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El 
 
 9ltttl|( 
 
PUBLIC SCHOOL 
 
 Euclid and Algebra 
 
 akutl)0Hteb for u»e in tit* lAublic $cl)Oi»l« of ftiniavio 
 bti ill* pctfaviment of aBbucaiion 
 
 PRICE 25 CENTS 
 
 \ 
 \ 
 
 TORONTO 
 THE HUNTER, ROSE COMPANY, Limited 
 
PS 
 
 ■^"^HHssasi— ^-- 
 
PREFACE 
 
 This little book has been prepared to present the work in 
 Geometry and Algebra laid down in the curriculum of the 
 Public Schools of Ontario. 
 
 These subjects are taugiit during only one year of the course, 
 and to pupils of whom the majority do not have opportunity 
 of proceeding further. Consequently, completeness of treat- 
 ment as well as simplicity has been considered. 
 
 The aim has been to present practical definite aspects of 
 the subjects rather than to seek to lay a foundation for 
 higher work in mathematics. 
 
 The problems an:? exercises are numerous, over fifteen 
 hundred in all. Many of these are suitable for viva voce 
 class work. 
 
CONTENTS. 
 
 6 
 7 
 
 8 
 
 48 
 
 Kuclid's Elements . 
 
 TV n XAOR 
 
 Definitions 
 
 Postulates * 
 
 Axioms 
 
 Symbols and Abbreviations 
 
 Propositions, Bk. I., 1-26 
 
 Exercises 
 
 Alobbra : 
 
 Chapter , ■M—ArithroeticalandAlgeuiuicalNum- 
 
 •^'^ 53 
 
 . 11. - Definitions, touostitutions. Combin- 
 
 ing Like Terms ^^ 
 
 III.— Addition, Subtraction, Brackets.. 66 
 IV. -Multiplication, Division, Miscel- 
 
 laneous Examples 71 
 
 . v.— Important Results in Multiplication 
 
 and Division. Factors, Multiples. 86 
 VI.— Simple Equations i aq 
 
 Chapter . Vll.-Solution of Problems, MlsceUaneous 
 
 Exercises. ... iqq 
 
 Chapter VIIT. -Fractions '. jgg 
 
 Chapter . IX.-Equations and Problems,' involving 
 
 Fractions, Miscellaneous Exer- 
 
 Chapter 
 
 Chapter 
 Chapter 
 
 Chapter 
 
 Chapter , 
 
 cises 
 
 Answers .. . ; 
 
 141 
 
EUCLID'S ELEMENTS 
 
 BOOK l.-PROPOSITIONS 1-26. 
 
 iluclid was the first teacher of mathematics in the fin^t g»^at 
 umversity of the world-that founded at Alexandria about 
 300 B. C. He was the author of works on Geometry, Aritli- 
 metic, Astronomy, Optics and Music. 
 
 His work known as Euclid's Elements consisted of thirteen 
 books, of which the first six and the last thi-ee treat of Geo- 
 metry ; the remaining books treat of Arithmetic. 
 
 The proofs of the propositions which are given in this edition 
 of a part of the first book of the Elements are considered to be 
 just about the same as those presented by Euclid to his classes. 
 They have been used ever since his time as models of deduc- 
 tive reasoning, and their form, as well as the geometrical facts 
 which they present, should be studied. 
 
 DEFINITIONS. 
 
 In the definitions, Euclid names the things with which he 
 proposes to deal, and states the distinguishing marks by which 
 these things are to be recognized. 
 
 The definitions should be carefully considered, and com- 
 mitted to memory as they are used, 
 
 1. A point is that which has position but has no maitni- 
 tude. * 
 
 A point is indicated by a dot with a letter attached, aa the 
 point A. A 
 
 2. A line is that which has length, but lias neither breadth 
 bor thickness. 
 
* EUCLID'S ELEMENTS. 
 
 A line ia indicated by a stroke, with a letter placed 
 at each end, as the line AB. A ^B 
 
 A letter placed at the intersection of two a^^ 
 
 lines indicates the point of intersection, as ^""^^--S-^^-^^ 
 
 the point 0. ^i---'^''^^"^-^ 
 
 d. A Straight line is one which lies evenly between its 
 extreme points. 
 
 Only one straight line can lie between two given points. 
 
 4. A S^rface is that which has length and breadth, but no 
 thickness. 
 
 6. A plane surface (or a plane) is one in which, if any 
 two points be taken, the straight line between them lies 
 wholly in that surface. 
 
 6. An angle is the incUnation of two straight Unes to one 
 another which meet together, but are not in the same straight 
 line. 
 
 The point at which the lines meet is called the vertex, and the lines 
 are called the arms of the angle. 
 
 An angle is named by three letters, one 
 placed at the vertex, and one on each of 
 the arms; but these must be arranged so 
 that the one denoting the vertex shall be 
 the middle letter. j»- ^ 
 
 ^^«8 the angle represented is called the angle ABC, or the angle 
 
 7. When a straight line standing on an- i 
 
 other straight line makes the adjacent angles I 
 
 equal to one another, each of the angles is 
 called a right angle, and the straight line 
 which stands on the other is called a perpen- 
 
EUCLID'S ELEMENTS. 
 
 8 
 
 8. An obtnse angle is one which 
 is greater than a right angle. 
 
 9. An acute angle is one wliich is less than 
 a right angle. 
 
 10. Any portion of a plane surface bounded (or contain- 
 ed) by one or more lines is called a plane figure. 
 
 11. A circle is a plane figure contained by one line which 
 is called the circumference, and is such that all-straight 
 lines drawn from a certain point within the figure to the 
 circumference are equal one another. This point is called 
 the centre of the circle. 
 
 A circle is usually named by three letters, each of which denotes a 
 point on the circumference. 
 
 12. A radius of a circle is a 
 straight line drawn from the 
 centre to the circumference. 
 
 13. A diameter of a circle 
 
 is a straight line drawn through 
 the centre, and terminated both 
 ways by the circumference. 
 
 Thus ABG iB a circle of which 
 is the centre ; OA, OB and 00 are radii, and AG in a diameter. 
 
 14. Parallel straight lines are such as are in the same 
 plane, and being produced ever so far both ways, do not meet. 
 
 Thus AB and GD are parallel straight 
 lines. 
 
 A- 
 C- 
 
 -B 
 
 15* A rectilineal figure is one which is contained b« 
 straight lines. 
 
EUCLID'S ELEMENTS. 
 
 to^r,tti^^^ " " "'""« ««- -stained by „„„ than 
 
 I 
 
 \ 
 
 three etSs'*''*^ ^"^'^"^^^ ^« «- ^hat has 
 
 20. An isosceles triangl 
 equal sides. 
 
 e 18 one that has two 
 
 21. A scalene triangle is 
 
 unequal sides. 
 
 one that has three 
 
 
EUCLID'S ELEMENTS. 
 
 23. An obtuse-angled triangle is 
 one that has an obtuse angle. 
 
 24. An acute-angled trianrie is one that 
 has three acute angles. 
 
 25. A rhombus is a quadrilateral that 
 has all its sides equal. 
 
 26. A square is a quadrilateral that has 
 all its sides equal, and all its angles right 
 angles. 
 
 27. A parallelogram is a quadri- /"' / 
 
 lateral whose opposite sides are parallel. / / 
 
 28. A rectangle is a quadrihiteral whose 
 opposite sides are parallel, and whose angles 
 are right angles. 
 
 that 
 
 29. A trapezium is a quadrilateral f 
 
 at hftfl t-AMrn aiAa€, ..- 11.1 / 
 
 A 
 
6 
 
 IBUOLID'S ELEMENTO. 
 
 POSTULATES. 
 
 He makes thei rea« Jl !u ^ ^^'^^idered successful. 
 i<et it be granted ; 
 
 AXIOMS. 
 In the axioms, Euclid makes twelve aimnl. .* . 
 the truth o£ which is self.vident tLvT™ T^"^ 
 tion on which the whole science of ge^:f JrbuUt "S. 
 
 2. If equals be added to equals, the sums are equal 
 
 3. If equals be taken f^mequals, theremainde« a,« eq»d 
 
 4. If equals be added to unequals, the sums are un«,.,.I *i. 
 grater sum being obtained froL th^ grea Ju„^„T^ *"' 
 
 unLil T''" "? **"""' *™'" "'"^"'"»- th" «n.ai„der, are 
 -nequal, the greater remainder being obtained from the ^It:: 
 
Euclid's elements. 
 
 one point 
 duced to 
 'tre, and 
 
 6. Things which are doubles of the same thing are equal 
 to one another. 
 
 7. Things which are halves of the same thing are equal 
 to one another. 
 
 8. Magnitudes which coincide with one another are equal 
 to one another. 
 
 9. The whole is greater than its part, and equal to the 
 sum of all its parts. 
 
 10. Two straight lines camnot enclose a space. 
 
 11. All right angles are equal to one another. ^ 
 
 12. If a straight line meets two other straight lines, so as to 
 make the interior angles on one side of it together less than two 
 right angles, these two straight lines will meet if continually 
 produced on the side on which are the angles which are 
 together less than two right angles. 
 
 SYMBOLS AND ABBREVIATIONS. 
 
 The following symbols and abbreviations are used in the 
 propositions :-:— 
 
 = stands for *is equal to,' 'are equal to,' or * be equal to.' 
 
 / 
 
 ({ 
 
 * angle.' 
 
 A 
 
 (€ 
 
 * triangle.' 
 
 
 
 l( 
 
 * circle.' 
 
 o~ 
 
 , i( 
 
 ' circumference.' 
 
 • 
 • • 
 
 « 
 
 * therefore.' 
 
 Def. 
 
 i( 
 
 * Definition.' 
 
 Post. 
 
 « 
 
 * Postulate.' 
 
 Ax. 
 
 « 
 
 * Axiom.' 
 
 Prop. 
 
 (( 
 
 * Proposition.' 
 
 Hyp. 
 
 i« 
 
 ' Hypothesis.' 
 
 Oonsti 
 
 • 
 
 * Construction.' 
 
8 
 
 EUCLID'S ELEMENTS. 
 
 THE PROPOSITIONS 
 
 INTRODUCTION TO PROPOSITION 1 
 
 WIawh.tU«.do.ll,uohp„i„t,i,,j A ^ 
 
 '7 '^V' » "»• ""-y point of which will b. .t ,1.. 
 
 fd) Find . point that i.'.,uidi.t«t from a «.d iS. 
 wrth th. dutanoo of th. point Tfron. ^hJ^^ 
 
 PROPOSITION 1. P„„B„,„, 
 To ^oriJ, „„ e,uila^^ triple on a yi^n .traight lin. 
 
 Let AB he the given straight line 
 It is required to describe an equilateral triangle on AB. 
 
EUCLID'S ELEMENTS. 9 
 
 With centre A and distance AB, describe £CJ). Pose 3 
 WHh centre £ and distance BA, describe ACJ:;. Post. 3. 
 Let the circumferences intersect at the point C. 
 
 Join AC and CB. p^gf 1, 
 
 -45C shall be an equilateral ^. 
 For, because A is the centre of BCD 
 
 And because B is the centre of ACE, 
 
 .-. BC^AB Wo/0. 
 
 Now, since ^C and BC are each equal to AB, 
 
 .'. AC = BC '^2x 1 
 
 . -^^T "tl'J^ *''^ ^^ *™ *" ^^"*^' »«<^ ^'^ equilateral 
 triangle ABC has been described on^A Def ofequUateraZ A- 
 
 QUESTIONS ON PROPOSITION 1. 
 
 1. If the two circumferences interaect also at J*, what kind of atrianirle 
 
 wul be formed by joining AFarndBF? 
 
 2. What kind of a quadrilateral is the figure AOBF? 
 
 INTRODUCTION TO PROPOSITION 2. 
 
 1. Define a circle Is it possible to draw a circle on a plane surface 
 
 with a pair of compasses ? 
 
 2. Stete the postulates. State Ax. 2 and Ax. 3. 
 
 3. (a) Find a point equidistant from A and B. a- fi 
 (h) Show how to describe a circle that will pass through A and B. 
 
 i. -^^CandD^^are concentric circles, having 
 as common centre the point 0. OD and OB 
 are rad-. .* the circle DEF which cut the 
 circumference of the circle ABO at the 
 points A and B respectively. 
 (a) Show that AD equals BE. 
 
 \ -J w .rom w, a point oa the circumfer* 
 
 Mioe ABOt a straight line equal to AD. 
 
10 
 
 kucud's elemento. 
 
 5. A radius of the circle ABO in p,x,duoed to 
 D, Show how to draw from B a .traight CI 
 Iwe equal to AD. "^gni; q 
 
 . VVhen AD i« not in the «me straight line a. C/"'^^ 
 
 PROPOSITION 2. Problem. 
 
 equal to a giv«n 
 
 Frmna. gi»n point to draw a straigU line 
 atrmgli. U„,, 
 
 On ^5 d«oribe th, equa.ter<U A DAS p!l , 
 
 W.th cent™ i, .„d disunc i»(7 6^XqcMF Z ' 
 
 Produce 2)5 to meet the 0« (7i?/. i^^^^^^- f f " f" 
 
 With centre D anW w,-o*„ « rr , '^-^ "* •«^' Poat. 2. 
 
 ni,re 2/ and distance DB describe jrfirjy r,^, .. 
 
 Rxxiuoe /)^ to meet fch« Oca ^p^^t .._ ^ ^' V^'' ^- 
 
 -ftwt 2l 
 
EUCLID'S ELEMENTS. 
 
 11 
 
 ) ^.— 
 
 2> 
 
 « fftv&n 
 
 ThenwiW AG ^ BC. 
 For, because D is the centre of EGJI, 
 
 .'. DE = DO. Df^f Qf /7) 
 
 Of these, the parts i)^ and i)^ are equal. Def. ofequUaieral A.' 
 .'. the remainders BE and AG are equal. Ax. 3. 
 
 And, because B is the centre of CEF, 
 
 • • ^^ = ^^- ' Def. 0/0. 
 
 But AG ^ BE. 
 
 ••• ^^ - -5C'. ^^. 1. 
 
 Thus from the point A a straight Hue AG has been drawn 
 equal to BC. 
 
 QUESTIONS ON PROPOSITION 2. 
 
 1. Under what circumstances would the point D lie outaide of the 
 ourcle CBFf 
 
 On the circumference of the circle OEF ? 
 ^' "2)^7" '^*^°"*' *^* °''°^' ^^^' ''°"^*^ ^* **** n^o^ry to produce 
 
 3. Could th: problem be solved by producing BD inatead of DB? 
 
 4. Could the problem be solved by joining A G instead of ^ 5 ^ 
 
 '' ^the Iw/^..*'?^:"' '" ^'^^ "*"* ^^'*°*^°"' ^° »•**«' ^»»i«h 
 of the above i.>ethod8 of construction is used ? 
 
 line. 
 oBC. 
 °08t. 1. 
 
 ^08t. 2. 
 
 W. 3. 
 
 '08t.2. 
 
 INTRODUCTION TO PROPOSITION 3. 
 
 AR^,AAG are two straight lines of which AB is the irreatw 
 Show how to cut oflf from AB a part equal to AG. 
 
 B 
 
12 
 
 Euclid's elements. 
 
 4 
 
 PROPOSITION 3. Problem. 
 
 to cut of h part 
 
 Front the greater of iwo given straight lines 
 eqxuu to the lets. 
 
 til 
 
 I i .- 
 
 A^«f}f ""'f^ ^ "" *"" «^™° '*'*«''* «»~. ot which 
 AJf IS the greater. 
 
 It is required to cut off from A£ a part equal to CD. 
 From A draw the straight line AF, equal to CD. Prop. 2 
 With centre A and distance AF describe ^/'^r Post 3 
 antting A£ in F. ' ' ' 
 
 Then will AF = CD. 
 For, because -4 is centre of £FG 
 
 .-. AF :=. AF. De/.o/G). 
 
 BntCD^AF. . '^^2. 
 
 .-. AF = Ci>. ^^ 2 
 
 Thus from ^^ a part AF has been cut off equal to CD.' ' 
 
 QUESTIONS ON PROPOSITION 3. 
 
 2. Cmi the required part be out oflF f«,m either end of the line ^5 / 
 f^^f -- ^^"' ^"**'°^ ^ *'** «>^truotion neoeMary tp draw .iJ& 
 
EUCLID'S ELEMENTS. 
 
 18 
 
 ines, of which 
 
 EXERCISES. 
 
 Find the solutions of the following exercises, and write the 
 proof of each solution in a form similar to that used by 
 Jl<uclid in his propositions. 
 
 length double that of A B. J 
 
 ^' ^t^r """"f^?. *"*°«'" °" ' «^^*" '•^"^»^* «°«. -"^J^ that 
 each of the equal .ide. .hall be twice a. long a. the ba«e. 
 
 0^ • «^^~ -J";i«ht line de«,ribe an i.o«M»le. triangle having each 
 of the equal sides equal to another given straight line. 
 4. Draw a straight line three time. a. long a. a given .traight line. 
 
 IZ ^ ^ '\7t^ ""' ^~°"^ '" "°^«»«- '""^gl* having each of 
 the equal .ide. three time, as long as the third side. 
 
 6. From a giv«n point C7, in a given .traight line AB, nS- . ,trai«ht 
 
 line equal to ^5. ■'•'raignii 
 
 7. Produce the leu of two given .traight line., making it equ«l to the 
 
 greater. ^ 
 
 INTRODUCTION TO PROPOSITION i. 
 
 1. 8Ute Ax. 8, and Ax. 10. 
 
 2. Whati. the meaning of 'coincide'? 
 
 ^' "^^Z?*^*" nece«anly equal, if the .traight line, which form the 
 angle, are equal, each to each T 
 
 4. Two circles have equal radii : .how 
 
 that they have eqaal area, and 
 equal circumference.. 
 
 5. Two niuare. have the .ide. of the one 
 
 "Sl,^-Stl"" "''"• ^'"' •-' '^•' •»" •»^«" 
 
14 
 
 EUCLID'S ELEMENTS. 
 PROK.HTTION 4. Thkorem. 
 
 Iw r '': "^ '^^"^ ^'^^' '""'y *^^ "'^ ^- '^^v 
 
 L rr.f "^ " f^' '^ ^'"^'^ *«^^ '^ --^' n«Ly Mot 
 to m;AwA <A« equal sideB are opposite. 
 
 In^the £^H ABC and /JiF/; 
 
 AC - /)/; 
 and ^ ^^(7 - ^ jp/jj'. 
 It is required to prove that BG » ^/^ 
 
 A ABC - A D^/; 
 / ABC - / />^/; 
 and Z -<lC-fi = / i)/'^. 
 If A 45C' be applied to A DEF, so that ^ falls on D, and 
 -i^ falls on DE, then ^ will coincide with E, 
 
 because AB - DE. 
 And because ^5 coincides wi+l i)JE', 
 and / BAC=l EDt\ 
 
 .*. ilC will fall oiiX'/'. 
 
 And because AC >m J)E, 
 
 .'. C will coincide with F. 
 
 And because B coincides with E, and C with /' 
 
 .-. i?C will coincide with EF. 
 
 For, if not, let it fall otherwise, as EGF 
 
 Hyp. 
 
 Bjp. 
 
 Hyp. 
 
ETTCLIDS ELEMENTS. 
 
