2 = sum of numerators, .*. Sum of the fractions = x(2,x-l)(2cc + l)' (2) Simplify 1 + ,, 1, . + ' a{a~h){a — c) b (b — a) (h — c) c (c — a) (c ~ h) Change the sign of the factor (5 — a) in the denominator of the second fraction, and change the sign before the fraction. Then change the signs of the factors (c — a) and (c — h) in the de- nominator of the third fraction. The result is, 1 1^1 a(a — h)(a — c) h {a — b) (h — c) c(a~c)(b~ c)' in which the factors of the several denominators are written in sym metrical form. The L. 0. D. = abc {a --b){a- c) (b - c). be (b — c) = b^c — b(? = 1st numerator, — ac (a — c) = — OJ^c -\- a(? = 2d numerator, ab (a — b) = a^b — ab^ = 3d numerator. a% — a^e — ab^ + ac^ + b^o — bc^ = sum of numerators, = a'{b-c)-a (P -c') + bc(b- c), = [a^~a(b + c) + bc][b-cl = [a^ ~ab — aC'\- be] \b — c], — [(o? — ac) — (ab — be)] [b ~ c\ ==\_a{a — c) — b{a — cy\\b — c], = {a — b){a — c) (b — c). FRACTIONS. 119 Sum of the fractions - («-*)(»- ') (* - «) abc {a — h)(a~ c) (b — c) 1 abc Exercise LIX. Simplify : 1. ^ I ^~y ' x—y y—x 3 + 2^ , 3a;-2 . 16a;-a:^ 2-a; "^ 2 + a: "^ a;2_4 ' 3?-\ X^\ \-X s. 3- 3y2 2-2y Gy + S 1 2 (2-m)(3-w) (m-l)(m-3) (m-l)(m-2) 6. ., A . . + 1 {b-a){x^a) (a-b)(x + b) a^-i-b' 2ab' 2a^b ' a'-b''^ b'-a''^ a^-\-b^' Q h — a a — 2 b Sx (a — b) a? — 6 b + x b^ — a^' g 3 + 2a: 2 — 3a; . 16x — aP 2-x 2 + x 3^-4: ' la — ^_ _ 7 _ 4 -20a; l-2a; 14-2a; 4.x'-l' 120 ALGEBRA. - - a~\-h I h-}- c I c-{-a (b — c){c — a) {b — a) (a — c) {a — b){h~ c)' 12 a^ — hc , h^-{-ac . (? -\- ah {a-h){a-cy {b + c){b-a) {c-a){c'{-b) 13. y + g , z + x x + y {x-y){x-z) {y-x){y-z) {z~x){z~-y) 3 4 6 14. {a — b)(b — c) (b — a) {c — a) (a — c) (c — b) 15. - f- x{x — y){x — z) y{y — x){y — z) xyz Multiplication of Fractions. 175. Hitherto in fractions, equal parts of one or more units have been taken. But it is often necessary to take equal parts oi fractions of units. Suppose it is required to take |- of f of a unit. Let the line AB represent the unit of length. ^1 I I I I I I I I I I I I 1 I U C D E F Suppose ^^ ^divided into 5 equal parts, at (7, D, E, and F, and each of these parts to be subdivided into 3 equal subdivisions. Then one of the parts, as A C, will contain 3 of these sub- divisions, and the whole line AB will contain 15 of these subdivisions. That is, -|- of -J- of the line will be -^ of the line ; ■^ of |- will be -jig- + y^ + ^ig- + T5-' ^r -5^, of the line ; and •| of f will be twice -j^, or ^, of the line. FRACTIONS. 121 Suppose it is required to take - of - of the line A B. d >t l I I I I I I I I I » i I t 1 1 ^ C D E F Let the line AB he divided into h equal parts, and let each of these parts be subdivided into d equal subdivisions. Then the whole line will contain hd of these subdivisions, and one of these subdivisions will be -— of the line. od If one of the subdivisions be taken from each of a parts, they will together be ^ of the line. That is, od "'7 0^T=r"7+r-7 + T3 isken a times, =^, do od od od od and - of - will be c times -— , or -— of the line. do od od Therefore, to find a fraction of a fraction, Find the product of the numerators for the nicmerator of the product, and of the denominators for the denominator of the product. 176. Now, -x^ means -off- do do Therefore, to find the product of two fractions, Find the product of the numerators for the numerator of the product, and of the denominators for the denominator of the product. The same rule will hold when more than two fractions are taken. If a factor exist in both a numerator and a denominator, it may be cancelled ; for the cancelling of a common factor hefore the multiplication is evidently equivalent to cancel- ling it after the multiplication ; an'd this may be done by § 163. 122 ALGEBRA. Division of Fractions. 177. Multiplying by the reciprocal of a number is equiv- alent to dividing by the number. Thus, multiplying by ^ is equivalent to dividing by 4, The reciprocal of a fraction is the fraction with its terms interchanged. Thus, the reciprocal of -f is f , for f X f = 1. § 42. Therefore, to divide by a fraction, Interchange the terms of the fraction and multiply hy the resulting fraction. Thus, a\ 2a_^_L^2a 3^^2a ^ ^ Zx^' Zx Zx^ \ x' The common factor cancelled is 3 a;. ^ 21y'' ' 9y 27.y'^ 7x Zy The common factors cancelled are 9y and 7 re. Ck\ a^ ^ cib o^ {a -{- x)(a — x) ^ ^ (a-xy ' a'-y~ (a-x)(a~x) , ab __ x(a-\-x) b (a — x) The common factors cancelled are a and a—x. If the divisor be an integral expression, it may be changed to the fractional form. § 162. 2. Exercise LX. a ^cx bx^ d 3. 2x ^ Zab ^Zac a c 26 4. Zp . 2p 2p-2 ' j9-l' 8a:V . 23^ Ibab^ ' Zab' FRACTIONS. 123 %aW Ibxi/" 9mV bp^q 24r'y^ 4.b3^y 24:a'P' 8ff ^ 2xy ^ 90mn' 9:x^y^z 20 a^b^c 25 Fm^ lOn^q Zpm 10 a'b^c ^ 18xfz' ' Uri'q^ ^ 7b fm ^ TR 7. ?fV X ^ X - i^. 10. ^-^ X ^'-^\ 4a;2r^ Qocy 2xy^ * o?-\-ah a? — ah* a^ + h^ ^ a~h 3^-\-x-~2 a:^~l3ar + 42 a^-b^ * a + ^>' a?-lx a?-{-2x. ' a;2-6:c4-9 :r*-5a;' «» + a:^ ^ (a - a:/ 15. ^^(^-3/^)' X ^ a! + 2a5 a^-25^ ^ + ^ ^ (^-yV ar^-4 .. a;2_25 _ ^2 17. -^ r^ X -^ =^. 19. rrr — n" n — rrt 3?-\-hx x^i-2x &^ + d^ ' c + d' a^-Aa + S a^-9a + 20 s. «'-7a • a^-ba + 4: a'-l0a + 2l a^-ba l^-^h + e ^>^+10^ + 24 ^ l' + 6b ' 52 + 35-4 52_i45_|.48 • 53_852- 3^—'Zxy-\-2y^ o?-\-xy {x~yY a^^Sa^b + Sab^-b^ 2ab ~ 2b^ ^^ a^ + ab a^-b' ' 3 a-b' 124 ALGEBRA. 24. a?~{b-~cf ' c'-la-by ' (x-by-a" a^-{a-bf {aJ^bf~{c-^df . {a-cf-~{d-bf ' (a-\-cY-{b-^df ' {a-bf~{d~cf x^-\-2xy + f — z^ x — y + z Complex Fractions. 178. A complex fraction is one which has a fraction in the numerator or in the denominator, or in both. . 179. A fraction may be regarded as the quotient of the numerator divided by the denominator. This is the simplest meaning of a complex fraction. Therefore, to simplify a complex fraction, Divide the numerator by the denominator. (1) Simplify 4. ¥ _ i-^l = 4X^ = |. (2) Simplify ||. • 2|_ 1 _ = l^¥ = fx A = AV. (3) Simplify ^^ Zx 3a: 4 _3a: . 4a; -1 1 ' 4 12a; _3a; 1 >=j^ -1 4a:-l FRACTIONS. 125 It is often shorter to multiply both terms of the fraction by the L. 0. D. of the fractions contained in the numerator and denominator. Thus, in (1), multiply both terms by 6 ; in (2), both terms by 24; in (3), by 4. The results obtained are |, ■^, 12x respectively. 4.X-1 (4) Simplify l-\-x^ l-x + x" x(l — X-}-C(^) X x~a^-\-a? x{l + x + x^) l-\-x-\-:^-{x-2(^-\-^) X-\-0(?-\-X^ l-\-x' The expression is reduced to the l-x-\-^ c x(l—x-}~x^ 1.-, x — x^-}-a^ form ^^— ! ^- , which = ' — -. (l+x){l~x + 3^) + x l + x + a^ The expression — — — is reduced to the form 1 X X^ ~j~ Xi xili-x + a^) which-^ + ^ + '^* l+x + s^-ix-x' + a^) 1 i-ar" 126 ALGEBRA. Exercise LXI. Simplify : Sx , X— 1 ' T + -3" . , 1 8. 1- |(^ + l)-|-2i 1 + 1 X 1 . 6 x~l 2. 1^ 9. 1 + x—o 1 —c 3 2:r-l 10. ^+1 ^+f_i 1- 2 2 1 + 1 a: 4. ^-" u. L ^ + « l + x+-^^ 1 —a; f«_£y2+5) f«+f_2y«+?+2 \x a) \x a) -jft \^ ^ / v^ ^ \a: a/ 1- x — a x-\-a 1 X x — y ^-if X y xy + f x^-\-xy x+l x~l x~l ' x+l x+l x~l 13. ;^— y^ a:+?/l (:r— y)^ x—y) x—y ( :^-y')(2x^-2xy) Hx-yY 7. I — :!: — Z.±-l 14. - xy x~l xTT ^ + 2/ FRACTIONS. 127 ab ac IK 0^ + (cb -{■ h) X -\- ab 3? -\- {a -\- c) X -{- ac 16. 1 b — c a?-\'{b + c)x-\-bc a-\-h , 16. _£_ + !_ 1 17. _*_£±i X a 2m-3 + - ^ + 1 + ^ la ^ 19. ^ ^" ^" 20. ^ 2m-l g'^ - (5 4- g)' T , 3 m a5 11^ 1-x Exercise LXII. miscellaneous examples. ^ ^- Simplify ^,_^^^_^^_^^_^^ . ; 2. Find the value of '^' + f ~^ + ^^^ when « = 4, 5 = i, * _, a^ — c>^ — )+(?-' :i(? -^xy ) 10. Simplify a^ — a;^ 1 + ^:::^ i 11. Simplify 1±£ -^ 1 _^~^ 1 ■?• a 4- ^ a^ 4- ^ 12. Divide ^ + ^-3^J-^^) + 4(^ + J)by.: + ^. 2:i:y 1 13. Simplify 2a;y 1- 1+' 4a5 when rr {x-yj- \ X} 14. Find the value of --±^ + ^^^^ ^j 25-a; 2^> + a; 452-:r2 ^ a + h' 15. Find the value of ^-^V-'^ when a; -= -^i^- and 16. Simplify 1,1.1 + + a{a — b){a~c) b{b — c)(b — a) c(c — a)(c~b)' Sabc g-l ^>-l c~l 17. Simplify 6c -j- ca — ab a b c FRACTIONS. 129 ■ m la Simplify ^ 1_1 m»4-7i8 n m 20. Simplify 3a-[J + i2a-(6-c)i] + i + |^ _l ?_j___? y 21. Simplify "-^ ''-y ^"-^y ^("-y)' (a — y)(a-i)^ (a — a;)(a — y/ 22. Simplify I 23. (^-.V-) (2^- 2^3/) 3 -a; 24. Simplify ^__ - ^ ^ ^-i^ + ^ j 25. Simplify ^ 1^ - + , ^^ -+- ^±^ (^-3/) (^-2) (y-a:)(y-z) (z-a;)(2;-y) 26. Simplify a(a— 5)(a— c) h(h — a){h — c) abc ^-4 + -^ 1 ^ + 5 27. Simplify ^^ix "^ ^ 6 (x-l)(;!;-2) CHAPTER IX. Fractional Equations, to reduce equations containing fractions. 180. (1) ^ + f=12. 2 4 Multiply both sides by 4, the L. C. M. of the denominators. Then, 2a; + a; = 48, 3a; = 48, .-. a; = 16. (2) ?_4 = 24--. 6 8 Multiply both sides by 24, the L. C. M. of the denominators. Then, 4x - 96 - 576 - Sx, 4x + 3a; = 576 + 96, 7a; = 672, .-. a; = 96. ^ I Multiply by 33, the L. C. M. of the denominators. is Then, 11 a;- 3a; + 3 = 33 a; -297, | 11a; -3a; -33a; = -297 -3, I -25a; = -300, J .-. a; = 12. ^ Since the minus sign precedes the second fraction, in removing the denominator, the + (understood) before x, the first term of the numerator, is changed to — , and the — before 1, the second term of the numerator, is changed to +. 181. Therefore, to clear an equation of fractions, Multiph/ each term hy the L. 0. M. of the denominators. FRACTIONAL EQUATIONS. 131 If a fraction is preceded by a minus sign, the sign of every term of the numerator must be changed when the denominator is removed. Exercise LXIII. Solve the equations : 1. bx~^ + 2_r,-^ ^ bx 5a:_9 2> - x 2 ' '2442 2 X ^- -x_\\ 5 2:. ^^-4^7 ^-2^ 2. X ^ 3 '• -"^ 6 ' 5 b~2x , o ^ 6a;-8 ^ x + 2 14 34-5a; '• 4 '^ "" 2 ■ ^' 2 9 4 7. 5a: + 3 3-4a: a: 31 9~bx 8 3*22 6 8. 10^ + 3 6^-7__^o(^ ^^ . 3 2 9. 5.-7 2. + 7^3^_^4^ 2 3 10. 7a;4-5 5.'r-6_8-5a; 6 4 12 11. a: + 4 a;-4_o , 3a:-l 3 5 " ' 15 12. 3.^ + 5 2x+l . ^Q 3.2:_Q 13. i(3:r-4) + |(5a: + 3)-43-5:r. 14. l(27-2x) = |-i(7^-54). 132 ALGEBRA. 15. 5a:-|8iP-3[16-6a;-(4-5a;)]J=6. 7 3 2 6^ ^ ^ 2x+1 9:g-8 ^ a;-ll • 7 11 2 • „ 8^-15 lla:-l_7a; + 2 18. —^ _______ 19 7a: + 9 3a7 + 1 ^ 9a;-13 249-9a; ' 8 7 4 14 • 182. If the denominators contain both simple and com- pound expressions, it is best to remove the simple expressions first, and then each compound expression in turn. After each multiplication the result should be reduced to the simplest form. 837 + 5 ■ 7a;-3 _^ 4a: + 6 ^^ 14 "^6:r + 2 7 ' Multiply both sides by 14. Then, 8a; + 5 + ^^^~^^ = %x + 12. 3a; + l Transpose and combine, — ^^^^ — = 7. ^^ ^ ' Sx + 1 Multiply by 3x + l, 49a; -21= 21a; + 7, 28 a; = 28, .-. a; = 1. 3_i^ !^_3 9 19 (2) . — .^ • ' ^ 4 4 10 Simplify the complex fractions by multiplying both terms of each fraction by 9. Then, 27-4.. 1 7a; - 27 36 4 90 • Multiply both sides by 180. 135- 20a; = = 45 -14a: + 54, -Qx- = -36, FRACTIONAL EQUATIONS. 133 Exercise LXIV. Solre tlie equations : J 9a; +20 ^ 4:(x-S) . x 36 5a; -4 4' 9(2a;-3) . 11a; - 1 _ 9a;+ 11 ." 14 '^3a;+l 7 ' 10 ^ + 17 12a; + 2 __ 5a;-4 18 13a;- 16 9 * 6a;+13 3a; + 5 _2a; 15 5a; -25 5' ^ 18a;-22 , o ,1 + 16^_^5 101-64a; '• 3grr^+^'^+~^^~^^ 24~"- 6-5a; 7-2a;^ ^ l + 3a; 10a;-ll . 1 * 15 14 (a; -1) 21 30 "^ 105' 9a; + 5 8a;- 7 _ 36a; + 15 41 * 14 "^6a; + 2 56 "^56* 6a;+7 2a; — 2 ^ 2a; + l 15 7a;-6 5 * 6a; + l 2a;-4 _ 2a;-l 15 7a; -16" 5 ' 7a; — 6 a; — 5 _x 35 6a;-101~5* 8. 9. 10. 183. Literal equations are equations in wticli all the numbers are represented by letters ; the numbers regarded as known numbers are usually represented by tne first let- ■ ters of the alphabet. 134 ALGEBRA. (1) (a-x)(a + x) = 2a^ + 2ax — a^. Then, a^ -a^ = 2a'' + 2ax~x', ~2ax = a^, a ... = --. (2) (x — a) {x — h) — (x — h) {x — c)=2(x — a) (a — c). {a^ — ax — hx + ah) — {3^ — bx — cx + bc) = 2 {ax — cx — a^ + ad), a^ — ax — hx-{-ab — 3^ + bx + cx — bc = 2ax — 2cx — 2a^ + 2ac. That is, —3ax + 3cx = — 2a^ + 2ac — ab + be, — 3 {a — c) x = — 2a {a — c) — b {a — c), -3x = -2a-b, 2a + b Exercise LXV. Solve tlie equations : 1. ax -}- he = hx -{- ac. 2. 2a~cx = Sc — 5 hx. 3. a^x -{- hx — c = b^x -}- ex — d. 4. — a(? + h'^e -}- ahcx = ahe + emx — ac^x + Pc — mc, 5. (a -{- X -{- h) (a -{- h —■ x) = (a -{- x) (h -~ x) ■— ah. 6. (a^i-xy = x' + 4:a^-{-a*. 7. {a^-x)(a^ + x) = a* + 2ax--x^. 8. ^i^ + a-^±^. 10. ax-^^^^=l. ^^ aj^x±^_^^_^a^^ ^^ Q^__ iax--2h ^^ bx h 3 x^ — a a—-x 2x a hx h h x ,- 3 ah — o[^ __4:X ~ ac e ox ex 14. am — h — -:r--\ = 0. 7n FRACTIONAL EQUATIONS. 135 18. _„ Sax — 2b ax — a ax 2 . b S ,, ab + x V-x x~b ab — X ' • V a?b a' b' • X x 19. ^ = bG + d+l X X ao? . . ax ^ b — cx G 20. a_icP_t^^ ax dx d Exercise LXVI. Solve the equations : 1 a; — 3 _ x — b .1 4(a;-l)~6(:r-l) 9' X— 1 x-[- 1 3 7 ^ 6:^+1 3(1 + 2:?^ ) X— 1 x-{- 1 ^—1 a;~l 2 (a; -3) d>{x-2) {x-2){x-S) . 2( 2:r + 3) _ 6 5a:+l ' 9(7-:r) 7-^ 4(7-a:y 37 + 3 3:r + 9 ^ x—7 2x — \b 1 07 + 7 2.7; -6 2(rr+7) 3a; + 5 ^ 2^ + 3' 136 ALGCEBRA. ^ 132a; + l , Sx + 5_.^ 3:^7-1 407-2 1 • 2x-l ^x~2 G' 10. 2 1 _ 6 2x-d, x-2 Sx+2' ' x-1 x-1 1-x'' „ X — 4: x — 5 _x- -1 x-^ x — b 07 — 6 :^ — 8 ^ — 9 14. {x — a){x ~b) = {x — a — hf. 15. {a—h){x—c)--(b — c){x—a)—(c~a){x—h)=^(}. X— 1 37+1 4,7 37 17, x + 2 x + S x^ + bx + Q 18. (:^+l)2 = :^[6-(l-a7)]-2. • 19 25-ia7 16:r + 4i _ 23 ^ ^+1 3:r + 2 ^r+l ^ 3(25g , a^b^ , (2 a + 5) 5^a7 _ o ,bx a + b~^(a + bf'^ a(a + by "'^'''"^T 2j 4 , 3 29 2 07-8 2o:-16 24 3or-24 22. 5-.f?-gV^-^"-(^-H V2 o;y 2 4 oo 1 3 l~x 23. = 5 07-1 3 24. -^^^ 2L._L_^f 5_=l_j i . |(^-l) + |(x+l) 15(l-i) CHAPTER X. Problems. Exercise LXVII. Ex. Find the number the sum of whose third and fourth parts is equal to 12. Let X = the number. Then - = the third part of the number, and - = the fourth part of the number, .'. - + - == the sura of the two parts. 3 4 ^ But 12 = the sum of the two parts, 3 4 = 12. Multiply both sides by 4x + 3a: = 144, 12: 7x = = 144, .'. x = = 20f. 1. Find the number whose third and fourth parts together make 14. 2. Find the number whose third part exceeds its fourth part by 14. 3. The half, fourth, and fifth of a certain number are together equal to 76 ; find the number. 4. Find the number whose double exceeds its half by 12. 5. Divide 60 into two such parts that a seventh of one part may be equal to an eighth of the other. 138 ALGEBRA. 6. Divide 50 into two sucli parts that a fourth of one part increased by five-sixths of the other part may be equal to 40. 7. Divide 100 into two such parts that a fourth of one part diminished by a third of the other part may be equal to 11. 8. The sum of the fourth, fifth, and sixth parts of a cer- tain number exceeds the half of the number by 112. What is the number ? 9. The sum of two numbers is 5760, and their difference is equal to one-third of the greater. What are the numbers ? 10. Divide 45 into two such parts that the first part divided by 2 shall be equal to the second part mul- tiplied by 2. 11. Find a number such that the sum of its fifth and its seventh parts shall exceed the difference of its fourth and its seventh parts by 99. 12. In a mixture of wine and water, the wine was 25 gal- lons more than half of the mixture, and the water 5 gallons less than one-third of the mixture. How many gallons were there of each ? 13. In a certain weight of gunpowder the saltpetre was 6 pounds more than half of the weight, the sulphur 5 pounds less than the third, and the charcoal 3 pounds less than the fourth of the weight. How many pounds were there of each ? 14. Divide 46 into two parts such that if one part be divided by 7, and the other by 3, the sum of the quotients shall be 10. PROBLEMS. 139 15. A house and garden cost $ 850, and five times the price of the house was equal to twelve times the price of the garden. What is the price of each ? 16. A man leaves the half of his property to his wife, a sixth to each of his two children, a twelfth to his brother, and the remainder, amounting to $600, to his sister. What was the amount of his property ? 17. The sum of two numbers is a and their difference is h ; find the numbers. 18. Find two numbers of which the sum is 70, such that the first divided by the second gives 2 as a quotient and 1 as a remainder. 19. Find two numbers of which the difference is 25, such that the second divided by the first gives 4 as a quo- tient and 4 as a remainder. 20. Divide the number 208 into two parts such that the sum of the fourth of the greater and the third of the smaller is less by 4 than four times the difference of the two parts. 21. Find four consecutive numbers whose sum is 82. Note I. It is to be remembered that if x represent a person's age at the present time, his age a years ago will be represented by a; — a, and a years hence by a; + a. Ex. In eight years a boy will be three times as old as he was eight years ago. Hoav old is he ? Let X = the number of years of his age. Then a; — 8 = the number of years of his age eight years ago, and a; + 8 = the number of years of his age eight years hence, .-. a; + 8 = 3 (a; - 8), a; + 8 =3a;-24, iB~3a; = -24-8, -2a; = -32, X :■- 16. 140 ALGEBRA. 22. A is 72 years old, and B's age is two-thirds of A's. How long is it since A was five times as old as B ? 23. A mother is 70 years old, her daughter is half that age. How long is it since the mother was three and one- third times as old as the daughter ? 24. A father is three times as old as the son ; four years ago the father was four times as old as the son then was. What is the age of each ? 25. A 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. 26. The sum of the ages of a father and son is half what it will be in 25 years ; the difference is one-third what the sum will be in 20 years. What is the age of each ? Note II. If A can do a piece of work in x days, the part of the work that he can do in one day will be represented by J. Thus, if he can do the work in 5 days, in 1 day he can do -J of the work. Ex. A can do a piece of work in 5 days, and B can do it in 4 days. How long will it take A and B together to do the work ? Let X = the number of day;j it will take A and B together. Then ^ = the part they can do in one day. Now, I = the part A can do in one day, and I = the part B can do in one day. •'• i + ¥ = ^he part A and B can do in one day. ••. l + h-h 4a; + 5a; = 20, 9 a; = 20, a;=2|. Therefore they will do the Work in 2|- days. 27. A can do a piece of work in 5 days, B in 6 days, and C in 7J days ; in what time will they do it, all work- ing together ? PROBLEMS. 141 28. A can do a piece of work in 2^ days, B in 3i days, and C in 3| days ; in what time will they do it, all work- ing together ? ■ 29. Two men who can separately do a piece of work in 15 days and 16 days, can, with the help of another, do it in 6 days. How long would it take the third man to do it alone ? A can do half as much work as B, B can do half as much as C, and together they can complete a piece of work in 24 days. In what time can each alone complete the work ? A does |- of a piece of work in 10 days, when B comes to help him, and they finish the work in 3 days more. How long would it have taken B alone to do the whole work ? 32. A and B together can reap a field in 12 hours, A and C in 16 hours, and A by himself in 20 hours.. In what time can B and together reap it? In what time can A, B, and together reap it ? 33. A and B together can do a piece of work in 12 days, A and C in 15 days, B and in 20 days. In what time can they do it, all working together ? Note III. If a pipe can fill a vessel in x hours, the part of the vessel filled by it in one hour will be represented by |. Thus, if a pipe will fill a vessel in 3 hours, in 1 hour it will fill ^ of the vessel. 34. A tank can be filled by two pipes in 24 minutes and 30 minutes respectively, and emptied by a third in 20 minutes. In what time will it be filled if all three are running together ? 35. A tank can be filled in 15 minutes by two pipes, A and B, running together. After A has been running by 142 ALGEBRA. itself for 5 minutes, B is also turned on, and the tank is filled in 13 minutes more. In wHat time may it be filled by each pipe separately ? 36. A cistern could be filled by two pipes in 6 hours and 8 hours respectively, and could be emptied by a third in 12 hours. In what time would the cistern be filled if the pipes were all running together ? 37. A tank can be filled by three pipes in 1 hour and 20 minutes, 3 hours and 20 minutes, and 5 hours, re- spectively. In what time will the tank be filled when all three pipes are running together ? 38. If three pipes can fill a cistern in a, 5, and c minutes, respectively, in what time will it be filled by all three running together ? 39. The capacity of a cistern is 755^ gallons. The cistern has three pipes, of which the first lets in 12 gallons in 3^ minutes, the second 15^ gallons in 2^ minutes, the third 17 gallons in 3 minutes. In what time will the cistern be filled by the three pipes running together ? Note IV. In questions involving distance, time, and rate : Distance _ m- Rate Thus, if a man travels 40 miles at the rate of 4 miles an hour, — = number of hours required. Ex. A courier who goes at the rate of 31-^ miles in 5 hours, is followed, after 8 hours, by another who goes at the rate of 22|- miles in 3 hours. In how many hours will the second overtake the first ? Since the first goes 31^ miles in 5 hours, his rate per hour is 6^'^ miles. PROBLEMS. ' 143 Since the second goes 22J miles in 3 hours, his rate per hour is 7J miles. Let X = the number of hours the first is travelling. Then x — S = the number of hours the second is travelling. Then 6-^^ x = the number of miles the first travels ; {x — 8) 7J = the number of miles the second travels. They both travel the same distance, .'.G^^x = {x-S)7h The solution of whicli gives 42 hours. 40. A sets out and travels at the rate of 7 miles in 5 hours. Eight hours afterwards, B sets out from the same place and travels in the same direction, at the rate of 5 miles in 3 hours. In how many hours will B overtake A ? 