 15 
 
 T},en the two atra>ht line^ JlC ^nd EF will enclose a Hpaoc, 
 which is impossible. J(r., 10. 
 
 Hence BC coincides ith and .-. ^ EF Ax, H 
 
 ^ ^^c* ' . = A Iff, 
 
 ^ ^SC • ., ^ Dji;p 
 
 and Z ACB •« «. .. ... ^ ^ 2)W. 
 
 Questions on Proposition 4. 
 1. State the axioms uaed in the proof. 
 
 !' ^ ^'i*^*""**'' ' «»«h to each,' necewary in the enunciation ? 
 3. Doe.^(7nece8«arilyfallonZ)A; if ^fi coincides with i)^? 
 
 bIoT ^''°P^'''°'*' ^^•K*""''*^ the superposition by applying 
 
 EXERCISES. 
 1. The sides of the square A BCD are 
 
 equal to the sides of the square* 
 
 EFGH. '^ 
 
 Show that : 
 
 (a) The diagonals AC and EO are 
 
 equal 
 
 (b) The diagonals AG and i?/) are] 
 
 equal. 
 
 (c) The diagonal ^C bisects, that is, divides into two equt 1 part. 
 
 the angle BAD. ^ ^ ' 
 
 (d) The squares are equal in area. 
 
 2. A straight line AD bisects the vertical angle BAG of the isr^^eles 
 
 triangle A BC, and meets the base at the point D. She that 
 D is the middle point of the base, and that AD is perpei iicu- 
 lar to BO. 
 
 3. If two straight line, bisect each other at right angles, any point in 
 
 either of them is equidistant from the extremities of the other 
 The jniddle point, of the sides of a square are joined in order. 
 Show that the quadrilateral formed by these joining lines is equi- 
 
 5. ABGD is a square. ^ is a point in AB, and i?- is a point in CD 
 sucji that AS is equal to GF ; EF is joined. Show that the »nsll 
 .^JKi- i» equal to the angle OFE. 
 
Euclid's elements. ^ 
 
 Propoiition. «•• divided into two classes, theorem, and problem. 
 
 A tiheurem u a truth that require, to be nTt,vSy.S^^ "»'*»*'•»■• 
 truths already known The f^v^ a u ^ ^^ "**'" °' °**«' 
 
 theorems. ^^* *™<^^ •I'^y known are either axiom, or, 
 
 A problem i. a construction which in fn k. ^0.1 u 
 
 the"b;rorr.:rh;crstr;ra^^ 
 
 onXaUo' r«^efTth?oth '^ '"r^"*^" '^^^ *- "^^ ^'^^e 
 
 to which the equal sidef are opposfto ^ ' "" "" **"'* '^''^ '^~« 
 ^^^re. Which may be made to coincide, are «id to be "equal inall 
 
 "^:-ot:Se"S.gTe '>'*'' ^''^^ ^^- °^ * ^-^^^ - -"ed the 
 
 part, of the other. °°' *'* ^"*^ *« *^« corresponding 
 
 Introduction to Proposition 5. 
 
 1. Define isosceles triangle. Which side of an isowele. tria««l. • 
 
 called the base? "oweie. triangle is 
 
 2. In the accompanying figure point out and name 
 
 the angle which i. 
 
 il^i m?*' f ™ °^ ^^^ *"«^« ^^^ and C5i). 
 
 r^; The diflFerence of the angles ABD and CBD. ^i 
 
 3. I»»^he figure. AB is equal to AD, and AQ is equal 
 
 ''"^''t/f;?«^^u^- Name the partsoftheAs^Ci) 
 and ^ j&^, which are equal. 
 
 (h) Prove that CD is equal to ^^ 
 
 (c) Join 5Z> Name all the parts of the as 5(72) 
 
 and DEB, which are equal, 
 rd; What kind of a triangle is a ABD .? 
 
EUCLID'S ELEMENTS. 
 
 PROPOSITION 6. Thbobem. 
 
 X? 
 
 Theories at the base of an iooscles triangle are equal; and 
 '^^P^^\he ^^oduced, the a^les on the other 
 tnde of the base shall also ba eoual. 
 
 to 
 
 It is required to prove that / ABC - / ACS 
 
 and / DBC = / ECB, 
 In BD take any point F^ 
 
 ^^ irom AE cvitoS AG, ec^VLoX to AF. Prop, 3. 
 Join BG and C^. 
 In Ae^^C'and^G?^, 
 AF = ^6r, 
 ^C = AB, 
 and / iy!4C - / G^^J?, 
 
 Z ^Jf'C - z ^(?^, 
 and I ACF » / ABG. 
 Again, because ^4^ =r AG, 
 and^-S s AC, 
 
 .'.BF - CO. 
 And in A a ^/'C and CGB, 
 BF = C6r, 
 FC - C?i9. 
 
 Constr. 
 Hyp. 
 
 Prop. 4. 
 
 ^«, 3. 
 
 fH 
 
 fifl 
 
 I": 
 
 ■h 
 
18 
 
 EUCLID'S ELEMENTO. 
 A. 
 
 . s 
 
 and / BFG -= l CGB, 
 
 and / FBC = ^ G?CJ5. 
 Now, because / ABG _ / ACF 
 and Z CJfir « ^ ^(7^^ 
 
 t*- /^ ^^C = / ACB, 
 and these are the angles at the base. 
 
 It has also been proved that I FBC = i GCB, and these 
 are the anejles on the other side of the base. 
 
 Prop. 4. 
 
 Ax.Z. 
 
 It is evident that if a triangle has all its sides equal, it has 
 all Its angles equal ; that is, an equilateral triangle is equi- 
 angular. ® ^ 
 
 a ^rJ^lr'' '^if : **^^'' ""^^'^ ^ "*^*y '^^ '"""'^y i»^«"«d from 
 a proposition, m called a ooroUapy of that proposition. 
 
 Questions on Proposition 5 
 
 1. What ia the hypotheais of Proposition 6 ? 
 
 2. What is the conclusion of Proposition § » 
 
 3. Would it do equally well to say '« In AD take any point F"1 
 
 EXERCISES. 
 
 1. Prove that the diagonal of a rhombus divides it into two isoscel.. 
 vruuigies. 
 
EUCLID'S ELEMENTS. 
 
 19 
 
 2. Prove that the opposite angles of a rhombus are equal. 
 
 3. Prove that the diagonal ot a rnombu. bisects each o* the anffle« 
 
 through which it passes. 
 
 4. Two isosceles triangles, ABGB^d DBG, have the same base BC 
 (a) Prove that the angle ABD h equal to the angle ACD. 
 
 (h) Prove that the angle BAD is equal to the angle CAD. 
 (c) Prove that AD, or AD produced, bisects the base BC. 
 6. ABO is an isosceles triangle ; and in the base BG two points D E 
 are taken such BD = GE-, prove that ^i)Jgr is an isosceles triangle. 
 
 6. Prove that the diagonals of a square divide the figure into four 
 
 isosceles triangles. 
 
 7. Two equal circles, whose centres are A and 5, intersect at the point 
 
 G. Join GA and GB, and produce them to meet the oifcumfer- 
 ences at i) and .» respectively. Join i>JB'. Prove that the an«le 
 CD^ equals the angle C^Z). 
 
 8. ^5(7 is an equilateral triangle : D, E and ^are points in the sides 
 
 AB, BC and GA, such that AD=BE=CF. Show that the tri- 
 angle DJBF is equilateral. 
 
 
 «■ \ 
 . i 
 
 ! 
 
 Introduction to Proposition 6. 
 
 1. There are two straight lines, AB and CD. 
 
 (a) liAB'vi not greater than CD, must ifl be less than CD ? Why ' 
 i\ S .i ^ °°^ *^"*^ ^CD,\&AB necessarily greater than CD ? ' 
 (c) If AB 18 not greater than CD, nor less than CD, what rektion 
 must exist between AB and CD ? 
 
 2. In the figure, AB = CD, and l ABC = l 
 
 DCB, prove that AG « BD, and that 
 A ABC = A i>(7A 
 
 
 ■>' H 
 
 ■0 
 
 i4 
 
 
\ 
 
 20 
 
 EUCLID'S ELEMENTS. 
 
 PROPOSITIOIf 6. Theorem. 
 K 
 
 In A ABC let / ABC = / ACB 
 T# An ' ^^ '^ ^«^«ired to prove that AG = AB 
 IfAC^s not equal to AB, one of them must be the greater, 
 -t^t ^^ be the greater. 
 From BA cut oflF BD equal to AC. 
 Join DC. 
 In Ab DBC &nd ACB, 
 J>B = AC, 
 BC = CB, 
 andz BBC - ^^(7^, 
 
 But thxs .sunpossible since A 2)2.^ is a part of a ^cl 
 
 Sin,-, , . t ^^ '' ^^* «^**^^ *han ^(7. 
 
 Snnzlarly it may be shown that AB is not less than AC 
 
 .'. AB ^ AC. 
 
 Prop. 3. 
 Post. 1. 
 
 Conatr. 
 
 Syp. 
 Prop. 4. 
 
 CoroUary. An equiangular triangle is equilateral. 
 
 Questions on Proposition 6. 
 1. How would you proceed to show «« thaf j » • x 1 
 ^ Wh.t « the h, A..i. of ?*;:«■»« f * " ■'°' '""'" ^ ^"' 
 a WUt is tll« oonoliMior. of Propositioa 6 1 
 
Euclid's elements. 
 
 21 
 
 4. What relation exists between the hypotheria of Prop. 6 and the ook- 
 cluMonofProp. 6; and also between the first part of the conclu- 
 sion of Prop. 6 and the hypothesis of Prop. 6 ? 
 
 Two propositiona are said to be oonver«», when the hypothesis of 
 each IS the conclusion of the other. 
 
 EXERCISES. 
 
 1. The diagonal of the square A3CD intersect at K Use Prop 6 to 
 
 prove that the triangle EAB is isosceles. 
 
 2. Prove that the diagonals of a rhombus bisect each other at riaht 
 
 angles. ^ ® 
 
 3. Show that the straight lines which bisect the angles at the base of 
 
 an isosceles triangle, form with the base a triangle which is also 
 isosceles. 
 
 4. In the figure of Prop. 1. if the straight line J 5 be produced both 
 
 ways, to meet the one circumference at D and the other at E 
 show that the triangle ODE is isosceles. 
 
 SI 
 
 u:- 
 
 
 'i 
 
 I 
 
 » 
 
 
 •B 
 
 Introduction to Proposition 7. 
 
 1. AB and CD are two straight lines 
 
 CE = AB. A 
 
 (a) How does AF compare in length q 
 
 with CEl 
 (h) How does AF compare in length with CD ? 
 
 2. ABCoifxdi DBF are two angles, 
 
 and L ABC= l HEF, which 
 is a part of l DEF. 
 (a) How does z. OBC compare 
 
 in magnitude with iL ^^Z"? jg" q ^ 
 
 (h) How does L OBC compare in magnitude with z. DEF^ 
 6. Show that two isosceles triangles cannot stand on thesame base and 
 on the same side of it, unless the vertex of the one triangle falls 
 inside the other triangle. 
 
 _ „,„ „„^ ^^^, gj^jgg ^1 ^^^ ^^^ respectively equal to 
 the three sides of the other. Can they be made to coindde ? 
 
 Ji G 
 
 ^1 
 
 •if 
 
 
\ 
 
 22 
 
 EUCLID'S ELEMENTS. 
 
 PROPOSITION 7. Theorem. 
 
 Twotriar^les on the same ha.e aruion tJ. same side of it cannot 
 luxve tJmr conterminous sides equal. 
 
 If it is -possible let the ty^^o j<% A nr Ann xu 
 A R «^A ^1 ^^y^u AS ^/yo, A/i.O on the same base 
 
 Three cases may occur 
 
 2 S!! ^'T 1 "^'' "" ""y •« ""'«'''« the other A. 
 2 The vertex o( one A may be inaide the other A 
 
 In th!^ T °'"' ^ ""y '«' "» " »d« °f the other A 
 
 In the first case ioin CD • anH in +k^ j ' 
 
 .nd p^duce AC Ji AD to /a"^" ' """"" '^'' '"» ^^ 
 
 Because AC mt AD 
 
 But / BCD is greater than ^ BCD. 
 
 AT L " "^ ^^^ ^® greater than / BCD 
 Much more then is / DDC greater than / BCD 
 But because 5C = BD 
 
 that is, Z i?Z>(7 is greater than and e'qual to / BC^' 
 which is impossible. 
 The third case needs no proof, because BC is not equal to 
 
 f^-of frn:::::;^^- -r — ^- -^ - thTi: 
 
 cannot ha^.^ their conterminous sides equal. 
 
 Prop. 5. 
 Ax. 9. 
 
EUCLID'S ELEMENTS. 
 
 28 
 
 mt 
 
 PROPOSITION 8. Theorem. 
 l/two triangles have two sides of the one equal to two sides of 
 tite other, each to each, and have likeunse tlieir bases equal, 
 the angle which is contained by the two sides of the one shall 
 be equal to the angle contained by t/ie two sides^ equal tff 
 them, of the otJier. 
 
 In the A 8 ABC and DEF, 
 let AB « DE, 
 AC = DF, 
 and BC - EF. 
 It is required to prove that I BAC ^ i EDF, 
 Apply the A ABC to the A DEF, 
 so that B is on E, and BC on EF 
 Then, because BC »= EF, 
 C will coincide with F. 
 Then AB and AC will coincide with DE and DF. 
 For, if they do not, but fall otherwise, as GE and GF, 
 then on the same base EF, and on the same side of it, 
 there will be two As, DEF and GEF, having equal 
 pairs of conterminous sides, 
 
 which is impossible. Prop. 7. 
 
 .-. BA coincides with ED, and AC with DF. 
 And hence / BAC coincides with and .-. -i / EDF. Ax. 8. 
 
 Corollary. If two triangles have the three sides of the one 
 respectively equal to the three sides of the other, the triaugles 
 are equal in all respects. 
 
 ml 
 
 f;ll 
 .hi 
 
 
 
 M 
 
 
24 
 
 l( 
 
 EUCLIDTS ELEMENTS. 
 
 1 
 
 fi 
 
 CE 
 
 QuBSTioNs ON Proposition 8 
 Apply the trianglea ao that they 
 may f.U on opposite sides of ^ 
 
 the common base EF. Join 
 
 (a) What kind of » triangle is 
 EDO? FDQf 
 
 (h) Prove that the z. ^i)/* ^ 
 
 re; Prove the proposition in this way when DG does not «« 
 between J? and Jf ^^ **°* P**" 
 
 ^ ^:p.T "" ^^" ^"^ ^ ^« *- *--«'- considered in 
 4 /!^^^*P*'''^"'P«>ved equal? 
 
 ro; What parts are proVed equal ? 
 
 re; Are the triangles equal in all respects? 
 
 e^uiSr:rtrr;:t^^^^ -^^^ - -^P-tively 
 
 respects ? *^ ^^* '"**■' ^'^^ "^^^^ ««• »ot equal in all 
 
 EXERCISES. 
 
 (<0 The oppoaite anglM an cfnaL 
 m^;a^and5Grinterse;,tinjr. <?^ ««d 5/' int6««,t 
 
EUCLID'S ELEMENTS. 
 
 Prove thAt : 
 
 (a) ^iTbifleota L DAF. 
 (h) I. BDH = L CFH. 
 
 (c) EH = HG. 
 
 (d) -4, if and K are in the same straight line. 
 
 25 
 
 ,1 
 
 PROPOSITION 9. Problem. 
 To bisect a given rectilineal angle. 
 
 Let ACB be the given rectilineal angle. 
 
 It is required to bisect it. 
 
 In ^C take any point 2>, 
 
 and from CB cut off CE equal to CD. Prop. 3. 
 
 Join DEy and on DE^ on the side remote from C, 
 
 describe the equilateral A DEF, Prop. 1. 
 
 Join CF. 
 C^ shall bisect ZilCA 
 In ^a DCF and ECF, 
 
 J>0 ^ EC, G<m8tr. 
 
 CF - CF, 
 and DF - EF, Def. of equilateral a . 
 .-. Z DCF = Z ^Ci?: Prop. 8. 
 
 That is, CF bisects Z -^CB. 
 
 '■a 
 
 ^ ?l 
 
 st'tl 
 * If 
 
 
 M 
 
 
\ 
 
 ^ EUCLID'S ELEMENTS. 
 
 Questions on Proposition 9. 
 
 O, what different oases would arise ? 
 
 3 S!fr.r"r^°j';^'"'**°''' '"""^^ ^''^ construction fail f 
 A Show that C/* bisects Z)Jgr. 
 
 2. Show that CF also biseoU the angle DFE. 
 
 PROPOSITION 10. Problem. 
 
 ^o bisect a given straight line. 
 
 C 
 
 Let AB be the given stra.>bt line. 
 It is required to bisect it. 
 On AB describe an equilateral a AhV. Prop 1 
 Bisect / ACB by CD, which meets AB at A Prop! 9.' 
 AB shall be bisected at D. 
 In the As ^Ci> and BCD, 
 AC = ^C, 
 CD =r Ci), Def. of equilateral^. 
 
 Conatr. 
 
 Prop. ^.:. 
 
 and Z ilCi) . i BCD, 
 ,'. AD = BD. 
 That is, il^ is bisected at D 
 
 Questions on Proposition 10. 
 
 2. Show that GD is at right ar-les toAB. 
 
 3. Every point equidistant from A and 5 Ues in the line CD. 
 
EUCUD'S ELEMENTS. 
 EXERCISES. 
 
 27 
 
 1. IHvide a given angle into four equal parta. 
 
 2. Divide a given etraight line into eight equal parts. 
 
 3. On a given ba-e describe an isosceles triangle such that the sum of 
 
 it« equal sides may be equal to a given straight line. 
 
 4. Produce agiven straight line so that the whole line may be five times 
 
 as long as the part produced. 
 
 5. 2), ^and ^are the middle points of the sides of an equiangular tri- 
 angle, show that the triangle DEF is equiangular. 
 
 6. Two isosceles triangles, ^5Cand DBQ, stand on the same base RC, 
 
 but on opposite sides of it. E is the mmaie po nt of AB, and F 
 the middle point of ^ C ; and BF and GE intersect at G. 
 (a} Prove that DE = J)F. ^- 
 
 (b) Prove that i. EDB = l FDG. 
 
 (c) Prove that GE = BF. 
 
 (d) Prove that l DBF = l DCE. 
 
 fe> Prove that A CiBC? is isosceles. 
 
 (/) Prove that EO = OF. 
 
 (g) Prove that i. EOD = l FOD. 
 
 (hj Prove that ^, C and D are in tha same straight line. 
 
 Introduction to Proposition 11. 
 
 1. When is one straight line said to be perpendicular to another 
 
 straight line ? 
 
 2. ABO is an equilateral triangle, and AD is a straight line bisecting 
 
 the vertical angle BAG, and meeting the base in 7) 
 fa) Show that AD bisects the base. 
 (b) Show that AD ia perpendicular to the base. 
 
7 
 
 28 
 
 EUCfLID's ELEMENTO. 
 
 PKOPOSITION 11. Phoblem. 
 
 To <^^aserai,Ke lir. perpendicular to a gi^n straight lir, 
 jrom a given point in the same. 
 
 Constr. 
 
 t ih 
 
 ^Let ^i( be the given ,tr«ght Kne, «.d C the given poirit 
 
 It « required to draw from C a perpendicular to AB. 
 In j4C take any point D. 
 From Off cut off CB equal to CD. j>rop 3 
 
 On DB describe the equilateral A DBF PrL 1 * 
 ZOXD.CF. ' ^' ' 
 
 C^ shall be perpendicular to ^ A 
 In the A s DCF and £CF, 
 
 -DC ^ EC, 
 CF - CF, 
 
 ^dDF = FF, De/.o/equilateralA 
 . . C/ IS a perpendicular to AB. Def. o/pery^endicular. 
 
 Questions ON Proposition 11. 
 '■ '"liST' ""' "• *^" ^^"^ O" -O^ *ooId b. «,„i 
 
 «. Ctt «.. proiK-itlon b. prov«l withont th. „« of Ptop. 8 1 
 
JUCLin'S KLEMENT8. 
 
 EXERCISES. 
 
 *' '''p^ia*V'"^°*" ^'""^ "*' "^'"^ ''* 'quidi-tant from a given 
 
 2. Find the line in which all points lie which are equidistant from two 
 
 given pomtfl. 
 
 3. Find, if poesible, a point which is equidiaUnt from three given 
 
 pointSa 
 
 4. Give a oon.truction for finding the centre of the circle which pa«es 
 
 through the three angular points of a triangle 
 
 *■ ^ ttn ^'°* u?r **"'*°°^ ''""^ *-*•" 8'^«" '^^^ ^ i» "q^*! to a 
 given .traight line, and whose distance from a given point £ i. 
 
 equal to another given straight line. Is this always possible ? 
 
 Introduction to Proposition 12. 
 trumgle to the middle point of .theba«,, is perpendicular to the 
 
 2. Make an isosceles triangle whose vertex is 
 
 the point C, and whose base is a part of ^ 
 
 the straight line AB, which is not limited 
 
 in length. A —— B 
 
 PROPOSITION 12. Problem. 
 To draw a straight Ivm perpendicular to a given straight line 
 from a given point without it. 
 
 Let AB be the given straight line, and C the 
 
 without it. ^ ' '- «'™ P^*'^* 
 
 It is required to draw from C a DernendinnUr f^ 4B 
 
 • I 
 
 Mil 
 
 Take 
 
 any point D on the other side of AB, 
 
30 
 
 !i^ > 
 
 EUCLID'S ELEMENTS. 
 C 
 
 I'rop. 10. 
 
 Constr. 
 Prop. 8. 
 
 With centre (7 and radius CD describe the Q EDF, cutting 
 AB, or AB produced, at E and F, 
 Bisect EF at 6^, 
 and join CG, 
 CG shall be perpendicular to AB, 
 Join C^, CF. 
 In the As Cfii'^and CGF, 
 EG = ^6?, 
 6^C = GC, 
 ^d C J' = CF, 
 
 r.lCGE^ltGF ^rop.^. 
 
 "CGiB perpendicular to AB. Def. of perpendi^uh^. 
 Questions on Proposition 12 
 
 1. Prove the proposition without the use of Prop. 8. * 
 
 2. Prove the proposition without the use of Prop. 8 or Prop. 10 
 
 3. Is there any objection to takmg the point i) ^ 
 (a) In the line AB ? 
 
 (h) On the same side as C? 
 
 EXERCISES. 
 
 oHf ' *'' '""T^"^ *"*°«^"' ^*^« «i^«" the base and the length 
 of the perpendicular drawn from the vertex to the b«e *^ 
 
 gi:e^plrftbt^'"' '^ ^^* ^'•^^ ^ ^^^^^^^'^ ^- two 
 
 given points. Is this always possible ? 
 3. From two given points, on opposite aides of a given straiaht Hn. 
 
 A f^f ^'^^^' Is this always poflsible? ^ 
 
 sfd!s7c7%7':'r>?p^' "^ '"^ ^ '^'^ *^« "id<"« points of the 
 Biaes, 2/0, C74 and AB respectively. From D a «tw.i»»,^ r 
 
 d^^ perpendicular ^ i^^fTnd iLiZoL^^^uZu 
 d^wn perpendicular to CA, meeting the former line ^0 Show 
 th»t O.P u perprndioukr to AB. 
 
EUCLID'S ELEMENTS. 31 
 
 PROPOSITION 13. THteOEEM. 
 77t6 angles which om straight line makes with another on one 
 side of it are together equal to two dght angles. 
 
 ^ E 
 
 c — 6 
 
 Let AB make with CD on one side of it, the Zs ABC ABD 
 It is required to prove / ABC and l ABD t<^gether 
 = 2 right ^ s. ® 
 
 (1) If L ABC = L ABD, 
 each of them is a right angle ; Def. of right l . 
 .-. L ABCaxid L ABD together = 2 right lb 
 
 (2) If L ABC be not = l ABD, 
 from B draw BE perpendicular to CD. 
 Then L 8 EBC, EBD are 2 right l s. 
 But L ABC and l ABD together = l 
 and L -fi^^Z) together ; 
 
 Pr&p. II. 
 Constr. 
 EBC 
 
 Ax. 8. 
 
 •I 
 
 .'4 - 
 
 fi I 
 
 ^ ^^C and L ABD together = 2 right ^s. Ax. 1. 
 
 CoroUary 1. If two straight lines intersect, the four angles 
 which they make at the point where they cut are Wether 
 equal to four right angles. 
 
 CoroUary 2. All the successive angles made by any 
 number of straight lines meeting at one 5)oint are together 
 equal to four right angles 
 
 Oorollary 3- Two st»««>»'"hf h — ^ ' 
 
 -W4J tp. Awi. esi,..,.gn^ iiucc c-ttxiuut nave a common 
 segment. 
 
 ' If 
 
 - r-»ti 
 
 *if 
 
32 
 
 Euclid's elements. 
 
 EXERCISES. 
 
 1 The .mgles ABC and ABD, whic* are made by the straight line AB 
 
 2 If <tr? : ? "" *^** *^* *°«^« ^^^ " a right angle. 
 
 L.U *''**"°' *"«**« '^^'^'^d by producing a aide of a trianal« 
 both way. are equal, .how that the\riangle i.*i.o^les * 
 
 3. Show that the angle, of a triangle formedby a diZnl 'and two of 
 
 4 r V r °^ * '^"*"' ^««*^*' ^'^ *^o right a^r 
 
 4. Con.truot an angle equal to half a right angle. 
 
 6. Make an i««cele. triangle having eaoh of it. baw angle, equal to 
 
 PROPOBITION U. Theorem. 
 
 <^posUes^des ofu make th. adjacent angles together eqUal 
 to two r^ght angles, these two straight angles shall be in or^ 
 and the same straight line. 
 
 A^JlT^l''' i^' ''' ^^ '^^^ ^^' -^PP-te sides of 
 AB, make z ABC and Z ABD together = 2 right /s 
 
 It IS required to prove i?i> in the same straight line with BG. 
 
 ^t Jijj be not in the same straight line with BC, 
 
 produce C^ to ^ ; p^^^ 2 
 
 then BE does not coincide with BD. 
 
 Now, since (7^^ is a straight line 
 
 Z ...... axxd ^ ^i?^ rogetner = 2 right ^ s. Prop. 13. 
 
[ht line AB 
 raight lines 
 i. 
 a triangle 
 
 uad two of 
 
 » equal to 
 to a given 
 
 ines make 
 angles. 
 
 liiMs on 
 er equal 
 •e in one 
 
 sides of 
 
 ith JiC. 
 
 I 
 
 ^08t. 2. 
 
 op. 13. 
 
 Euclid's elements. 33 
 
 But / ABC and /. ABD together = 2 right ^ s ; Hyp 
 
 .-. Z ABC and / ABE together . ! ABC Ind z fl!>' 
 together. 
 
 Take away from these equals / ABC, which is coinniot- ^ " 
 
 which is impossible. 
 .*. BE must coincide with BD ; 
 that is, BD is in the same straight line with BC. 
 
 Questions on Proposition 14. 
 
 1. What relation does Prop. 14 bear to Prop 13 ? 
 
 2. Show^«je necessity of the words 'on opp<.i,« .ides' in the enun- 
 
 When two straight lines intersect each other, the opposite 
 
 angles are called verticaUy opposite angles. 
 
 t 
 
 PROPOSITION 15. Theorem, 
 
 // two straight lines cut one another, the vertically opposite 
 <ingle8 sJudl be equal. ^^ 
 
 -B 
 
 Let ^^ and CD cut one another at E. 
 It IS required to prove that z AEC -= / BED 
 
 and z BEC = / AED. 
 Because CB stands on AB, 
 •. Z AEC and / ^jrc together -= 2 right / s. Prop. 13. 
 ••^'vxnuac juM, Hcanas on VD 
 .-. ZBECa^/ BED together = 2 right / a. />n>p. 13. 
 
 WrH 
 
 '■I 
 
 *.-? 
 
 ni 
 
 
84 
 
 Euclid's elements. 
 
 .-. L. AEC and / BEG together = ^ BEG and / BED 
 together. ^^ ^ 
 
 Take away from these equals Z BEG, which is common. 
 
 .-. Z AEG = z ^^i). Ax. 3. 
 
 Similarly, / J5J^C = ^ ^JET). 
 
 EXERCISES. 
 
 1. If four Btraight lines m^t at a point so that the opposite angles are 
 
 j^aal, these straight lines are two and two in the same straight 
 
 2. What reUtion does the above theorem bear to Prop. 16? 
 
 3. Show that the bisectors of either pair of verticaUy opposite angles. 
 
 in Frop. 15, are in the same straight line. 
 
 4. Show that ii AB is perpendicular to the straight line OD, which it 
 
 meets at 5, then if AB is produced to JP, BE is aleo perpendicular 
 
 6. From two given points on the same side of a given straight line, 
 ■how how to draw two straight lines which shaU meet at a point 
 m the given straight line and make equal angles with it. 
 
 6. In the figure of Prop. 16. made EB equal to ED, and ^C equal to 
 
 i?^. and jom^A DBt^dBC, Then prove the angle ^igr2 equal 
 
 7 T^ A^^n^l' T^^^°"*"""'°^K»°yP'oP«'itionafterProp. 6. 
 
 7. The side AOoi the tnangle ABO is bisected at S, and BE is drawn 
 
 and produced to F, making ^Z* equal to EB, 
 Show that: 
 
 (a) FO = AB. 
 
 (b) I. FCE = £. BAB. 
 
EUCLID^S ELEMENTS. 
 
 PROPOSITION 16. Theorem. 
 
 35 
 
 If one. side of a triavgU he produced, the exterior angle shall he 
 greater than either o/t/.e interior opposite angles^ 
 
 Let ABC be a triangle, and let BC be produced to D. 
 
 It is required to prove l ACD greater than l BAG, 
 
 and also greater than l ABC. 
 
 Bisect AC at U. Prop. 10. 
 
 Join BE, and produce it to F, making FF = FB. Prop. 3. 
 
 Join CF. 
 In As AFB and CFF, 
 AF = CF, 
 FB = FF, 
 and L AFB = l CFF^ 
 
 .'. L FAB = L ECF. 
 But L ACD is greater than ^ FCF; 
 .: L 4 CZ) is greater than l FAB. 
 
 Also, if ^C be produced to G, 
 
 L BCG is greater than l ABC. 
 
 But I. ACD = L BCG; 
 
 .'. L ACD ia greater than ^ ABC. 
 
 Constr. 
 
 Constr. 
 
 Prop. 15. 
 
 Prop. 4. 
 
 Ax. 9. 
 
 Prop. 15, 
 
 
 
 .M 
 
 ?'.r^ 
 
 "M 
 
 im 
 
 
^^ EUCLID'S ELEMENTS. 
 
 PROPOSITION 17. Theorem 
 
 Let ^5C be a triangle. 
 
 It .« required to prove the sum of any two of its angles 
 
 less than two ri«;ht angles. 
 
 Produce BC to D. 
 
 . acb'Z^' ^'^ -' "«^*- '- »•- ^ ^^^ -a • 
 
 V /f»-^ i"^*^^ "■"* *°«''"'^' >«^ *an two right /s 
 Now ^ ^5(; ,„a / ^(7i» are any two angles of thetL^ie • 
 
 Questions on Pbopositiok 17 
 
 2. State Axiom 12. 
 
 3. Enunciate Prop. 17 and Axiom 12, so as to show thatth.,,. 
 
 vewe theorems. . " •« i;o snow that they are oon- 
 
 EXERCISES. ^ 
 I. A iB a given point and BO a given straight line. 
 (a) Fmd a point in BC, whose distance from A is eaual t^ f h« l ^u 
 of another straight line BE ^ *''* ^*°«*^ 
 
 r^; Show t^t two, and not more than two. such straight linesoanbe 
 
EUCLID'S ELEMENTS. 
 
 87 
 
 (c) Show that only one perpendicular can be drawn from A to BC. 
 
 3. P is any point within the triangle ABO, and PA and PB are joined. 
 
 Show that the angle APB is greater than the angle AOB. 
 
 4. Show that two angles of every triangle must be acute angles. 
 
 6. Show that two exterior angles of every triangle must be obtuse angles. 
 
 Of what triar ^le will the throe exterior angles be obtuse ? 
 6. In the figure of Pwp. 16, show that the area of the triangle ABC is 
 
 equal to the area of the triangle FBC. 
 
 PROPOSITION 18. Theorem. 
 The greater ai(h of a triangle hxa tits greater angle oppointe to it 
 
 Let ^^C be a triangle, having ^C greater than ^ J?. It is 
 required to prove / ABC greater than / ACB. 
 
 From AG cut oflP AD equal to AB, * Prop. 3. 
 and join ^Z>. 
 Because I ADB is an exterior angle of A BCD, 
 
 .'. Z ADB is greater than Z ACB. Prop. 16. 
 
 But Z ADB = z ABD, since AB = AD ; Prop. 5. 
 .-. Z ABD is greater than I ACB. 
 Much more, then, is Z ABC greater than z ACB. 
 
 Questions on Proposition 18. 
 
 1. State the hyjMthesis and the conclusion of Prop. 18. 
 
 2. Is Prop. 18 equivalent to the theorem, "the greatest sic'w of a triangle 
 
 has the greatest angle opposite to ft " ? 
 
 EXERCISES. 
 
 1. Prove Prop. 18 by producing the shorter side. 
 
 2. A BCD is a quadrikteral of which AD is the longest side, and BC 
 
 1. , „„...„ „„;,„ „,_ oj.g,o ^ixft/ IB greater than the angle 
 
 ADO, and the angle BCD greater than the angle BAD. 
 
 V 
 
 » ' i 
 
 I. 
 
 ,,( 
 ^ 1 
 
 * 
 
 ll 
 
 IL ll 
 
 ? 'If '■•if 
 
 !( SB;! 
 . .4 
 
 4-- 
 
EUCLID*S ELEMENTS. 
 
 PROPOSITION 19. Theorem. 
 The yrea^r anyle of a triar^le J^s tk. greater si,e ^.U. 
 
 Aj 
 
 Let ABO be a triangle ha™>g ^ ^W greater than / ACB 
 
 It « required to prove ^C greater than ^A 
 
 If ^C be not greater than AB 
 
 then AC must bo equal to AB, or less tha. AB 
 
 liAC^AB, 
 
 theji I ABC = I ACB. 
 
 But it is not. 
 .*. AC is not equal to AB. 
 If -4 C be less than AB, 
 then z ABC is less than ^ ACB. 
 But it is not. 
 . : . -4 C is not less than AB. 
 Hence AC must be greater than AB. 
 
 Questions on Proposition 19. 
 
 1. Enunciate the converse of Prop. 19. 
 
 2. laProp. 19 equivalent to the theomm ««♦».- 
 
 "gle h« th, gr.at«t .id. op^^II i/*? «~""' -«'"" • *"■ 
 
 Prop. 6. 
 Hyp. 
 
 Prop. 18. 
 Hyp. 
 
 EXERCISES. 
 1. In an obtuse-angled triangle the areateafc r.-H« {- .. , 
 
 angle ; and int right-anried tri^rT^^ " ""^"^^^ '^« *'»''««« 
 the right angle. ^ ^ * **"* «'"***«''* "^^^ " opposite 
 
 "^ '!:!: *'!^'_'"^ ^'l"*^ -*-'«h* «-« cannot be drawn W . ...^ 
 j^u*b w n {{tven siraigjit line, b"^»» 
 
Euclid's elements. 
 
 89 
 
 fe opposite 
 
 ZACB. 
 
 3. 
 
 Prop. 6. 
 Hyp. 
 
 'rop. 18. 
 Hyp. 
 
 of a tri- 
 
 I obtuse 
 ppoBite 
 
 
 3. The perpendicular is the shortest line that can be drawn from a 
 given point to a given straight line: and of the others that which 
 is nearer the perpendicular is less than the one more remote. 
 
 :. Any straight line drawn from the vertex of an isosceles triangle to 
 a point in the base is less than either of the equal sides. 
 
 5. Enunciate and prove a theorem similar to Ex. 4, when the point is 
 taken in the base produced. 
 
 6 The vertical angle ABG of the triangle ABO is bisected by the 
 stnight line BD, which meets the base in D. Show that ^^ is 
 greater than AD, and GB is greater than CD. 
 
 PROPOSITION 20. Theorem. 
 
 Any two aides of a triangle are together greater tJian the third 
 side. 
 
 Let ABG be a triangle. 
 It is required to prove that any two of its sides pre together 
 greater than the third side. 
 
 Produce BA to D, making AD equal U> AC. Prop. 3. 
 
 Join CD. 
 Then Z ACD = i ADC, since AD = AC. Prop. 5. 
 But Z BCD is greater than Z ACD ; 
 
 ,'. Z BCD is greater than / ADC. 
 . '. BD is greater than BC. Prop. 1 9. 
 
 But BD = BA and AG together ; 
 •. BA and AC are together greater than BG» 
 Now BA and AG are any two sides ; 
 
 an^ 
 
 
 third sida, 
 
 aes of a ti iaugle are together greater thau the 
 
 / 
 
 '■) 
 
 
 -N 
 
 1 4 
 
 
 . V i 
 
 •^ "(i 
 
 4. '. 
 
 P -*M 
 
 ' 
 
 , 
 
 ']. 
 
 
 i 
 
 E 
 
 'f 
 
 ffl 
 
 1 
 
 4* ''.fi 
 
 m 
 
 Stniii 
 
 
40 
 
 EUCLID'S ELEMENm 
 EXERCISES. 
 
 ' '':Z!Z^l^I,^''''' ''' ''-''-' -«^« ^^ • '^-^^Ht line 
 2. P^e Prop. 20. by drawing a perpendicular from the vertex to the 
 
 3. (a) In the figure of Prop. 16, prove that OF in equal to AB 
 
 rt; Henoe prove thatthesumof any two .ide. of^ triangfetgreater 
 
 A f . J** ^'^"*'«*o the opposite vertex. i~«6 0i 
 
 ^''' J^*** *''.!;"™ °^ *^" "*^- «^ *°y quadriUte ral is greater 
 than twice either diagonal. greaiw 
 
 5. Tdj.^„, potot 0, <urf join t. tt. „g^ potat. „e tt, triugl. 
 luiguiar points, show that the sum of these ioininff lin«« {. ♦»,« i« * 
 
 ul.BdM^B»ndJCi.gM.tertI.«itheKniiofi>S«ndi>(7 
 
EUCLID'S ELEMENTS. 
 