41. A person walks to the top of a mountain at the rate of 2^ miles an hour, and down the same way at the rate of 3-|- miles an hour, and is out 5 hours. How far is it to the top of the mountain ? ; 42. A person has a hours at his disposal. How far may he ride in a coach which travels b miles an hour, so as to return home in time, walking back at the rate of c miles an hour ? ; 43. The distance between London and Edinburgh is 360 miles. One traveller starts from Edinburgh and travels at the rate of 10 miles an hour ; another starts at the same time from London, and travels at the rate of 8 miles an hour. How far from London will they meet ? 44. Two persons set out from the same place in opposite directions. The rate of one of them per hour is a mile less than double that of the other, and in 4 hours they are 32 miles apart. Determine their rates. 144 ALGEBRA. 45. In going a certain distance, a train travelling 35 miles an hour takes 2 hours less than one travelling 25 miles an hour. Determine the distance. Note V. In problems relating to clocks, it is to be observed that the minute-hand moves twelve times as fast as the hour-hand. Ex. Find the time between two and three o'clock when the hands of a clock are : I. Together. II. At right angles to each other. III. Opposite to each other. Fig. 1. Fig. 2. Fig. 3. I. Let (7-H"and CM (Fig. 1) denote the positions of the hour and minute hands at 2 o'clock, and CB the position of both hands vtien together. Then arc HB = one-twelfth of arc MB. Let X = number of minute-spaces in arc MB. Then — = number of minute-spaces in arc HB, and 10 = number of minute-spaces in arc MS. Now arc MB =■ arc MH + arc IIB. That is. a;=10 + 12 The solution of this equation gives x =■ 10|^. Hence, the time is 10-}-f- minutes past 2 o'clock. II. Let CB and CD (Fig. 2) denote the positions of the hour and minute hands when at right angles to each other. PROBLEMS. 145 Let X =• number of minute-spaces in arc MHBD. Then — = number of minute-spaces in arc HB, and 10 = number of minute-spaces in arc MH. 15 = number of minute-spaces in arc BD. Now arc MEBD = arcs ME + HB + BD. That is, x = \0-\-^+ 15. The solution of this equation gives x — 27^. Hence, the time is 27-j^ minutes past 2 o'clock. III. Let CB and CD (Fig. 3) denote the positions of the hour and minute hands when opposite to each other. Let X = number of minute-spaces in arc MHBD. Then — = number of minute-spaces in arc HB, and 10 = number of minute-spaces in arc MH, 30 = number of minute-spaces in arc BD. Now arc MEBD - arcs MH + HB + BD. That is, a; = 10 -f — -f 30. 12 The solution of this equation gives x = 43^. Hence, the time is 43^^ minutes past 2 o'clock. 46. At what time are the hands of a watch together : I. Between 3 and 4 ? IL Between 6 and 7? III. Between 9 and 10? 47. At what time are the hands of a watch at right angles : I. Between 3 and 4 ? II. Between 4 and 5 ? III. Between 7 and 8? 18. At what time are the hands of a watch opposite to each other : I. Between 1 and 2? II. Between 4 and 5 ? III. Between 8 and 9? 146 ALGEBRA. 49. It is between 2 and 3 o'clock ; but a person looking at his watch and mistaking the hour-hand for the minute hand, fancies that the time of day is 55 minutes earlier than it really is. What is the true time? Note VI. It is to be observed that if a represent the nninber of feet in the length of a step or leap, and x the number of steps or leaps taken, then ax will represent the number of feet in the distance made. Ex. A hare takes 4 leaps to a greyhound's 3 ; but 2 of the greyhound's leaps are equivalent to 3 of the hare's. The hare has a start of 50 leaps. How many leaps must the greyhound take to catch the hare ? Let 3 a; = the number of leaps taken by the greyhound. Then 4a; = the number of leaps of the hare in the same time. Also, let a denote the number of feet in one leap of the hare. Then — will denote the number of feet in one leap of the grey- hound. That is, 3 a; X — = the whole distance, 2 and (50 + 4 x) a :-= the whole distance, .-. ^ = (50 + 4a;) a. Divide by a, — = 50 + 4 a;, 9x=100 + 8a;, a; =100, .-. 3 a; = 300. Thus the greyhound must take 300 leaps. 50. A hare takes 6 leaps to a dog's 5, and 7 of the dog's leaps are equivalent to 9 of the hare's. The hare has a start of 50 of her own leaps. How many leaps will the hare take before she is causht ? PROBLEMS. 147 k 51. A greyhound makes 3 leaps while a hare makes 4 ; but 2 of the greyhound's leaps are equivalent to 3 of the hare's. The hare has a start of 50 of the greyhound's leaps. How many leaps does each take before the hare is caught ? 52. A greyhound makes two leaps while a hare makes 3 ; but 1 leap of the greyhound is equivalent to 2 of the hare's. The hare has a start of 80 of her own leaps. How many leaps will the hare take before she is caught ? Note VII. It is to be observed that if the number of units in the breadth and length of a rectangle be represented by x and x + a, respectively, then x{x + a) will represent the number of surface units in the rectangle, the unit of surface having the same name as the linear unit in which the sides of the rectangle are expressed. 53. A rectangle whose length is 5 feet more than its breadth would have its area increased by 22 feet if its length and breadth were each made a foot more. Find its dimensions. 54. A rectangle has its length and breadth respectively 5 feet longer and 3 feet shorter than the side of the equivalent square. Find its area. 55. The length of a rectangle is an inch less than double its breadth ; and when a strip 3 inches wide is cut off all round, the area is diminished by 210 inches. Find the size of the rectangle at first. 56. The length of a floor exceeds the breadth by 4 feet ; if each dimension were increased by 1 foot, the area of the room would be increased by 27 square feet. Find its dimensions. Note VIII. It is to be observed that if b pounds of metal lose a pounds when weighed in water, 1 pound will lose i of a pounds, or I of a pound. 148 ALGEBRA. 57. A mass of tin and lead weighing 180 pounds loses 21 pounds when weighed in water ; and it is known that 37 pounds of tin lose 5 pounds, and 23 pounds of lead lose 2 pounds, when weighed in water. How many pounds of tin and of lead in the mass ? 58. If 19 pounds of gold lose 1 pound, and 10 pounds of silver lose 1 pound, when weighed in water, find the amount of each in a mass of gold and silver weighing 106 pounds in air and 99 pounds in water. 59. Fifteen sovereigns should weigh 77 pennyweights ; but a parcel of light sovereigns, having been weighed and counted, was found to contain 9 more than was sup- posed from the weight ; and it appeared that 21 of these coins weighed the same as 20 true sovereigns. How many were there altogether ? 60. There are two silver cups, and one cover for both. The first weighs 12 ounces, and with the cover weighs twice as much as the other without it ; but the sec- ond with the cover weighs one-third more than the first without it. Find the weight of the cover. 61. A man wishes to enclose a circular piece of ground with palisades, and finds that if he sets them a foot apart he will have too few by 150 ; but if he sets them a yard apart he will have too many by 70. What is the circuit of the piece of ground ? 32. A horse was sold at a loss for $ 200 ; but if it had been sold for $250, the gain would have been three-fourths of the loss when sold for $200. Find the value of the horse. 63. A and B shoot by turns at a target. A puts 7 bullets out of 12, and B 9 out of 12, into the centre. Be- tween them they put in 32 bullets. How many shots did each fire ? PROBLEMS. 149 64. A boy buys a number of apples at the rate of 5 for 2 pence. He sells half of them at 2 a penny and the rest at 3 a penny, and clears a penny by the tran- saction. How many does he buy ? 65. A person bought a piece of land for f 6750, of which he kept f for himself. At the cost of $ 250 he made a road which took j^-g- of the remainder, and then sold the rest at 12-|- cents a square yard more than double the price it cost him, thus clearing his outlay and $500 besides. How much land did he buy, and what was the cost-price per yard ? 66. A boy who runs at the rate of 12 yards per second starts 20 yards behind another whose rate is 10-| yards per second. How soon will the first boy be 10 yards ahead of the second ? 67. A merchant adds yearly to his capital one-third of it, but takes from it, at the end of each year, $5000 for expenses. At the end of the third year, after de- ducting the last $5000, he has twice his original capital. How much had he at first ? 68. A shepherd lost a number of sheep equal to one-fourth of his flock and one-fourth of a sheep ; then, he lost a number equal to one-third of what he had left and one-third of a sheep ; finally, he lost a number equal to one-half of what now remained and one-half a shaep, -after which he had but 25 sheep left. How many had he at first? 69. A trader maintained himself for three years at an ex- pense of $ 250 a year ; and each year increased that part of his stock which was not so expended by one- third of it. At the end of the third year his original stock was doubled. What was his original stock ? 150 ALGEBRA. 70. A cask contains 12 gallons of wine and 18 gallons of water ; another cask contains 9 gallons of wine and 3 gallons of water. How many gallons must be drawn from each cask to produce a mixture contain- ing 7 gallons of wine and 7 gallons of water ? 71. The members of a club subscribe each as many dollars as there are members. If there had been 12 more members, the subscription from each would have been $10 less, to amount to the same sum. How many members were there ? 72. -A number of troops being formed into a solid square, it was found there were 60 men over; but when formed in a column with 5 men more in front than before, and 3 men less in depth, there was lacking one man to complete it. Find the number of troops. 73. An officer can form the men of his regiment into a hol- low square twelve deep. The number of men in the regiment is 1296. Find the number of men in the front of the hollow square. 74. A person starts from P and walks towards Q at the rate of 3 miles an hour; 20 minutes later another person starts from Q and walks towards P at the rate of 4 miles an hour. The distance from P to Q is 20 miles. How far from P will they meet ? 75. A person engaged to work a days on these conditions : for each day he worked he was to receive h cents, and for each day he was idle he was to forfeit c cents. At the end of a days he received d cents. How many days was he idle ? 76. A banker has two kinds of coins : it takes a pieces of the first to make a dollar, and h pieces of the second to make a dollar. A person wishes to obtain c pieces for a dollar. How many pieces of each kind must the banker give him ? CHAPTER XI. Simultaneous Equations of the First Degree. 184. If one equation contain two unknown quantities, an indefinite number of pairs of values may be found that will satisfy tlie equation. Thus, in the equation x -\- y = 10, any values may be given to x, and corresponding values for y may be found. Any pair of these values substituted for x and y will satisfy the equation. 185. But if a second equation be given, expressing differ- ent relations between the unknown quantities, only one pair of values of x and y can be found that will satisfy both equations. Thus, if besides the equation a: + y = 10, another equa- tion, x — y~2, be given, it is evident that the values of X and y which will satisfy both equations are x = ^\ for 6 + 4 = 10, and 6 — 4 = 2; and these are the only val- ues of X and y that will satisfy both equations. 186. Equations that express different relations between the unknown quantities are called independent equations. Thus, a; 4- y = 10 and a; — y = 2 are independent equa- tions; they express different relations between x and y. But x-\-y^ 10 and 3a;-f3y=30 are not independent 152 ALGEBRA. equations; one is derived immediately from the other, and both express the same relation between the unknown quantities. 187. Equations that are to be satisfied by the same val- ues of the unknown quantities are called simultaneous equations. 188. Simultaneous equations are solved by combining the equations so as to obtain a single equation containing only one unknown quantity ; and this process is called elimination. Three methods of elimination are generally given : I. By Addition or Subtraction. II. By Substitution. III. By Comparison. Elimination by Addition or Subtraction. (1) Solve: 2x -3y = = 32/ (1) Zx + 2y = (2) Multiply (1) by '. I and (2) by 3, 4a;- 6y = 8 (3) 9a; + 6y = 96 (4) Add (3) and (4), 13 a; 104 .-. a; -8. Substitute the value of x in (2), 24 + 22/ = 32, .-. 2/ = 4. In this solution y is eliminated by addition. (2) Solve; 6^7 + 352/ = 177-) 8a7-21y= 33j (1) (2) Multiply (1) by 4 and (2) by 3, 24a; + 140y = 708 24a;- 63y= 99 Subtract (4) from (3), 203 y ^ 609 . (3) (4) SIMULTANEOUS EQUATIONS. 153 Substitute the value of y in (2). 8a; -63 = 33. .-. X = 12. In this solution x is eliminated by subtraction. 189. Hence, to eliminate an unknown quantity by addi- tion or subtraction, Multiply the equations hy such numbers as will make the coefficients of this unknown quantity equal in the resulting equations. Add the resulting equations, or subtract one from the other, according as these equal quantities hoA)e unlike or like signs. Note. It is generally best to select that unknown quantity to be eliminated which requires the smallest multipliers to make its coeffi- cients equal ; and the smallest multiplier for each equation is found by dividing the L. C. M. of the coefficients of this unknown quantity by the given coefficient in that equation. Thus, in example (2), the L. G. M. of 6 and § (the coefficients of x), is 24, and hence the smallest multipliers of the two equations are 4 and 3 respectively. Sometimes the solution is simplified by first adding the given equations, or by subtracting one from the other. (3) a; + 492/= 51 (1) 49a; + .v= 99 (2) Add (1) and (2), Divide (3) by 50, Subtract (4) from (1), Subtract (4) from (2), 50a; + 50y = 150 a; + y = 3. 48y = 48, .-. y = i. 48 a: = 96, .-. a; = 2. (3) (4) Exercise LXVIII. Solve by addition or subtraction : 1. 2:r + 3y = 7| 3. 7x + 2y = 30| 5. 5a; + 4y=58| 4:r-5y = 3j y-2>x= 2J 3a:+7y=67J 2. a;-2y = 4| 4. 3:r-5y = 51| 6. ?>x + 2y=m\ 2x- y = 5J 2a;+72/= 3J 3y-2ar=13/ 154 ALGEBEA. 7. 3:r-4y = -5| 11. 12:r+ 7y = 1761 4a;-5y= 1) 3y-19a;= 3i 8. lla; + 3y = 100| 12. 2a;-7y== 81 4a;_7y= 4/ 4y-9a; = 19J 9. a: + 49y = 693) 13. 69y-17a;= 103 j = 357/ 14:p-13v = -41J 49 a; + y = 357J 14a; -13y 17a;-f3y = 57| 14. 17a; + 30y 16y-3a; = 23J 19a; + 28y Elimination by Substitution. (1) Solve: 2a: + 3y = 8| 3a;+7y = 7J 2x + 3y = 8 (1) 3a; + 7y = 7 (2) Transpose 3 y in (1), 2x = 8 — 3y. Divide by coefficient of x, x= ~ ^ (4) Substitute the value of x in (2), 3 ( ^~^A + 7y = 7, 24 -9?/ + 142/ = 14, 5y = -10, .-. y = -2. Substitute the value of y in (1), 2 a; — 6 = 8, .-. x = 7. 190. Hence, to eliminate an unknown quantity by sub- stitution, From one of the equations obtain the vahce of one of the unknown quantities in terms of the other. Substitute for this unknown quantity its value in the other equation^ and reduce the resulting equation. SIMULTANEOUS EQUATIONS. 155 Exercise LXIX. Solve by substitution : 1. 3a;-4y = 2| 8. 3a:-4y = 181 7a;-9y=7J 3a; + 2y= OJ 2. 7a;-5y = 24| 9. 9a;-5y = 52| 4a;-3y = llJ 8y-3a7= S) 3. 3a; + 2y = 321 10. 5a;-3y= 4) 20a;-3y= 1) 12y-7a; = 10J 4. lla;-72/ = 37\ 11. 9y-7a; = 13| 8a; + 9y = 41j 15a;-7y 7a;+ 5y = 60| 12. 6x-2y= 511 13a;-lly = 10i 19a;-3y = 180J 6. 6a;-7y = 421 13. 4:x+ 9y==106| 7a:-6y = 75J 82; + 17y = 198J 7. 10a;+ 9y-290'l 14. 8:r + 3y = 3" 12a; - lly = 130 J 12a; + 9y = 3 i Elimination by Comparison. Solve: 2a;-9y = ll| 3a;-4y = 7 J 3a;-4y = 7. Transpose 9y in (1) and 4y in (2), 2x = ll+9y, 3a; = 7 + 4y. (1) (2) (3) (4) Divide (3) by 2 and (4) by 3, x = ^i-^' 2 (5) 3 (6) Equate the values of x, ^^^ = ^-^. W 156 ALGEBRA. Reduce (7) 33 + 272/ = 14 + 8 y> 193/ = _19. .-. y = -i. Substitute the value of y in (1), 2 a; + 9 = 11, .-. x = \. 191. Hence, to eliminate an unknown quantity by com- parison, From each equation obtain the value of one of the unknown quantities in terms of the other. Form an equation from these equal values and reduce the equation. Note. If, in the last example, (3) be divided by (4), the resulting equation, - = — - — ^, would, when reduced, give the value of y 3 7 + 42/ This is the shortest method, and therefore to be preferred. EXEE€ISE LXX. Solve by comparison : 1. a; + 15y = 531 8. 3y-7a;= 4| Zx-\- 2/ = 27J 2y + 5a; = 22J 2. 4:r+ 9y = 51| 9. 21y + 20a;= 165 | 8:77-132/= 9 J 772/ -30a; = 295 J 3. 4a; + 32/ = 481 10. 11a;- 10^/ = 14 | 5y-3a; = 22i 5a;+ 1y = ^l) 4. 2a; + 32/ = 43| 11. 7y-3a; = 139| 10a:- 2/= 7J 2a; + 5y= 91 J 5. 5a;- 7y= 331 12. 17a;+12y= 59 1 11a; +122/ = 100 J 1^^- 4y = 153i 6. 5a; + 7y = 431 13. 24a;+ 7y= 27 1 lla;+9y = 69J 8a;-33y = 115i 7. 8a; 8a;-21y= 331 14. a; = 3y-19| 6a;-f 35y=177J 3/ = 3a; -23 J SIMULTANEOUS EQUATIONS. 157 192. Each equation must be simplified, if necessary, be- fore the elimination is performed. CI) Solve: (x-~l) (y + 2) = {x-S) {y-l) + 8] 2rg-l 3(.y-2) _. \ 6 4 ) {x~l){y + 2) = {x-3)(y-l) + 8 (1) 2^:zi- % -^ ) =.l (2) 5 4 ^ ^ Simplify (1), xy + 2x-y-2 = xy-x-Sy + 3 + 8. Transpose and combine, 3a; + 2y = 13. (3) Simplify (2), 8 a; - 4 - ISy + 30 = 20. Transpose and combine, 8a; — 15y = — 6. (4) Multiply (3) by 8, 24 a; + 16 3/ = 104. (5) Multiply (4) by 3, 24a; - iby = - 18. (6) Subtract (6) from (5), 61 y = 122, .'.y = 2. , Substitute the value of y in (3), 3 a; + 4 = 13, .-. a; = 3. Exercise LXXI. Solve : 1. x(^+7) = ^(x-\-l)) 3. 2^3 I 5(a;+3) = 3(2/-2) + 2j 5 I 5 10 I 3y + ^-9 = oJ E + ^Zl2 = 3 j . 3 6 4 5. (a; + l)(y + 2)-(a; + 2)(y + l) = -ll 3(a; + 3)-4(y + 4) = -8 J 6 ^-2 10-a: _ y-10 5 3 4 2y + 4 2a;-fy _ a7 + 13 3 8 4 158 ALGEBRA. 3 4 5 4 3 ^ . -^ 15 x~A_y + 2] ' 5 10 6^ 4 ^' ^5 +~3 :a; + l 2a;-3y 4a;-3y ^ ^ 3 2 "^ 16. 3^i-12^ = 9 l-3a7 _ ll-3.y 7 5 9 2:^;-~y+3 ^-2y+3 ^^ 3 4 4 "^ 3 10. lix = iy + 4.^\ ^\x=^\y~2\i^\ 17. 5:r-i(5y + 2)=:32| I 3y + -|-(a; + 2)=9 J ? U. 13 a; + 2y + 3 4:i;-5y + 6 3 _ 19 6^-5y + 4 3a: + 2y + l 18. 3a;-.25y=28 .12:r+.7y=2.54 12. £±^==15 y~x 8 9^_3y+ii = 100 7 13. 3a;-5y ^ 3_ 2^-f y ' 2 o a?-2i _ 4 2 " 3 8--—^=.? + ^ 19. 7(:r-l) = 3(y + 8) 4a; + 2 ^ 5y+9 9 2 20. 7a; + i(22/+4) = 16 1 = 8 J 3y-i(^ + 2) 14. 4a;-3y-7 _3:r 2y_5 5 10 15 6 3 2 20 15 6^10 SIMULTANEOUS EQUATIONS. 159 ► 21. ^^~^y -\-Zx = ^y-^ 13 ^ 5a;-j-6y 3a; — 2y 22. 23. 24. 6 5:r-3 2y-2 3a:-19 ^. 3y 2 2 3 2a:+y 9a:-7 __ 3(y + 3) 4a: + 5y 2 8 4 16 3y + ll= ^^-y(^ + ^y) + 31-4a; (a:+7)(y-2) + 3 = 2^y-(y-l)(.;+l) 6a; + 9 . 3;r + 5y _oi i 3a; + 4 ^ 4 "^ 4a;-6 ~ "^ 2 8y+7 , 6^-3y ^^ ^ 4y-9 10 2y-8 25. X 2y-~x_ 20- 59-2:r y 23 -a: 2 y-3 _^3Q 73-3.y -18 3 Literal Simultaneous Equations. 193. The method of solving literal simultaneous equa- tions is as follows ; Solve : ax-\-hy = m\ cx~{- dy = n ) Multiply (1) by c, Multiply (2) by a, Subtract (4) from (3) Divide bv coefficient ax + by==m (1) cx + dy = n (2) aex + bey = em (3) aex + ady = an (4) {bo — ad) y = cm~an ofv. v-"^^^^ be — ad 160 ALGEBRA. To find the value of x Multiply (1) by d, Multiply (2) by 5, Subtract (6) from (5), adx + hdy = dm hex + hdy = hn {ad — bc)x = dm — hn dm — hn (5) (6) Divide by coefficient of cc, ~ , , ^ ad -he Exercise LXXII. Solve: 1. a; + y = a ") 3. mx -\-ny ^=a\ 5. tux — ny=r \ x — y = h} px-{- qy=^h ) m'x + n'y = / J 2. ax-\-'by = c^ 4. ax-\~hy = e'\ 6. ax-\-hy = c\ px-\- qy = T ) ax-\- cy = d i dx -\-fy = c^ S a h ' a 8. abx + cc?y = 2 d-h x-y-l rc + y— 1 13. ao; = 5y a^ + J^ ax — cy = hd (a — h)x — {a-{'h)y 9. 10. Z> + y 3a + a: aa; + 2 5y = c? 14. ax-\-hy^= (? a h h+y a+x = x JL 1 a-\-h a — b a-\-h X , 2L 1 15. a-\-h a — h x — y _ x-\-y 2ah ~ a^-\-W ^ = 2al a-\-h a — h a — h 11. a{a—x) = h(x^y — a)^ 16. 6a; — 5(? = ay — ac | a{y—h—x)—h{y—h)) x — y = a — h ) SIMULTANEOUS EQUATIONS. 161 17. ^-^ = c y-h a (x — a) -\- b (y ~ h) -{- ahc = 18. {a-{-h)x — {a — h)y = 4:ab | {a-h)x + (a + h)y = 2oj'-2h^] 19. (^x-\-a){y-\-h)-{x~a)(:y-h) = 2{a-hf\ x-y-\-2(a-h)^0 J 20. {a + h){x-\-y)-{a~h)(x-y) = o?-) (^a-h){x-\-y) + {a-^b){x~y) = hn 194. Fractional simultaneous equations, of which the de- nominators are simple expressions and contain the unknown quantities, may be solved as follows : (1) Solve: - + - = m ' X y c , d x y o- h '- + i = n. (2) Multiply (1) by c. "^^^-^^mn. (3) X y Multiply (2) by a, ^ + ^=an. (4) X y Subtract (4) from (3), ^o-ad _ ^^ _ ^^ y Multiply both sides by y, be — ad= {cm ~ an) y, be — ad .'. y = cm — an Multiply (1) by i, ^ + M= dm. (5) X y ^ ^ Multiply (2) by 6, ^^^=hn. (6) X y ^ 162 ALGEBRA. Subtract (6) from (5), ad— he = dm — hn. Multiply both sides by x, ad—bc = {dm — hn) x, ad — he (2) Solve dm — hn _7_ l_=c &x lOy 3 a; 62/ ' Multiply (2) by 4, Add (1) and (3), Divide both sides by 19, 1_ ^ 6a; lOy 14 2_ 3a; by 3. 12. li=19. 3a; 3a; (1) (2) (3> Substitute the value of x in (1), Transpose, Divide both sides by 2, 5 + Solve : X y X y a X y X y 5y • 5y 5y Exercise LXXIII. 3. 2_A=1 32/ X y 27 n 72 5. 4=5 i-5 = 6 aff -+ X y h a: y a -< SIMULTANEOUS EQUATIONS. 163 ^2,3 . ax by 1_1 = 3 ax hy 8. ™+» -^ + n] nx my x^y x^y J h a_^ X y "" 195. If three simultaneous equations are'given, involv- ing three unknown quantities, one of the unknown quanti- ties must be eliminated between two pairs of the equations ; then a second between the resulting equations. 196. Likewise, if four or more equations are given, involving four or more unknown quantities, one of the unknown quantities must be eliminated between three or more pairs of the equations ; then a second between the pairs that can be found of the resulting equations; and so on. Solve: 2a;-3y-f4z= 4] 3:r + 5y-7z = 12 bx— y — Sz= SJ Eliminate z between two pairs of these equations. Multiply (1) by 2, 4x-6y + 82= 8 (3) is bx- y-Sz= 5 Add, 9a;-7y =13 Multiply (1) by 7, 14a:-2l3/ + 28z = 28 Multiply (2) by 4, 12a; + 20y-28z = 48 Add, 26a;- y =76 Multiply (6) by 7, 182 a; -73/ = 532 (5) is 9a;-7v= 13 Subtract (5) from (7), 173 a; =519 • r — ^ Substitute the value oi . . X — 0. 'xin(6), 78-2/ = 76 Substitute the values of x and y in (1), 6 — 6+4z = 4, .-. z = 1. (1) (2) (3) (4) (5) (6; CO 164 ALGEBRA. o I . Exercise LXXIV. 1. 5a; + 3y-62; = 4 ^ 10. Sx — 7/ + z = 17 ^ Sx-7/-Jr2z = S i 6x + Sy — 2z = 10 I x-2^ + 2z = 2 J 7a: + 4y-5z = 3 J 2. 4^ — 5y + 20 = 6 -\ 11. x-\-y-{-z = 5 -\ 2x + Sy~z^20 I 3:r-5y + 70 = V5 [ 7a;-4y + 3z==35 J 9:r- II2 + 10 = J 3. ^ + y + z = 6 -j 12. rc + 2y + 30 = 6 "i 5a: + 4y + 3^ = 22 i 2a; + 42/ + 23 = 8 15a; + 10y + 62 = 63 J 3^ + 2^ + 8^ = 101 J 4. 4:r-3y + z = 9 "j 13. :r-3y-2z = l 9:r + y-5z = 16 I 2:r- 3y + 5z = - 19 a;-4y + 32 = 2 J 5a; + 2y-z = 12 5. 82; + 4y-3z = 6^ 14. 3x-2y = 5 ^ 4:^-5y + 4z = 8j x-2y--6z = -1 J 6. 12a; + 5y-4z = 29 -j 13a;-2y + 50 = 58 I 17a; — y — 2 = 15 J 7. y-5; + z = -5 ^ 16. 2:i;-3y = 3 2 — y — a; = — 25 I 3y — 40=7 :r + y + z = 35 J 42;— 6^7 = 2 J 8. a; + y + 2 = 30 ^j 17. 3:r-4y + 6z = l 8:r + 4y + 22 = 50 I 2:r + 2y-z = l 27^ + 9y + 32 = 64 j 7:z;-6y+70 = 2 9. 15y = 24z-10a; + 41 ^ 18. 7:r-3y = 30 15:r = 12y-162 + 10 I 9y-5z = 34 18^~(7z-13) = 14y J a: + y + z = 33 SIMULTANEOUS EQUATIONS. 165 19. ^+M=^ ^+1+1=-^ .+1 + 1=17 20. ^+?=5 1 X y y 2 Z X 21. 1,1 1 ' 1-1+1=5 1+1-1=0 y z X 22. hz~{-cy=^a ' az + ca; = 5 If ay-^hx = c . 23. § 4 1_, 1 a; 5y z h-h-h^'^ 4 1 , 4 lAi 24. 2_§ + l = 2.9 ■ o: 2/ z ^-^-^ = -10.4 X y z ? + L0_§ = i4.9 y z X \ 25. 2+1-3=0^ ^ y z ^-2_2==0 z y 1.14^ -+-_ =0 X z Z J 26. ax-^hy -\-cz^=a ax — by — cz= h ax -\- cy -{- hz = c 2x — y ^ Sy + 2z ^ x — y—z ^. 3 4 5 28. y y~z x-\-z X a + ^+ c * Subtract from the sum of the three equations each equation separately, f Multiply the equations by a, b, and c, respectively, and from the sum of the results subtract the double of each equation separately. CHAPTER XII. Problems producing Simultaneous Equations. 197. It is often necessary in the solution of problems to employ two or more letters to represent the quantities to be found. In all cases the conditions must be sufficient to give just as many equations as there are unknown quan- tities employed. If there be more equations than unknown quantities, some of them are superfluous or contradictory ; if there be les8 equations than unknown quantities, the problem is in- determinate or impossible. (1) When the greater of two numbers is divided by the less the quotient is 4 and the remainder 3 ; and when the sum of the two numbers is increased by 38, and the result divided by the greater of the two numbers, the quotient is 2 and the remainder 2. Find the numbers. Let X = the greater number, and y = the smaller number. x-S , Then 2/ + ?/ + 38 - 2 and .^ -r y ^ o. - ^ _ 2. X From the solution of these equations x = 47, and y = 11. (2) If A give B $10, B will have three times as much money as A. If B give A $10, A will have twice as much money as B. How much has each ? PROBLEMS. 167 Let X = number of dollars A has, and y = number of dollars B has. Then y + 10 =■ number of dollars B has, and a; — 10 == number of dollars A has after A gives 1 10 to B. .-. y + 10 = 3 (a; - 10), and re + 10 = 2 (y - 10). From the solution of these equations, a; = 22 and y = 26. Therefore, A has $22 and B $26. EXEECISE LXXV. 1. The sum of two numbers divided by 2 gives as a quo- tient 24, and the difference between them divided by 2 gives as a quotient 17. What are the numbers ? 2. The number 144 is divided into three numbers. When the first is divided by the second, the quotient is 3 and the remainder 2 ; and when the third is divided hj the sum of the other two numbers, the quotient is 2 and the remainder 6. Find the numbers. 3. Three times the greater of two numbers exceeds twice the less by 10 ; and twice the greater together with three times the less is 24. Find the numbers. ; 4. If the smaller of two numbers be divided by the greater, the quotient is .21 and the remainder .0057 ; but if the greater be divided by the smaller, the quotient is 4 and the remainder .742. What are the num- Bers? 5. Seven years ago the age of a father was four times that of his son ; seven years hence the age of the father will be double that of the son. What are their ages? 6. The sum of the ages of a father and son is half what it will be in 25 years ; the difference between their ages is one-third of what the sum will be in 20 years. What are their ages ? 168 ALGEBRA. 7. If B give A f 25, they will have equal sums of money ; but if A give B $22, B's money will be double that of A. How much has each ? 8. A farmer sold to one person 30 bushels of wheat and 40 bushels of barley for $67.50; to another person he sold 50 bushels of wheat and 30 bushels of barley for $85. What was the price of the wheat and of the barley per bushel ? 9. If A give B $5, he will then have $6 less than B ; but if he receive $5 from B, three times his money will be $20 more than four times B's. How much has each ? 10. The cost of 12 horses and 14 cows is $1900; the cost of 5 horses and 3 cows is $ 650. What is the cost of a horse and a cow respectively ? Note I. A fraction of which the terms are unknown may be rep- resented by -. y Ex. A certain fraction becomes equal to |- if 3 be added to its numerator, and equal to f if 3 be added to its denominator. Determine the fraction. Let - = the required fraction. y By the conditions ^ = A, y and -V^ = f •From the solution of these equations it is found that Therefore the fraction = y\. 11. A certain fraction becomes equal to 2 when 7 is added to its numerator, and equal to 1 when 1 is subtracted from its denominator. Determine the fraction. PROBLEMS. 169 12. A certain fraction becomes equal to -J when 7 is added to its denominator, and equal to 2 when 13 is added to its numerator. Determine the fraction. 13. A certain fraction becomes equal to |- when the denom- inator is increased by 4, and equal to J^ when the numerator is diminished by 15. Determine the f^-ac- tion. 14. A certain fraction becomes equal to -f if 7 be added to the numerator, and equal to f if 7 be subtracted from the denominator. Determine the fraction. 15. Find two fractions with numerators 2 and 5 respectively, V whose sum is 1^, and if their denominators are inter- changed their sum is 2. 16. A fraction which is equal to J is increased to -^j when a certain number is added to both its numerator and denominator, and is diminished to |- when one more than the same number is subtracted from each. De- termine the fraction. Note II. A number consisting of two digits which are unknown may be represented by 10 a; + y , in which x and y represent the digits of the number. Likewise, a number consisting of three digits which i^e unknown may be represented by 100 a; + lOy + z, in which x, y, and z represent the digits of the number. For example, consider any number expressed by three digits, as 3G4. The expression 364 means 300 + 60 + 4 ; or, 100 times 3 + 10 times 6 + 4. Ex. The sum of the two digits of a number is 8, and if 36 be added to the number the digits will be inter- changed. What is the number ? Let X = the digit in the tens' place, ^ and y = the digit in the units' place. », Then lOx + y = the number. K.. By the conditions, x + y = S, (1) ^L and 10a; + y + 36 = lOy + a;. (2) 170 ALGEBRA. From (2), 9a;-% = -36. Divide by 9, x — y = ~ i. Add (1) and (3), 2x = i, :.x = 2. Subtract (3) from (1), 2y = 12, :.y = Q. Hence, tlie number is 26. 17. The sum of the two digits of a number is 10, and if 54 be added to the number the digits will be inter- changed. What is the number ? 18. The sum of the two digits of a number is 6, and if the number be divided by the sum of the digits the quo- tient is 4. What is the number ? 19. A certain number is expressed by two digits, of which the first is the greater. If the number be divided by the sum of its digits the quotient is 7 ; if the digits be interchanged, and the resulting number diminished by 12 be divided by the difference between the two digits, the quotient is 9. What is the number ? 20. If a certain number be divided by the sum of its two digits the quotient is 6 and the remainder 3 ; if the digits be interchanged, and the resulting number be divided by the sum of the digits, the quotient is 4 and the remainder 9. What is the number ? 21. If a certain number be divided by the sum of its two digits diminished by 2, the quotient is 5 and the re- mainder 1 ; if the digits be interchanged, and the resulting number be divided by the sum of the digits increased by 2, the quotient is 5 and the remainder 8. Find the number. 22. The first of the two digits of a number is, when doubled, 3 more than the second, and the number itself is less by 6 than five times the sum of the digits. What is the number ? PROBLEMS. 171 23. A number is expressed by three digits, of which the first and last are alike. By interchanging the digits in the units' and tens' places the number is increased by 54 ; but if the digits in the tens' and hundreds' pUces are interchanged, 9 must be added to four times the resulting number to make it equal to the original number. What is the number ? 24. A number is expressed by three digits. The sum of the digits is 21 ; the sum of the first and second exceeds the third by 3 ; and if 198 be added to the number, the digits in the units' and hundreds' places will be interchanged. Find the number. 25. A number is expressed by three digits. The sum of the digits is 9 ; the number is equal to forty-two times the sum of the first and second digits ; and the third digit is twice the sum of the other two. Find the number, 26. A certain number, expressed by three digits, is equal to forty-eight times the sum of its digits. If 198 be subtracted . from the number, the digits in the units' and hundreds' places will be interchanged ; and the sum of the extreme digits is equal to twice the mid- dle digit. Find the number. Note III. If a boat move at the rate of a^ miles an hour in still water, and if it be on a stream that runs at the rate of y miles an hour, then a; + y represents its rate down the stream, x — y represents its rat« wp the stream. 27. A waterman rows 30 miles and back in 12 hours. He finds that he can row 5 miles with the stream in the same time as 3 against it. Find the time he was rowing up and down respectively. 172 ALGEBRA. 28. A crew which can pull at the rate of 12 miles an hour down the stream, finds that it takes twice as long to come up the river as to go down. At what rate does the stream flow ? 29. A man sculls down a stream, which runs at the rate of 4 miles an hour, for a certain distance in 1 hour and 40 minutes. In returning it takes him 4 hours and 15 minutes to arrive at a point 3 miles short of his starting-place. Find the distance he pulled down the stream and the rate of his pulling. 30. A person rows down a stream a distance of 20 miles and back again in 10 hours. He finds he can row 2 miles against the stream in the same time he can row 3 miles with it. Find the time of his rowing down and of his rowing up the stream ; and also the rate of the stream. Note IV. When commodities are mixed, it is to be observed that the quantity of the mixture = the quantity of the ingredients; the cost of the mixture = the cost of the ingredients. Ex. A wine-merchant has two kinds of wine which cost 72 cents and 40 cents a quart respectively. How much of each must he take to make a mixture of 60 quarts worth 60 cents a quart? Let X = required number of quarts worth 72 cents a quart, and y = required number of quarts worth 40 cents a quart. Then, 12x = cost in cents of the first kind, 40 3/ = cost in cents of the second kind of wine, and 3000 = cost in cents of the mixture. .-. x + y = 50, 72 ^ + 40 y = 3000. From which equations the values of x and y may be found. PROBLEMS. 173 31. A grocer mixed tea that cost him 42 cents a pound with tea that cost him 54 cents a pound. He had 30 pounds of the mixture, and by selling it at the rate of 60 cents a pound, he gained as much as 10 pounds of the cheaper tea cost him. How many pounds of each did he put into the mixture ? 32. A grocer mixes tea that cost him 90 cents a pound with tea that cost him 28 cents a pound. The cost of the mixture is $61.20. He sells the mixture at 60 cents a pound, and gains $3.80. How many pounds of each did he put into the mixture ? 33. A farmer has 28 bushels of barley worth 84 cents a bushel. With his barley he wishes to mix rye worth $1.08 a bushel, and wheat worth $1.44 a bushel, so that the mixture may be 100 bushels, and be worth $1.20 a bushel. How many bushels of rye and of wheat must he take ? Note V. It is to be remembered that if a person can do a piece of work in x days, the part of the work he can do in one day will be represented by -. Ex. A and B together can do a piece of work in 48 days ; A and C together can do it in 30 days ; B and C to- gether can do it in 26| days. How long will it take each to do the work? Let X = the number of days it will take A alone to do the work, y = the number of days it will take B alone to do the work, and 2 = the number of days it will take C alone to do the work. Then, _, _, _, respectively, will denote the part each can do X y z . , ^ ma day, and - + - will denote the part A and B together can do in a day; but — will rlenote the part A and B together can do in a day. 174 A.LGEBRA. Therefore, 1 + 1 = 1 'x y 48 (1) Likewise, 1 1 1 a; "^ 2 ~ 30 (2) and 1113 t/^2~26f~80 (3) Add (1), (2), and (3), 2 2 2 11 x^y'^z 120 (4) Multiply (1) by 2, 2 2 1 ~x'^y ^24 (5) Subtract (5) from (4), 2 1 2 ~20 .-. 2 = 40. Subtract the double oi •(2) from (4), ^^ ^ y 40 .-. y = 80. Subtract the double of (3) from (4), ? = -1 X 60 /. X = 120. 34. A and B together earn $40 in 6 days; A and C to- gether earn $54 in 9 days; B and G together earn $80 in 15 days. What does each earn a day? 35. A cistern has three pipes, A, B, and C. A and B will fill it in 1 hour and 10 minutes ; A and in 1 hour and 24 minutes ; B and in 2 hours and 20 minutes. How long will it take each to fill it ? 38. A warehouse will hold 24 boxes and 20 bales ; 6 boxes and 14 bales will fill half of it. How many of each alone will it hold ? 37. Two workmen together complete some work in 20 days ; but if the first had worked twice as fast, and the sec- ond half as fast, they would have finished it in 15 days. How long would it take each alone to do the work ? 38. A purse holds 19 crowns and 6 guineas ; 4 crowns and 5 guineas fill \-\ of it. How many of each alone will it hold ? PROBLEMS. 175 39. A piece of work can be completed by A, B, and C to- gether in 10 days ; by A and B together in 12 days ; by B and C, if B work 15 days and 30 days. How long will it take each alone to do the work ? 40. A cistern has three pipes, A, B, and G. A and B will fill it in a minutes ; A and in 5 minutes ; B and m c minutes. How long will it take each alone to fill it? Note VI. In considering the rate of increase or decrease in quan- tities, it is usual to take 100 as a common standard of reference, so that the increase or decrease is calculated for every 100, and there- fore called per cent. It is to be observed that the representative of the number result- ing after an increase has taken place is 100 + increase per cent ; and after a decrease, 100 — decrease per cent. Interest depends upon the time for which the money is lent, as well as upon the rate per cent charged ; the rate per cent charged being the rate per cent on the principal for one year. Hence, p,. 1 • ^ , Principal X Rate X Time bimple interest = , ^ 100 where Time means number of years or fraction of a year. Amount = Principal + Interest. In questions relating to stocks, 100 is taken as the representative of the stock, the price represents its market value, and the per cent represents the interest which the stock bears. Thus, if six per cent stocks are quoted at 108, the meaning is, that the price of $100 of the stock is $108, and that the interest derived from $100 of the stock will be jf^ of $100, that is, $6 a year. The rate of interest on the money invested will be \^^ of 6 per cent. 41. A man has $ 10,000 invested. For a part of this sum he receives 5 per cent interest, and for the rest 4 per cent; the income from his 5 per cent investment is $50 more than from his 4 per cent. How much has he in each investment ? 176 ALGEBRA. 42. A sum of money, at simple interest, amounted in 6 years to $26,000, and in 10 years to $30,000. Find tlie sum and the rate of interest. 43. A sum of money, at simple interest, amounted in 10 months to $26,250, and in 18 months to $27,250. Find the sum and the rate of interest. 44. A sum of money, at simple interest, amounted in m years to a dollars, and in n years to b dollars. Find the sum and the rate of interest. 45. A sum of money, at simple interest, amounted in a months to c dollars, and in b months to d dollars. Find the sum and the rate of interest. 46. A person has a certain capital invested at a certain rate per cent. Another person has $ 1000 more capital^ and his capital invested at one per cent better than the first, and receives an income $80 greater. A third person has $1500 more capital, and his capital in- vested at two per cent better than the first, and re- ceives an income $ 150 greater. Find the capital of each, and the rate at which it is invested. 47. A person has $12,750 to invest. He can buy three per cent bonds at 81, and five per cents at 120. Find the amount of money he must invest in each in order to have the same income from each investment. 48. A and B each invested $1500 in bonds; A in three per cents and B in four per cents. The bonds were bought at such prices that B received $5 interest more than A. After both classes of bonds rose 10 points, they sold out, and A received $50 more than B, What price was paid for each class of bonds? PROBLEMS. 177 49. A person invests $10,000 in three per cent bonds, $ 16,500 in three and one-half per cents, and has an income from both investments of $1056.25. If his investments had been $2750 more in the three per cents, and less in the three and one-half per cents, his income would have been 62^ cents greater. What price was paid for each class of bonds ? 50. The sum of $2500 was divided into two unequal parts and invested, the smaller part at two per cent more than the larger. The rate of interest on the larger sum was afterwards increased by 1, and that of the smaller sum diminished by 1 ; and thus the interest of the whole was increased by one-fourth of its value. If the interest of the larger sum had been so in- creased, and no change been made in the interest of the smaller sum, the interest of the whole would have been increased one-third of its value. Find the sums invested, and the rate per cent of each. Note VII. If x represent the number of linear units in the length, and y in the width, of a rectangle, xy will represent the number of its units of surface ; the surface unit having the same name as the linear unit of its sides. 51. If the sides of a rectangular field were each increased by 2 yards, the area would be increased by 220 square yards; if the length were increased and the breadth were diminished each by 5 yards, the area would be diminished by 185 square yards. What is its area ? 52. If a given rectangular floor had been 3 feet longer and 2 feet broader it would have contained 64 square feet more ; but if it had been 2 feet longer and 3 feet broader it would have contained 68 square feet more. Find the length and breadth of the floor. 178 ALGEBRA. 53. In a certain rectangular garden there is a strawberry- bed whose sides are one-third of the lengths of the corresponding sides of the garden. The perimeter of the garden exceeds that of the bed by 200 yards; and if the greater side of the garden be increased by 3, and the other by 5 yards, the garden will be en- larged by 645 square yards. Find the length and breadth of the garden. Note VIII. Care must be taken to express the conditions of a problem with reference to the same principal unit. Ex. In a mile race A gives B a start of 20 yards and beats him by 30 seconds. At the second trial A gives B a start of 32 seconds and beats him by 9^^ yards. Find the rate per hour at which each runs. Let X = number of yards A runs a second, and y = number of yards B runs a second. Since there are 1760 yards in a mile, — = number of seconds it takes A to run a ^ mile, and — li = number of seconds B was running in the y y first and second trials, respectively. Hence, 1115 _ 1^ = 30. and l!i_l!62„32. y ^ The solution of these equations gives x = 5if and y == 5^^. That is, A runs — i^, or — , of a mile in one second ; 1760 300 and in one hour, or 3600 seconds, runs 12 miles. Likewise, B runs lOjW miles in one hour. 54. In a mile race A gives B a start of 100 yards and beats him by 15 seconds. In the second trial A gives B a start of 45 seconds and is beaten by 22 yards. Find the rate of each in miles per hour. PROBLEMS. 179 55. In a mile race A gives B a start of 44 yards and beats him by 51 seconds. In the second trial A gives B a start of 1 minute and 15 seconds and is beaten by 88 yards. Find the rate of each in miles per hour. 56. The time which an express-train takes to go 120 miles is -f^ of the time taken by an accommodation-train. The slower train loses as much time in stopping at different stations as it would take to travel 20 miles without stopping ; the express-train loses only half as much time by stopping as the accommodation- train, and travels 15 miles an hour faster. Find the rate of each train in miles per hour. 67. A train moves from P towards Q, and an hour later a ~ second train starts from Q and moves towards P at a rate of 10 miles an hour more than the first train ; the trains meet half-way between P and Q. If the train from P had started an hour after the train from Q its rate must have been increased by 28 miles in order that the trains should meet at the half-way point. Find the distance from P to Q. 58. A passenger-train, after travelling an hour, meets with an accident which detains it one-half an hour ; after which it proceeds at four-fifths of its usual rate, and arrives an hour and a quarter late. If the accident had happened 30 miles farther on, the train would have been only an hour late. Determine the usual rate of the train. 