 41 
 
 PROPOSITION 21. Theorem. 
 If^ from the ends of a aide of a triangle, ttiere he drawn two 
 straight lines to a point within tlte triangle, these will he 
 together less than the other sides of the triangle, hut will 
 contain a greater angle. 
 
 Let ABC be a A, and from B and C let BD and CD be 
 drawn to any point D within the A. 
 
 It is required to prove that BD and DC are together less 
 than HA and AC, but that Z BDC is greater than / BAC. 
 Produce BD to meet AC in E. 
 Then BA and ^i^ are together greater than BE. Prop. 20. 
 
 Add to each EC. 
 Then BA and AC are together greater than BE and EC. 
 Again, DE and EC are together greater than DC. Prop. 20. 
 
 Add to each BD. 
 Then BE and EC are together greater than BD and DC. 
 And it has been shown that BA and ^C are together greater 
 
 than^^and jFC; 
 .*. BA and AC are together greater than BD and DC. 
 Next, because Z BDC is greater than I DEC, Prop. 16. 
 and I DEC is greater than iBAC^ Prop. 16. 
 
 •. I BDC is greater than ^ BAC, 
 
 EXERCISES. 
 L In the figure ^f Prop. 21, join DA, and ahow that the Bum of DA, 
 
 DB and DO la less than the sum of the aides of the triangle ABC, 
 
 but greater than half the sum. 
 — «a i^n^ uguxc ut i ivp. ii, suuTT ;.ziax) cne angle i^i^O' la greater than 
 
 the angle BAO, by joining AD and producing it towards the baae. 
 
 r, 
 
 ■{It 
 
 
42 
 
 EUCUD'S ELEMENTS. 
 
 PROPOSITION 22. Phoblem. ^ 
 
 ^trarght Unes, any tu,o o/u^hich are greater than ths Zl 
 
 A- 
 
 B — 
 C- 
 
 Let A,B,Ch^ the three given straight lines, any two of 
 which are greater than the third. ^ 
 
 It is required to make a triangle the sides of wh,V>, »K ii u 
 respectively equal to 4, ^ and ^ ^ '*'*" ^" 
 
 Jr^r/""''* "^^ ''^ ^^""^^"^^ '^^ ^' b"t unlimited 
 From it cut off 2)/^ = ^ , i.G' = B, GH = ^ P.o« 3 
 With centre F and radius FD, describe the DK^' 
 With centre G^ and radius GH, describe the HKL 
 cutting the other circle at K. ' 
 
 Join iT/; KG, 
 
 KFG is the triangle required. 
 
 Because FK = *7) 
 
 .-. fkIa. 
 
 Because GK = GH, 
 .'. Gfir= a 
 
 And FG was made equal to B. 
 .-. A ^^G' has its sides respectively equal to ^, ^ and C. 
 Questions on Prop. 22. 
 1. Why doea the enunciation state that anv two of <ih« „,• 
 
 together greater than the third f ^ * *™ ^^ *^« 
 
 '^ ^l^l^'J^T^^ °!/ - --»- -t'-ct. if the sum of their 
 
 ^«/o/0. 
 ^e/o/0. 
 
'',). I'- 
 
 three gimn 
 ■ the third' 
 
 E 
 
 y two of 
 
 shall be 
 
 nlimited 
 
 Prop. 3. 
 
 id a 
 
 inea are 
 9f their 
 
 43 
 
 EUCLID'S ELEMENTS. 
 
 PROPOSITION 23. Problbm. 
 At a given point in a given straight line, to make an angle 
 equal to a given angle. 
 
 c ^ 
 
 Let AB be the given straight line, A the given point in it, 
 and / DCF the given angle. 
 
 It is required to make at A an angle eqiwl to / DCU, 
 In CD, CE take any points 2), E. 
 SovaDE. 
 Make A AFG, «nch that AF = CD, FG = DE, and 
 
 • ^^ = ^^ Prop. 22. 
 
 i^^(y is the required angle. 
 In the As AFG and CDE, 
 AF = CD, 
 AG = CE, 
 and FG = DE, 
 
 .: Z FAG = z DCE. Prop.^. 
 
 EXERCISES. 
 
 1. Prove Prop. 23, giving all the oonfltruotiou, instead of aasumina 
 
 Prop. 22. * 
 
 2. Conatruct a triangle, having given two sides and the angle between 
 
 them. 
 
 3. Construct a triangle, having given the base, and having the angles 
 
 adjacent to the base ennal fn fwn /*i^on o..»i»- 
 Is tni8 always possible ? 
 
 -i 
 
 
44 
 
 EUCLID'S ELEMENTS. 
 
 //• . . .' , ^^^POSITION 24. Theorem. 
 
 the other, each to ea^h, hut the contaM armies ur^J 
 
 TJZ "^^'"^ '^^^ ^*' ^ '^ ~ -" 
 
 aw^^e 8h^.. _ greater uAaw //^e hase of the other. 
 A 
 
 AC = DF, but Z ^^C greater than L EDF 
 It 18 required to pr<^ve BC greater than EF. 
 
 At i)make Z iE'i?6' equal to / BAG. 
 Cut off i)6? equal to AG or 2)^?: 
 Join EG. 
 Bisect z /7)6r by i)^, j^eeting ^6^ at ^ . 
 and if F does not lie on EG, join J^ir. 
 In the As i5C7and 2>^G?, 
 5^ = ED, 
 AC = i>cr, 
 and I BAG ^ i EDG, 
 
 .-. BG = ^G^. 
 Again in the As FDR and G^iJjy, 
 FD = 6^2), 
 i?Zr = DH, 
 and Z i^^^ = I QDH, 
 
 ^"Zeh ""h ^J ^"^^ = "^^ -d e/r togethe/'r'i.^; 
 But iJ^^dZ-ir are together greater than ^^; p^.^ 
 .*. -£'61^ is greater than EF ; 
 ''. BCiB greater than EF, 
 
 Prop. 23. 
 Prop. 3. 
 
 Prop. 9. 
 
 Hyp. 
 
 Corutr. 
 
 Constr. 
 
 Prop. 4. 
 
 Conatr. 
 
 Corutr. 
 Prop. 4. 
 
1$ 
 
 EUCLID'S ELEMENTS. 
 PROPOSITION 25. Theorem. 
 
 45 
 
 r/two trtanffles have two aides of tU one respectively equal to 
 two sides of the other, hut tJieir hoses unequal, tJte angle 
 contained hy the two sides of the triangle which has the 
 greater hase shall he greater thmi tlie angle contained by 
 the two sides oftlie other. 
 
 Let ABC, DEF be two triangles, of which AB = DE, 
 AC ^ DF, but base BC is" greater than base EF. 
 It is required to prove zBAC greater than Z EDF. 
 If Z BAC be not greater than /. EDF, it must be eitner 
 equal to it or less than it. 
 
 But z BAC is not = z EDF, 
 for then base BG would be equal to base EF. Prop. 4 
 But it is not. Uyp^ 
 
 And Z BAC is not less than Z EDF, 
 for then baae BC would be less than base EF Prop. 24. 
 
 But it is not. Hyp 
 
 .'. Z EAC must be greater than Z EDF. 
 
 EXERCISES. 
 
 1. Show thftt Prop. 24 and Prop. 26 are converse propoBitions. 
 
 2. Aasnming the truth of Prop. 26, deduce the truth of Prop 24 
 
 S. DiB the middle pomt of the side BG of the triangle ABC. ' Prove 
 that the angle ABB is greater or less than the angle ADG, accord- 
 uigMtABia greater or less than A O. 
 
 4. State and prove the converse of the preceding theorem. 
 
 j. I 
 
 
\ 
 
 46 
 
 EUCLID'S ELEMENm 
 
 PROPOSITION 26. Theorem. 
 
 If two trmr^les Juwe two avgles of ths ovs equal to two <mgU, 
 of tJui other, each to each, and one aide equal to one aide 
 namely either th^ sides adjadent to the equal amgles, or the 
 aides opposite to equal angles in each ; then ahall the other 
 Sides be eqwa, eaeh to each, and also the third angle of tJie 
 one equal to the third angle of the other. 
 
 l?n AaABC and DBF, 
 let L ABC = L DBF, and l ACB = l DFE 
 
 First let the sides adjacent to the equal ^ s in each be equal, 
 
 chat IS, let BC = FF. 
 It is required to prove that AB ^ DE, AG =^ DF 
 
 and L BAG = l EDF. * 
 
 For if^^be not equal to M, let AB be the greater, and make 
 GB=.DE, uad join GC. 
 
 Then in As GBG and DEF, 
 GB =^ DEy 
 BC = EF, 
 and L GBG = l DEF, 
 
 .'. L GOB = L DFE. 
 But z. ACB = L DFE, 
 
 .-. L GGB = L ACB; 
 which is impossible. 
 .'. AB is not greater than DE. 
 Similarly it may be shown that AB is not less than DE. 
 
 Prop. 4. 
 jffyp- 
 
EUCUD'S ELEMENTS. 
 
 47 
 
 Then in ^B ABO and DJSF, 
 because ili?= 2) j; BC^£F,and l ABC^ l DBF, 
 
 .'.AG^DF.fuadLBAG'^LEDF, Prop. i. 
 Next, let the sides which are opposite to equal angles in each 
 triangle be equal, that is, let AB^DE, 
 
 It is required to prove that AC » DFy BC = EF 
 a.nd/:BAC = Z EDF. \ 
 
 *Z DEF, 
 
 Frop. 4. 
 Hyp. 
 
 F 3C be not equal to EF, let BG be the greater, and make 
 Jll = i^i?; and join ^iJ. 
 
 Then in As ABH and DEF, 
 AB^DE, and BH^EF, and / ABB 
 .-. lAEB^ Z.DFE. 
 But L AGB^ L DFEy 
 .'. L AHB=^ L, ACB; 
 that is, the exterior ^ of A AEG is equal to the interior and 
 opposite L AGB, which is impossible. ' Prop. 16. 
 
 .-. BG is not greater than EF. 
 Similarly it may be shown that BG is not less than EF. 
 
 .'.BG^EF. 
 
 Then in As ABC and DBF, 
 
 because AB^DE, BG^EF.ixnd l ABC^ l DEF, 
 
 .-. AG^DF, and Z BAG^ / EDF. Prop. 4, 
 
 
is 
 
 Euclid's elemeni^l 
 exercises. 
 
 I. Tha angle ^^C is bisected by the line 42). Prom 2) the line. i)5 
 
 w /f« ' ni'*'^ "^^^ *^* *°«^*' ^ ^^^^ ^/>C equal. Prove 
 viiat i^if, I/O are eqoaL 
 
 '' ]L*'l)**Prov«tlI^.'-.'^ "^le^^^of the triangle :45(7meet.5C7 
 
 ^* SLTn ^*•",,'^^*" ^°^*^'' triangle J5(7 are billeted by the 
 line. £D, GU, which meet the opposite ddes in J) and B r Jpec- 
 tively ; prove that BD and OE are equal 
 
 4. ^7 point in the bisector of an angle is equidistant from the arm. 
 of the angle. 
 
 ^" ^tid^ ^^* ^ *^^ ^^ °^ * ^'^^^^ ""^^"^ ^ equidistant from 
 
 6. Prove Prop. 26 by superposition. 
 
 ^' nf !r "'^f" °^ V'^'^»« beproduoed. prove that the two bi^ctors 
 
 lit Till '^ r^ "^^ ** " ^^^ •qoWirtant from the three 
 ude. of the tnangle. 
 
 EXERCISES ON PROPOSITIONS 1-26. 
 
 ^' 'ZVn'^^ '^leBACof an isosceles triangle ABO is bi««ted 
 b^i). If any pomti^ be taken in AD, prove that EB equals 
 
 2. ABO is a triangle such that AB equals JC. A line OD is 
 drawn to cut^5 in 2>, the point ^ is taken in AC such that AE 
 equals ^A Rove that th« angled JS^5 i. equal to the angle ADC. 
 ITJ!^T ^^^-^^•^^-^t^-l- equal, the op^te an- 
 
 '' It^r/T '*^!'* "°" "^ ^ " * ^^^^-^ Po^<^ J d'*^ through 
 -4 and C a straight Une AD whose length is four times AB. 
 
 5. Describe an equilateral triangle having a given point A for the 
 
 m^dlepointof oneside, and having each of th'sides eq^m 
 
 length to a given straight line 5a 
 
 U *t^C- - ?. - «'"^''" 5 „d (7 in th. oirouorf^L 
 
EUCLID'S ELEMENTS. 
 
 40 
 
 7 Upon the baM BG describe a triangle such that the perpendionlar 
 from the vertex on the base equals a given straight line EF, and 
 outa the base in 2>. 
 
 8. Describe a quadrilateral having given the length of each eide. and 
 of one diagonal. 
 
 9. (3onstruot a quadrilateral having each of its sides equal to a given 
 straight line, and having one of its angles equal to a given angle. 
 
 10. If thf-dxagonab of a quadrilateral bisect each other, the opposite 
 sides and angles are equal. 
 
 11. Any three sides of a quadrilateral are together greater than the 
 fourth side. 
 
 12. If a quadrilateral has four equal sides and equal diagonals, it is a 
 square. , 
 
 13. If two right angled triangles have equal hypotenuses, and a^ide of 
 
 ^A ?D *^')J ^ * "*** °^ *^® ****"*''' *^®y "« ^"^^l i^ e^«ry respect. 
 
 14. AB, AO are the equal sides of an isosceles triangle ; BD and CD 
 are drawn perpendicular respectively to AB and ^Ctomeetin 
 ^. Prove that ^2) equals CA-and that AD bisects the angle 
 
 15. ABOiB a triangle. BG is the base, and ^(7 is greater than ^5. 
 A wjorned to A the middle point, of BG. Prove that the angle 
 ^DO* 18 obtuse. * 
 
 16. If B is any point in AD, of Ex. 15, and ^ be joined to 5 and (7. 
 prove that ^fl is less than ^a 
 
 17. P is any point in the plane of the angh BAG; show how to draw 
 through P a straight line, which wiU make an isosceles triangle 
 with the arms of the angle BA G. 
 
 18. The base of a triangle is produced both ways, and the exterior an- 
 gles are bisected. Prove that the point of intenection of these Unes 
 « eqm^tant from the sides of the triangle ; also, that the line 
 jouung this pomt of intersection to the opposite angle of the tri- 
 angle bisects the angle. 
 
 19. If one side of a triangle be less than another, the angle opposite the 
 less side must be acute. etr^y^ vue 
 
 20. ftove that the sum of the perpendiculars, diuwn from the vertices 
 triL iT ''^^'** "****' " ^^ *^*'' *^* perimeter of t' a 
 
 21. IJ~ve that tfie sum of the three lines joining the middle points of 
 
 the three sides of a iriA-aoiof^i^^i,. :^_ ::_..-, . , V 
 
 peiaruneter of the triangle. 
 
 -:i! 
 
 
60 
 
 EUCLID'S ELEMENTS. 
 
 26. 
 
 ^' ^D^ ^^" 1f ^' "" " ^^ ^*"* ^ '''' ^ ^^' P'ove that 
 
 5io^. rn ^"^^ ^' '"^ ^'^^ "«^" ^^^ " «q'«^ *o the angle i^C7i? 
 Show that A Bin equal to ^ C ^ 
 
 ^l.th.n..ddl.pou.toJ .la,„d Othemiddl. point of a<7 JIF 
 0^«,„»I to ifft Show tut S.B.MK U, in the „„, .t;™:^ 
 
 How many solutions are generally poHsible ' 
 the base, and the sum of the siles. 
 
 ^* ^Ta .V").^/'' h'^vinggiven the b«. ,, one of the angles at the 
 base, and the diflTerence of the sides. 
 
 ^' Sirr*^" K !,*'^"5^* " "^"** ^ *^« «"™ of the othe. two. the 
 tf^glo can be divided into two isosceles triangles. 
 
 o/Zm r f "".1* ^"•^<Jrilavera, and tne angles adfacent U> one 
 of them; construct the quadruaterai. 
 
 ^* rf Jh?^ ^ ^^n^' *"*?«^' ^^" • " J'^^^d *o A the middle point 
 J 1a^' • / ^^Pfoduced to E, making 2>i2? equal to ^ ofLd 
 l^^d iET joined. Prove that the angle EBD is equal to the ingle 
 
 33. i>isthe middle point of the hypotenuse AB of the right-angled 
 triangle ABC. CD is drawn and produced to J^, mfkinri^ 
 equal to CD Show that the angle AEB ^s a right a^gle. ^ 
 ^"^"^otarhombushavingitsdiagonalsequaltotwo^ven straight 
 
 36. ^^OZ), EFOff are two squares. If they be placed so that F falls 
 on a and ^JK? along CD, show that FG will either faU along CA 
 or in the same straight line with it. ' 
 
 ^. The three exterior angles of a triangle, made by producing the sides 
 m succession, are together greater than three right angles. 
 
 <*7* Through a given |wint draw a straight line, such that the per- 
 
EUCLID'S ELEMENTS. 
 
 51 
 
 Prove that 
 C. 
 
 luoed toi> 
 Jigla JSCB. 
 
 re A BCD, 
 iBC. EF 
 iced, mak- 
 le straight 
 
 l^enuBe and 
 
 ' the base 
 
 » opposite 
 
 angles at 
 
 (les at the 
 
 • two, the 
 
 int to one 
 
 die point 
 ADt and 
 bhe angle 
 
 it-angled 
 Eing DS 
 
 straight 
 
 .ti^ falls 
 ong CJ5, 
 
 bhe sides 
 
 the per- 
 
 pendioulars drawn from two given points to it may be equal, and 
 on opposite sides of it. 
 
 38. In a given straight line find a point the diflferenoe of whose dis- 
 tances from two given points on the some side of the line may be 
 the greatest possible. 
 
 Examine the case where the given points are situated on opposite 
 sides of the given line. 
 
 39. Three equal straight Imes, OA, OB and OC are equally inclined to 
 each other ; if their extremities, ^, JS and C be joined, prove that 
 ABG is an equilateral triangle. 
 
 49. If the angles, AOB, BOO e^nd GOA, in Ex. 39, be bisected bv the 
 lines OD, OJUhad 0^ respectively, which meet AB, BG and CA 
 mD,E and F; prove that OD, OE and OF are equal and^ually 
 inclined to one another. 
 
 41. If through the extremities of the lines in Ex. 39, perpendiculars be 
 drawn, provfc that the triangle formed by these perpemtioulum ia 
 equilateral, and that ^, 5 and C bisect the sides. 
 
 42. On the sides of a square^ and remote from it, equilateral triangle* 
 are d -scribed, and their vertices joined in order. Prove that the 
 quadrilateral so formed is equiUteral, and that its diagonals and 
 those of the square intersect in the same point. 
 
 43. Will Ex. 42 be true, if, instead of describing equiUteral triangles, 
 w6 describe equal isosceles triangles ? 
 
 Note.— In the following deductions, assume as the definition of a 
 
 rectangle, that it is "a quadrilateral, with its opposite sides equal, 
 
 and all its angles right angles." 
 
 In a rectangle, prove : 
 
 (a) That the diagonals are equal. 
 