59. A passenger-train after travelling an hour is detained 15 minutes ; after which it proceeds at three-fourths of its former rate, and arrives 24 minutes late. If the detention had taken place 5 miles farther on, the train would have been only 21 minutes late. Deter- mine the usual rate of the train. 180 ALGEBRA. 60. A man bought 10 oxen, 120 sheep, and 46 lambs. The cost of 3 sheep was equal to that of 5 lambs ; an ox, a sheep, and a lamb together cost a number of dol- lars less by 57 than the whole number of animals bought; and the whole sum spent was $2341.50. Find the price of an ox, a sheep, and a lamb, respec- tively. 61. A farmer sold 100 head of stock, consisting of horses, oxen, and sheep, so that the whole realized $11.75 a head ; while a horse, an ox, and a sheep were sold for $110, $62.50, and $7.50, respectively. Had he sold one-fourth of the number of oxen that he did, and 25 more sheep, he would have received the same sum. Find the number of horses, oxen, and sheep, respectively, which were sold. 62. A, B, and together subscribed $ 100. If A's sub- scription had been one-tenth less, and B's one-tenth more, C's must have been increased by $ 2 to make up the sum ; but if A's had been one-eighth more, and B's one-eighth less, C's subscription would have been $ 17.50. What did each subscribe ? 63. A gives to B and C as much as each of them has ; B gives to A and as much as each of them then has ; and C gives to A and B as much as each of them then has. In the end each of them has $6. How much had each at first ? 64. A pays to B and C as much as each of them has ; B pays to A and one-half as much as each of them then has ; and pays to A and B one-third of what each of them then has. In the end A finds that he has $1.50, B $4.16|, C $.581 How much had each at first? CHAPTER XIII. Involution and Evolution. 198. The operation of raising an expression to any re 3[uired power is called Involution. Every case of involution is merely an example of multi- plication, in which the factors are equal. Thus, {2ay = 2a^x2a^ = ^a\ 199. A power of a simple expression is found by multi- plying the exponent of each factor by the exponent of the required factor, and taking the product of the resulting factors. The proof of the law of exponents, in its general form, is : (^^ry ^ ^r. y^ ^m ^^ ^^m -^ ^^ ^ factors, Hence, if the exponent of the required power be a com- posite number, it may be resolved into prime factors, the power denoted by one of these factors may be found, and the result raised to a power denoted by another, and so on. Thus, the fourth power may be obtained by taking the sec- ond power of the second power ; the sixth by taking the second power of the third power ; the eighth by taking the second power of the second power of the second power. 200. From the Law of Signs in multiplication it is evi- dent that, I. All even powers of a number are positive. II. All odd powers of a number ha,ve the same sign as the number itself. 182 ALGEBBA. Hence, no even power of any number can be negative ; and of two compound expressions whose terms are identical but have opposite signs, the even powers are the same. Thus, Q,_af=\-{a-h)Y={a- bf. 201. A method has been given, § 83, of finding, without actual multiplication, the powers of binomials which have the form (a zh ^). The same method may be employed when the terms of a binomial have coefficients on exponents. (1) {a-hf = a^~?>a% + Zah''-b^. (2) (bx''-2ff, = {bxj - 3 {bx''f{2f) ^?>{bx''){2ff-(2ff, = 125 jc« - 150^y + 60 ^y - ^y\ (3) {a-hY=^a^-^a^b^Qa'h^-4.ah^ + b\ (4) (p^-iy)\ ^{x'y-^{:^)\',y)-\'(S(xy{}yf-^x'Qiyf+(}y)\ = x'-2x'y-\-ixy~\xy-\-i-^y\ 202. In like manner, a polynomial of three or more terms may be raised to any power by enclosing its terms in par- entheses,'SO as to give the expression the form of a binomial. Thus, (1) {a-\-b^cf=\a+{b-\-c)Y, -: a^ + 3a' (J + ^) + 3a (^ + cf+ {b + c)\ ■=-a3 + 3a25 + 3a2^ + 3a52+6a5c ^2>ac^ + b^-{-Wc^^bc'^c\ INVOLUTION AND EVOLUTION. 183 (2) (x'-2x' + Sx-{-4:y, = \(x'-2x') + (dx + i)Y, ■ ^ix"- 2xy + 2(x'- 2x') (3:r + 4) + (3^ + 4)^ = x^-4:a^ + 4:x* + 6x* — 4:x^—16x''+9x''+24:x-\-l6., = x^~ix'i- 10:r* - 4r^ - 7x' + 24:x + 16. . Exercise LXXVI. Write the second members of the following equations : 1. (aj= 11. (2a'bc'y= 21. (-Ba'b'cf = 2. (xj= 12. (-5axyy= 22. (-3x1/^ = 5. (xyy= 13. {~1m'na^yy= 23. (-5a'ba^f = «!^'Y= 14 f-^^Y= 24 ^-M!Y= 2 y 'V Sa^^y * V 4cV 6. (a; + 2/== 16. (2x — ay= 26. (l-a-a7 = 7. (a; -2)*= 17..t3:r + 2ay = 27. (2-3.r + 4.r7'- 8. (:r + 3y= 18. (2x-yy= 28. (l-2;r+a;7=- 9. (l+2a;)^= 19. {x'y-2xy'y=- 29. (l-a: + .r7 = { 10. (2m- 1)^=20. (a^-3y=: 30.(1 + ^+^')*-= Evolution. 203. The operation of finding any required root of an expression is called Evolution. Every case of evolution is merely an example of factor- ing, in which the required factors are all equal. Thus, the square, cube, fourth roots of an expression are found by taking one of the two, three, four equal factors of the expression. 184 ALGEBRA. 204. The symbol whicli denotes that a square root is to be extracted is^Vj ^^^ ^^^ other roots the same symbol is used, but with a figure written above to indicate the root, thus, -J/, -y/, etc., signifies the third root, fourth root, etc. 205. Since the cube of a^ = a^, the cube root of a^ = a^. Since the fourth power of 2a^ = 2'*a^, the fourth root of Since the square of abc — a^b^(f, the square root of a^5V — a5c. o- ,1 /.«& a^^^ ,-1 J. pO^b^ ah Since the square oi — = -^-5, the square root 01 — .-^ = — xy ory^ or^f xy Hence, the root of a simple expression is found by divid- ing the exponent of each factor by the index of the root, and talcing the product of the resulting factors. 206. It is evident from § 200 that I. Any even root of a positive number will have the double sign, ±. II. There can be no even root of a negative number. III. Any odd root of a number will have the same sign as the number. Ttus,^^ = i|^; ■>5^-27mV = -3mn2; But -yj—oi? is neither + ^ nor — x, for (+ x^ = + ^, and {-x)^-^-\-x^. The indicated even root of a negative number js called an impossible, or imaginary, number. INVOLUTION AND EVOLUTION. 185 207. If the root of a number expressed in figures is not readily detected, it may be found by resolving the number into its prime factors. Thus, to find the square root of 3,415,104 : 2^ 3415104 2" 426888 ^' 53361 7 5929 7 847 11 121 3,415,104 = 11 2« X 3^ X .-. V3.415,104 = 2^x3 x7 xll = 1848. Exercise LXXVII. Simplify : 1. -Va', -v/?, V4^^ ^64, -^/aVy^ ^16a^^6V, ~s/-S2a'\ 2. ■y/^^rf2S?dVf, ^/33756V, ■v/3111696cV. ; 3. V533616*cVV^ \- l 3 I 4. 5. 6 7. 8, 9. 216 5V^ 64: x'' 343z^*' \729z^«' a/27^ X VMS^ X Vl6^^ When a=l, b = S, x = 2, y==6, find the values of; 4 V2ri; — ■\/abxy + b^a^h^xy. A a'w o ax + hVl2hy + 4:ahx-\/bxy. Va^ + 2a5 + ^2 X -v^a' + 3a^5 + 3a^'^ + 6'. 186 ALGEBRA. Square Roots of Compound Expressions. 208. Since tlie square oi a -}- b is a^ -{- 2 ah -}- P, the square root of a^ -{- 2ab -{- P is a -\~ b. It is required to find a method of extracting the root xi-\-b when a^ ~{- 2 ab -{- b^ is given : Ex. The first term, a, of the root is obviously the square root of the first term, a^, in the expression. o? -\~2ab -\~b^\a-\-b ^^ ^^® ^^ ^® subtracted from the 2 given expression, the remainder is o \ Tr\ o 7. I 72 2ab-\-h^. Therefore the second term, J a + 6 Zab-\-b 6, of the root is obtained when the Ago -\- £j.g^ iexTQ. of this remainder is di- vided by 2 a, that is, by double the part of the root already found. Also, since 2 a& + 6' = (2 a + 5) &, the di- visor is completed by adding to the trial-divisor the new term of the root. (1) Find the square root of 26 x^ — 20 a^i/ + 4: x^if. 26x'-20a?y-\-4:xy \6x~2c(^y 2b x" ^20s^y-{-4:xy -~20:^y-\-4xY I0x~2x^y The expression is arranged according to the ascending powers of x. The square root of the first term is 5 a;, and 5 a; is placed at the right of the given expression, for the first term of the root. The second term of the root, — 2ar^y, is obtained by dividing — 203;^^ by lOx, and this new term of the root is also annexed to the divisor, 10 x, to complete the divisor. 209. The same method will apply to longer expressions, if care be taken to obtain the trial-divisor at each stage of the process, by doubling the part of the root already found, and to obtain the complete divisor by annexing the new term of the root to the trial-divisor. INVOLUTION AND EVOLUTION. 187 Ex. Find tlie square root of 16x^ Sa^-Sx" -24: 0^+25 X* -24:3^4- 9x* 8a;3-6a:24-2a: 16a;*-20;r»+102:2 16a;*-12:r»+ 4:3^ 8r^-6^-f4a:~l - 8a^+ 6a^-4cx+l - 8^+ Qx'-Ax-^l The expression is arranged according to the descending powers of X. It will be noticed that each successive trial- divisor may be obtained by taking the preceding complete divisor with its last term doubled. Exercise LXXVIII. Extract the square roots of : 1. a* + 4a^ + 2a^--4a + l. 2. x^-2a^y + 3xy-2xf + y\ 3. 4a«-12a^a; + 5aV+6aV + aV. 4. 9x^-12a^y^+16x'y*-24xy + 4:y^-i-l&xi/', 5. 4a8+16c«+16aV-32aV. 6. 4:X* + 9-S0x-20a^ + S1x^. 7. 16x*-16abx'+16b^x' + 4:a^b''-8ab^ + 4b\ 8. x^ + 25x^+10x*-4:x'-~203^+16-24x. 9. x^-i-Sxy-4a^i/-4:xf + 8xy-10xy-{-y^ 10. 4-12a-lla* + 5a2-4a^ + 4a«+14a^ 11. 9a^-6ab + 30ac + 6ad+b^--10bc-2bd 'i-2bc^+10cd+d\ 188 - ALGEBRA. 12. 2bx' - 31xy + S4:xy—S0x^i/ + y' — Sxy^ + lOa^y. 13. 771^ - 4m^ + 10 m^ — 20 m^ ~ 44 m^ + 35m* + 46m2 - 40m + 25. •7 14. X* — x^y — -x^y^ + xf + y\ 15. rr'* — 4A + 6a;y — 6:r?/^ + 53/*— ^+^. 16. ^'_^ + 43^,^_3 ^ ^* 9 2 48 4 4 ,^ T , 4 , 10 , 20 , 25 , 24 16 18. ^^__2a 3_25 ^ ^^^ ^4 _^ ^ _ 5^^_ ^ _^ 1^ 62 6 a a^ 12 3 9 Squaee Boots of Arithmetical Numbers. 210. In the general method of extracting the square root of a number expressed by figures, the first step is to mark off the figures in periods. Since 1 = 1^ 100 = 10^, 10,000 = 100', and so on, it is evident that the square root of any number between 1 and 100 lies between 1 and 10 ; the square root of any number between 100 and 10,000 lies be- tween 10 and 100. In other words, the square root of any number expressed by one or two figures is a number of one figure ; the square root of any number expre&sed by three or four figures is a number of two figures ; and so on. If, therefore, a dot be placed over the units figure of a square num- ber, and over every alternate figure, the number of dots will be equal to the number of figures in its square root. Find the square root of 3249. 3249 (57 In this case, a in the typical form a^ + 2ab + b^ 25 represents 5 tens, that is, 50, and b represents 7. 107) 749 The 25 subtracted is really 2500, that is, a^ and 749 " the complete divisor, 2 a -h &, is 2 x 50 + 7 = 107. INVOLUTION AND EVOLUTION. 189 211. The same method will apply to numbers of more than two periods by considering a in the typical form to represent at each step the part of the root already found. It must be observed that a represents so many tens with respect to the next figure of the root. Ex. Find the square root of 5,322,249. 5322249(2307 4 43)132 129 4607)32249 32249 212. If the square root of a number have decimal places, the number itself will have tiaice as many. Thus, if .21 be the square root of some number, this number will be (.21)2 _ 21 X .21 = .0441 ; and if .111 be the root, the number will be (.111)2 = .111 X .111 = .012321. Therefore, the number of decimal places in every square decimal will be even, and the number of decimal places in the root will be half as many as in the given number itself. Hence, if the given square number contain a decimal, and a dot be placed over the units' figure, and then over every alternate figure on both sides of it, the number of dots to the left of the decimal point will show the number of integral places in the root, and the number of dots to the right will show the number of decimal places. Ex. Find the square roots of 41.2164 and 965.9664. 4l.2i64(6.42 965.9664(31.08 36 9__ 124)521 61)65 496 61 1282)2564 6208)49664 2564 49664 It is seen from the dotting that the root of the first example will have one integral and two decimal places, and that the root of the second example will have two integral and two decimal places. 190 ALGEBRA. 213. If a number contain an odd number of decimal places, or if any number give a remainder when as many- figures in the root have been obtained as the given number has periods, then its exact square root cannot be found. We may, however, approximate to its exact root as near as we please by annexing ciphers and continuing the operation. Ex. Find the square roots of 3 and 357.357. 3.(1.732 1 357.3570(18.903 1 27)200 28)257 189 224 343)1100 369)3335 1029 3321 3462) 7100 37803)147000 6924 113409 Exercise LXXIX. Extract the square roots of: 1. 120,409; 4816.36; 1867.1041; 1435.6521; 64.128064. 2. 16,803.9369 ; 4.54499761 ; .24373969 ; .5687573056. 3. .9; 6.21; .43; .00852; 17; 129; 347.259. 4. 14,295.387; 2.5; 2000; .3; .03; 111. 5. .00111; .004; .005; 2; 5; 3.25; 8.6. a 1. 16. 100. 169. 2 89. 400 ^' ¥» 49"' T?4» 2T3"' 3T¥ ' "62T- '^' "2 ■' 3 ; 4 ; 3T ; rfc ; tts ; t ; tt- Cube Koots of Compound Expressions. 214. Since the cube oi a + h i^ a? -{-?>o?b + ?> ah'^ + h^, the cube root of a^ + 3 a?b + 3 a^^ + Z*^ is a + h. It is required to find a method for extracting the cube root a-{-b when a? + 3 a^6 + 3 ah"^ + ^^ is given : INVOLUTION AND EVOLUTION. 191 (1) Find the cub'e root of a« + 3 a^b -\-2>ab'-\- b\ aJ'-\-%a'b^Zab^-{-b'\a±b 2>d + 3a5 + Z)2 2>a}-\-^ab-\-b' Za'b^Zab^ + b^ 3a^Z> + 3a^>'^ + 6' The first term a of the root is obviously the cube root of the first term a? of the given expression. If a? be subtracted, the remainder is 3 a^6 + 3 ab^ + &' ; therefore, the second term 6 of the root is obtained by dividing the first term of this remainder by three times the square of a. Also, since 3 a'^S + 3 ah^ + 6^ = (3 a^ + 3 a6 + h"^) b, the complete divisor is obtained by adding Sab + b^ to the trial-divisor 3a^ (2) Find the cube root of 8a;' + d6x'y -f b4:xf + 21 f. 83^+36 x''i/+64:xf +27 f{2x±Sy (Qx-]-Sy)3y= 18a;y+9y' 12a;"'+18^3/+9y 36a:V+54a;3/'+273/' 36a;V+54a:y^+2V The cube root of the first term is 2x, and this is therefore the first term of the root. The second term of the root, 3y, is obtained by dividing 36x^y by 3(2x)'^= 12a;^ which corresponds to Sa"^ in the typical form, and is completed by annexing to 12a;2 the expression {3(2 a;) + Sy}3y = 18 xy + 9y^, which corresponds to Sab + b^, in the typical form. 215. The same method may be applied to longer expres- sions by considering a in the typical form S a^ ~{- S ab -\- b^ to represent at each stage of the process the part of the root already found. Thus, if the part of the root already found be x + y, then 3 a' of the typical form will be represented by 3 (x + y)' ; and if the third term of the root be + z, the Sab -{-b"^ will be represented by S{x + y)z ■\-z^. So that the complete divisor, 3a'^-3a6 + &^ will be repre- sented by 3 (x + 3/)2 + 3 (x + 2/)2 + ^. 192 ALGEBRA. Find the cube root of x^ — Sx^ -\-bx^ — 'Sx — 1. x^ 3x-l {3x^-x){-x) = 3a^ + 3a;*-3a;3^ ^2 3a^ +3x* x3 S{x''-xy = 3x^-63^ + Sx'' {Sx''-3x-l)i-l)= -3a;'^ + 3a; + l 3a:* -6a^ + 3x + l 3x* + Qa^-Sx-l - 3a;* + 6x3- 3a; -1 The root is placed above the given expression for convenience of arrangement. The first term of the root, x^, is obtained by taking the cube root of the first term of the given expression; and the first trial-divisor, 3 a;*, is obtained by taking three times the square of this term of the root. The first complete divisor is found by annexing to the trial-divisor (3a;^ — a;) (— a;), which expression corresponds to {3a + b)b in the typical form. 4 The part of the root already found (a) is now represented by a:^ — a;; therefore, Sa"^ is represented by 3(a;^ — a;)"'^ = 3 a;* — Ga;^ + 3a;^ the sec- ond trial-divisor; and {3a + b)h by (3x2 — 3 a; — 1) (— 1) ; therefore, in the second complete divisor, 3 a^ + {S a + b)b is represented by (3a;* - Bar' + 3a;2) 4- (- Sa;^- 3a; - 1) X (- 1) == 3a;* - 6a^ + 3a; + 1. Exercise LXXX. Find the cube roots of : 1. x' + 6x'y + 12x7/' + Sy\ 3. x''-\-12x^ + ASx-\-64:. 2. a'-9a' + 21a-27. 4 . x^—S ax^-j-b a^x^-S d'x- a^ 5. x^-\-2>3(^^^x'-^1x^^i6x'-\-^x + l. 6. 1 - 9a; + 39a;' - 99a;'+ 156a;* - 144a;^ -f 64a;^ 7. a'' - 6^5 + 9a* + 4a' -9a'- 6a -1. 8. 64a;« + 192:^-^ + 144a;* - 32a;' - 36a;' + 12a; - 1. 9. l-3a; + 6a;'-10a;'-fl2a;*-12a;^+10a;«-6:i-^-f3:i-«-a;^ INVOLUTION AND EVOLUTION. 193 10. a^ + 9 a'b - 135 aW + 729 ab' - 729 h\ 11. c' - 12 he" + 60 5V - 160 ¥c^ + 240 5V - 192 h'c + 64 h\ 12. 3 a« + 48 a^Z> + 60 a^h^ - 80 a^^^s _ qq ft2^4_|_ ^Qg ^^s. 2756, Cube Roots of Arithmetical Numbers. 216. In extracting the cube root of a number expressed by figures, the first step is to mark it ofi" into periods. Since 1 = 1^, 1000 = 10^, 1,000,000 = 100^, and so on, it follows that the cube root of any number between 1 and 1000, that is, of any num- ber which has one, two, or three figures, is a number of one figure ; and that the cube root of any number between 1000 and 1,000,000, that is, of any number which has four, five, or six figures, is a number of two figures ; and so on. Hence, if a dot be placed over every third figure of a cube num- ber, beginning with the units' figure, the number of dots will be equal to the number of figures in its cube root. 217. If the cube root of a number contain any decimal figures, the number itself will contain three tiTnes as many. Thus, if .3 be the cube root of a number, the number is .3 X .3 X .3 Hence, if the given cube number have decimal places, and a dot be placed over the units' figure and over every third figure on both sides of it, the number of dots to the left of the decimal point will show the number of integral figures in the root ; and the number of dots to the right will show the number of decimal figures in the root. If the given number be not a perfect cube, ciphers may be an- nexed, and a value of the root may be found as near to the true value as we please. 218. It is to be observed that if a denote the first term of the root, and h the .second term, the first complete divisor '^ 3a2 + 3a6 + Z>2, and the second trial-divisor is 3 (a + ^)^, that is, 194 ALGEBRA. which may be obtained from the preceding complete divisor by adding to it its second term and twice its third term, 3a' + 6a6 + 3^>^ a method which will very much shorten the work in long arithmetical examples. 219. Ex. Extract the cube root of 5 to five pi decimals. of 5.000(1.70997 1 3 X 10' = 300 3 (10x7) = 210 7^ = _49 559 259 4 000 3 913 3 X 1700' = 8670000 3(1700x9)= 45900 9'= 81 8715981 45981 3x1709^ = 8762043 87 000 000 78 443 829 8 556 1710 7 885 8387 670 33230 613 34301 After the first two figures of the root are found, the next trial-di-vi- sor is obtained by bringing down the sum of the 210 and 49 obtaineci in completing the preceding divisor ; then adding the three lines con- nected by the brace, and annexing two ciphers to the result. The last two figures of the root are found by division. The rule m such cases is, that two less than the number of figures already obtained may be found without error by division, the divisor to be employed being three times the square of the part of the root already found. involution and evolution. 195 Exercise LXXXI. Find the cube roots of : 1. 274,625. 7. 1601.613. 13. 33,076.161. 2. 110,592. 8. 1,259,712. 14. 102,503.232. 3. 262,144. 9. 2.803221. 15. 820.025856. 4. 884.736. 10. 7,077,888. 16. 8653.002877. 5. 109,215,352. 11. 12.812904. 17. 1.371330631. 6. 1,481,544. 12. 56.623104. 18. 20,910.518875. 19. 91.398648466125. 20. 5.340104393239. 21. Find to four figures the cube roots of 2.5 ; .2 ; .01 ; 4 ; .4. 220. Since the fourth power is the square of the square, and the sixth power the square of the cube ; the fourth root IS the square root of the square root, and the sixth root is tne cube root of the square root. In like manner, the ••ighth, ninth, twelfth roots may be found. Exercise LXXXII. Find the fourth roots of: I. 81a*-54Oa33 + 135Oa2^2_i50Oa53_|_5255^ Find the sixth roots of : 3. 64 - 192a; + 240:^2 - 1600;^ + 60a:* - 12^^ + x\ 4. 729 a;« - 1458^^+ 1215a;* - 5400;^ -^I2>ba^ -ld>x-{-l. Find the eighth root of : 5. l-8y + 282/2-567/« + 70y*-562/'4-283/«-8/ + 3/'. CHAPTER XIV. Quadratic Equations. 221. An equation which, contains the square of the un- known quantity, but no higher power, is called a quadratic equation. 222. If the equation contain the square only, it is called a pure quadratic ; /but if it contain the first power also, it is called an affected quadratic. Pure Quadratic Equations. Solve the equation 5^ — 48 = 2a^. 5^2 — 48 = 2^ It will be observed that there are two roots of g^ —. ^g equal value but of opposite signs ; and there are 2 ___ 1 /^ only two, for if the square root of the equation, . . rc^ = 16, were written ± re = ± 4, there would be only two values of x ; since the equation — x = + 4 gives ic = — 4, and the equation — re = — 4 gives a; = 4. Hence, to solve a pure quadratic, Collect the unknown quantities on one side, and the Jcnown quantities on the other; divide hy the co-efficient of the un- known quantity ; and extract the square root of each side of the resulting equation. Solve the equation Srr^— 15 = 0. ^•^ ~ l'^ = ^ It will be observed that the square root of 5 3^7^ = 15 cannot be found exactly, but an approximate x^ =1^^ value of it to any assigned degree of accuracy . ^ __ _J-a/5 Diay be found. QUADRATIC EQUATIONS. 197 223. A root whicli is indicated, but which can be found only approximately, is called a Surd. Solve the equation 3^+ 15 = 0. t^:?r+iO = U It will be observed that the square root 3^:^^= — 15 of— 5 cannot be found even approximately ; o?- ^=. — 5 for the square of any number, positive or • ^ _- j.-^/ 5 negative, is positive. 224. A root which is indicated, but which cannot be found exactly or approximately, is imaginary. § 206. o 1 . Exercise LXXXIII. 1, :t^-3 = 46. 6. 5a;2-9 = 2a;2_|_24. 2. 2(a;2_i)_3(^_^l)^14^0. 7. (:r + 2)2 = 4^ + 5. 3 0?-^ 2:r^+l^l g 'J? 3?-Vd ^^ SO+or' 3 6 2' '5 15 25 * 4. ^^4 ^ 35^2_7 90 + 4£^^^ .3 17 ,. Q , 7 65^ '• 4^-67^=3- ''• '" + i = — 4a:'+5 2:r^-5 _ 7^^-25 10 15 20 ^,^ 10^;^ + 17 V2.x^-^2 _bx'-^ 18 ' ll:r^-8 9 ^3 14a;^ + 16 23;" + 8 _2a:^ 21 8a;^-ll 3* 14. x^ -{-hx -\- a^^hxiX — hx). 16. mx'^-\-n = q. 16. a:^ — aa;+i> = «^(^ — 1). 198 ALGEBRA. Affected Quadratic Equations. 225. Since (aa;i5)^ = a^a;^i2a52: + 5^ it is evident that the expression c^x^ dr 2 ahx lacks only the third tej^m, h^, of being a complete square. It will be seen that this third term is the square of the quotient obtained from dividing the second term hy twice the square root of the first term. 226. Ever J affected quadratic may be made to assume the form of a^x^ i 2 ahx = c. The first step in the solution of such an equation is to complete the square; that is, to add to each side the square of the quotient obtained from dividing the second term by twice the square root of the first term. The second step is to extract the square root of each side of the resulting equation. The third and last step is to reduce the resulting simple equation. (1) Solve the equation 16 a;* + 5 ^ — 3 = 7a;* — a: + 45. ]6a;* + 5ir - 3 = 7x* - a; + 45. Simplify, 9 a:* + 6 a; = 48. Complete the square, 9a:;* + 6a; + l = 49. Extract the root, 3 a; + 1 = ± 7. Reduce, 3a; = — 1 + 7 or — 1 — 7, 3 a: = 6 or - 8, .-. a; = 2 or - 2|. Verify by substituting 2 for x in the equation 16a;2 + 5a;-3 = 7a^-a; + 45, 16(2f + 5(2) - 3 = 7(2)2 _ (2) + 45, 64 + 10 - 3 = 28 - 2 + 45, 71 == 71. QUADRATIC EQUATIONS. 199 Verify by substituting — 2| for x in the equation 16x2 + 5x-3 = 7x^-x + i5, 16(-|)=^ + 5(-|)-3 = 7(-ff-(-|) + 45; ^f^ - ¥ - 3 = ^f^ + I + 45, 1024 - 120 - 27 = 448 + 24 + 405, 877 = 877. (2) Solve the equation So^-~4:X = S2. Since the exact root of 3, the coefficient of a?, cannot be found, it is necessary to multiply or divide each term of the equation by 3 to make the coefficient of x^ a square number. Multiply by 3, Ox^ _ i2a; = 96. Complete the square, Ox* — 12a; + 4= 100. Extract the root, 3 r — 2 = ± 10. Reduce, 3a; = 2 + lOor 2- 10 ; 3a; =12 or -8. /. a; = 4 or — 2%. Or, divide by 3, « 4x 32 Complete the square. x-»- 4a; 4 32 4 100 3 9 3 9^ 9 • Extract the root, 3 3 . ^ 2 ±10 3 ' = 4or-2§. Verify by substituting 4 for X in the original equation, 48 - 16 = 32, 32 = 32. i Verify by substituting — 2f for x in the original equation, 2H-(-10f) = 32, 32 = 32. 200 ALGEBRA. (3) Solve tlie equation — Sa^-}-5x==-~2. Since the even root of a negative number is impossible, it is necessary to change the sign of each term. The resulting equation is, 3ar^-5a; = 2. - Multiply by 3, 9a;2_i5a._6. Complete the square, 9x^-15.+ 25^49_ 4 4 Extract the root, 3.-1=4. Reduce, -=¥• 3a; = 6 or -1. .-. 0^ = 2 or -i. Or, divide by 3, o 5a; 2 3 3' Complete the square ^ 5a; 25__49 3 36 36" Extract the root, 6 6 6 = 2or-i If the equation 3a;* — 5a; = 2be multiplied hj four times the coeffi- cient of x^, fractions will be avoided : 36 a;' -60 a; = 24. Complete the square, 36a;2 - 60 a; + 25 = 49. Extract the root, 6 a; — 5 = ± 7, 6a; = 5±7, 6a; =12 or -2. .'. a;= 2 or — ^. It will be observed that the number added to complete the square by this last method is the square of the coefficient of x in the original equation 3 o;*^ — 5 a; = 2, QUADRATIC EQUATIONS. 201 3 1 (4) Solve the equation r = 2. 5 — a; 2x—5 Simplify (as in simple equations), 4x'-23a; = -30. Multiply by four times the coefficient of c^, and add to each side the square of the coefficient of x, 64^^ - ( ) + (23)2 = 529 - 480 = 49. Extract the root, 8 a; — 23 = ± 7. Eeduce, 8cc = 23±7; 8a; = 30 or 16. .-. a; = 3|or 2. If a trinomial be a perfect square, its root is found by taking the roots of the^rsi and third terms and connecting them by the sign of the middle term. It is not necessary, therefore, in completing the square, to write the middle term, but its place may be indicated as in this example. (5) Solve the equation 12a^—S0x = ~1. Since 72 = 2^ X 3', if the equation be multiplied by 2, the coeffi- cient of ar^ in the resulting equation, 144 x^ — 60 a; = — 14, will be a square number, and the term required to complete the square will be (If)' = (i)^ = -¥"■• Hence, if the original equation be multiplied by 4x2, the coefficient of x^ in the result will be a square number, and fractions will be avoided in the work. Multiply the given equation by 8, 576 ar^- 240 a; = -56. Complete the square, 576 a;^ — ( ) + 25 = — 31. Extract the root, 24 a; — 5 = iV— 31. Reduce, 24 a; = 5 ± V-31. /. a; = ^V(-5±V^31). Note. In solving the following equations, care must be taken to select the method best adapted to the example under consideration. Solve: KUibJ^ UJs^J^J^L V . 1. ^ + 4^7 = 12. 4. :ir2-7^==8. 7. a^-x = 0. 2. a^-6x = 16. 5. 3x''-4:x = 1. 8. 5x'~Sx=2. 3. a^~-12x-{-6 = h 6. 12:r2+:z;-l = 0. 9. 2x'~27x=U 202 ' ALGEBRA. 10. ^-2^ + 1 = 0. 13. ^-+i = 2^izi. 3 ^12 x + 4: x + 6 11. ^-^ = 2(a: + 2). 14; -^ ^+A_=,_A. 2 3 ^ ^ ^ x + 1 2(x-^4:) 18 12.3^ + ^ = ^. 15. -^=.-^ + -2_-. 4 3a; 6 x-1 x-2 x-A 16. 5a;(a;-3)-2(:r2-6) = (a; + 3)(:r + 4). 3a: 5_^ Sx" 23 * 2(a: + l) 8 x^-l 4:(x~l)' . 18. (x-2)(x--4:)~2(x-l)(x-S)==0. 19. l(a;-4)-?(a;-2) = i(2a; + 3). 7 o a; 20. -(^dar'-x-5)-l(x'-l) = 2(x-2y. 5 3 2a; 3a; -50 _ 12a;+70 • 15~^3(10 + a;) 190 ' 22 ^^ - 15 - 7a; ^^ ^ 14a;-9 _a:^-3 • ^2_i 8(l-a;)' ' 8a;-3 a; + r 2a:-l ■ 1^ 2a;-3 . a; + 5 ^ a;-6 a;-l "^6 a;-2' * 2a; + l a; - 2' 24. ^-i-^ = I. ^. ^ + 1^=2^V a; — 1 2a; 3 7~x x 2g 2.r + 3 7-a; ._ 7-3a.- 2(2a;-l) 2(a;+l) 4 -3a;* 12a;^-lla;^ + 10:^-78 _-., .1 ^- 8r^-7:. + 6 '^"^ 2- 30. _J^ 1^ = _! 8_ a;— 1 a; + 5 a;-|-l x~b QUADRATIC EQUATIONS. 203 227. Literal quadratic equations are solved as follows : (1) Solve the equation aa? ■i-hx = c. Multiply the equation by 4 a and add the square of 6, 4oV + ( )+b^ = i a c + b^. Extract the root, 2ax + b = ± V'4 ac + b^. Reduce, 2ax = — b± V4 ac + b*. — 6±V4ac + 6* .*. X =• 2a (2) Solve the equation adx — acoi? = hex — hd. Transpose bcx and change the signs, acx^ + bcx — adx = bd. Express the left member in two terms, ac3? + {be — ad) x='bd. Multiply by 4 ac, 4 a^c^x^ + 4 ac (6c — ac? ) a; = 4 abed. Complete the square, 4 aVa;2 + ( ) + (6c - a^ )2 = Z>V + 2 abed + a^^. Extract the root, 2 acx + {be — ad) = ± {be + ad). Reduce, 2 acx =- — {bc — ad)± {be + ad ) = 2 ac? or — 2 be. d b .'. a? = -or . c a pa (3) Solve the equation pot? —px + qx^ -\- qx— _r . Express the left member in two terms, {p + q)ar'-{p-q)x==^^. Multiply by four times the coefficient of a^, 4:{p + qfc^ — 4:{p^ — ^x = ipq. Complete the square, 4:{p + qf3i^-{ ) + {p-qf =p^ + 2pq + q^. Extract the root, 2 {p + q) x — {p — q) = ± {p + q). Reduce, 2 {p + q) x = (p — q) i: {p + q), = 2p or —2q. p q .•. x=- — ■ — or ; — . p+q p+q Note. The left-hand member of the equation when simplified must be expressed in two terms, simple or compound, one term con- taining a^, and the other term containing x. 204 ALGEBRA. c. 1 . Exercise LXXXV. 1. x^-i-2ax = 0?. 14. oi?-\-ax^^a-\-x. 2. x^^^^ax-\- 7(2^. 15. o(? -^ax — hx-\- ah. 3. ^=7»i'_3»^. 16. f + ? = ? + l 4 a X b X 4. ^_5«5_M = 0. 17. 1+ 1 1-1 2 2 xx-\-baa-\-b 5 _^^=:— i!_- 18 ^ + 5^_Z!=:0 ' (x-\-af {x-af ' 3"^ 4 3a 6. cx^ax' + bx'--^^. 19. ^+^ = a + ^Zl?. a + 6 x — ?> :r + 3 7. ^ + ^]^2a^. 20. m:.^-l==£Mz:^. 8. (a^ + 1) :r = a:^;^ + a. 21. (ao; — b) (bx — a) = c^. ♦q a , 5 2c aa;4-^ '^^ + ^ x — a x — b x~c bx-{-a nx-\-m 10. 1^=1+1 + 1. 23. ^^ + -J!^ = c. a-\-o-\-x a X m-^-x ra — x 11 1 1 _3+^ (^-iy^2^2(3a-l)a; _.. a — rr a-\-x a^ — xr 4 a— 1 ^2^ g!±gaHa-+^^)^2. 25. (^^-^^)(^+l)^2.. a^+^2 ■ a^ + S^ 2a;-a+25 {m-\-nf ^ 27 ^ ■ a-b ^ \^a^-^ab-\^b'' {2a~Zb)x ' "^ a^>2 18 a^^^ "*■ 2a5 * QUADRATIC EQUATIONS. 205 c & c ' 29. ^ 2>m — 2a 4:a — 6m 2 30. Q:c + (^±^=b(a-b) + ^^. 31. i(x' + a^ + ah) = ix(20a + 4:b). 32. x^ — (b — a) c = ax — bx-\- ex. 33. x^~2mx~ (n—p-\-m)(n~p~7n). 34. x^ — (m -}- n) X = ^ (p + q -{- m + n) (p -i- q ~ m ~ n). 35. wna;^ — (m + w) ('^w -{- 1) x -}- (m -j- ny = 0. 2^— rr — 2a . 4-^ — 7a _ a: — 4a bx ax — bx ab —b^ 37. 2x'(a'-b') - (Sa' + b') (x-l) = (SP + a') (x+l). „^ a — 2b — x 6b — X , 2a — x — 19b 39. 0^2 _ 4 ^2 aa; + 2 5^7 2 5a; — aa; x + lSa + Sb ^_ a — 2b ha-Zb-x x-\-2b' = 0. * ' 8a2-12a5 9Z^2_4^2 (2a + 35)(a;-35) 41. na;^ +J9a; — ^:^;^ — -ma; + '??^ — w = 0. 42. (a + & + c)a;2_(2ct-f-6 + c)a; + a = 0. 43. {ax — b){c — d)=^{a — b) (ex — d) x. 44. 2r.- + l 1/1 2\ 3a;+l 5 x\b aj a ^g 1 I 1 _. a 2bx-{- b 2x'-\-x-l 2x'-2>x-\-l 2bx-b ax^-a' 206 ALGEBRA. 228. An affected quadratic may be reduced to the form a^-i-px-\- q = 0, in which p and q represent any numbers, positive or negative, integral or fractional. Ex. Solve : x^ -{-px-]- q = 0. 2x-\-p = ± Vp^ — 45', 1 By this formula, the yalues of x in an equation of the form x^ +px + g- = 0, may be written at once. Thus, take the equation Divide by 3, ar^-fa; + | = 0. Plere, p = - f , and g' - |. = f ± i. = lori 229. A quadratic which has been reduced to its simplest form, and has all its terms written on one side, may often have that side resolved by inspection into factors. In this case, the roots are seen at once without com- pleting the square. (1) Solve ^ + 7:?;- 60 = 0. Since or^ + 7a; - 60 = (a; + 12) (a: - 5), the equation ar' + 7a; — 60 = may be written (a; + 12) (a; — 5) = 0. It will be observed that H either of the factors a; + 12 or a; — 5 is 0, \i}a.e product of the two factors is 0, and the equation is satisfied. Hence, x + 12 = and x — 5 = 0. :.x=-. — 12, and a; = 5. f QUADKATIC EQUATIONS. 207 (2) Solve ^+ 7a; = 0. The equation a^ + 7a; =« becomes a; (a; + 7) — 0, and is satisfied if x = 0, or if a; + 7 =• 0. .*. the roots are and — 7. It will be observed that this method is easily applied to an equation all the terms of which contain x. (3) Solve 2r'-;r2- 6a; = 0. The equation 2a;3_ajf_g2.„0 becomes x (2 a;* — a; — 6) = 0, and is satisfied if s; = 0, or if 2 a;' — a; — 6 = 0. By solving 2a^ — x— 6 = the two roots 2 and — f are found. .•. the equation has three roots, 0, 2, — f . (4) Solve a;« + a;2-4a:-4 = 0. The equation a^ + x* — 4a: — 4 = becomes ar^ (a; + 1) — 4 (x + 1) = 0, (x»-4)(a; + l) = 0. .*. the roots of the equation are — 1, 2, — 2. (5) Solve a;«-2a;2_i;L^^12=0. Since 2^Z^2^z:Lli^±i2^^_^_12, X —1 the equation a;'-2a;* — lla; + 12 = may be written {x — l){a^ — x ~ 12) = 0. The three roots are found to be 1, — 3, 4. An equation which cannot be resolved into factors by inspection may sometimes be solved by guessing at a root and reducing by divi- sion. In this case, if a denote the root, the given equation (all the terms of the equation being written on one side), may be divided by a. 208 ALGEBRA. Exercise LXXXVI. Find the roots of : 1. (xi-l)(x-2)(x^+x~2)=0. 7. x'~a^-x + l = 0. 2. (x'-Sx+2)(x'-x-12)=0. 8. 8a^-l=:0. 3. (^+l)(^-2)(a;+3) = -6. 9. Src^ + l^O. 4. 2.a^ + 4a^~10x = 0. 10. :r«-l = 0. 5. (x''-x-~6)(x'-x~20) = 0. 11. x(x-a){x''-h')^0. 6. x(xi-l)(x+2)=(a+2)(ai-l)a. 12. w(a:2+l)+:r + l = 0. 230. If r and / represent two values of x, then a; — r = 0, and . a: — / = 0, .*. (a; — r) (a; — 7^') = 0. This is a quadratic equation, as may be seen by performing the indicated multiplication. Now r and r' are roots of this equation ; for, if either r or r' be written for x, one of the factors, x — r, x~r', is equal to 0, and the equation is satisfied. Also r and r' are the only roots, for no value of a;, except r and r', can make either of these factors equal to 0. Since r and r' may represent the values of x in any quadratic equation, it follows that every quadratic equation has two roots, and only two. Again, if r, r', r", represent three values of x, then, {x -r){x- r') {x - r") = 0. This is a cubic equation, as may be seen by performing the indi- cated multiplication. Hence, it may be inferred that a cubic equa- tion has three roots, and only three; and so, for any equation, that the number of roots is equal to the degree of the equation. It may also be inferred that if r be a root of an equation, x — r will he a factor of the equation when the equation is written with all its terms on one side. QUADRATIC EQUATIONS. 209 If r and r' represent the roots of the general quadratic equa- tion, a;2+pa; + 2 = 0. This equation may be written (a; — r ) (x - r^) = 0, or, x'- — (r -f- r'') a; + rr^ = 0. A form which shows that the sum of the roots = — p, and the product of the roots = q. 231. It will be seen from § 230 that an equation may be formed if its roots be known. If the roots of an equation be — 1 and ^, the equation will be (x + 1) (a; — ^) = 0, or, a;2 + ^-l = 0, 4 4 or, by multiplying by 4, 4a;2 -h 3 a; — 1 = 0. If the roots of an equation be 0, 1, 5, (the equation will be (a; — 0) (a; — 1) (a; — 5) = ; that is, x{x—l){x — 5) = 0, or, a^ -Qx^ + 5x = 0. If x occur in every term, the equation will be satisfied by putting X = 0, and may be reduced to an equation of the next lower degree by dividing every term by x. I 232. By considering the roots of x"^ -{- px -{- q = 0, namely, r = —■- + ^ V^^ ~^9, and / = — ^ — 2 Vp'^ ~4:q, it will be seen that the character of the roots of an equation may be determined without solving it : I. As the two roots have the same expression, Vjt?^ — 4:q, both roots will be real, or both will be imaginary. If both be real, both will be rational or both surds, accord- ing as^^ — 4^ is or is not a perfect square. 210 ALGEBRA. II. When ]p^ is greater than 4 q, the two roots will be real, for then the expression p^ — 4:q is positive, and therefore Vp^ — ^q can be found exactly or approximately. Since also its value in one root is to be added to — -^j and in the other to be subtracted from ~-~, the two roots will be different in value. III. When p^ 'is equal to 4 q, the roots will be equal in value. IV. When p^ is less than 4 q, the roots will be imaginary, for then the expression j9^— 4 g" will be negative, and therefore ■y/p^ — 4:q represents the even root of a negative number, and is imaginary. V. If 2' (— ■^ X /) be positive, the roots, if real, will have the same sign, but opposite to that of^ (since r-{-r' ==~p). But if q be negative, the roots will have opposite signs. 233. Determine by inspection the character of the roots of: (1) oc'-bx + ^^O. In this equation j3 is — 5, and q is 6. .-. \//-4^=V25-24 = 1. .*. the roots will be rational, and both positive. In this equation, jo is 3, and q is 1. ... Vp^~'i:q=V9^=^VE. .'. the roots will be surds, and both negative. In this equation p is 3, and q is 4. ... V/-4g=\/£nri6=,>/r7. .'. the roots will be impossible. quadratic equations. 211 Exercise LXXXVII. Form the equations whose roots are : 1. 2.1. 6. -5, -f 9. 0,-i,f.-l. 2. 7,-3. 6. — -J, f 10. a-25, 3a-f2^>. 3. |, J. 7. 3, -3, f, -f. 11. 2a-^, ^-3a. 4. I, -f. 8. 0,1,2,3. 12. a(a+l), 1-a. Determine by inspection the character of the roots of : 13. a;'-7a; + 12=0. 17. ^' + 4a;+l = 0. 14. a;'^-7a;-30 = 0. 18. a:''-2a; + 9 = 0. 15. x'i-4:x-b = 0. 19. 3a:2-4:r-4 = 0. 16. 5a;' + 8=-0. 20. a;' + 4a; + 4 = 0. 234. It is often useful to determine the maximum or min- imum value of a given quadratic expression. (1) Find the maximum or minimum value of 1 -{- x — x^. Let 1 + X — x"^ = m; then, x^ — X == 1 — m, and 4 x-2 - ( ) + 1 = 5 - 4w, 2.r — 1 =-= ±\/5 — 4 m. .'. a; = I ± ^ V5 — 4 m. Now, for all possible values of a, 5 — 4 m cannot be negative ; that IS, m cannot be greater than f ; and for this value x is |. Therefore, I is the maximum value of the given expression. ^2) Find the maximum or minimum value oi x^ -}-Sx -\- 4. Let x^ + 3x + 4: = m; then, x^ + 3 X = m — 4, and, 4a;2+( ) + 9 = 4m-7, 2a; + 3 = +V4 m X = — ^V4m-7. For all possible values of a;, 4m — 7 cannot be negative ; that is, m cannot be less than ^ ; and for this value a; = — |. Therefore, ^ is the minimum value of the given expression. 212 ALGEBRA. Exercise LXXXVIII. Find tlie maximuin or minimum value (and determine wHcli) of: 1. 4: + 6x-a^. 4. (a-x){x-h). 7. a^-2x + 9. 2^ (^+a/_ g^ X ^ ^ X ' ' 1+^* (x-{-a)(x—b) 3. ^+^ . 6. ^ + 8^7+20. 9. -^. 10. Divide a line 20 in. long into two parts so that tlie sum of the squares on these two parts may be the least 11. Divide a line 20 in. long into two parts so that the rect- angle contained by the parts may be the greatest possible, 12. Find the fraction which has the greatest excess over its square. 235. Two other cases of the solution of equations 5y covi- pleting the square should be noticed. I. When any two powers of x are involved, one of which is the square of the other. II. When the addition of a numher to an equation of the fourth degree will mahe both sides complete squares. (1) Solve 8a;« + 63a;' = 8. In this equation the exponent 6 is the double of 3, hence o^ is the square of a;^. 8 a;« + 63 a.-^ = 8, 256a;6+() + (63)2 = 4225, 16a;3 + 63 = ±65, 16 :r3 =. 2, or - 1 28, x^ = \, or - 8. By taking cube root, x = J, or — 2. QUADRATIC EQUATIONS. 213 The other roots of the equation are found by finding the remain- ing roots of the equations, a? =^ I, and a;^ = — 8. Since, x^ = i^ .-. 8 a;^ - 1 = Now, by I 230, 8 a^ - 1 = (2rK - 1) (4 x2 + 2a: + 1) .•.(2a:-l)(4a;2 + 2a; + 1) = and is satisfied if4a;2 + 2a; + l = as well as if 2 a; — 1 = The solution of 4a;2 + 2 a; + 1 = gives a; = i (— 1 ± V^). Since, a^ = - 8, .-.o^ + % = Q Now, by I 230, a:» + 8 = (a; + 2) (a;2 - 2 a; + 4) .•.(a: + 2)(x2_2a; + 4) = and is satisfied if a^ — 2a; — 4 = as well as if a; + 2 = The solution ofar^-2a; + 4 = gives x=\ ±V— 3. '. the roots are |, - 2, 1 ± V^, ^(—1 ± V— 3). (2) Solve x' - lOa^ + 35^^ - 50a; + 24-0. Take the square root of the left side. ^x'-bx - 10 x' + 35a;2 _ 5q^ .^ 24|£^_-5^+_5 -10:^ -^bx-" 2x' -lOx + b 10^^- 60a; + 24 10a;2-50a; + 25 - 1 It is now seen that if 1 were added, the square would be complete and the equation would be x' - 10 x^ + 35a;2 - bOx + 25 = 1. Extract the square root, and the result is. a;2 — 5a; + 5 = ±l. That is. 5a; 4, or - 6, 4a;2-() + 25 = 9, or 1, 2a;-5 = ±3, or ± 1, 2.x = 8, 2, 6, or 4. .-.a; -4, 1, 3, or 2. 214 ALGEBRA. Exercise LXXXIX. Find the possible roots of: 1. x''+73^ = 8. 8. (a^-9y = S-{-ll(3^--2). 2. X*-53r^ + 4: = 0. 9. X^ +14:0^ + 24: = 0. 3. S1x^-~9==4:x\ 10. 19x^ + 216x'^ = x. 4. 16a;»=17a;*-l. 11. a:« + 22^* + 21 =- 0. 5. S2x^^-SSx^i-l = 0. 12. a;2'« + 3a;'"-4 = 0. 6. (:r2-2)2=i(:r2+12). 13. 4x'-20x^+2^ar'-i-5x=6. 7. ^4«__5^_25^Q ^^^ 1 4.1_20 = 0. 3 12 ctr"^ x^ 15. x*-4:S^-10a^ + 2Sx-15==0. 16. :r^-2a;3-13a;24-14:r = -24. 17. 108a:^ = 20a;(9:r2-l)-51:r2+7. 18. (a;2_i)(^__2) + (:i-2_3)(^_4^)^^^5_ Problems Involving Quadratics. 236. Problems which involve quadratic equations have apparently two solutions, as a quadratic has two roots. Sometimes both will be solutions ; but generally one only will be a solution, and the other be inconsistent with the conditions of the problem. No difficulty will be found in selecting the result which belongs to the problem, and sometimes a change may be made in the statement of a problem so as to form a new problem corresponding to the solution which was inapplicable to the original problem. / QUADRATIC EQUATIONS. 215 (1) The sum of the squares of two consecutive numbers is 481. Find the numbers. Let X = one number, and a; + 1 = the other. Then a^ + (a; + If = 481, or 23? -v2x + l== 481. The solution of which gives, x = 15, or — 16. The positive root 15 gives for the numbers, 15 and 16. The negative root — 16 is inapplicable to the problem, as consecu- tive numbers are understood to be integers which follow one another in the common scale, 1, 2, 3, 4 (2) "What is the price of eggs per dozen when 2 more in a shilling's worth lowers the price 1 penny per dozen ? Let X = number of eggs for a shilling. Then, - = cost of 1 egg in shillings, 12 and — = cost of 1 dozen in shillings. But, if X + 2 = number of eggs for a shilling, 12 cost of 1 dozen in shillings. a; + 2 12 12 — (1 penny being ^ of & shilling). X x + 2 12 The solution of which gives x = 16, or — 18. And, if 16 eggs cost a shilling, 1 dozen will cost ^| of a shilling, or 9 pence. Therefore, the price of the eggs is 9 pence per dozen. If the problem be changed so as to read : What is the price of eggs per dozen when two less in a shilling's worth raises the price 1 penny per dozen ? the algebraic statement Will be _1^_12^^ x~2 X "^12' The solution of which gives x = 18, or — 16. Hence, the number 18, which had a negative sign and was inappli- cable in the original problem, is here the true result. 216 ALGEBRA. Exercise XC. 1. The sum of the squares of three consecutive numbers is 365. Find the numbers. 2. Three times the product of two consecutive numbers exceeds four times their sum by 8. Find the numbers. 3. The product of three consecutive numbers is equal to three times the middle number. Find the numbers. 4. A boy bought a number of apples for 16 cents. Had he bought 4 more for the same money he would have paid i of a cent less for each apple. How many did he buy ? 5. For building 108 rods of stone- wall, 6 days less would have been required if 3 rods more a day had been built. How many rods a day were built ? 6. A merchant bought some pieces of silk for $900. Had he bought 3 pieces more for the same money he would have paid $ 15 less for each piece. How many did he buy ? 7. A merchant bought some pieces of cloth for $168.75. He sold the cloth for $12 a piece and gained as much as 1 piece cost him. How much did he pay for each piece ? 8. Find the price of eggs per score when 10 more in 62 i cents' worth lowers the price 31 i cents per hundred. 9. The area of a square may be doubled by increasing its length by 6 inches and its breadth by 4 inches. De- termine its side. 10. The length of a rectangular field exceeds the breadth by 1 yard, and the area is 3 acres. Determine its dimensions. QUADRATIC EQUATIONS. 217 11. There are three lines of which two are each f of the third, and the sum of the squares described on them is equal to a square yard. Determine the lengths of the lines in inches. 