 (5) That the angles made by a diagonal and a pair of opposite sides 
 are equal, and that either diaeonal bisects the rectangle. 
 
 (c) That the diagonals bisect eacn otner. 
 
 (d) That the diagonals and the sides make four isosceles triangle •. 
 Let ABGD be a rectangle, ^and ^the middle points of ^Z? and 
 GB respectively. Join ED, EG and EF, and prove : 
 (a) That ED equals EG. 
 
 (6) That ^i^is perpendicular to GD. 
 
 _s ...... „„t„v ^i- io pjjj-p-mjj^jjjjjj. x^ JIJ;^^ 
 
 (c) That AEFD and EBGF are equal rectangles. 
 
 44 
 
 45 
 
 ^.«_■' J 
 
52 
 
 Euclid's elements. 
 
 47. "-^iathemiddlepoiiitof <7i)m Vr An. • • itto 
 
 '*■ J°itV!.'f,'*'7'"'!" " »A«nd /-tb. „Mdl. point „,OT 
 
 height and th, J'.f Sf^lSlAtder""^ °' '" '"'^'"-'" 
 50. Show how to cut a rho.bu. «, a. to fo™ out of th. part, . ^ 
 
*' 
 
 
 bi:.emit:nta.iiy ^t^oebii^. 
 
 CHAPTER I. 
 
 Arithmetical Numbers. 
 
 We measure and compare concrete quantities of tlie^ same 
 kind by stating how many times a certain definite quantity 
 of that same kind, called the unit of mectstirement, must be 
 repeated to make up each of the given quantities. We indicate 
 the number of times by the symbols— 
 
 ], 2, 3, 4, 5, 6, 7, 8, 9, 
 and combinations of these, according to the decimal system of 
 notation. 
 
 When there is a remainder, and we cannot arrive at the 
 quantity to be measured by repetitions of the unit of measure- 
 ment, we divide the unit into halves or thirds or tenths, etc., 
 and then measure the remainder in terms of these subdivisions 
 of the unit. We indicate the result by such symbols as — 
 h h ru or -7, etc. 
 
 These symbols, used to represent numbers, are dealt with in 
 arithmetic. 
 
 Algebraical Numbers. 
 
 There exist pairs of quantities of such a nature that the 
 one counted in with the other annuls the other, in whole or 
 in part. Such as — 
 
 loss €md gain, 
 distance up and distance down, 
 addend and subtrahend. 
 
 ft 
 
 if 
 
 Pu,. h 4\ 
 
04 
 
 ELEMENTARY ALGEBRA. 
 
 
 These quantities can be measured and compared in tenns of 
 related units, such that one unit of the one kind destroys or 
 cancels one unit of the other kind. 
 Such related units are— 
 
 one dollar loss and one dollar gain, 
 one yard up and one yard down, 
 1 to be added and 1 to be subtrai ted. 
 It has been found convenient to denote these related units 
 called respectively the positive unit and the negative Unit' 
 by the symbols — ' 
 
 ♦ +1 a/nd -1. 
 
 + 1 represents the unit first chosen, and consequently is 
 the fundamental unit of algebraical numbers. 
 
 Repetitions of these ynits (and of subdivisions of them) are 
 indicated by the symbols— 
 
 + 1, +2, -f 3, +4, etc., +^, +1, etc., 
 -1, -2, -3, -4, etc., -|, -|, etc. 
 
 These are the symbols used to denote the numbers of element 
 ary algebra. 
 
 The mark + is caUed the positive sign, and is read plus 
 The mark - is called the negative sign, and is read minus 
 Algebraical numbers which differ only in sign are called 
 
 complementary numbers. 
 
 The sign + is often omitted, and when neither + nor - 
 IS placed before an arithmetical number, the sign + is to be 
 understood. 
 
 In dealing with these algebraical numbers, arithmetical 
 resulte are included, for either of the above series of numbers 
 includes all the numbers of arithmetic. The value of an alge- 
 braical number, considered independently of its sign, is called 
 
 Its absolute, or arithmetical, value. 
 

 ELEMENTARY ALGEBRA. 
 
 66 
 
 +1 indicates that the positive unit is taken once, and is 
 read plus one. , 
 
 +2 indicates that the positive unit is taken twice, and is 
 read plits two. 
 
 — 1 indicates that the negative unit is taken once, and is 
 read minus one. 
 
 - 2 indicates that the negative unit is taken twice, and is 
 read mimi8 ttoo, 
 
 + 1 and +1 counted together give +2. 
 - 1 and - 1 counted together give - 2. 
 -I- 1 and - 1 counted together give 0. ^ - 
 In arithmetic, when one yard is the unit, the number 3 
 indicates the measurement of the quantity three ya/rds. 
 
 In algebra, when one yard up is the positive unit, the 
 number -f 3 indicates the measurement of the quantity three. 
 yards up, and - 3 the measurement of the quantity tJi/ree yards 
 doton. 
 
 Exercise 1. 
 Find the algebraical numbers which represent the following 
 measurements : 
 
 1. Thirty feet up when the positive unit is one foot up. 
 
 2. Thirty feet down when the positive unit is one foot up. 
 
 3. Thirty feet up when the positive unit is one foot down, 
 
 4. Thirty feet down when the positive unit is one foot down. 
 
 6. A gain of 25 dollars, the positive unit being a gain of 
 5 dollars. 
 
 6. A loss of 40 cents, the jositive unit being a gain of 
 10 cents. 
 
 7. Ten miles east, when the positive unit is one hundred 
 yards west. 
 
 8. If a debt of 5 dollars is represented by + 5,, what wiU 
 + 3 represent 1 What will - 4 represent ? 
 
 ^ 
 
 
 1 
 
 f 1 
 
 
 1 
 
 i i:?" 
 
56 
 
 ELEMENTARY ALGEBRA. 
 
 9. What is the poBitive unit when a debt of 10 dolUirs is 
 represented by + 2 ? "oiiara is 
 
 10. What is the negative unit when an expansion of 3 
 inches is denoted by - 2 ? 
 
 ADDITION. 
 
 u Ju "^S "• '*' """**" "' --«»» <»««'-er the 
 
 th„^^T'"i" "?*!'"! ™ "*■ """'«" •'y ""»«»« together 
 the umt. indicated ; but we m«at remember that complement^ 
 
 «7 number^ «uch as + land - 1, or +2 and -2.„r +3«.d 
 - 3, etc., when added cancel each other. 
 
 Thus, + 2 and - 2 when added give 0. 
 
 (( 
 
 « 
 
 
 + 5. 
 -6. 
 -i. 
 + 1. 
 
 + 2 and +3 
 -2 and -3 
 + 2 and' -3 
 - 2 and + 3 
 
 EXBRCISB 2. 
 
 Add the numbers — 
 
 1. +2, +3, +5. 
 
 2. -2, -e, -4. 
 
 8. +1. -2,+3, -4, H-5,-6. 
 
 4. -12, -15, +14, +20, -36, +60. 
 
 6. +h -I +1, -f 
 
 6- + n, - 2f, +6^, - 2,!^,. 
 
 7. 32, -16,14, -28, +30, -60,60. 
 
 8. 1600, -18-27, +131-4, -1416, -1513 
 
 9. State a rule for the addition of two numbers which hay 
 like signs. 
 
 10. State a rule for the addition of two numbers which have 
 unlike signs. 
 
 11. Add the algebraical numbers which represent the quaa. 
 titles, 15 dollars loss, 18 dollars gain and 20 dolkrs gain, 
 ^hen one dolkr loss is the poMtivs aait. 
 
ELBMENTART ALOIBRA. 
 
 57 
 
 SUBTRACTION. 
 
 By the subtraction of one number from another we mean 
 the taking away, or counting out, from -Uie latter the units 
 indicated by the former. 
 
 If we wish to subtract +3 from a given number we can do 
 so by counting in - 3 with the given number, since this will 
 have the effect of cot':>? ig out + 3. 
 
 To subtract -\-4 horn -f , we add +7 and - 4. 
 
 To subtract -< f i m -I V ; we add + 7 and + 4. /" 
 
 To subtract - 4 ft . i^ - 7, we add - 7 and +4. 
 
 To subtract +4 from - 7, we add - 7 and - 4. 
 
 That is, to subtract an algebraical number we add the com- 
 plementary number. 
 
 Exercise 3. 
 
 1. Subtract +3 from +6. 
 
 2. Subtract - 3 from 4-5. 
 
 3. Subtract +3 from -5. 
 
 4. Subtract - 3 from - 6. 
 
 5. From - 36 take - 20, and add - 30 to the result. 
 
 6. From the sum of - 28 and - 32 subtract the sum of - 60 
 and +30. 
 
 7. Add - 13, - 12, +18, ^d subtract the result from the 
 sum of 8, - 5, - 20 and + 30. 
 
 8. From the Bum of - 12, +15, - 13 and 10 take the sum 
 of -18 and +27. 
 
 9. What must beadded to the sum of - 4, - 6 - 6 and + 8 
 to give +131 
 
 10. "What number must be subtracted from the sum of - 19 
 — 20 and - 5 to give a remainder + 17 ? 
 
 I ■ 
 
 
 "-■j'f" 
 
58 
 
 ELEMENTARY ALGEBRA. 
 
 MULTIPLICATION. 
 
 In arithmetic we multiolv 7 hv <» K„ 
 and 7 and 7, and obtaj 21 th„f • ""^r""'""* '"'^"•^ ^ 
 
 we did with the unuS !L^3 V :'*' ^ """ ""'"* 
 bv « »• j; J d • ^' -*''»' when we multiolv « 
 
 ParS th»r * '^*° ''^ '^•»' P"*« ""'J t^k" four of tt2 
 n*:^ f "^ "' '" ^■*'' * '-'" -"-- - ^i<l with the^ 
 
 Similarly, in algebra, we multinlv + 7 hv j. <i i, j • . , 
 + 7 just tl«.t which we did with*^ "^ I to^et + 3^ fh7 "'^ 
 count together + 7 and + 7 and +7 and fb Jn 1 2^ V 
 we multiply + 7 by - 3, by counting toget^er^" l!^ ,^ 
 - 7, and obtain - 21 ; for wo oKf^,-« S <• , , 
 
 -'rdt^n'l'f ^'~' -««"•« -rand -7and 
 
 .'°w"::hfc?J,-^----«-'"--rand .7and 
 
 Exercise 4, 
 
 1. Multiply +4 by +2. 
 
 2. Multiply - 4 by +2. 
 
 3. Multiply + 4 by _ 2. 
 
 4. Multiply - 4 by - 2. 
 
 5. Multiply the sum of +10, -8 and iflK. *u 
 -4,-6 and +8. and - 16 by the sum of 
 
 I' m!t itt^ "*" f u"" P"^""' °' '"" P-'*-o --be- ) 
 
 J: .r:: ' '«" "' *"" j"^"°^°' *^° »»»•«,, wuch 
 
BLEMENTARY ALGEBRA. 
 
 59 
 
 DIVISION. 
 The division of one number by another means the finding 
 of that number which multiplied by the latter number gives 
 the former. 
 
 We wish to divide +6 by +2. Now we counted +1 
 twice to obtain + 2 ; hence to divide + 6 by + 2 we must find 
 that number which counted twice will give +6: therefore 
 + 3 is the result. 
 
 To divide +6 by -2 we must find that number which, 
 after having its sig i changed, and being counted twjce, wiU 
 give + 6 : therefore - 3 is the result. 
 
 Similarly, - 6 divided by + 2 gives - 3, 
 and - 6 divided by - 2 gives + 3. 
 
 EXERCiSE 5. 
 
 1. Divide +12 by +4. 
 
 2. Divide -12 by +4. 
 
 3. Divide +12 by -4. 
 
 4. Divide -12 by -4. 
 
 5. Divide the sum of - 20 and - 40 by the sum of + 30 
 and -20. 
 
 6. From the product of - 40 and - 40 subtract the pro- 
 duct of -30 and -30, and divide the result by the sum of 
 + 40 and -30. 
 
 7. What is the sign of the result of dividing a positive 
 number by a positive number ? 
 
 8. What is the sign of the result of dividing a positive 
 number by a negative number 1 
 
 9. What is the sign of the result of dividing a number 
 by another of diflferent sign 1 
 
 10. What is the sign of the result of dividing a number 
 by another of the same sign ? 
 
 
60 
 
 ELEMENTARY ALGEBRA. 
 
 nu!^ll i "'"^ "^ distinguish the two series of algebraW 
 
 of ^dV ?V '^"^ - -«--^ Vindicate the o^lt^ 
 of addition and subtraction respectively, as in arithm^ic 
 
 The signs x and - are also used to indicate multiplication 
 and division respectively. "^wpucatjon 
 
 The bar —. also indicates division. 
 
 An algebraical number is sometimes enclosed in brackets 
 to distinguish the two uses of + and - . 
 Thus: ( + 5) + (-3) « +2. 
 (-7) - ( + 3)= _io. 
 ( + 5) X (-3) = -!6. 
 (+12) 4- (-6)= -3. 
 + 12 
 ♦ - 6 
 
 -2. 
 
 Tj,. , ^, , , Exercise 6. 
 Find the value of ; 
 
 i. ( + 5) + (-6) + ( + 7) + (-8). 
 
 2. ( + 4) -(-4) + (-4) -( + 4). 
 
 3. ( + 5) X (-6) + ( + 3) X (-2). 
 
 4. (-10) + ( + 2) - ( + 12) + (-3). 
 6. ( + 5) X (-5) + ( + 6) + (-2). 
 
 6.( + 1^2) + (^-4) + (-3)x(.2)-( + 6)x(_n 
 7. 
 
 8. 
 
 -60-30 -40 
 -12 -5 "^Ti' 
 -20 _40 60 
 + 4 -4 " "e 
 
 9. (-15) + (-3) + (-40) + ( + 4) - (-20) 4- (-5). 
 
 10. 
 
 -52 
 • 4 
 
 
 -60 
 4 
 
 __10 
 10 
 
ELEMENTARY ALGEBRA. 
 
 61 
 
 CHAPTER IT. 
 
 Definitions, Substitutions, Combining Liks Terms. 
 
 In algebra we deal not only with definite numbers, such as 
 those of the preceding chapter, but we reason also about gene- 
 ral numbers, which we represent by letters. 
 
 Thus, no matter what number is denoted by either a or 6, 
 we have 
 
 a-\rh-h = a. 
 
 When we use letters to represent numbers, we indica^i the 
 operation of multiplication by writing one letter immediately 
 after the other. Thus ah means the product of the number a 
 and the number h. 
 
 We also indicate the product of a definite number and a 
 general number by writing the general number after the other 
 with no sign between. Thus - 2a means the product of - 2 
 . and a. 
 
 When a number is expressed as the product of two numbers, 
 each of these numbers is called the coefficient of the other, and 
 each is also called a factor of the product. 
 
 A coefficient is called a numerical coefficient when it is a 
 definite number, and a literal coefficient when it is a general 
 number. In the product - 4a;, - 4 is the numerical coefficient 
 of aj; in the product of ax, a is the literal coefficient of x. 
 
 EXEBOISE 7. 
 
 1. What is meant by wwi ? -6m; +4w; dm? -4mn? 
 
 2. If mis +6and«is - 3, what ia the value of m»i ; -Qn? 
 + in? Sm9 -imnf 
 
 3. State the meaning of each of the following sets of sym- 
 bols : oftc, o6 + c, a + 6c, -2 + bc, -26 + c, -26c. 
 
 4. If * is + 2, 6 is +3, and c is - 4, find the value of 060 
 
62 
 
 ELBMENTABY ALGEBRA. 
 
 5. Find the value of each of the following expressions, in 
 
 r-iVi"*"^' *""^' '"'^' '^"■*"^' ''"■*■'' y=-^' 
 
 (l)a + b+c + d. (2) a + b + cx + dy. 
 
 (3) ab~cd+ax^yz. (4) ac -bd+2x^3i/ + 4z. 
 
 (5) abc + abd+ bcd+xyz. (6) obex - abdy + acdx-. bcdu, 
 (7) abcdxyz. 
 
 6. If the value of + 3a; is + 30, what must be the value of x f 
 
 7. When - 3a; is + 30, what is the value of a; ; 
 
 8. When ~ 3a> is - 30, what is the value of x ? 
 
 9. If the value of a6 is +40, and the value of a is - 10 
 what is the value of 6 ? ' 
 
 10. liab ^ -50, and 6 = + 5, what is the value of a ; 
 
 11. Ifo^c- +60^anda= +2,and5- -3,whati8the 
 value of c ^ 
 
 12. Ifafic- -90,anda- - 9, what is the value of 6<j ; 
 
 13. abed is equal to 40, when a = - 1, 6 = - 2, and c - 
 + 6, what is the value oi d? 
 
 14. If yz is equal to - 10, what is the value of abyz when 
 a " - 3 and b == +2? 
 
 16. If a + bis equal to + 20, and b is equal to + 6, what is 
 the value of a / 
 
 16. Itta + cd is equal to 100, and ab is equal to + 40. what 
 isthe valueof CO?/ 
 
 17. If a6 + c(i - 60, and ab - 20, and c - - 4, what is the 
 value of d? 
 
 18. Ifaa;+6y- -50,anda- -4,6= -10,anda;« -& 
 what is the value of y ; 
 
 19. a6-crf-60,a- +2,c- +3,rf.. + 4, wlmt must be 
 the value ot b ? 
 
 30, -3«+40 - 70. what is the value of a;f 
 
38810ns, in 
 y=-6, 
 
 + 4«. 
 lue otx9 
 
 is - 10, 
 
 ofa; 
 at is the 
 
 of 6c; 
 a.nd ■■ 
 
 '^z when 
 what is 
 0, what 
 t is the 
 - -^ 
 lustbe 
 
 ELEMENTARY ALGEBRA. 
 
 63 
 
 A collection of algebraical symbols, that is, of letters, figures 
 and signs, which, when ail the operations indicated are carried 
 out, results in a number, is called an algebraical expres- 
 sion, or, shortly, an expression. 
 
 The parts of an algebraical expression which are to be used 
 as addends in finding the final value of the expression are 
 called teims. 
 
 Thus ax-b^ + 2z-36ia an expression of which the terms 
 areaa;, -by, +2z, -36. 
 
 Those terms which are written with the sign + (or with- 
 out the sign) are called positive tenn>J. 
 
 Those written with the sign - are called negative terms. 
 
 Terms which differ only in the sign are called comple- 
 mentary terms. 
 
 Thus in the expression a+'dx-by + bi/-3x, + 3a; and -3a; 
 are complementary terms. Such also are - bi/ and + 6y. 
 
 An expression consisting of one term is called a monomial : 
 as iah. 
 
 An expression of two terms is called a binomial : as 2a + 3c. 
 
 An expression of three terms is called a trinomial : as 
 
 a + b + c. 
 
 All expressions of more than one term are called multi- 
 nomials, or polynomials. 
 
 Combining Like Terms. 
 
 Two terms which contain the same letters involved in the 
 same way are called like tetms. 
 
 Thus, + 5m, + 3w, - 6m are li c^^ terms. 
 abc, 4a6c, - 5abc are also like terms. 
 