12. A grass plot 9 yards long and 6 yards broad has a path round it. The area of the path is equal to that of the plot. Determine the width of the path. 13. Find the radius of a circle the area of which would be doubled by increasing its radius by 1 inch. 14. Divide a line 20 inches long into two parts so that the rectangle contained by the whole and one part may be equal to the square on the other part. 15. A can do some work in 9 hours less time than B can do it, and together they can do it in 20 hours. How long will it take each alone to do it ? 16. A vessel which has two pipes can be filled in 2 hours less time by one than by the other, and by both to- gether in 2 hours 55 minutes. How long will it take each pipe alone to fill the vessel ? 17. A vessel which has two pipes can be filled in 2 hours less time by one than by the other, and by both to- gether in 1 hour 52 minutes 30 seconds. How long will it take each pipe alone to fill the vessel ? 18. An iron bar weighs 36 pounds. If it had been 1 foot longer each foot would have weighed ^ a pound less. Find the length and the weight per foot. 19. A number is expressed by two digits, the second of which is the square of the other, and when 54 is added its digits are interchanged. Find the number. 20. Divide 35 into two parts so that the sum of the two fractions formed by dividing each part by the other may be 2^^. 218 ALGEBRA. 21. A boat's crew row 3^ miles down a river and back again in 1 hour 40 minutes. If the current of the river is 2 miles per hour, determine their rate of row- ing in still water. 22. A detachment from an army was marching in regular column with 5 men more in depth than in front. On approaching the enemy the front was increased by 845 men, and the whole was thus drawn up in 5 lines. Find the number of men. ^. A jockey sold a horse for $ 144, and gained as much per cent as the horse cost. What did the horse cost ? 24. A merchant expended a certain sum of money in goods, which he sold again for $ 24, and lost as much per cent as the goods cost him. How much did he pay for the goods ? 25. A broker bought a number of bank shares ($100 each), when they were at a certain per cent discount, for f 7600 ; and afterwards when they were at the same per cent premium, sold all but 60 for $5000. How many shares did he buy, and at what price ? 26. The thickness of a rectangular solid is I of its width, and its length is equal to the sum of its width and thickness ; also, the number of cubic yards in its vol- ume added to the number of linear yards in its edges is I" of the number of square yards in its surface. Determine its dimensions. 27. If a carriage-wheel 16 J feet round took 1 second more to revolve, the rate of the carriage per hour would be 1 J miles less. At what rate is the carriage travelling ? CHAPTER XV. Simultaneous Quadratic Equations. 237. Quadratic equations involving two unknown quan- tities require different methods for their solution, according to the /orm of the equations. 238. Case I. When from one of the equations the value of one of the unknown quantities can be found in terms of the other, and this value substituted in the other equation. Ex. Solve: 3:^-2.y = 5| (1) x-y==2 f (2) Transpose x in (2), y = x — 2. Substitute in (1), 3 a;* - 2 x (x - 2) = 5. The solution of which gives a; = 1 or — 5. .•.y = -lor-7. Special methods often give more elegant solutions of examples than the general method by substitution. I. When equations have the form, x ± y = a, and xy = b ; x* ± 3/* = a, and xy => b ; or, x ± y = a, and a? + y^ =^ b. (1) Solve- ^ + y = 40| (1) Square (1), ar^ + 2 xy + 3/* = 1600. (3) Multiply (2) by 4, 4xy = 1200. (4) Subtract (4) from (3), x^ - 2 xy + y« = 400. Extract root of each side. x — y = ± 20. (6) Add (1) and (6), 2x = 60or20, .-. x = 30 or 10. Subtract (6) from (1), 2y = 20 or 60, .•.y = 10 or 30. 220 ALGEBRA. (2) Solve: 5;-y = 4 \ (1) a^ + f = iO) (2) Square (1), a^-2xy +y^=16. (3) Subtract (2) from (3), - 2 «y = - 24. (4) Subtract (4) from (2), s^ + 2xy +y^==6i. Extract the root, a; + y = ± 8. (5) By combining (5) and (1), cc = 6 or — 2. y = 2or-6. 1+1 = 1 x^y 20 C3) Solve: ^ 1 _ 41 S"^^~400> (1) (2) Square (1), A + - + -, = :^- (3) ^ ^ ^' 3? xy y^ 400 ^ ^ Subtract (2) from (3), 1 = i5.. (4) icy 400 Subtract (4) from (2), i - A + i = -L. ^ ^ ^ ^' a? xy y^ 400 Extract the root, = ± t^t.- (5) a? y 20 By combining (1) and (5), « = 4 or 5. y = 5 or 4. II. When one equation may he simplified by dividing it by the other. ■Ave: T fy C (1) (2) Divide (1) by (2), x^-xy + y^== 13. (3) Square (2), x" +2xy +f== ^d. (4) Subtract (3) from (4), 3 xy = 36. Divide by -3, -a;y = -12. (5) Add (5) and (3), (t^~2xy +f==l. Extract the root, a; — y = ± 1. (6) By combining (6) and (2), a; = 4 or 3. y = 3 or 4. SIMULTANEOUS QUADRATIC EQUATIONS. 221 Solve : 1. X + 7/ = lS') X2/ = S6 ) Exercise XCI. 11. a;4-y = 49 | 2. a: + y = 291 x^ = 100 j 12. c(^ + f = Ml^ x-\-y = \l J 3. a;-y=19| ^y = 66 / 13. 0:3 + 2/3^10081 a; + y=:12 J 4. a: — y = 45 I .^y = 250 J 14. a;3_2^ = 98 a;-y 3 = 98 I = 2 J 5. a; — y=10 1 15. a;3_2/J = 279 6. a; — y = 14 "I a;2_|_ 2^ = 436/ 16. a;-3y = l| ^y + y^ = si 7. a; + y = 12 1 a;2-fy2=104i 17. 4y = 5:r + l 1 2a;y = 33-a,^j a; y i + l = A a;2"^y2 16 J a 1+1=5 a; y ■ 10. 7ar^-8a:y=:159| 5a; + 2y = 7 J 18. X B X y 1 1 = 21 19. 1 X 1_ 2i 1 a;2" 1 = 81 20. ar^-2a:y-y2=lj 222 ALGEBRA. 239. Case IL When each of the two equations is homo- geneous and of the second degree. Ex Solve- 2y^- 4x2/ + 3^ =171 « Jix. boive. y2_^=i6 / (2) Let y = vx, and substitute vx for y in both equations. From (1), 2vV - 4va^ + Sa;^ - 17, . ., 17 ••^ -2v^ -4v + 3 From (2), v'x' -x^ = 16, Equate the values oi 17 16 22;2-4v + 3 r'-l' 32^2-64^+48 = 17^2-17, 15t;«-64v = -65. The solution gives, v = ! or !;'. Substitute the value of v in 16 v^ then, / a» = 9 or — , y /. a: = ± 3 or ± -, 3 J ;^ 13 and 1/ = va; = ± 5 or ± ~. „ 1 Exercise XCII. Solve : 2x' + 2xi/ + y'-=13i xg + 2y'-^ = 0) 2. a^ + xg-^Ay^ = Q\ 5. r^-xy- 35 = ) 3r^ + 8y2=14 J rry + y'-lS rr2-a:y+y2=:3 2n 6. a^ + xy + 2y' = U f-2xy = -lb J 2a;'-a;y + y2 = 16 SIMULTANEOUS QUADRATIC EQUATIONS. 223 7. a^ + x7/-15 = 0^ 9. 2ar' + Sx^ + y' = 10') = J Qx' + xv-i/^bO J x^ — xy -{- 7/^ = 7 1 10. a.-^ — rry — 3/^ = { 35r^+13:ry + 8y2 = 162J 2 a;^ _|_ 3 ^^ _|_ ^ 240. Case III. When the two equations are symmetrical with respect to x and y ; that is, when they have x and y similarly involved in them. Thus, the expressions 2 a:^ + 3 ar'y' 4- 2 y^, 2xy — Zx — Zy i-l, ar* — 3x^2/ — 3 a;3/^ + y* are symmetrical expressions. (1) Solve: :^ + ^ = 18.y1 (1) a: + y = 12 J (2) Put M + V for X, and w — v for y, in (1) and (2). (1) becomes (w + vf + (w - vf - 18 (w + v) (m - v), or V? + Zw^ = ^ {u* ~ 'i^). (3) (2) becomes (w + y) -f (w — v) = 12, or 2w = 12, .-. M = 6. Substitute 6 for u in (3). (3) becomes whence. 216 + 18v« = 9 (36 - 1^, r2 = 4, .•.v = ±2, .-. x = w + V = 6 ± 2 = 8 or 4, and y = u-v = 6T2 = 4or8. Solve : a;4-y = 8 ) ^' + y' = 706i (1) (2) Put w + V for (1) becomes a:, and w - -y for y, in (1) and (2). {u +v) +{u-v) = 8, .-. w = 4. (2) becomes u* + 6uV + 1;* = 353. (3) Substitute 4 for u in (3), 256 + 96 V* + r* = 353, or, t;4 + 96v« = 97. (4) The solution of (4) giyes v = ± 1 or ± v'^^~97. Taking the possible values of v, a; = 5 or 3, and y = 3 or 5. 224 ALGEBRA. Q 1 Exercise XOIII. bolve : 1. 4^y = 96-ar^y2| 4^ 4:{x + y) = 2>xy 1 a: + y = 6 J a: + y + ^ + 2/' = 26J 2. a?-\-f--=l^-x-y^ 5. 4:r2 + a:y 4-42/2=z 58 \ xy-=Q J 5a;2_f_5^^e5 3. 2(:p2_|_^>)^5^^-| Q xy{x-\-y)=^2>0 4:{x — y)=xy ) rc^ + y^ =^ 35 241. The preceding cases are general methods for the solution of equations which belong to the kinds, referred to ; often, however, in the solution of these and other kinds of simultaneous equations in- volving quadratics, a little ingenuity will suggest some step by which the roots may easily be found. c, , . Exercise XCIV. 1. x-y=l \ 8. x-y^l 1 a?-\-xy-^f = lZ\ :r2 + y2 = 8^J 2. ar^ + :ry = 351 9. a? -\-4:xy=^2> ) xy-f = ^ ] 4:xy + y^=2i) 3. a:y-12 = 0| 10. x'-xy + f = ^S^ x-2y = 5 ) x--y — 8 = ) 4. ^y_7 = \ 11. ar' + Bxy + y'=l 1 x' + y^^bO) Sx' + xy + Sy'^lSy 5. 2a;-5y = 9 ) 12. a^ -2xy + Sy^ ^li') a^-xy-{-f=-7) o?-\-xy-f^\. J 6. x-y = ^ I 13. a; + y = a ) a:y + 8 = 0i 4:r2/-a2 = -4^)2j 7. 5:c — 7y = \ \\, x — y=\ \ 5^_13^^4_7^ ^ + ^ = 2i 4 ) y X ) SIMULTANEOUS QUADRATIC EQUATIONS. 225 15. x^ + 9x2/ = M0) 16. /+y = 6 1 V7. 3:ry + 2a; + y = 485| 3a:-2y = i 18. rr — y = 1 1 a;3-y3 = 19i 19. ^•3 + 3/^ = 2728 -) r^-a:y + y2 = 124J 20 21. :i^-f = 3:i-2-4a:y + 5y2==9 22. 23. a; — y :r + y 3 ar» + y2 = 45 ^ y 1 17 x-{-l y+1 12 24. :r^ — a:y + y^ = 7 a;* + ^y2 + y* = 133 25. a; + y = 4 :r* + y* = 82 a J 26. :i--3 — y3 = a^ x-^j 27. a--^ — a:y = a^ + Z>^ ^y — y^ = 2 a^ 28. ^-y2 = 4aZ>| xy = d^ — h^ i 29. a:y = 1 ar' + y2=16j 30. a:2 + 5:y + y2 = 37 1 .'C* + :ir2^ + y^ = 48lJ 31. a;^ = aa; + Jy ~l i/^ = ay-\-bxi 32. a; — y — 2 = | 15(:r2-y2) = 16a:yJ 33. 34. rr + y a; 40 x — y x-{-y 6a: = 20y + 9 a . h X y 35. x'-^f^l + xy I «} rr^ + y^ = 6 a7y — 36. a;^-y^ = 3093| x-y=Z ] 37. f(a:-l)-|(^ + l)(y-l) = i(y + 2) = i(a; + 2) 38. l0x^-\-\bxy = ^ah-2aJ'\ 10/+15a7 = 3a6-262J "} 226 ALGEBRA. Exercise XCV. 1. If the length and breadth of a rectangle were each in- creased by 1, the area would be 48 ; if they were each diminished by 1, the area would be 24. Find the length and breadth. 2. The sum of the squares of the two digits of a number is 25, and the product of the digits is 12. Find the number. 3. The sum, the product, and the difference of the squares, of two numbers are all equal. Find the numbers. Note. Represent the numbers hy x +y and x—y, respectively. 4. The difference of two numbers is f of the greater, and the sum of their squares is 356. What are the num- bers? 5. The numerator and denominator of one fraction are each greater by 1 than those of another, and the sum of the two fractions is ly^^ ; if the numerators were in- terchanged the sum of the fractions would be 1|-. Find the fractions. 6. A man starts from the foot of a mountain to walk to its summit. His rate of walking during the second half of the distance is J mile per hour less than his rate during the first half, and he reaches the summit in bi hours. He descends in 3 1 hours, by walking 1 mile more per hour than during the first half of the ascent. Find the distance to the top and the rates of walking. Note. Let 2 a; = the distance, and y miles per hour == the rate at first. Then ^+—^ = 5* hours, and -^ = 3| hours. y y-J y+1 ^ SIMULTANEOUS QUADRATIC EQUATIONS. 227 7. The sum of two numbers which are formed by the same two digits in reverse order is -f f of their difference ; and the difference of the squares of the numbers is 3960. Determine the numbers. 8. The hypotenuse of a right triangle is 20, and the area of the triangle is 96. Determine the sides. Note. The square on the hypotenuse = sum of the squares on the sides ; and the aroa of a right triangle = ^ product of sides. 9. Two boys run in opposite directions round a rectangular field the area of which is an acre ; they start from one corner and meet 13 yards from the opposite cor- ner ; and the rate of one is -f of the rate of the other. Determine the dimensions of the field. 10. A, in running a race with B, to a post and back, met him 10 yards from the post. To make it a dead heat, B must have increased his rate from this point 41-f- yards per minute ; and if, without changing his pace, he had turned back on meeting A, he would have come 4 seconds after him. How far was it to the post ? 11. The fore wheel of a carriage turns in a mile 132 times more than the hind wheel ; but if the circumferences were each increased by 2 feet, it would turn only 88 times more. Find the circumference of each. \ 12. A person has $6500, which he divides into two parts and loans at diffen^ent rates of interest, so that the two parts produce equal returns. If the first part had been loaned at the second rate of interest, it would have produced $ 180 ; and if the second part had been loaned at the first rate of interest, it would have pro- duced % 245. Find the rates of interest. CHAPTER XVI. Simple Indeterminate Equations. 242. If a single equation be given whicli contains two unknown quantities, and no other condition be imposed, the number of its solutions is unlimited; for, if any value be assigned to one of the unknown quantities, a corresponding value may be found for the other. Such an equation is said to be indeterminate. 243. The values of the unknown quantities in an inde- terminate equation are dependent upon each other ; so that, though they are unlimited in number, they are confined to a particular range. This range may be still further limited by requiring these values to satisfy some given condition ; as, for instance, that they shall be positive integers. 244. The method of solving an indeterminate equation in positive integers is as follows : (1) Solve 3^; + 4y = 22, in positive integers. Transpose, 3x = 22 — 4y, ... H .. , 1-v " 3 ' the quotient being written as a mixed expression. ... a; + 2/ - 7 = i^. Since the values of x and y are to be integral, a; + y — 7 will be integral, and hence, -^ will be integral, though written in the form of a fraction. Let ~ y = m, an integer ; SIMPLE INDETERMINATE EQUATIONS. 229 Then 1-2/ = 3m, .\y =\ — 3m. Substitute this value of y in the original equation, 3a; + 4- 12w = 22, .-. a; = 6 + 4m. The equation y = 1 — 3 m shows that m in respect to y may be 0, or have any negative value, but cannot have a positive value. The equation a; = 6 + 4m shows that m in respect to x may be 0, but cannot have a negative value greater than 1. .'. m may be or — 1, and then a; = 6, 3/ = 1, or a; = 2, y = 4. (2) Solve 5a; — 14y=ll, in positive integers. Transpose, 5 a; = 11 + 14y, ,.._2y-2 = liii, .•. — — — 2l must be integral. o Now, if — -— ^ be put = m, then y = ^~ , a fraction in form. 5 4 To avoid this difficulty, it is necessary in some way to make the coefficient of y equal to unity. Since — - — ^ is integral, any multiple of — ^'^ — ^ is integral. Multiply, then, by such a number as 5 will make the coefficient of y greater by 1 than some multiple of the denominator. In this case, multiply by 4. Then i+A^U. or 32/ + ^4-^ is integral 5 5 .'. — i-2^ = m, an integer ; 5 .•. y = 5m — 4. Since x = ^ (11 + 142/), from the original equation, .-. X = 14m — 9. Here it is obvious that m may have any positive value, and a; = 5, 19, 33 y = 1, 6, 11 230 ALGEBRA. The required multiplier can always be found when the coefficients are prime to each other, and it is best to divide the original equation by the smaller of the two coefficients, in order to have the multiplier as small as possible. 245. The necessity for a multiplier may often be obviated by a little ingenuity. Thus, The equation 43/ = 29 — 7a; may be put in the form of 4y =- 29 -Sx + X, .■.y^7-2x + 'L±^, in which the fraction is of the required form. The equation 5a; = 18 + 133/ gives a; = 3 + 2y + ^^^ ^ ^\ 1 + V ^ in which - ^^^-^ is of the required form. 5 246. It will be seen from (1) and (2) that when only pos- itive integers are required, the number of solutions will be limited or unlimited according as the sign connecting x and 2/ is positive or negative. [3) Find the least number that when divided by 14 and 5 will give remainders 1 and 3 respectively. If N represent the number, then Let Ifm = l, N-l 14 ^.,and^-'^=.,. 5 N= Ux + 1, and N=5y + 3, .-. 14 X + 1 = 5y + 3. 5y = 14a; -2, 5y = 15x- 2-a;, 2 + a; 5 m, an integer ; ,'. X = 5m -2. y = |^(14a; — 2), from original equation, ''y = 14m - 6. X = 3, and y = S, .-. N = 14.r + 1 = 53/ + 3 = 43. Ans. SIMPLE INDETERMINATE EQUATIONS. 231 (4) Solve 5a; + 6y = 30, so that x may be a multiple of y, and both positive. Let X — my. Then (5m + 6)y = 30, 30 y = and 5w + 6 30 m 5m + 6 Ifm = 2, a; = 3|, y = 1^. Ifm = 3, a; = 4f, 3/ = If (5) Solve 14 a; + 22y = 71, in positive integers. a; = 5 — v H ^. 14 If we multiply the fraction by 7 and reduce, the result is — 4y + J, a form which shows that there can be no integral solution. There can be no integral solution oi ax ±hy ^ c li a and h have a common factor not common also to c ; for, if c? be a factor of a and also of 6, but not of c, the equation may be written, mdx ± ndy = c, or mx ± ny = —, a. fraction. Exercise XCVI. Solve in positive integers : 1. 2a; + lly-:49. 5. 3a; + 8y = 61. 2. 7a; + 3y = 40. 6. 8a; + 5y = 97. 3. 5a; + 7y = 53. 7. 16a;+7y=110. 4. a; + 10y = 29. 8. 7a;+10y-=206. Solve in least positive integers : 9. 12a;-72/ = l. 12. 23a;-9y = 929. 10. 5a;-17y = 23. 13. 23a;-33y = 43. 11. 232/ -13a; = 3. 14. 555 a; — 22 y-- 73. 232 ALGEBRA. 15. How many fractions are there with, denominators 12 and 18 whose sum is |-|- ? 16. What is the least number which, when divided by 3 and 5, leaves remainders 2 and 3 respectively ? 17. A person counting a basket of eggs, which he knows are between 50 and 60, finds that when he counts them 3 at a time there are 2 over ; but when he counts them 5 at a time there are 4 over. How many are there in all ? 18. A person bought 40 animals, consisting of pigs, geese, and chickens, for $40. The pigs cost $5 a piece, the geese $1, and the chickens 25 cents each. Find the number he bought of each. 19. Find the least multiple of 7 which, when divided by 2, 3, 4, 5, 6, leaves in each case 1 for a remainder. 20. In how many ways may 100 be divided into two parts, one of which shall be a multiple of 7 and the other of 9? 21. Solve 18:r — 5y = 70 so that y may be a multiple of x, and both positive. 22. Solve 8:r + 12y = 23 so that x and y may be positive, and their sum an integer. 23. Divide 70 into three parts which shall give integral quotients when divided by 6, 7, "8, respectively, and the sum of the quotients shall be 10. 24. Divide 200 into three parts which shall give integral quotients when divided by 5, 7, 11, respectively, and the sum of the quotients shall be 20. 25. A number consisting of three digits, of which the mid- dle one is 4, has the digits in the units' and hundreds' places interchanged by adding 792. Find the number. SIMPLE INDETERMINATE EQUATIONS. 233 26. Some men earning each $2.50 a day, and some women earning each. $1.75 a day, receive altogether for their daily wages $44.75. Determine the number of men and the number of women. 27. A wishes to pay B a debt of £1 12 s., but has only half- crowns in his pocket, while B has only 4 penny-pieces. How may they settle the matter most simply? 28. Show that 323 a; — 527y = 1000 cannot be satisfied by integral values of x and y. 29. A farmer buys oxen, sheep, and hens. The whole num- ber bought is 100, and the whole price £100. If the oxen cost £5, the sheep £1, and the hens Is. each, how many of each did he buy ? 30. A number of lengths 3 feet, 5 feet, and 8 feet are cut ; how may 48 of them be taken so as to measure 175 feet all together ? 31. A field containing an integral number of acres less than 10 is divided into 8 lots of one size, and 7 of 4 times that size, and has also a road passing through it containing 1300 square yards. Find the size of the lots in square yards. 32. Two wheels are to be made, the circumference of one of which is to be a multiple of the other. "What cir- cumferences may be taken so that when the first has gone round three times and the other five, the differ- ence in the length of rope coiled on them may be 17 feet? * • 33. In how many ways can a person pay a sum of £15 in half-crowns, shillings, and sixpences, so that the num- ber of shillings and sixpences together shall be equal to the number of half-crowns ? CHAPTER XVII. Inequalities. 247t Expressions containing any given letter will liave their values changed when different values are assigned to that letter ; and of two such expressions, one may be for some values of the letter larger than the other, for other values of the letter smaller than the other. Thus, 1 + X + a^ will be greater than 1 — a; + a^ for all positive values of x, but less for all negative values of x. 248t One expression, however, may be so related to an- other that, whatever values may be given to the letter, it cannot be greater than the other. Thus, 2 X cannot be greater than sc'. + 1, whatever value be given to X. 249. For finding whether this relation holds between two expressions, the following is a fundamental proposition : If a and h are unequal, a^ ■\-h^>2 ah. For, (a — hf must be positive, whatever the values of a and h. That is, {a-hf>0, or a2-2a6+62>0; 250. The principles applied to the solution of equations may be applied to inequalities, except that if each side of an equality have its sign changed, the inequality will be reversed. Thus, if a > 6, then - a will be < - 6. INEQUALITIES. 235 (1) If a and h be positive, show that o?-{-h^is, > o?h-{-ah^. a^ + £3 > ^25 ^ ^12^ if (dividing each side by a + 5), a2 - a6 + 62 > ah, if a2 + 62>2a6. But a' + &2 is > 2 ab, | 249. .•.a3 + &3>a26 + a62. (2) Show that a' + 5' + c' is > a6 + ac + he. Now, a2^62is>2a6, a^ + c^ is >2ac, ^249. J2 + c2 is > 26c. By adding, 2a2 + 26^ + 2c2 is > 2a6 + 2ac + 26c, /. a^ + 62 -I- c2 is > a6 + ac + 6c. Exercise XCVII. Show that, the letters being unequal and positive : 1. a'^ + 35Ms > 25(a+ 5). 2. a'^» + ah^ is > 2a''Z>l 3. (a^-\- b') (a' + b') is > (a' + P)\ 4. a^^, _^ ^2^ ^ ab^ ^ i^q ^ ^^2 _^ j^2 -g ^ g ^j^^ 5. The sum of any fraction and its reciprocal is > 2. 6. If a;* = a^ + 5^ and y^ = c^ + (i ^rcy is >ac+ 5c?, or ac?+ 5c. 7. ab-\-ac-\-bc<{a-\-b-cy-\-{a-\-c-by-\-{b^c~a)\ 8. Which is the greater, {6} + b^) {c^-\-d'') or {ac + ^c^)^ ? 9. Which is the greater, rri} -\-m ox rni? -\-\1 10. Which is the greater,^*— 5*or4a^(a— 5) whenais>5? 11. Which is the greater, -^^^ +'\/— or Va + V5 ? 12. Which is the greater, ^^-^ or -?^ ? ^ 2 a + 5 13. Which is the greater, -^ + 4- °^ - + ~ ? CHAPTER XVIII. Theory of Exponents. 251. The expression a**, when n is a positive integer, has been defined as the product of n equal factors each equal to a. §24. And it has been shown that a"* X a" = a"*+". § 66. That a" -^ a" = «'*"**, if m be greater than n\ § 93. or , if m be less than n. § 94. And that (a"*)« = a*"". § 199. Also, it is true that a*'xb'^ = (ahy ; for (ahy = ah taken n times as a factor, = a taken n times as a factor X b taken w times as a factor = a" X h"". 252. Likewise, Va, when n is a positive integer, has been defined as one of the n equal factors of a (§ 203) ; so that if Va be taken n times as a factor, the resulting product is- a ; that is, i^aT — a. Again, the expression Va^ means that a is to be raised to the mih power, and the nth rodt of the result obtained. And the expression (Va)"* means that the nth root of a is to be taken, and the result raised to the mth power. It will thus be seen that any proposition relating to roots and powers may be expressed by this method of notation. It is, however, found convenient to adopt another method of notation, in which fractional and negative exponents are used. THEORY OF EXPONENTS. 237 253. The meaning of a fractional exponent is at once sug- gested, by observing that the division of an exponent, when the resulting quotient is integral, is equivalent to extracting a root. Thus, a^ is the square root of a^, and 3, the expo- nent of a^ is obtained by dividing the exponent of a* by 2. If this division be indicated only, the square root of a^ will be denoted by a^, in which the denominator denotes the root, and the numerator the power. If the same mean- ing be given to an exponent when the division does not give an integral quotient, a^ will represent the square root of the cube of a ; and, in general, an, the nth root of the mth power of a. This, then, is the meaning that will be as- signed to a fractional exponent, so that in a fractional exponent 254. The numerator will indicate a power, and the de- nominator a root. 255. The meaning of a negative exponent is suggested by observing that in a series of descending powers of a, a" a^, a*, a^ a^ a\ the subtraction of 1 from the exponent is equivalent to di- viding by a ; and if the operation be continued, the result ^^ a\ a-\ a-\ a-\ a'' a"". Then a'^- = l; a'' = 1 -^ a = -; ' a a' -2 1 1 _„ 1 a or a" This, then, is the meaning that will be assigned to a neg- ative exponent, so that, 256. A number with a negative exponent will denote the reciprocal of the number with the corresponding positive exponent. 238 ALGEBRA. It may be easily shown that the laws which apply to pos- itive integral exponents apply also to fractional and negative exponents. 257. To show that a« X ^" = {ahy : m m = Va'^h'^, = y/{ahr, m. = (a&) « (by definition) Likewise 111 1 a" X &" X c** = {ahcY, and so on. 1 1 1 258. To show that («»«)« = a*"" : Let 1 1 Then 1 a;« = a*", and «"»" = a. 1 But .-. a; = a*"**. 1 1 X = (a'»)» (by supposition), 1 1 1 .*. (a»»)n =*a»»«. 259. To show that oT X a^** = a*""" : Now a"* X a-" = a*" x — > a" == — ■ = a*^*-** if m > n ; a** = if m < n, = a- (»*-"») (by definition), 260. In like manner the same laws may be shown to apply in every case. THEORY OF EXPONENTS. 239 261. Hence, whether m and n be integral ox fractional, positive or negative : I. «»" X a" = a*"-^". III. (a™)" = a""*. 11. a"* -^ a** = a"*-". IV. a*" X 5*" = (a5)^ Exercise XCVIII. Express with fractional exponents : 1. V?; V?] (V^)^ V^'\ Va'; {Vaf ; x-'y-^] 6:^-^; ^ V * "^^''^ ' . Write in the form of integral expressions : ' z' ' rry' be' a'b-'' ^-f ' ^i' Simplify : 6. a^Xa^ ; b^xb^ ] c^ X c^ ; d^ X d^. 7. m^ X m~* ; w^ X ti"^^ ; a^ X a^ ; a" X a"^ 8. a^ xVa; c~^ X V^; y^ X -^y \ x^ X V^~. 9. a5^c X a~^b^ ; a^5^c~^ X a^T^c^c^. ; 10. :^ y^ ^ X x~^ y~^ -f^ ; x^ y^ -^ X x~^ y~^ tT^ . 240 ALGEBRA. 12. a^-^a^ ; c^ ^ c^ ] n^^^ -^n^ ; J^ ^a^ 13. (a«)^ -^ (a«)^ ; (c~^)* ; (m"^)* ; (J)-' ; (x^)i 14. (p'^y^y (qb~^; (^~^/r^^; (a^xJy^i 15. (4«-V^ (27^-T*; (64c^«)-^; (32c-^*')i 262. The laws that apply to the exponents of simple expressions also apply to the exponents of compound ex- pressions. (1) Multiply y* + y^ + 3/^ + 1 by 3/^ - 1. yi + yi + 3/1 + 1 Vi -1 y ^ -yl + yi + yi - -yi- -yi -yi- -1 y -1 y — 1. Ans. (2) Divide x^ + x^- 12 by x^ - 3. x^+ x^ 12 f^'*-^ 4a;* -12 4a;* -12 xi + 4. Ans. ,, 1^. 1 Exercise XCIX. Multiply 1. x^P + x^f + y'^ by a:'^ — x'y^ + y'^. 2. a;*""-" — if by a;" + y"*"-". 3. a;^-2a:* + l by a:*-l. THEORY OF EXPONENTS. 241 5. l-j-ab~'-{-a'b-'hjl-ab-' + a'b-\ 6. a'b-' + 2 + a-'b' by a'b'^ - 2 - a-'b\ Divide : 8. a;*" — y by a;" — y". 9. a; + y -j- 2 — 3 ar^y* 2* by a;* + y* -f- 2^- 10. aj + y by rc^ — a:^3/^ + ^^2/^ — ^ 2/^ + 3/ . 11. a;^3/~^ + 2-f-^~^y^ by a:y~^-f-a;~^3/. 12. a-' + a-'b-' + 5"* by a"^ - a'^b-' + 5"^ Find the squares of : 13. 4:ab-^; J-b^; a + a'^; 2ah^-arh^. If a = 4, b = 2, c = l, find tbe values of : 14. ah] bab-^- 2{abf] Grh~'c^; 12 a-'b-\ 15. Expand (a^ - 5^7 ; (2a;-^ + a;)*; (ab'' - bi/-y. Extract the square root of : 16. 9x-' — 18ri--'y* + 15a:-'y - ^x'^y^ + y\ Extract the cube root of : 17. 8a;^ + 12a7'^ - 30a; - 35 + 45 a;"^ + 27a;-^ - 21x~\ Resolve into prime factors with fractional exponents : 18. V^, v'72, -V^, VM ; and find their product. Simplify : 19. K^"')'X(a;^*)-'l3"i^. 20. (a:'^'' X a;-^') i^2. 21. 3(a* + b^y - 4(a^ + b^) {a^ - b^) + {a^ - 2b^)\ 22. \i^arY-l\'^i. 24. [\{a-'^yY]^-^[\{a^y]-P]-\ 23 /^Y-/^— Y' 25 ^'^^'~'^-y''^^"'^ 242 ALGEBRA. Radical Expressions. 263. An indicated root that cannot be exactly obtained is called a surd, or irrational number. An indicated root that can be exactly obtained is said to have the form of a surd. 264. The required root shows the order of a surd ; and surds are named quadratic, cubic, biquadratic, according as the second, third, or fourth roots are required. 265. The product of a rational factor and a surd factor is called a mixed surd; as, 3V2, b^/a. 266. When there is no rational factor outside of the rad- ical sign, the surd is said to be entire; as, V2, Va. 267. Since Va X V^ X Vc=. Va^, the product of two or more surds of the same order will be a radical expression of the same order consisting of the product of the numbers under the radical signs. 268. Inlikemanner, V^^Va^X V^=aV6. That is, A factor under the radical sign whose root can be talen, may, by having the root taken, be removed from under the radical sign, 269. Conversely, since aVb = V^b, A factor outside the radical sign may be raised to the cor- responding power and placed under it Again: ^=^F>^ = 3v^4. (3) ^/l^-^ = ^Wy^ X 2/^ = yV'hff. 1 (5) * _5a_ 4 166V 26c v'40 a6c2. (6) V296352. 23 296352 2' 37044 32 9261 3 1029 V 343 7 49 Hence, 296352 = 2^ x 3^ x 7', .-. v^296352 = v^x c^-\/aFc] bahc^abc'^. 3- fVV; 16VIII; (^+y)^(=^. 244 ALGEBRA. Express as mixed surds : 4. V^; V8^; V)X («'^"V)^. 12. Show that V20, V45, Vf are similar surds. 1^ 13. Show that 2i/aW, VW, i^j are similar surds. 14. If V2 = 1.414213, find the values of 272. Surds of the same order may be compared by ex- pressing them as entire surds. Ex. Compare |V7 and fVlO. • ' fV7 =V^, I VIO == VV-. V^ = V^, and V^ = >/W-. As "n/^ is greater than V^-, f VlO is greater than f V7. RADICAL EXPRESSIONS. 245 273. The product or quotient of two surds of the same order may be obtained by taking the product or quotient of the rational factors and the surd factors separately. (1) 2V5x5V7 = 10V35. (2) 9V5--3V7 = 3V| = 3Vf| = fV35. Exercise 01. 1. Which is the greater 3V7 or 2Vr5? 2. Arrange in order of magnitude 9 V3, 6 V7, SVlO. 3. Arrange in order of magnitude 4 V4, 3 a/5, 5 V3. 4. Multiply 3 V2 by 4V6 ; ^VlO by -/^VIS. 5. Multiply 5 V| by f vT62 ; \-s/i by 2^2. 6. Divide 2V5 by 3 Vl5 ; f V2T by ^V^. 7. Simplify fV3x I V5--fV2. 8. Simplify ?:^x^^i:^. 3V27 5Vl4 15V21 9. Simplify 2^/4x5^32-- a/IOS. 274. The order of a surd may be changed by changing the power of the expression under the radical sign. Thus, V5=^25; Te=V^\ Conversely, V25 = V5 ; V? = Vc ; or, in general, "^v^ = V^. In this way, surds of different orders may be reduced to the same order, and may then be compared, multiplied, or divided. 246 ALGEBRA. (1) To compare V2 and V3. ^ = 3^ = 3l = ^« == v^. .-. V3 is greater than a/2. (2) To multiply "V^ by V6a;. .^ = (4a)J = (4a)l = V^i^^ _ «/l6^« ; = y/l6^trx2T6^, = v/2*a2x23x 3=*af', = v/2« X 2 X 3=^a2x=*, == 2V54a''ar'. -4ns. (3) To divide "v^ by VU. v^ = (3a)^ = (3a)t = v^(3^2 = -^^9^2. VOb = (66)i = (66)i = ^e^p? = ^^216^. el g^ ^ e/ g'^ ~ \24?;3~ A'2=*y 3i=*' 2463 \2ax36 6/23xl5!E=2 ^\,2«X3«6« 6/ ^1944g'^6=*. ^ns. Exercise Oil. Arrange in order of magnitude : 1. 2^3. 3V2, 1^4. 3. 2-\/22, 3a/7, 4V2. 2. Vf , ^f 4. 3Vl9, 5a/2, 3-\/3. RADICAL EXPEESSIONS. 247 Simplify : 5. 2 Vo^ X -VS^ X V2bi ; Va^ X A^a%. 6. 3(4a52)i -- (2a'b)i ; (2a%')^ X (a*Z>^)^ - (a^^^'')^ 7. (2ab)^ X (3a52)i - (6ab^)i ; 4vT2 -- 2V3. ' ©'Kf)'-(A)' 9. (7V2-5V6-3V8 + 4V20)x3V2. 10. V(ll7xV(||?; -^v^^I^ X ^J/(2^. IL (-V^bf X (a/^^/ ; J b-^ c^d-^- ai b'^"^ c~i^ d^i 275. In tKe addition or subtraction of surds, each surd must be reduced to its simplest form ; and, if the resulting surds be similar, Add the o'ational factors, and to their sum annex the com- mon surd factor. If the resulting surds be not similar, Connect them with their proper signs. 276. Operations with surds will be more easily performed if the arithmetical numbers contained in the surds be ex- pressed in their prime factors^ and if fractional exponents be used instead of radical signs. (1) Simplify V27 + ViS + Vl47. V27 = (3=^)5 = 3x35=3\/3; Vis = (2* X 3)^ = 22 X 35 = 4 X 3i = 4\/3 ; VIi7 = (7^ X 3)^ = 7 X 3^ = 7\/3. .-. V27 + \/i8 + VTl7 = (3 + 4 + 7) V3 = 14 VS. Ans. 248 ALGEBRA. (2) Simplify 2^v^0-3a/40. 2v^320 = 2 (2« X 5)^ = 2 X 2^^ X 5^ = 8\^; 3\/40 = 3 (23 X 5)^ - 3 X 2 X 5' = 6\/5. .•.2\/320-3\/i0 = 8v^-6^5 = 2\/5. Ans. (3) Find the square root of Vsl. The square root of V^ = (Sl^)^ = 81« = (3^)« = 3t = (3=^)^=^/9. (4) Find the cube of ^-\/2. The cube of 1^2^ {If X {2'f = i X 2^ = |\/2. Exercise OIII. Simplify : 1. V27 + 2V48 + 3V108; 3VI000 + 4V50 + 12V288. 2. ^T28+-v/686 + -\/l6; 7^/54 + 3^l6+ \/432. 3. 12a/72-3V128; 7-v/81 - 3-v^I029. 4. 2V3 + 3Vli- V5i; 2V|+V60-Vl5-V|. ■Vg;3V|+2VS-4VS. 6. V4^+ V25^3-(a-55)V^. 7. cV^m?-a-y/^''? + bV'^^b^\ 8. 2-^40 + 3-v^T08+ -^500 "320-2-^1372. 9. (2^3^/; (3-^3)1 10. (f ^|Y ; (V27)i 11. (^81)^ (^512)^ (-^256)-^; ^16; ^27. 12. ^4; 'Vm- ^32; ^243; ^W^ ; ^49. 13. VM"; V9^^ V16a^^ V32a^ EADICAL EXPEESSIONS. 249 14. (Vsy; (^27)*; (^64)^ (^f. 15. (a^ ^.0075433^ X 78.343 x 8172.4^ x 0.00052 64285.* X 154.27* X 0.001 X 586.79* 15.832^ X 5793.6* X 0.78426 80 \0.000327* X 768^942 X 3015.3 x 0.007*' J 7.1895 X 4764.2^ X 0.00326^ 'o.00048953 X 45 7^ X 5764.42* 82. A(3 84. 85. 86. 87. 1416 X 4771.21 X 2.7183* 30.103* X 0.4343* x 69.897*' .03 27P X 53.429 X 0.77542^ 32.769 X 0.000371* 732.0562 X 0.0003572* X 89793 42.27983 X 3.4574 X 0.0026518^* 3 / 7932X 0.00657x0.80464 S( 0.03274 X 0.6428 * 7.1206 X V0.13274 X 0.057389 4 f 3.0755262 X 5771.2* X 0.0036984' x 7.74 1^ I 72258 X 327.93^ X 86.97^ J * V0.43468 X 17.385 x VO.0096372 3.0755262 X 5771.2* X 0.0036984' x 7.74 1^ 276 ALGEBRA. 318. Since any positive number otKer than 1 may be taken as tbe base of a system of logarithms, the following general proofs to the base a should be noticed. I. The logarithm of the product of two or more numbers is equal to the sum of the logarithms of the numbers. . For, let m and n be two numbers, and x and y their logarithms. Then, by the definition of a logarithm, m = a* and n^oM. Hence, w X ri = a* X a^ = a*+y. .'. log {mxn) = x + y, = log m + log n. In like manner, the proposition may be extended to any number of factors. II. The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. For, let m and n be two numbers, and x and y their logarithms. Then m = a* and n = an. Hence, m -^ n = a'' -^ av -= a'^-v. .'. log {m -i-n) = X — y, = log m — log n. From this it follows that log — == log 1 — log m. But, since log 1 = 0, log — = — log m. III. The logarithm of a power of a number is equal to the logarithm of the number multiplied by the exponent of the power. For, let X be the logarithm of m. Then m = a*, and mP = {a^y = op*. .'. log mP = px, = p log m. LOGARITHMS. 277 IV. The logarithm of the root of a number is equal to the logarithm of the number divided by the index of the root. For, let X be the logarithm of w. Then m = a* and m/ = (a*/ = a*. _ log m .*. log m*" = - o r r 319. An exponential equation, that is, an equation in which the exponent is the unknown quantity, is easily solved by logarithms. For, let o* = TO. Then log a* = log m, .'. X log a = log TO, log a' Ex. Find the value of x in 81* = 10. 81* -10, ^_ log 10 log 81* .*. log X = log log 10 + colog log 81, = + 9.7193 - 10, .-.a; =0.524. 320. Logarithms of numbers to any base a may be con- verted into logarithms to any other base b by dividing the computed logarithms by the logarithm of b to the base a. For, let log TO = y to the base h, and log h = X to the base a. Then to = hv, and h = a", .-. TO = (a^y = a^. .'. log TO (to base a)'^xy = log h (to base a) X log to (to base h). ^ fi. -L •L\ lo^ wi (to base a) .'. log TO (to base h) = , ^ , ;, , (. log (to base a) log a TO This is usually written, logj to log a b' CHAPTER XX. Ratio, Proportion, and Variation. 321. The relative magnitude of two numbers is called tlieir ratio, and is expressed by the fraction which the first is of the second. Thus, the ratio of 6 to 3 is indicated by the fraction |, which is sometimes written 6 : 3. 322. The first term of a ratio is called the antecedent, and the second term the consequent. When the antecedent is equal to the consequent, the ratio is called a ratio of equality ; when the antecedent is greater than the conse- quent, the ratio is called a ratio of greater inequality ; when less, a ratio of less inequality. 323. When the antecedent and consequent are inter- changed, the resulting ratio is called the inverse of the given ratio. Thus, the ratio 3 : 6 is the inverse of the ratio 6:3. 324. The ratio of two quantities that can be expressed in integers in terms of a common unit is equal to the ratio of the two numbers by which they are expressed. Thus, the ratio of $9 to $11 is equal to the ratio of 9 : 11 ; and the ratio of a hne 2f inches long to a hue 3,} inches long, when both are expressed in terms of a unit j^j of an inch long, is equal to the ratio of 32 to 45. 325. Two quantities different in kind can have no ratio, for then one cannot be a fraction of the other. RATIO. 279 326. Two quantities that can be expressed in integers in terms of a common unit are said to be commensurable. The common unit is called a common measure, and each quantity is called a multiple of this common measure. Thus, a common measure of 2J feet and 3f feet is |^ of a foot, whicli is contained 15 times in 2J feet, and 22 times in 3f feet. Hence, 2\ feet and 3f feet are multiples of ^ of a foot, 2J feet being obtained by taking i of a foot 15 times, and 3f by taking i of a foot 22 times. 327. When two quantities are incommensurable, that is, have no common unit in terms of which both quantities can be expressed in integers, it is impossible to find a fraction that will indicate the exact value of the ratio of the given quantities. It is possible, however, by taking the unit suf- ficiently small, to find a fraction that shall difier from the true value of the ratio by as little as we please. Thus, if a and h denote the diagonal and side of a square, ?=V2. o Now VI - 1.41421356 a value greater than 1.414213, but less than 1.414214. If, then, a millionth part of h be taken as the unit, the value of the ratio I lies between l^^fiH ^^^ iUtM' ^^^ therefore differs from either of these fractions by less than xTn^^^nr- By carrying the decimal farther, a fraction may be found that will differ from the true value of the ratio by less than a billionth, tril- lionth, or any other assigned value whatever. 328. Expressed generally, when a and h are incommen- aurable, and h is divided into any integral number (n) of equal parts, if one of these parts be contained in a more than m times, but less than m-\-\ times, then ^-.^ \. i. ^ ^ + 1 . Y > — , but < , on n that is, the value of - lies between — and — ^^. b n n 280 ALGEBRA. The error, therefore, in taking either of these values for _ is < -. But by increasing 71 indefinitely, - can be made b n n to decrease indefinitely, and to become less than any as- signed value, however small, though it cannot be made absolutely equal to zero. 329. The ratio between two incommensurable quantities is called an incommensiLrable ratio. 330. As the treatment of Proportion in Algebra depends upon the assumption that it is possible to find fractions which will represent the ratios, and as it appears that no fraction can be found to represent the exact value of an incommensurable ratio, it is necessary to show that two incommensurable ratios are equal if their true values always lie between the same lim^its, however little these limits differ \from each other. Let a : h and c : c? be two incommensurable ratios. Suppose the true values of the ratios a : h and c : d \\q between ^ and !— 1_. Then the difference between the true values of these n n ^ -1 ratios is less than -, however small the value of - may be. § 328. n n But since - can be made to approach zero at pleasure, _ can n n be made less than any assumed difference between the ratios. Therefore, to assume any difference between the ratios is to assume it possible to find a quantity that for the same value of — shall be 1 •'^ both greater and less than - ; which is a manifest absurdity. n Hence, a:h ^ c: d. 331. It will be well to notice that the word limit means a fixed value from which another and variable value may be made to differ by as little as we please ; it being impos- sible, however, for the difference between the variable value and the limit to become absolutely zero. RATIO. 281 332. A ratio will not he altered if both its terms he multi- plied hy the same number. For the ratio a : 6 is represented by ~, the ratio ma : mh is repre- sented by — ; and since — - = -, .". ma : mh = a:b. mb mh . h 333. A ratio ivill he altered if different multipliers of its terms he taJcen ; and will he increased or diminished accord- ing as the multiplier of the antecedent is greater or less than that of the consequent For, ma : nh will be > or < a : 6 according as ^ is > or < ^ ( = ^\ nb b\nbj as ma is > or < na, as m is > or < n. 334. A ratio of greater inequality will he diminished, and a ratio of less ineqv/xlity increased hy adding the same num- ber to both its terms. For, a -^-x-.h ■\-x\s>> ox or < %, 6 +a; h as a6 + 5a; is > or < a6 + ox, as ia; is > or < ax, as 6 is > or < a. 335. A ratio of greater inequality will be increased, and a ratio of less inequality diminished, by subtracting the same number from both its terms. For, a — x:h—x will be > or < a : 5 according as ^~- is > or < -, h —X h as ah — hx\Q>ox or < hx, as a is > or < 5. 282 ALGEBRA. 336. Ratios are compounded by taking the product of the fractions that represent them. Thus, the ratio compounded of a : 6 and c : c? is found by taking the product of %nd - = ^. 6 d hd The ratio compounded oi a-.h and a : 5 is the duplicate, ratio a' ; 6', and the ratio compounded o^ a-.h, a:h, and a:b is the triplicate ratio 337. Ratios are compared by comparing the fractions that represent them. Thus, a:h ia> ov < c:d according as - is > or < — , d Cbd • ^ ^ be as acZ is > or < he. Exercise OXIII. 1. "Write down the ratio compounded of 3 : 5 and 8 : 7. Which of these ratios is increased, and which is diminished by the composition? 2. Compound the duplicate ratio of 4 : 15 with the tripli- cate of 5 : 2. 3. Show that a duplicate ratio is greater or less than its simple ratio according as it is a ratio of greater or less inequality. 4. Arrange in order of magnitude the ratios 3:4; 23 : 25 ; 10 : 11 ; and 15 : 16. 5. Arrange in order of magnitude a-{-h:a — h&nda^-\-b^:a^ — h^,i£a>h. "Find the ratios compounded of: 6. 3:5; 10:21; 14:15. 7. 7:9; 102:105; 15:17. RATIO. 283 a^ — a^ X -{- ax^ — x^ a-{-x 9 ^~Q^ + 2Q ^^^ x'-lSx + 4:2 a^ — 6x a^ — bx 10. a + 5:a-J; a' + b' : (a + hf; (a' - bj : a* - b\ 11. Two numbers are in tKe ratio 2:3, and if 9 be added to eacb, they are in tbe ratio 3 : 4. Find the num- bers. (Let 2 X and 3 x represent the numbers). 12. Show that the ratio a : Z> is the duplicate of the ratio a-{- c:b-\-c, ii c^ = ab. 13. Find two numbers in the ratio 3:4, of which the sum is to the sum of their squares in the ratio of 7 to 50. 14. If five gold coins and four silver ones be worth as much as three gold coins and twelve silver ones, find the ratio of the value of a gold coin to that of a silver one. 15. If eight gold and nine silver coins be worth as mucn as six gold and nineteen silver coins, find the ratio of the value of a silver coin to that of a gold one. 16. There are two roads from A to B, one of them 14 miles longer than the other ; and two roads from B to 0, one of them 8 miles longer than the other. The dis- tance from A to B is to the distance from B to C, by the shorter roads, as 1 to 2 ; by the longer roads, as 2 to 3. Find the distances. 17. What must be added to each of the terms of the ratio m : n, that it may become equal to the ratio p : q'^ 18. A rectangular field contains 5270 acres, and its length is to its breadth in the ratio of 31 : 17. Find its di- mensions. 284 ALGEBRA. Proportion. 338. An equation consisting of two equal ratios is called a proportion ; and the terms of the ratios are called propor- tionals. 339. The algebraic test of a proportion is that the two fractions which represent the ratios shall be equal. Thus, the ratio a : b will be equal to the ratio c : cZ if - = - ; and d the four numbers a, h, c, d are called proportionals, or are said to be in proportion. 340. If the ratios a : b and c : d form a proportion, the proportion is written . i. _ ^ . g (read the ratio of a to 5 is equal to the ratio of c to cT) or a'.h : \ c \ d (read a is to 5 in the same ratio as c is to d). The first and last terms, a and d, are called the extremes. The two middle terms, h and c, are called the means. 