 Consider the like terms +5m and +3nu +5m means m 
 
 repeated five timfts. anH j.3«, *»,«o„« ^ . ^ j ., 
 
 - -' ■ •'"" "i^o'na m icpuated tnree times. 
 
 aence, their sum is m repeatejj ei^ht times, that is + 8m. 
 
 sm-'- 
 
 i. ' 
 
 '1 . 
 
 V .^ 
 
 tt J 
 
 V,- > r] 
 it 
 
f \Y 
 
 ^i 
 
 ELEMENTARY ALGEBRA. 
 
 Cons-fler the like terms +8m and ~6rrr. .?- ^ m.w.s /•& 
 repeated eight timts, and - 6m means tha. tho sign oi' m is 
 chwiged, and the resulting number repeated inx Umm. H9n<^> 
 their sum is m repeated twice, that i«, + 2m. 
 
 From these examples w« see that tv^o like terms may be 
 oombmed into a like term, who. e numerical coefficient is the 
 sum of the numerical i»efficienib oi the given terms. 
 
 The sum of -Gadcand +9abch +[yabc, biijce the num of 
 - 6 and + 9 ir +3. 
 
 Exercise 8. 
 
 Simplify \h.P> follows 'ng expressions by combining like terms : 
 
 1. 4' <% -i~ tk- - 9a. 
 
 2. -[- iW'i; 4- '^abc - 9ahc. 
 
 3. -lUx-liy + i^-bx. ^ 
 
 4. 2y -Uy + \Zy-Ux. 
 
 5. ^x-^x + ^x-^x-^-lx, 
 
 6. |a6c + faftc - f oJc + f aJc. 
 
 7. -^+\0pq-1pq + x + y + z. 
 ^. pq-ipq + Qpq + Zp + 2q. 
 
 9. ap + bq + lZpq-2ipq-\Gpq, 
 
 10. 3a; - 4a; + 8a5 + 1 2xy - ixy - IQxy. 
 
 1 1. iOabcd - SSabcd - 1 Gahcd + 32abcd. 
 
 12. 12aj+13y+14« + 6a;-5y-8«-3a;+4y-5a, 
 
 13. 7a;-3« + 4a;+14y-20a-32a;-4y. 
 
 14. 27mw-36w^ + 40;m + 42w^-32mw. 
 
 , What does mn mean ? What does nm mean ? 
 
 15. +3mn + inm + 5mn. 
 
 16. 4a6c - 5acb + Qbca - 17c6a. 
 
 17. 13a^« + 1 2ayz - 20yzx - 1 Izay + 50«ya; + lOOaay. 
 
 18. 8a:+13r-167+14X-47. 
 
m."'«:n8 /,•& 
 
 ELEMENTARY ALGEBRA. 
 
 65 
 
 mt 
 
 m 
 
 CHAPTER in. 
 
 ADDITION. 
 
 To add + Sx and - 4y, we merely write the terms, one aiter 
 the other, with sign unchanged ; thus, + 3a:- 4y. 
 
 Also, to add +3a;+2y and +6x-5y, we write +3«+2v 
 + 6a; - 5y. ^ 
 
 This la^t result may be simplified by combining the like terms 
 giving +9a;-3y. 
 
 It is sometimes more convenient to write the expressions to 
 be added, one under the other, having V\e terms in vertical 
 columns. Thus, the above sum would be written— 
 
 + 3a: + 2y 
 
 + 6a;~5y 
 
 + 9a; - 32/ 
 
 Exercise 9. 
 Add together — 
 
 1. +5a + 66; +7a-2b. 
 
 2. 14a + 36/ -6a + 26; -3a -56. 
 
 3. 6a + 76; 96- 13a/ 14a + 66 + c. 
 
 4. -3a-26 + 4c/ -6a-76- 136/ 17a + 186 + 19a 
 
 5. 22a!-y + 6«/ 12y-13aj/ 40aj - 30« + 25y. 
 
 ^' «:+t/ + z; y + z-x; z + x-y; x + y-z; -a;-y-«. 
 
 7. «+2y+3«/4y + 5«-6a;/7« + 8a:-9y;10aj+lly-12* 
 
 8. 2a+6-c-c?/ 3a-66 + 14c + c;/ 12a-306/ 166 + 13c 
 
 9. a + 6+c + c?/ b + c + d-a; c + d-a-b; d+a-b-c. 
 10. lOxy+Uyz; lOyz + IZxy - 20zx ; 27xy+12yz + zx. 
 U. 7 + 13a6-166c/ 226c-16a6 + 13; -17+a6. 
 12.;,gr + a6c-100; 45a6c-16j^r/ l7-pqr-ahc. 
 
 13. 14a6 + J6c + ^crf; -|a6 + |6c- lOcrf. 
 
 14. •25a6 - -6360 + -dcd; '5bc - -iab + '75cd. 
 
 }fi fr**?"A''ti"^^' ^-y + z-xyz; ^x-y-^z + xyz, 
 ,16. l3aJbcd+Uabde~l5bcde; l2boda - 15bade - lObecd. 
 
66 
 
 ELKMENTARY ALGEBRA. 
 
 'iOpr~13pg; 12pg+13rg+Uqp, '^''''^» ''^'^^l^' 
 
 ^'.';.f X;.f''^^ 190^-300,...^, 
 
 20. b + lc + :^a; c + la-^b ; a + §6-|c; -2a-26-2c. 
 
 SUBTRACTION. 
 
 We have already shown that to subtract an algebraical 
 number is the sa,„e as to add the con^plementary'" 
 
 changing the sign of each of its terms : for. the sum of any 
 
 :rhTirt::,irr ""^" '^ ^^--^ ^^« ^ 
 
 ::^nif^rch^tiH^^^^^^^^ "^^^^ ^-^--^ ^^^ 
 I3^"^2t:r:r!r,^^ ^-- ^^« ^ ^^^^ - -- ^dd 
 
 This may be done by writing the terms in .succession, as 
 13a - 126- 6a - 4&, 
 and combining like terms. 
 
 Or, as in the case of addition, by writing the Jike terms in 
 vertical columns ; thus :— 
 
 13a - 126 
 -•6a - 46 
 
 7a - 166. ~ 
 When the subtraction is performed in the latter way, it is 
 usual to leave the signs of the expression to be subtracted 
 unchanged when writing it under the other. Then the signs 
 snust we ehaaged mentaUy when performing the addition. " 
 
ELBlMEN'TARY AtOIlBllA. 
 
 61 
 
 Thus, to subtract Sab - 6bc + ide from Uab - 8bc ^ Qde 
 we write 
 
 I3ab - 8bc - 6de 
 
 8ab - Qbc + 4de 
 
 and obtain 5ab - 2bc - lOde. 
 
 ^ , Exercise 10. 
 
 Subtract : — 
 
 1. 2a + 36 + 4c from 6a + 66 + 6c. 
 
 2. a + 5 + c from 2a - 6 + c. 
 
 3. a - b - c from a + b + c. 
 
 4. 2aj - 3y -{. 4« + 6w from IOt - y - « - 10«^ 
 6. a + 26 - 3c - 4c? from b + c - d - a. 
 
 6. 17aj - 14y + 20« from 40a! - y - « _ w. 
 
 7. w + n + jp from 2m. 
 
 8. 2m + 3n - 4^9 from - ,p + 5m. 
 
 9. 2a6 + 36c + 5ca from 4a6 + 46o + 4co. 
 
 10. lOxyz + 12y«w - 132;M>a;from - 5yzw. 
 
 11. Subtract 2xy - 3yz + 4saj f rt)m 3xy - 4yz + zx i^d 
 add ocy - yz ^ zx to the remainder. 
 
 12. Subtract the sum of 10^ - 7m + 3n - />and 3w - w 
 + ;> - 3Hrom 12/ - 11m + lOn. 
 
 13. Subtract from x + f/ + z the sum of 2a; + y + «, 
 aj + 2y - 3» and - aj - y + «. 
 
 14. Subtract from aa? - 36a; + 4ca;, the sum of - 2aa; + 
 46aj - 6ca;, oa; - 36a; + 6ca; and 2aa; - 26a; - 2ca;. 
 
 15. Subtract the sum of aa; - a + 1, 2aa! - 2a 2, and 
 - oa; - a - 4 from aa; + a + 1. 
 
 BRACKETS. 
 
 We indicate that an algebraical expression is to be added as 
 a whole by placing it in brackets with a + sign prefixed. . 
 
 Thus, a + (6 + c) means that the sum of 6 and c is to be 
 added t a. But this is the same as writing down the terms 
 of thtj expression to be added with their signs' unchanged. 
 Thatis, 
 
 a + (b + o) " a + +c. 
 
 '«wK! 
 
 w 'a 
 
V 
 
 68 
 
 J 
 
 ELEMENTARY ALQBBRA. 
 
 ■V ine sign • , . 1„„ ,ats may Im removed. 
 
 Thus, a - (A + o) means that f >><, exDression A j. . i- * u 
 subtracted from a. Bv^ ^ .« h expression ft + c is to be 
 
 »<.. of each te™ .Hieh was L^TJ^^re tZ:!"^"" ''" 
 bn«kets a« r.JoTmZLlu^''/'"^ «- the paU« <, 
 
 In removing these braoketa, it is best to begin with the 
 "me^ost pair, and «move only one pair at a tim? 
 Thus, 
 
 "■ ^ - {^ - 2r> + y + 2y.| 
 
 - - - a- + 2a: - y _ 2y. 
 = 2a: - 3y. 
 
 EXERCLU: 11. 
 
 Remove the brackets from the followim. «vT.r« • 
 combine the like terms : ^ expresBions and 
 
 .1. a; + (y _ z). 
 
 2. a: - (y > 4 
 
 3. 2a: - (y - 3^;) + ^4^ _ ^^^ 
 
 4. (3a: - 3y + 4:z)^2x 
 
 - 9, 
 
 Q-N 
 
 /-- 
 
 /-^w - y + «;t 
 
^h 
 
 ELEMENTARY ALGEBRA. 
 
 09 
 
 ^■(^ -V) + {y ' z) + (z - x) + (x + y + z), 
 
 «. a: - (y -«)-(«_ a;) - (a; _ y). 
 
 7- (x - y + z) + Sx - (2y - 2a:) - (« - y + ,). 
 
 8. 20 + (13 - a) + 4 - I 5 - (6 -»}. 
 
 9. « - \(^ - z) - {z - X)} - (« - y + «). 
 
 10. 2;, - (3^ + 4r) - {2p - (3q + 4r)} + p + g + r. 
 
 11. a - {26 - 3a - (4a - 26)} + 26 - 3a - (4a - 26) 
 
 12. 3a + [76 - j5c- (2a -6)+ 3c}]. 
 
 13. a + 6 + e + d-''i _ (ft - (<. . rf)}J 
 
 14. [2a: - y - {3x + 2y - (y - «:)}] . [{ ^^ _' ^^ 
 
 - a: } + y]. 
 
 15. 2 - [2 - {2 - (2 - a) - 2} - 2], 
 
 A bar is sometimes placed over the terms which are lo be 
 considered as one quantity. Thus, a - F~~c means the 
 same as a - (/> - c). ^ * 
 
 16. 2a: - [2y- {2a; - y - (2a: - y - r^^) + gy} _y]. 
 
 1 <=^ + b + e + d-[a-b+{c + d~(b- r^Td 
 _- h~d))-]. 
 
 18. 2x - 3y + 4» + 3w - a; - 2y - 3a - 4m,. 
 
 19. a-[-2a- {26 - (2c - S^TTl,, _ 4c)-3a-6i -cl. 
 
 20. 1 ^2 - 3—^} ] - 2 - [ .. \A _ (5 - 
 
 Evidently any number of terms of an expression may be 
 enclosed within a pair of brackets, p. ceded by the sign + 
 without changing the signs of the terms ; and within a pair of 
 brackets preceded by the sign - , if t:.d signs of the terms are 
 changed. 
 
 '!■■ 
 
70 
 
 ELEMENTARY ALGEBRA. 
 
 EXERCIS£ 12. 
 
 Enclose the third, fourth and fifth terms of each of the' 
 
 ^' 2a - b - 3c + id - e. 
 
 2. a-h-e-d-e+f. 
 
 3. 2a: - 3y + 4« - Zx - 3y + 3«. 
 
 4. 2a; - y - 3a; - 4y - 4a: + y. 
 6. a + 6_c-o + 6-o. 
 
 6. 2z - 3y + 42 - 5w + 6m. 
 
 7. 3a - 36 - 4c - 4t« - *. 
 
 8. 26 - 4 - 2c - y + 3«. ' 
 
 9. - as + y - « _ ^ ^ 6^ 
 
 10. - h^ + \y -\z + \w ~ fw. 
 
 Enclose the third and fourth terms of the following in a 
 pair of brackets preceded by the sign -.and then all of the 
 expression after the first term in another pair of b ackets 
 preceded by the sign - : oracicets 
 
 11. o-6-c-fc?4-c. 
 
 12. 2a:-3y-4«H-5M;. * 
 
 13. x-y-i-z-B. 
 
 14. ab-j-bc-\-ca - abc. 
 
 15. -x+y-z-^tv. 
 
 16. -ia+^ft>^c+Jrf. 
 
 17. (a-6)-(c-c?) + («+^), 
 
 18. a:-.(y-2«)-(«-2x)-(ar_2.y)+«, 
 
BLEMENTARV ALGEBRA. 
 
 71 
 
 ich of the 
 (1) by the 
 
 «^mg in a 
 all of the 
 brackets 
 
 CHAPTER IV. 
 
 MULTIPLICATION. 
 
 We know that the product of two arithmeticai numbers is 
 the same, no matter which number is taken as multiplier. 
 
 Thus 3 multiplied by 4 gives the same result as 4 multiplied 
 by 3; and § multiplied by * gives the same result as * 
 multiplied by §. 
 
 Evidently the same will be true of algebraical numbers : for 
 the absolute value of the product is independent of the signs, 
 and the sign of the product is independent of the order of the 
 factors. 
 
 MULTIPLICATION OP MONOMIALS. 
 
 We have already learned that the product of two general 
 numbers is indicated by writing the letters one immediately 
 after the other. In finding the product of two monomials, we 
 have then only to multiply the numerical factors of the terms 
 and annex the literal factors. 
 
 Thus 
 
 And 
 
 (+ 4a;) X (+ 3y) 
 = + \2xy. 
 
 - 3xv X - iab 
 - + \.2xyab. 
 
 When a literal factor is repeated in a product, the number 
 of times tha factor occurs is indicated by a small figure 
 written a little above and to the right of the letter. 
 
 Thus the product of - 4a and - 5a is written + 20a». 
 
 N 
 
 m 
 
 I ' 
 
 ■'!■■ • 
 
 . n 
 
 
72 
 
 ELEMENTARY ALGEBRA. 
 
 
 Instead of aa we write a\ 
 
 " aaa «• 
 
 •' " aaaa " 
 and so on.- 
 
 (a + b) multiplied hy (a + 6), indicated by (a + b) (a + b) 
 IS also written (a + by. fy*^»h 
 
 (a + 6) (a + b) (a + 6), is also written (a + 6)3. 
 7 X 7 X 7 X 7 X 7, is also written 7«. 
 
 io2t ^'^"'*'' " ' " ' " ' "^^ ^"^^ *^^ ««««-d, third, 
 
 fourth, ....... powers of a. 
 
 The number which indicates the power is called the index 
 or exponent. When a letter has no index, 1 is to be under- 
 stood ; thus, a = a}. 
 
 The product of a^ and a' is a" + ^ ; 
 
 for a* SB a a, 
 and a? = a aa, 
 
 .'. a* X a^ 
 
 Similarly, a? X a? ^ 
 for or* ^ 
 
 and a^ = 
 
 .'. a^ X afi ^ 
 
 ■ a a a a a 
 '■ a? 
 
 • a^ ; 
 
 X X X X X, 
 
 xxxxxxxx 
 
 a*. 
 
 Hence we see that the index of the productof two powers 
 
 Multiply: 
 
 1. - 3 by + 4. 
 3. - 2 by - 5. 
 5. - 5a; by - 3y. 
 
 Exercise 13. 
 
 2. - 
 4. - 
 6. - 
 
 5 by - 3. • 
 3J by + 4a. 
 2m by - bn. 
 
, ELEMENTARY ALGEBRA. 
 
 7. - 36 by + 46. 8. - 5xy by - 3x. 
 
 9. - 2mn Ly - 5mii. 10. - 36' by + 46«. 
 
 11. - 5a:»by - 3a:*. 12. - 2m»» by - 5m«. 
 
 13. - 4 a 62 c' by - 5 aH^ c. 
 
 15. - I 0^ 6» c* c?2 by + 1 a 6^ c d' e. 
 
 15. _ 3 X y 22, 8 a:^ ^2 ^^j _ ^6^ ^3 3,2^ 
 
 16. - 2 m" w^^*, 3 m np q and ;? y. 
 
 17. - Xy - X and - a;. 
 Find the value of : 
 
 18. (- zy. r9. (- 2a6y. 
 20. (- 2aHy. 21. (- 3f. ' 
 22. (-2a2 6)«. 23. (-«62(,7. 
 
 24. (- ^x^yf X (- 2a5/)2 X (- xyz)\ 
 
 25. - (c? hcf X {- a^ bcf. 
 
 78 
 
 
 ii 
 
 i 
 
 ;^^'l 
 
 MULTIPLICATION OF A MULTINOMIAL AND A MONOMIAL. 
 
 We indicate the product of a + 6 and c by placing a + 6 in 
 brackets and writing c either immediately after it or before it. 
 Thus (a + 6) c, or c (a + 6). 
 
 To multiply c by a + 6, we must use the quantity c as the 
 unit in forming the expression a + b. 
 
 That is, we must do with c that which we did with + 1 to 
 obtain a ; and also we must do with c that which we did with 
 -f 1 to obtain 6 ; and then add the two results. 
 
 Hence to find the product of (a + b) and c, we must add the 
 product of a and c and the product of 6 and c. 
 
 Thus (a + b) c = ac + bc. 
 
 This result is true, no matter what the values of a and 6 
 
 may 
 
 be. 
 
74 
 
 ELEMENTARY ALGEBRA. 
 
 Similarly it may be shown that 
 
 Now, every eip^ssion may be written in tl,e fom, 
 
 . , . « + * + <!+.... 
 
 by enclosing each term in a braclcet »„w i • , 
 
 before it. oracRet and placing tlie sign + 
 
 Thus 2a! - Sj, + 4^^ _ ^^ 
 
 And th„ J^^"^''^~^*^*< + ''"^'^ + (-«"^)' 
 -t;ti,r^— :V--;- a ~ai, sa. 
 
 the quantities enclosed in tli^ "Sts td'^^^C "' ^""^ °' 
 The product is ^* 
 
 multinomial by the mono Jal'^^*' "^^ *'™ "^ *•>« 
 Multiply : Exercise 14. ' 
 
 7. 2a + 36 bv- 23- R <l, , kjl "•• ^o + 5rf by 10„,. 
 
 13..' + :rby2x. ' i7 ". !-« + <; by rf. 
 15. a^+2a,+ lby^ , « f ;^-" ""^ 3»^- 
 
 17. =,^ + ^ + y'byx.' 18 ^. „,^ ,7 
 
 19. ^rHay +y by y». 2O ^tT/^/ ^^ "^■ 
 
 b."rath:th:::;:.r-"^— ''^*-- 
 
 Thin Itf fht^l^' "^ ^; aw by + ,,, „„ by ^. 
 