341. When four numbers are in proportion, the product of the extremes is equal to the product of the means. For, if a.h :: c: d, 4.1 a c """ -r-d- By multiplying by bd, ad = be. 342. If the product of two numbers be equal to the product of two others, either two may be made the extremes of a pro- portion and the other two the means. For, if ad= be, by dividing by bd, -— = — , bd bd a c .-.a. b .: c: d. pROPonTioN. 285 343. The equation ad = be gives he J ad a = — ; h = — ; a c so that an extreme may be found by dividing the product of the means by the other extreme ; and a mean may be found by dividing the product of the extremes by the other mean. 344. If four quantities, a, b, c, d, be in proportion, they will be in proportion by : I. Inversion. That is, b will be to a as c? is to c For, if a:h :: c:d, h'd' and a ^ c or b ^d a c' .*. 5: a : : d: c. 345. II. Composition. That is, a-\-b will be to 5 as d? + c? is to d. For, if a:b :: c:d, then a c b'd' and h'-2-^- or a+b c+d b ~ d ' .a + b:b::c + d:d. 346. III. Division. That is, a — b will be to 5 as c — c? is to d. For, if a:b :: c: d, XI, a c then T- = 3i a 2^6 ALdEBRA. and ,- — 1 - -- — 1, a a — b e~ d —I d-' .\a — b:b::c — d:d. 347. IV. Composition and Division. That is, o + ^ will he to a — h £is c-\-d is to c — d. For, from II., ^-±i«^ b d ' and from III., ^^^^Zli. b d By dividing, 9l+1=.L±A^ a—b c—d .'.a + b:a—b : : c ^\- d:c — d. 348. When the four quantities a, b, c, d are all of the same hind, they will be in proportion by : V. Alternation. That is, a will be to c? as 5 ' is to d. For, if a:b: :c:d, then a b' c 'd' By multiplying by -, ab be' be 'cd' or a c ' b . a-.c : :b:d. 349. From the proportion a: c :: b : d may be obtained by: VI. Composition. a-{- c : c ::b-{- d: d. VII. Division. a — c : c : : b — d : d. VIII. Composition and Division. a-{-c:a—c ::b-\-d:b~d. PROPORTION. 287 350. In a series of eqv/xl ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its e^nsequent. For.if ^=.£. = 1^£. b d f h r may be put for each of these ratios. Then -- = r, i=r, 1 = r, I = r, b d f h .'. a-^'br, c = dr, « =/r, g = hr. .'. a + c -^ e ^ g = {b + d +f + h) r. . a-\-c-ir6-irg_ _a 6 + cZ+/+ A ~ ~b' .'.a -\- c + e + g : b + d +f + h ;: a : b. In like manner, it may be shown that ma + nc + pe + qg : mb + nd + pf + qh : : a : b. 351. If a, b, c, d be in continued proportion, tliat is, if a:h = h:c — c:d, tHen will a: c = a^ :h^ and a: d=a^:b^. For, Honce. So a b c b c d a V b _ J c a c b b a? .-. a :c = a^:b\ a b C X c d~ a d~ a a b^'b a? -I ,\ a :d^ ■.a?:b\ 352. If a, 5, c be proportionals, so that a:b::b:c, then b is called a mean proportional between a and c, and c is called a third proportional to a and b. 288 ALGEBRA. li a:h wh : c, then h = -\/ac. For, if a-.h ■.-.h-.c, then ^ = ^, h c and h^ == oc, .•. & = Vac. 353. ^A^ products of the corresponding terms of two or more proportions are in proportion. For, if a-.h :: c:d, e:f:: g:h, and h:l:: m:n, then ^ = ^, ^=1, h.^.'UL. h d f h' I n Hence, by finding the product of the left members, and also of the right members of these equations, aek _ cgm bfl dhn .•, aeh : bfl : : cgm : dhn. 354. lAke powers, or like roots, of the terms of a propor- tion are in proportion. For, if a:b::c:d, en a c '"d By raising both sides to the nth power, a" .-. a^:b^: ■ C* : C?". By extracting the nth root, a* 0^ .-.J^-.h^: ■.i^:i\ PROPORTION. 289 355. If two quantities he increased or diminished hy like parts of each, the results will be in the same ratio as the quantities themselves. a \ n) For, h 1^ . n., ^ that is, n m '.\a± — a:o±—o. n n 356. Tlie laws that liave been established for ratios should be rememembered when ratios are expressed in their fractional form. oc^^x + l x^-x + 2 2^ 2o^ CI) Solve: By I 347 and this equation is satisfied, when a; = ; or, diviaing by • — , = — , *" -^ 2' x + 1 2-x \2) li a : h : : c: d, show that a^-\-ab :b^ — ah :: c^ + cd: d^— cd. If a c b~d' then a+b c+d a — b c — d' and a c -b -d' . a ^^a + b c ^^c^-d , — 6 a — —d c — d that is, a^ + ab (? -\- cd y^-ab d^-cd! or a^ + ab-.b^-ab-.-.c' + cd-.d'^-cd. §347. 353. 290 ALGEBEA. (3) When a \ h : : c \ d, and a is tlie greatest term, show that a-\-di^ greater than h-[-c. Since ?- = -, and a>c, b d .'. h>d. Also, since ^^ = ^^, ^ 346. and 6 > c?, .*. a — 6 > c — c?. By adding, Z> + <^ = Z> + c?, a -{■ d>h + c. Exercise OXIV. li a\h wc-.d, prove that : 1. ma :nb : : mc : nd. 4. a^ : P : : c^ : d^. 2. 3a+5:5 : : 3c + c?:d 5. a : a + 5 : : c : ^ + c?. 3. a + 2b:b :: c + 2d:d. 6. a: a~h : : c : c — d. 7. ma + ^^ ; '^a — nh : : mc + wc? : mc — ?2C?. 8. 2a + 3Z>:3a-4Z» :: 2c + 3c?:3e-4c?. 9. ma^ + n1 10. ma^ + 7ia5 + J9&^ : mc^ + wcc? +i9c?* : : 5^ : d^. If a:b : : b : c, prove that : 11. a + b:b + c :: a:b. 12. a2_|_^5 . J2_|_ j^ . . ^ . ^^ 13. a:c :: {a-\-bf : (b -{-c)\ 14. "When a, 5, and c are proportionals, and a the greatest, show that a-\-c'> 2b. 15, If — -^ = ^ = , and x, y, z be unequal, then Z4-m + w = 0. PROPORTION. 291 16. Find X when a; + 5:2a; — 3::5:c+l:3:r — 3. 17. Find x when x-\-a:2x — h :: 2>x-{-h :^x — a. 18. Find X when Va; + V6 : V.'?; — V^ :: a\h. 19. Find a; and y when a; : 27 : : y : 9, and a; : 27 : : 2 : x—y, 20. Find 5; and y when a;-fy+l:a; + y + 2 :: 6:7, and when y-\-2x:y — 2x : : 12x-\-Qy — 2) : 6y — 12a: — 1. 21. Find X when :r2_4a:-{-2 : a;2_2a;_l ; : a^-4:x : a;2_2a;-2. 22. A railway passenger observes that a train passes him, moving in the opposite direction, in 2 seconds ; but moving in the same direction with him, it passes him in 30 seconds. Compare the rates of the two trains. 23. A and B trade with different sums. A gains $200 and B loses $50, and now A's stock : B's : : 2 : J. But, if A had gained $100 and B lost $85, their stocks would have been as 15 : 3i. Find the original stock of each. 24. A quantity of milk is increased by watering in the ra- tio 4 ; 5, and then 3 gallons are sold ; the remainder is mixed with 3 quarts of water, and is increased in the ratio 6:7. How many gallons of milk were there at first ? 25. In a mile race between a bicycle and a tricycle their rates were as 5:4. The tricycle had half a minute start, but was beaten by 176 yards. Find the rates of each. 26. The time which an express- train takes to travel 180 miles ie to that taken by an ordinary train as 9 : 14. The ordinary train loses as much time from stopping as it would take to travel 30 miles ; the express-train loses only half as much time as the other by stopping, and travels 15 miles an hour faster. What are their respective rates ? 292 ALGEBRA. 27. A line is divided into two parts in the ratio 2:3, and into two parts in the ratio 3:4; the distance be- tween the points of section is 2. Find the length of the line. 28. A railway consists of two sections ; the annual ex- penditure on one is increased this year 5%, and on the other 4%, producing on the whole an increase of 4x%%. Compare the amount expended on the two sections last year, and also this year. 29. When a,h,c,d are proportional and unequal, show that no number x can be found such that a-\-x, h-\-x, c-\'X, d-\-x shall be proportionals. Variation. 357. Two quantities may be so related that, when one has its value changed, the other will, in consequence, have its value changed. Thus, the distance travelled in a certain time will hQ^doubled if the rate be doubled. The time required for doing a certain quantity of work will be doubled if only half the number of workmen be employed. 358. Whenever it becomes necessary to express the gen- eral relations of certain kinds of quantities to each other, without confining the inquiry to any particular values of these quantities, it will usually be sufficient to mention two of the terms of a proportion. In all such cases, however, four terms are always implied. Thus, if it be said that the weight of water is proportional to its volume, or varies as its volume, the meaning is, that one gallon of water is to any number of gallons as the weight of one gallon is to the weight of the given number of gallons. VAEIATION. 293 359. Quantities used in a general sense, as distance, time, weight, volume, to which particular values may be assigned, are denoted by capital letters. A, B, C, etc. ; while as- signed values of these quantities may be denoted by small letters, a, 5, c, etc. The letters A, B, Cwill be understood to represent any numerical values that may be assigned to the quantities ; and when two such letters occur in an ex- pression they will be understood to represent ang corre- sponding numerical values that may be assigned to the two quantities. 360. When two quantities A and B are so connected that their ratio is constant, that is, remains the same for all corresponding values of A and B, the one is said to vary as the other ; and this relation is expressed by ^ oc ^ (read A varies as B). Thus, the area of a triangle with a given base varies as its altitude ; for, if the altitude be changed, the area will be changed in the same ratio. A If this constant ratio be denoted by m, then -— = m, or ^ = mB. ^ From this equation m may be found when two corre- sponding values of A and B are known. 361. When two quantities are so connected that if one be changed in any ratio, the other will be changed in the inverse ratio, the one is said to vary inversely as the other. Thus, the time required to do a certain amount of work varies in- versely as the number of workmen employed ; for, if the number of workmen be doubled, halved, or changed in any ratio, the time re- quired will be halved, doubled, or changed in the inverse ratio. 362. If A vary inversely as B, two values of A have to each other the inverse ratio of the two corresponding values of ^ ; oil a\o! \\V \h\ that is, ah = dV , 294 ALGEBRA-. Hence, the product AJ5 is constant, and may be denoted by m. That is, A£ = m. If any two corresponding values of A and J5 be known, the constant m may be found. The equation AJB = 7n may be written A=~, and as m is constant, A is said to vary as the reciprocal of J5, or 363. The two equations, A = mB (for direct variation), A = -- (for inverse variation), furnish the simplest method of treating Variation. If A = mBC, A is said to Yd^rj jointly as B and C. If J. = -— -, A is said to vary directly as B and inversely as (7. 364. The following results are to be observed : I. If ^ oc ^ and ^ oc C, then A ex: C. For A = m^, where m is constant, and B = nC, where n is constant. .". A = mnC. .•.A oc C, since mn is constant. In like manner, if ^ oc J? and -S oc — , then Ace —. C G II. If^aCand^oc(7,then^±^oc(7,andVZgoc(7. For J[ == m(7, where m is constant, and B = nC, where n is constant. :.A±B = {m±n)a .'. A ± B cc C since m ± ?i is constant. Also, VAB== VmCx nC= VmnC^== Cy/mn. :. Vud-S oc C, since y/rrm is constant. VARIATION. 295 III. If ^ oc^and CocZ), then ^(7oc^i). For A = mB, where m is constant, C = nD, where n is constant. .\AC=mnBD. .'. AC oc BD, since mn is constant. IV. If ^ oc 5 then ^" oc B\ For A = mB, where m is constant. .•. J." = m^B^. .". J." oc 5" since m" is constant. V. li A cc B when (7 is unchanged, and A dz C when ^ is unchanged, then A cc ^Cwhen both B and (7 change. For -4 = mB, when ^ varies and C is constant. Here, m is constant and cannot contain the variable B, :. A must contain B, but no other power of B. Again, A = nC, when C varies and B is constant. Here, n is constant and cannot contain the variable C, :. A must contain C, but no other power of C. Hence, A contains both B and C, but no other powers of B and C, and therefore, A ■ — = », or -4 = pBC, where p is constant. BO ^ \ . .'. A oc BO, since p is constant. In like manner, it may be shown that if A vary as each of any number of quantities B, C, D, etc., when the rest are unchanged, then when they all change, A oc BCD, etc. Thus, the area of a rectangle varies as the base when the altitude is constant, and as the altitude when the base is constant, but as the product of the base and altitude when both vary. The volume of a rectangular solid varies as the length when the width and thickness remain constant; as the width when the length and thickness remain constant ; as the thickness when the length and width remain constant; but as the product of length, breadth, and thickness when all three vary. 296 ALGEBRA. (1) If A vary inversely as B, and when A = 2 the corre- sponding value of £ is 36, find the corresponding value of B when A =^9. Here A = _ m B' or m = /. m = ^AB, = 2 X 36 = 72 A] id if 9 and 72 be substituted for A and m respectively in A- m = — B' the result is 9 = .9^ = .:B = 72 B' = 72. = 8. A71S. (2) The weight of a sphere of given material varies as its volume, and its volume varies as the cube of its diam- eter. If a sphere 4 inches in diameter weigh 20 pounds, find the weight of a sphere 5 inches in diam- eter. Let TT represent the weight, V represent the volume, ' D represent the diameter. Then TFocFand Foe Z)3, .-. Woe i)3. Put W= mD^, then, since 20 and 4 are corresponding volumes of IF" and D, 20 = m X 64, .•.m = ff:=TV .'. when i) = 5, W= j\ of 125 = 39^. Exercise CXV. 1. If^oc^, and ^ = 4when5 = 5, find ^ when ^ = 12. 2. If ^ oc B, and when B=i, A = i, find A when B = i. 3. If A vary jointly as B and C, and 3, 4, 5 be simulta- neous values of .4, JB^ O, finc^ 4 when B= C=10. V 2^J3iXJX XJLV. )iS. ^1/ 1 4. If ^ oc —, and when A = B when A ==4. 10 ,B = = 2, find the value of 5. If ^ oc — , and when A = 6 the value of A when B 6, B^ = 4, and (7 = = 3, find = 5andC = 7. 6. If the square of Xvary as the cube of Y", and X= 3 when y=4, find the equation between Xand Y. 7. If the square of X vary inversely as the cube of F, and Jf = 2 when ]r= 3, find the equation between X and Z 8. If ^vary as X directly and F inversely, and if when ^=2, X=3, and F= 4, find the value of ^ when X- 15 and r- 8. 9. If ^ oc ^ + ^ where c is (Constant, and if -4 = 2 when B = 1, and if J. = 5 when ^ = 2, find A when ^ = 3. 10. The velocity acquired by a stone falling from rest varies as the time of falling ; and the distance fallen varies as the square of the time. If it be found that in 3 seconds a stone has fallen 145 feet, and acquired a velocity of 96f feet per second, find the velocity and distance at the end of 5 seconds, 11. If a heavier weight draw up a lighter one by means of a string passing over a fixed wheel, the space de- scribed in a given time will vary directly as the difference between the weights, and inversely as their sum. If 9 ounces draw 7 ounces through 8 feet in 2 seconds, how high will 12 ounces draw 9 ounces in the same time ? 12. The space will vary also as the square of the time. Find the space in Example 11, if the time in the lat- ter case be 3 seconds. 298 ALGEBItA. 13. Equal volumes of iron and copper are found to weigh 77 and 89 ounces respectively. Find the weight of 10 i feet of round copper rod when 9 inches of iron rod of the same diameter weigh SIj^q- ounces. 14. The square of the time of a planet's revolution varies as the cube of its distance from the sun. The distances of the Earth and Mercury from the sun being 91 and 35 millions of miles, find in days the time of Mer- cury's revolution. 15. A spherical iron shell 1 foot in diameter weighs ^^ of what it would weigh if solid. Find the thickness of the metal, it being known that the volume of a sphere varies as the cube of its diameter. 16. The volume of a sphere varies as the cube of its diame- ter. Compare the volitme of a sphere 6 inches in diameter with the sum of the volumes of three spheres whose diameters are 3, 4, 5 inches respectively. 17. Two circular gold plates, each an inch thick, the diam- eters of which are 6 inches and 8 inches respectively, are melted and formed into a single circular plate 1 inch thick. Find its diameter, having given that the area of a circle varies as the square of its diameter. 18. The volume of a pyramid varies jointly as the area of its base and its altitude. A pyramid, the base of which is 9 feet square, and the height of which is 10 feet, is found to contain 10 cubic yards. What must be the height of a pyramid upon a base 3 feet square, in order that it may contain 2 cubic yards ? CHAPTER XXI. Series. 365. A succession of numbers which proceed according to some fixed law is called a series; and the successive numbers are called the terms of the series. Thus, by executing the indicated division of •, the series 1 +x 1 — x + ar' + a^ + is obtained, a series that has an unlimited number of terms. ^ 366. A series that is continued indefinitely is called an infinite series ; and a series ihat comes to an end at some particular term is called a finite series. 367. When a; is < 1, the more terms we take of the infi- nite series 1-}-^ + ^ + ^+ , obtained by dividing 1 by 1—x, the more nearly does their sum approach to the value 1—x Thus, if ic = i, then = == -, and the series becomes 1 + ^ 1—x 1—^ 2 + i + iV + ' ^ ^^^^ which cannot become equal to f however great the number of terms taken, but which may be made to differ from f by as little as we please by increasing indefinitely the number of terms. 368. But when x, is > 1, the more terms we take of the Faeries l-\-x-{-oi?~\-a?-{- the more does the sum of the series diverge from the value of \-x ^_1 2' S + 9 + 27 + a sum which diverges more and mo^e from — i Thus, if X = 3, then = = — , and the series becomes 1 -f 1 —X 300 ALGEBRA. the more terms we take, and which may be made to increase indefi- nitely by increasing indefinitely the number of terms taken. 369. A series whose sum as the number of its terms is in- definitely increased approaches some fixed finite value as a limit is called a converging series ; and a series whose sum increases indefinitely as the number of its terms is increased, is called a diverging series. 370. When x = l, the 'division of 1 by l~x, that is, of 1 by 0, has no meaning, according to the definition of divi- sion; and any attempt to divide by a divisor that is equal to zero leads to absurd results. Thus, 8 + 4 = 8 + 4; by transposing, 8 — 8 = 4 — 4; or, dividing by 4 — 4, 2 = 1; a manifest absurdity. 371. When x=l very nearly, then the value of X. X will be very great, and the sum of the series 1 + ^ + ^ + o? -\- will become greater and greater the more terms we take. Hence, by making the denominator \-~x approach indefinitely to zero, the value of the fraction may be made to increase at pleasure. 372. If the symbol o be used to denote a quantity that is less than any assignable quantity, and that may be con- sidered to decrease without limit, not, however, becoming 0, and the symbol Q? be used to denote a quantity that is greater than any assignable quantity, and that may be con- sidered to increase without limit, not, however, becoming oo, then 1 _=00 O In the same sense -— = Q?^ where a represents any value that may be assigned. o SERIES. 301 1 — X^ 373. If X in the fraction :, be equal to 1, the numer- 1 —X ator and denominator will each become 0, and the fraction will assume the form -. 374. If, however, x in this fraction approach to 1 as its limit, then the denominator 1 — x, inasmuch as it has some value, even though less than any assignable value, may be used as a divisor, and the result is l-{-x-{'X^-{-x^-\-x*. Hence, it is evident that though both terms of the fraction become smaller and smaller as 1 — a; approaches to 0, still the numerator becomes more and more nearly five times the denominator. It may be remarked that when the symbol § is obtained for the value of the unknown quantity in a problem, the meaning is that the problem has no definite solution, but that its conditions are satisfied if any value whatever be taken for the required quantity ; and if the symbol %, in which a denotes any assigned value, be obtained for the value of the unknown quantity, the meaning is that the condi- tions of the problem are impossible. 375. The number of different series is unlimited, but the only kinds of series that will be considered at this stage of the work are Arithmetical, Geometrical, and Harmonica! Series. Arithmetical Series. 376. A series in which the difference between any two adjacent terms is equal to the difference between any other two adjacent terms, is called an Arithmetical Series or an Arithmetical Progression. 377. The general representative of such a series will be a, a-\-d, a-\-2d, a + 3c? , in which a is the first term and d the common difference ; 302 ALGEBRA. and the series will be increasing or decreasing according as d is positive or negative. 378. Since each succeeding term of the series is obtained by adding d to the preceding term, the coefficient of d will always be 1 less than the number of the term, so that the nth term = a-{- (n—V) d. If the nth term be denoted by /, this equation becomes l=.a + {n-l)d. (1) 379. The arithmetical mean between two numbers is the number which stands between them, and makes with them an arithmetical series. 380. If a and h denote two numbers, and A their arith- metical mean, then, by the definition of an arithmetical ^^"*^' A^a = b-A, .■.A = ?^+^. (2) A 381. Sometimes it is required to insert several arithmeti- cal means between two numbers. If -^ = the number of means, then m + 2 = w, the whole number of terms ; and if m + 2 be substituted for n in the equation l = a + (n-V)d, the result is Z = a + (^ + 1) '^. By transposing a, / — a = (m + 1) J, . I — a m-f 1 = d. (3) Thus, if it be required to insert six means between 3 and 17, the 17 — 3 value of d is found to be = 2 ; and the series will be 3, 5, 7, 9, 6 + 1 11, 13, 15, 17. SERIES. 303 382. If I denote the last term, a the first term, n the number of terms, d the common difference, and s the sum of the terms, it is evident that s= a -f-(a+c?) + (a+2c^) + -^(l-d)-\- I, or _s = I +(l-d)+(l--2d) + + (a + d)+ a .■.2s = (a+l) + (a + l)+(a+l) + + (« + /) + (a-j-l) = n(a-\-l) .•.s=|(«+0- w 383. From the two equations, l=a + (n-~l)d, (1) s = l(a + l), [2] any one of the quantities a, d, I, n, s may be found when three are given. Ex. Find w when d, I, s are given. Frora(l), a = Z-(n-])cZ. 'y