ELEMENTARY ALGEBRA. 
 
 76 
 
 MULTIPLICATION OP MULTINOMIALS. . ' 
 
 To find the piquet ota + ba^de + d. Enclosing each 
 expression m brackets, we have 
 
 (a + b)(c + d) = a(c + d) + b{c + d). 
 But a{c + d) = ac + ad, 
 and b (c + d) = bc + bd. 
 
 , •''(a + b)(c + d) ^ ac + ad+bc + bd. 
 Again, 
 
 (a + b + c) {d+e-\-/) 
 
 ^ ^(d+e+f) +b{d+e+/) +c(d+e+/) 
 - <^+ae + a/ + bd+be + b/ + cd+ce + c/. 
 From these examples we conclude that the product of two 
 multinomials may be found by taking the sum of the products 
 formed by multiplying each term of the one expression by 
 each term of the other. 
 The product of 
 
 2a: + 3y and 3aj + 4y 
 is obtained thus : 
 
 (2a; + 3y) (3aj+4y) 
 - 2« (3a; + 4y) + 3y (3a: + 4y) 
 "60^+8x1/ + 9xy+l2y^ 
 -6ar»+17a3/+12y*. 
 And the product of a - 36 and 2a + ib: 
 (a- 36) (2a + 46) 
 -a (2a + 46) -36 (2a + 46) 
 = 2a'+4a6-6a6-1262 
 = 2a'»-2a6-126». 
 The work is commonly arranged thun 
 2a + 46 
 a-36 
 
 ^*' + ^«* product of 2a + 46 and a. 
 
 ;"• ' ~ ^ ^'-^ pi wiuci oi 2a + 46 and - 36. 
 
 p 2a*~2a6-126' 
 
 .4* * 
 
 ^..'1 
 
 f ■ 
 
 IS)* 
 
 ''j^^3<i 
 
 ii: !■: 
 
re 
 
 ELEMENTARY ALGEBRA. 
 
 Exercise 
 Find the product of : 
 1. 2a; + 3y and x + 2y. 
 3. 2a + 36 and a + 26. 
 5. 2/n + 3n and m + 2n. 
 7. 2a -6 and a + 26, 
 9. 2in-n and m + 2n. 
 11. a + 6 and 2a + 5. 
 13. a;+12anda; + 5. 
 15. a; + 4anda;+15. 
 17. x + 2 anda; + 30. 
 19. aj-12 and a; -5. 
 21. a; -4 and a; -16. 
 23. a:- 2 and a- Sp. 
 25. X- 12andaj + 5. 
 27. a;-4anda;+15. 
 29. a;-6andaj+10. 
 31. a: -3 anda;+20. 
 33. iP-2anda;+30. 
 35. a;-l anda;+60. 
 37. 3x + 24and3a;+l. 
 39. 3a:+12and3a:+2. 
 41. 3a; + 8 and 3a: + d. 
 43. 3a; -8 and 3a; -3. 
 45. ar+yand aj-y. 
 47. m + n and m-n. 
 49. 2a; + y and 2a; -y. 
 61. a6 + c and a6-c. 
 63. a' + 6»anda»-6l 
 65. 2a' + 36» and 2a»- 36'. 
 
 16. 
 
 2. 
 4. 
 6. 
 8. 
 
 10. 
 
 12. 
 
 14. 
 
 16. 
 
 18. 
 
 20. 
 
 22. 
 
 24. 
 
 26. 
 
 28. 
 
 30. 
 
 32. 
 
 34. 
 
 36. 
 
 38. 
 
 40. 
 
 42. 
 
 44. 
 
 46. 
 
 48. 
 
 60. 
 
 52. 
 
 54. 
 
 56, 
 
 3a + 6 and 2a + 66. 
 
 3a; + y and 2a; + 5y. 
 
 Sp + q And 2p + 5q. 
 
 3x - 2y and 2a; - 3y. 
 
 3m - 2n and 3m - 3n. 
 
 7a; -4 and 3a; + 6. 
 
 a;+6 and a;+10. 
 
 a; + 3 anda;+20. 
 
 a;+l anda; + 60. 
 
 a: -6 and a;- 10. 
 
 « - 3 and x - 20. 
 a; - 1 and x - 60. 
 a;+12 and a; -5. 
 a; + 4 and a;- 15. 
 a; + 6 and a;- 10. 
 a; + 3 and a; - 20. 
 a; +2 and a; -30. 
 a; + 1 and a; - 60. 
 3a; -24 and 3a; +1. 
 3a;+12and 3a; -2. 
 3a; -8 and 3a;+3. 
 3a; - 10 and 3a; - 2. 
 a + 6 and o - 6. 
 p + q and p-q. 
 2a; + 3y and 2a; -3y. 
 pq + bm and pq - bm. 
 o' + 6candrt»~6c. 
 Imn -j-p^ i^jjd Imn -pqr. 
 
BLKMENTARY ALGEBRA. 
 
 77 
 
 Find the product of: ^™^"« ^6. 
 
 1. 2aj +y + 3« and 3x. 
 
 2. 2a; +y + 3« and 2y. 
 
 3. 2x +y + 3»and3a; + 2y. 
 
 4. 6c + ca + a* and a + 6 + c. 
 5-x + y + zaadx + y + z 
 
 6. a + J + c and a + 6 + c. 
 
 7. 2a; + 3y + 4«and2«; + 3y +4«. 
 
 8. Jb3 + y2 + 2^ _ 
 
 yz 
 
 - fix - xy and x + yj^z. 
 
 11. 2a'»-4 + 6a + 3a3and2« + 3. 
 
 12. 3* - 4ar' + aj3 _ g ^^j 2^ ^ g 
 
 13. a;» + ar»y + ay + y' and^a; - y. 
 
 14. 3m» + 4 + 2m + 5^^ and m^ - 2 + m 
 
 15. 1 + 2. + 3.^ + 4^ and 3.=^ + 2z +1 ' 
 
 It »ill usually be found convenient to arrange all such 
 «pre«.ons accorfing to descending or ascendJ power of 
 ^tter. And in multiplying two such e.plToIco: 
 ^powers of the same letter, it is advisable to ar^n^ 
 «5oth expressions m the same order 
 
 of leTrf^^ """""^"^^ "^ ^-^o.nding powers of m, example 1 4 
 of ohe previous exercise would be worked in the following way 
 5m8 + 3m^ + 2m + 4 ^* 
 
 w' + m - 2 
 
 
 Sw* -r 3m*^ 2m» + imf 
 
 + 5m* + 3m8 + 2m» + 4m 
 
 10m» ~ 6m^ - 4 m - 8 
 
 ow» + 8m* - 5m3 
 
 like t«rm« ,•„ " *• 1 ""' ,'"'' *^'"''* ""^^^^ ^3^ *fae placing of 
 iute terms in vertical columm. 
 
 » .^ ii 
 
 
78 
 
 BLBMENTABY ALGEBRA. 
 
 Exercise 17. 
 
 Find the produdt of ; 
 1. ^a^ + Sxy - 4y» and a?" - 2xy + 
 
 + 3? + a? + 
 
 y' 
 
 aj + 1 and a:^ - ar* + a; - 1 
 
 3. 4a» + W + c» - 36c - 2ca - 
 
 4. «» - 
 
 a;y + y' + a: + y + 1 and a; + 
 
 6a6 and 2a + 36 
 
 + c. 
 
 5. a» 4- 3a»6 + 3a6» + i^ and a« - 
 
 y - 1. 
 
 2a6 + h^ 
 
 6. 2** - 3ar» + a:» - 2x + 1 and «" + 2ar' + 2a; - 1 
 
 7. a;' + 3«2y + Szy* + y^ q,xm^'^ - 
 
 ®- (a^ + y\ {y + «) and (« + aj). 
 9. (a + 6), (6 + c) and (c + a). 
 
 10. (a; + 2), (ar + 3) and {x + 4). 
 
 11. (x - 2), (^ - 3) and (a; - 4). 
 
 12. (x + 2y), (aj 4 3y) and (a: + 4y). 
 
 13. {x - 2y), (x - 3y) and (x - 4y). 
 
 14. a:^ + jB + 1 and a;* - a; + 1. 
 
 3«V + 3a;y» -yS. 
 
 15. a;» + 
 
 a;y + y« and «» - ay + 
 
 .x. .^ -r *y -r y- ana ar - ay + y'. 
 Simplify, by removing brackets and combining like 
 
 i6. xiic" + X + I) - (x^ + x + I 
 
 terms 
 
 ). 
 
 17. x^ix^ + '«y + y')+f{x^ + xy + f). xy(x^ + 
 
 + fy 
 
 xy 
 
 !q' /V^"*' m fr -^ ^> ^ ^« (« - 2) + 2a» + 9a - 3. 
 
 19. (3a + 6) (2a + 36) + (a + 26)(2a - 66 ) + 7( 
 
 20. 2[2«-3{a-2(a + 7)-l}+l]. 
 (a + 6){aa-(a-6)-6}-a(a + 6). 
 
 y + 6'). 
 
 21. 
 
 + 2c). 
 
 22. a[a + a{a-(a+3)-2}+2]. 
 
 23. 3(a + 2£. + 3c)-4(2a + 36 + c)+2(3a + 6 
 
 24. 2 {3 (a-6) a; + 4 (6-c) y} - {2 (a+ 6).r- 3 (6 + c) y}. 
 
 86. 4 {(«-6)(ar + y)-(a + 6) {x-y)} -2 {(a + iX-^-^v)- 
 (a-6)(a:-y)}. 
 
 Vt 
 
ELEMENTARY ALGEBRA. 
 
 79 
 
 When a factor is common to all the terms within a pair of 
 brackets, that factor can be removed from each term and 
 placed outside the brackets. 
 
 Thus : 
 
 (ax + hx) = {a + b)x, 
 and im'n - 6mn' + s^nn = (4m - 6n + 3) mw. ' 
 
 Exercise 18. 
 Place in brackets the terms which contain like powers of x, 
 and then remove that power of x outside of the brackets : 
 
 1. 4ma;2 + 6na^. ^■ 
 
 2. 2ax + 5603 + ex. 
 
 3. ax Ar hx -{■ cx^ + da?. 
 
 4. 2aj + 3a:2 + 5caj + ^ma?, ^ ■ ' 
 
 5. mx? + 4mnx + na^ + pqx. 
 
 6. 2x + c«^ + dx -^ ex} + /a;3 ^ 4^_ 
 1' a}hx^o?hx^^cC-x^^aWx^a}?o^. 
 
 9. <X'^^-(hx^c^)-{ax-^h:^)^{a:t?-cx). 
 10. 3a!-a(2:cH3a^) + 6(a^-a;). 
 
 Use brackets to indicate : 
 
 11. The product of x + y and z. 
 
 12. Theproductof ;r+yanda + 6. 
 
 13. That the product of « + 6 and c + rf is to be added to x. 
 
 14. That the product of m - ^ and m + n is to be subtrax^ted 
 from m^. 
 
 15. The third power of the sum of a and h. 
 
 16 That the product of the sum and difference of a and h is 
 equal to the difference of their squares. 
 
 
 
 ,< r- 
 
 II' 
 
80 
 
 ELEMENTARY ALGEBRA. 
 
 DIVISION. 
 WVI8I0N0FAMON0MIAL BY A MONOMIAL. 
 
 product the lite^al factors. 1^^^ ^LluhTt t; "^ 
 one monomial by another we must di^TI '^^ 
 
 coefficient of the dividend bv fL , *^^ numerical 
 
 Ai ' ^ uiviaena by the numerical coefficienf nf +k 
 
 dmsor, and write after this the hfpmi * ^ ^oe^^cient of the 
 
 To divide - 20ab by + 56 
 
 Divide -20 by +5, and we obtain -4 
 Remove 6 from a6, and we obtain a. 
 Hence the quotient is - 4a. 
 To divide - 16abchy ~ iac. 
 
 - 16 divided by - 4 by gives + 4, 
 
 and ac removed from a6c gives 6. 
 Hence the quotient is + 46. 
 
 To divide a^ by a\ 
 Since a^ = oaoaa, 
 and a^ = acw, 
 ••• the quotient obtained by dividing a« by a' 
 
 To divide - 20a' a^ y by + 5a^ x. 
 - 20 divided by + 5 gives - 4 
 
 Hence the quotient is - 4aa^ 
 
ELEMENTARY ALGEBRA. 
 
 81 
 
 -if 
 
 Divide . 
 
 1. ab by a. 
 
 4. Sab by a. 
 
 7. a'bya' 
 10 
 12, 
 
 Exercise 19 
 
 3. ab by ab. 
 6. - 5ab by a6. 
 
 9. aJ*3^«* by a;^^*. 
 
 14. 
 16. 
 18. 
 20. 
 
 2. a6 by 6. 
 5. + 4a6 by ft. 
 8. ar'y*by £ir»y2. 
 
 72«Wby-9ai.o. 17. I^ by -„J*^' 
 -«Wd>by-aAcd. 19. 72«y by -3^' .' 
 
 BIVMION OF A MUITIKOMUL BT A MOHOMUI. 
 
 We found the product of « multinomial and a monomial b, 
 multiplymg each term of the multinomUl by the monomij 
 H»ce .t « eT>dent that we can find the quotient of a TZ- 
 
 b;':^];^~'" "^ "^^^^-^ -*■ ^-^ »' ^-^ *vidtd 
 
 To divide 2ab - Sa^c by - 2a. 
 
 + 2a6 divided by - 2a gives - b 
 and - 6a»c divided by - 2a gives + ^ ; 
 .-. the quotient is - 6 + 3ac. 
 
 Exercise 20. 
 Divide : 
 
 !.««« + 6^+ car by a:. 2, aar^ 4 6a^ + car» by «:». 
 3. 2«^ + 36a: + 4cx by ar. 4. aa^ + 26ar» + 3ca/by a:» 
 o. - 3aa; + 56a; - 46ca; by - a;. ' 
 
 6. ^403?^+ 86ar» - Gcai^ by - 2ar». 
 
 7. - lOabxy + 5a^bx - 15a6Vy2 by _ 5ax 
 
 8. - 8aVy* + l2aVy* - lOaVy^* by - 2aa:'.» 
 ». - 5adc - lOaW - 15bcdh^r _ KA 
 
 iO. Ua'b^cd - 4aWd - 80^3 ^ ^abcdhj iahcd. 
 
 w: 
 
 mi 
 
82 
 
 ELEMENTARY ALGEBRA. 
 
 DmsiON OP A MULTINOMIAL BY A MULTINOMIAL. 
 
 We found the product of two multinomials by multiply W 
 
 Turn' TfTe ! 77 '^J^^' *^"" ^^*^«^^^- -^ ^'^^S 
 
 sum of the partial products. Hence it is evident that in 
 finding the quotient of one multinomial by another, for every 
 term of the quotient found we must remove from th^ dividend 
 the product of that term and each term of the divisor 
 
 Thus, to divide a^ + 7a; + 1 2 by a; + 3. 
 
 It is evident that x is one term of the quotient 
 
 leaveTir; T "''"^^ ^ " '" '"" ^^ + '"^ ^ '' '' *^- 
 
 The next term of the quotient is + 4. 
 The product of a; + 3 and +4 is 4a; + 12, 
 .'. ^he quotient is a; + 4. 
 The work U conveniently arranged thus : 
 
 a; + 3)ar»+7a;+12(a; + 4 
 5'+ 3a; 
 
 4a;+12 
 4a; +12 
 
 bo^ *^f*^^ '''^°*r' "^^ "^"y ^ *'""«l « the terms of 
 tea. the d™or and the dividend are arranged in descending 
 
 term of the d.v,dend divided by the first term of the divisor 
 will give a term of the quotient. 
 
 usfl'ir"''f.'"° *^''' '""' »»>t»<'"^l by another we 
 usually proceed tnus : 
 
 1. Arrange divisor and dividend in descending or ascending 
 
 powers of some common letter, and keep the remainder 
 atter each subtraction in this same order 
 
 2. The first 3nn of the dividend divided by the first term 
 
 ot the divisor will give a term of f h« *;.„* 
 
ELBMENTART ALGEBR> 
 
 88 
 
 3. Multiply each term of tne divisor by this term of the 
 
 quotient, and subtract the product from the dividend 
 
 4. Repeat these operations, taking the remainder as a new 
 
 dividend, until there is no remainder, or one in which 
 the highest power of the common letter is lower than 
 that in the divisor. 
 
 To divide «♦ - 4ar'y + 6xy - ixif' + '^y a?-2xy + y^. 
 x'-2xy + y^)x*-ix'y + esc'y^ + y* (ar» - 2ay + y" 
 
 x*-2x'y + xY 
 
 - 2a^y + bx'^y'^ - ixy^ 
 -2a*y + ixy-2xy^ 
 
 «*y2 - 2xy^ + y* 
 x'y'-2xy^ + y* 
 The quotient is ar* - 2xy +y\ 
 
 It will be noticed that the term +y* was not brought down 
 with the remainder after the first subtraction, as there was no 
 like term in the expression to be subtracted. 
 
 Exercise 21. 
 Divide : 
 
 1. a^+3a; + 2by «+!. 
 
 3. a;2 + 5a;+6 by a; + 2. 
 
 5. ac» + 7x + lOhyx+5. 
 
 7. a'-3a + 2by a-1. 
 
 9. a^-7a + 12 by a -3. 
 11. m*-m-2bym+l. 
 13. ay^+2xy + y^hy x + y. 
 
 2. sc^ + ix + i by x + 2. 
 
 4. a' + 5a + 4 by a + 4. 
 
 6. aP + 7x + Qhy x+l. 
 
 8. ar»-6a;-14by x + 2. 
 10. aj2_7a._i8by x-9. 
 12. c2-3c-18byc + 3. 
 
 14. a^ + 3xy + 2y^hy x + 2y. 
 15. a^ + 5xy + Qy^hyx + 3y. 16. x"" - 5xy - Uy^ hy x - 7y. 
 17. a'-lla&-1262by« + 6. 18. m^- 12m« + 35n2byw-5n. 
 
 19. 3a^y + 8xy + iy by xy + 2y. 
 
 20. x^ + ax + hx + ah hy x + a. 
 
 21. a^ + 5a^y+lOaV+10a!2y3 4.5a^4^^byar»+2a;y + y». 
 
 
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 >^>A'Q 
 
 
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84 
 
 ELEMENTARY ALGEBRA. 
 
 24. V-isy+iey-: by 2y-v-i' 
 
 26. a^ + a^y + ybya^+jy + yii. 
 
 26. 2^/-3aV-9«V'+9a,y-2by2arV' + 3a,„-2 
 
 9t^V.t^-'*^+"^^ ■•"^by l+3a,-6a^-4a^. 
 
 «". ^ ^ „".':?' - '"'''-^ + ^»'»'«' - 3«iV+ 26V by 
 
 32. a' + 6»-2aA + 2a-26 + l bya-6+1 
 
 33. a«-oA-26»-36c-c»bya-26-c 
 
 34. 2a^'' + 5ay-3y»-3a:-9yby2a;-y-3 
 
 36. «^+y»+«»+2y«;-2««-2aybya:-y_;, 
 
 28. 
 29. 
 
 30. 
 
 Miscellaneous Examples. 
 Exercise 22. 
 
 I'tf^^^^' -7, -6, -13 and +9. 
 
 3 a„hT Xl ' '''^*'*"* - ^^ ^^-^ *h« product. 
 3. Subtract the sum of «-?6-,«,d2a + 6-2c fn>m the 
 sumof6 + 2c-aandc-a-26. . 
 
 t' i^u^TV^*^"*^" +2,findthevalueofa:«+2a», + V« 
 6. Multiply a^ + a. + ^ + ^^^^^^jj^^^_2^+;^ + y- 
 
 6. Add2a + 36 + 4c, -3a + 6-5c,2«-26 + 2c. 
 
 7. Multiply the sum of «« + aA + i» and a^'-oi + fi^^ by a + 6. 
 
 then multiply the result by a-6 ' 
 
 8. I>mdea^ + 4a:*+16byar»^2aj + 4 
 
 _ _ii_ ^n« r «uc „x ^3a! + 2y/ + (2a: + 3y)» when « - - y. 
 
ICLEMENTARY ALQEBKA. 35 
 
 11. Find the expression which, added to 2a^- 3x- 4. irives 
 
 the sum -Q-ix-ae^ 
 
 12. Multiply {^-(2a=-3y) + 4y} bjr« + y. 
 Id. Divide l-a^hyl-x. 
 
 14. Find the value of 
 
 ^(b-e) + b'(c-a) + c'(a-b) 
 
 (a-b)(b-e)(c-a) 
 whenassl, 6-2, c-S. 
 16. If «a:+6y- .10, when a- +2. 6- -2, and x^+3, 
 una the value of y. 
 
 
 I .^ I 
 
 r ■ 
 
 17. Find the product of a: - 2y, a: - y, a: + y and a: + 2y. 
 
 18. Simplify a [1 - 2{2 - 3 (1 - 46)}]. 
 
 19. Divide «:»-. Say + 4y» by aj''-2ay + y«. 
 
 20 Tf ** ^ ^ 
 
 a + 6 --60, whena«4and6--2, find the value 
 
 of c. 
 
 21. Divide a:« + 5a-y+10a:»y»+10a;»y3., 5^ ^.^ y^^ ^, 
 
 +y*. 
 
 22. Multiply 2a - 36 - 3 (a + 6) + 4 {2a-(«-6)} by 
 
 «-6{2-(4-6)}. ^ y n y 
 
 23. Find the value of the product of a^-^ay+y^ and a^ - ary 
 
 -f r, when a:= - 1, and y= + 2. 
 
 '24. ^ndthevaluecf |4^£ ^hena- + 2, 6= -2 
 
 ,u 
 
 Mi 
 
■UUIENT/Sy ALGKBBA. 
 
 CHAPTER V. 
 
 IMPORTAWT RESULTS IN MULTIPLICATION AWD DIVISIOIT. 
 FACTORS AND MULTIPLES. 
 
 To find the square of a binomial : 
 
 a-b 
 
 «' - 2ab T¥. 
 
 From the first exabple we see that thel^tiai^* of the sum 
 of two numbers ,s equal to the sum of the sqlres of the nZ 
 bera, increased by twice their product. 
 
 d J~°" *^« •^"<J «"«»Ple we see that the square of the 
 d^e«nce of two numbers is equal to the sum of 2e squa Jof 
 the numbers, diminished by twice their product. 
 
 Or. rememberin »t the sign f.rms a part of the tenn we 
 tt^, """^ r.. "^' *^** *^' ^"''"^ «* - binomial is e;ual 
 
 Exercise 23. 
 
 1- « + y. 
 3. m + n. 
 6. 2y + 3*. 
 V. «-y. 
 9. m-n. 
 
 2. y + «. 
 4. '£x-{.y, 
 6. 3m-i-4n. 
 ^. y-z. 
 10. 2aj-y. 
 
ELEMENTARY ALGEBRA. 
 
 87 ' 
 
 IVUIOV. 
 
 F the sum 
 ' the num- 
 
 kre of the 
 iquaresof 
 
 term, we 
 >1 is equal 
 roduot of 
 
 f each of 
 
 11. 2y-3s 
 
 13. 2p + q. 
 
 15. ab+2c. 
 
 17. 2xy + 6. 
 
 19. x + l 
 21. 
 23. 
 
 2ar + |. 
 
 2a + #6. 
 
 ¥^ 
 
 z 
 
 T 
 
 12. 
 
 3wi-4n. 
 
 14. 
 
 ah + e. 
 
 16. 
 
 7a6 + 3. 
 
 18. 
 
 3mn - 4a 
 
 20. 
 
 «+«. 
 
 22. 
 
 2x-l 
 
 24. 
 
 3m +1. 
 
 26. 
 
 7a; -f. 
 
 
 «' 
 
 28. 
 
 2- 2- 
 
 30. 
 
 30 + 2. 
 
 32. 
 
 100 + 3. 
 
 34. 
 
 106. 
 
 36. 
 
 99^. 
 
 38. 
 
 2000J. 
 
 25. 4a^ + 
 
 27. 2a:«-f 
 
 29. 20 + 3. 
 
 31. 60 + 1. 
 
 33. 102. 
 
 35. 100^. 
 
 37. 1003. 
 
 39. What term must be added to ar^ + y" to form the square 
 otx + yl 
 
 40. What term must be added to ar' + y' to form the square 
 of a; - y ? 
 
 41. What term must be added to a:' + 2ay to form the 
 square of a; + y ? 
 
 42. What term must be added to ar* + ixy to form the 
 square of a; + 2y ? 
 
 43. Form a complete square by adding a term to ar* - 10a;. 
 
 44. Form a complete square by adding a term to 4a;'' + 12a;. 
 
 Exercise 24. 
 Express each of the following as the square of a binomial ; 
 1. ar'+2a;y + y». 2. ar'-2a^ + y«. 
 
 3. a» + Ja6 + b\ 4. c* - 2cd+(P. 
 
 f > 
 
 
 1 '■ 
 
 .■il 
 
 - ' f .'I 
 
 o. sr -r ^xy + 4y' 
 
 6. x'-ixy + iy. 
 
88 
 
 BLBMENTARY ALGEBRA. 
 
 ^ 4«" + 4aj+l. 
 9. a^ + 2abc + b*e'. 
 11. 16y«-8y+l. 
 
 17. p+ 2 +1,. 
 
 19. 26a«-2a + ^. 
 
 21. («+y)'+2(a;+y)« + «a. 
 
 8. a»+4a6 + 46» 
 10. V-12jt»^ + 4^> 
 12. a:» + a; + f 
 , . 4m" 4m 
 
 9*." 
 
 16 
 
 18. 16-8a;+!e'. 
 
 8U 
 
 of 
 
 EXEROISB 25. 
 
 1. w»»+2mn. 
 
 4. m' + 6m. 
 
 7. »»»-2mw. 
 10. m»-6m. 
 13. a^- IQx. 
 16. x»+3x. 
 19. aV+4ay. 
 
 2. ^+2py. 
 
 6. p' + 8jD. 
 
 8. p'-2pq. 
 ll.y-8p. 
 14. y' + 2y. 
 17. y»-6y. 
 20. 4a'b^+8ab. 
 
 22. 16«y-2«y. 23. !*+«,. 
 
 25. m* + n\ 
 28. l-2a. 
 31. 4mn+l. 
 34. af + bx. 
 37. 4a:» + 6x. 
 40. i + 4««. 
 
 26. y + y». 
 29. 9 + 6m. 
 32. 6m*w + m'«'. 
 35. a^+2cx. 
 38. 4a?'+l. 
 41. -2+ar>. 
 
 3. 4a:»+4ay. 
 
 6. 4ix^+l2x. 
 
 9. 4x>-4ay. 
 12. 4a^-.12«. 
 16. ««-4«. 
 18. 4«« + 5«. 
 21. y^-5py. 
 
 ^*- ^'4^- 
 
 27. l+o« 
 30. 64a»+16a. 
 33. a^+ax. 
 36. 4a:>+a«. 
 39. i + :r. 
 42. « + |, 
 
ELEMENTARY ALQEBRA. 
 
 89 
 
 )e + <^. 
 
 following, 
 
 luare. 
 
 f. 
 
 e. 
 f. 
 
 a. 
 
 To find the product of the sum and differencQ of two 
 
 numbers. 
 
 « + * sum of a and b. 
 
 ^'-^ diflTerence of a and b. 
 
 a^ + ab 
 -ab-b* 
 
 *' - *' difference of squares of a and b. 
 
 From the above example we see that the product of the 
 sum and difference of two numbers is equal to the difference 
 of the squares of the numbers. 
 
 Exercise 
 Write the product of : 
 
 1. x + 1/ and x~y. 2. 
 
 3. 6 + c and b-e. -4. 
 
 6. 2m + n and 2m - n. 6. 
 
 7. x + 7 and a; -7. 8. 
 9. x+l andaj-1. 10. 
 
 11. l + 2ajand 1 -2x. 12. 
 
 13. 2a6 + c and 2a6 - c. 14. 
 
 1 5. 4hm + 5p and ihm - 5p. 1 6. 
 17. a» + 6»anda»-6». 18. 
 
 19. (a + ft) + cand (o + 6)-c. 20. 
 
 21. (2a + 6) + 2c and (2a + 6) -2c, 
 
 22. 2a 4- 36 + 5c and 2a + 36 -2c. 
 
 23. l+m + n And I- m+7i, 
 2i. 2a-6 + 3cand2a + 6 + 3c. 
 
 25. a-6 + c and a + 6-c. 
 
 26. 2a-6 + 3cand2a + 6-3c. 
 
 27. aj»+aj+land«»-aj+l. 
 
 28. aV + aa;+l anda»a:»-aa:+l. 
 
 29. «' + ajy + y'andaj»-«y+y». 
 
 30. (!+»+«»), (l-x + x") and (1 -aP+sx*), 
 
 26. 
 
 m + n and m — n. 
 b + 2c and b - 2c. 
 2»n + 3nand 2m -3n. 
 £6 + 4 and 07-4. 
 1 + a; and \-x. 
 3 - 2a; and 3 + 2a;. 
 oft + 2c and ab - 2c. 
 2a6 + J and 2a6 - ^. 
 2a» -6c and 2a» + 6c. 
 a + 6 + c and a+6-c. 
 
 ■I 
 
 1 1 I'l 
 
 •■ ,<' 
 
 i- 
 
 If «ii 
 
 J' ' It 
 
 
 w 
 
 « 
 
 v^ 
 
 illl- ^H 
 
 i 
 
 \r 1 
 
 t 
 
 f ' B 
 
 > 
 
 ^ 
 
 %' 
 
 1 
 
 ~* 
 
 9 
 
 
 1 
 
 m 
 
 
 - 1 
 
 ■ 
 
 mi 11 
 
 
90 
 
 BLEMENTARY ALGEBRA. 
 
 two numbe™. '^ ' ** "^ "«" ««*"«"» of tho« 
 
 = (« + 6 + c)(a + 6-c). 
 And a*-6* = (a«+6') (a«-i«) 
 
 £SZBR0I8B 27. 
 
 Knd the factors of ^ 
 
 4. (^» «-y« 
 
 7. 4»H«-n» 
 10. 4-«». 
 
 13. 4aV-«y. 
 16. 49a»6V-9. 
 19. 26-16p«^ 
 22. a«-26i». 
 
 6. a^-y»«». 
 8. V-4y» 
 
 II. d-4y». 
 
 14. a^-9. 
 17. a»6V-rf«. 
 20. l-26py. 
 23. a*-6«. 
 
 26. 16 -a*. 26. 625a^-y4. 
 
 28. 6V-266. 
 
 31. 27»-23». 
 
 34. (a + 4)»_c». 
 36. 4a» + 4a4 + 6a_ca. 
 38. a»-(5 + c)». 
 40. a»-62 + 26c-c= 
 42. (a+y)»-(a.-y)». 
 
 29. «»-.y-. 
 
 32. 103»-97> 
 
 3. f-fi, 
 
 6. «y-«v. 
 
 9. 16a:* -9y>. 
 
 12. 1--^. 
 4 
 
 16. 16 -y«. 
 18. l-16a»6V. 
 21. 81a»-266«. 
 24. a«-i«. 
 
 30. a»6»-c». 
 33. 220»-26. 
 
 36. a'+2aJ + 6a-c«. 
 
 37. a*-4ad + 462-4c« 
 
 39. a»-5»-26c-c». 
 
 41- 9«^-I2ay + 4y»-26««. 
 43. (a + 6)»-(c+rf^. 
 
ELEMENTARY ALGEBRA. 
 
 44. a* - 2a6 + 6» - c» - 2crf - rf'. 
 46. a* - 4a6 + 46» - 4c' + ied - rf>. 
 
 46. 9-4a* + ixy-y\ 
 
 47. a»-&« + o«-rf»_2ao-26rf. 
 
 48. 4a> - 96» + «« - 26rf« + 4ac + 30irf. 
 
 49. («» + y«)«-ajy. 
 
 60. «* + «y + y«. 61. flj» + jK«4.i. 
 
 91 
 
 To find the product of two binomials which have a com- 
 mon term. 
 
 Consider the following products : ^'^ 
 
 X +6 
 X +7 
 
 p^+5x 
 + 7aj + 36 
 
 a:'+12« + 35. 
 
 x-5 
 
 X +7 
 
 as* - 6aj » 
 + 7aj-36 
 
 X -6 
 
 ac^-bx 
 • -7aj + 35 
 
 «'~12« + 35. 
 
 X +6 
 
 
 x-7 
 
 
 ac^ + bx 
 
 
 - iX- 
 
 -35 
 
 «»+2«-35. «»-2a;-35. 
 
 We see that the pro^'i t consists of three terms : 
 
 (1) The square of the common term. 
 
 (2) The product of the common term and the sum of 
 
 the unlike terms. 
 
 (3) The product of the unlike terms. 
 
 EXEBGISB 28. 
 
 Find, without ordinary multiplication, the product of: 
 1. a: + 2anda;+3. 2. a; + 3 andaj + 5. 
 
 3. a; + 7andaj+ll. 4. a;- 2 and*- 3. 
 
 6. x-B and a; -5. 
 
 6. as- 7 and «- 11. 
 
 Ml 
 
 Ullj 
 
92 
 
 ILEMENTART ALGEBRA. 
 
 7. »+2«id«-3. 
 9. «+7»ndaj-ll. 
 11. «-3 andfl;f5. 
 13. m + 9andm-7. 
 10. y + 9andy-7. 
 17. «fe-9andA + 7. 
 19. 2a; + 3and2«-5. 
 21. 3«-6 .uid3«-8. 
 23. Sabc+7 aaid3abc-9. 
 
 26. 3aV + 7aud3aV+17. 
 
 27. tn + U »ndm + ll. 
 
 8. x + 3 and a -5. 
 
 10. a;-2andfl; + 3. 
 
 12. «-7anda:+ll. 
 
 14. rf+9andrf-7. 
 
 16. mn + 9 and mn- 7. 
 18. y« + 9andy«-7. 
 20. 3y+lland3y-7. 
 22. 4mn-6and4mn-f 12. 
 24. isx^+ 13 And 4x^-6, 
 26. a: + 3anda; + 3. 
 28. 2a5+6and2a!+5. 
 
 29. 2m + 7and2m + 7. 
 
 Thus: 
 
 «'+(« + 6)aj + a6 
 
 «'+(a-6)aj-a* 
 
 Write the product of : 
 1. « + aandifc + 2a. 
 3. x+2a and x - ia. 
 5. 2aj + aand2aj + 3a. 
 
 ^ + (-a-b)x+ab 
 or,a?-(a + b)x+ab 
 x-a 
 x + b 
 
 ot^-ax ■ 
 + bx-a b 
 
 ^ + (-a + b)x-ab 
 or, «»-(a-6)a:-a6 
 Exercise 29. 
 
 2. a? +36 find aj + 46. 
 4. a? -36 and a: +46. 
 6. 3a:-46and3aj+7& 
 
ELEMENTARY ALOEfiRA. 
 
 8S 
 
 -7. 
 
 r. 
 
 •7. 
 n+12. 
 
 their sum 
 >Iacmg in 
 
 lb 
 
 7. 4mn +;> and 4mn - 2/). 8. ahc - 2x and ahe + 6«, 
 9. a + 2y and a; - 5y. 1 0. 3aj + 2y and 3a; - 5y. 
 
 11. 2«» - 9a and 2aJ - 7a. 12. 3a«+ 106 and 3a*- 106. 
 
 13. 10/mn + 5x and 10/mn - 9x, 
 
 14. sey + « and xy + u;. 
 
 16. (0 + 6) + 2 and (a + 6) 4- 3. 
 
 16. (m + n) + 6 and (m + n)- 6. 
 
 17. (2m + n) + 10 and (2m + n) - 16. 
 
 18. a + 6 + 3anda + 6 + 4. 
 
 19. m + n + lOandm + n-7. 
 
 20. -a; + 7and -as- 9. 
 
 21. -4mn+ll and ~4mn-6. * 
 
 22. -a + 64-3and -a + 6-10. 
 
 23. a; + 2y + 3« and a: + 2y - 8«. 
 
 24. 2a;+2« - 3y and 2x-f)z- 3y. 
 
 25. 2mn + 4 + 3pand 2mw-7 4-3p. 
 
 Conversely, we can find the two factors which give a pro- 
 duct of the form jb' + 9aj+ 14, if we can find two terms which 
 (1) added together give the co-efficient of Xy (2) multiplied to- 
 gether give the third term. 
 
 Thus, to find the factor of «» -|- 9* -f- 14, we must find two 
 numbers which, added together, give +9, and multiplied to- 
 gether give -1-14. 
 
 By trial, we find the numbers to be -i- 2 and + 7. 
 
 Hence, «»-f 9a; -H4 - (x -H 2) (a -i- 7). 
 
 Again, to find the factors of aj» - 9a; - 22, we must find two 
 numbers whose sum is - 9, and whose product is - 22. 
 
 Since the product is a negative number, the required num- 
 bers have different signs. 
 
 And since their sum is - 9, the absolute value of the mega 
 tive one is greater than that of the other by 9. 
 
 Hence the numbers are - 11 and |- 2. 
 
 Therefore, ob" - 9a! - 22 - (a? - 11) (a?.*. 2). 
 
 •^.i 
 
 v!i 
 
KLEMBNTAHY ALOEBRA. 
 
 To And the fwton, ota^^5ax^ S6a». 
 
 The sum of the two unlike tenn. is - 5«. 
 •nd their product is - 36a« ^ 
 
 Hence the two terms are -DaanJ^ 4a. 
 
 i^itr^rd ri?2^^^^^^ the.or...s.. 
 
 •xamples, where we write thi? ^ "^ " ^'^ **^" '"""'^i"* 
 diffe«.oe of two IquC '"""" "^ ^ '^^^-^ - the 
 
 «* + 8aj+12 
 
 -a:>-^8«+16 + 12-l6 
 -(«: + 4)»-2* 
 
 -(aj + 4 + 2)(«+4-2) 
 -(« + 6)(« + 2). 
 
 «'-17aj-60 * 
 
 -«'-17« + (Y.)._60-(V-)» 
 
 -(«-V-+-V-)(«-V--y-) 
 
 -(« + 3)(a.-20). 
 Find factors of: ^™<'»« 30. 
 
 1- as»+4a; + 3. 
 
 3. »'+10ar+16. 
 
 5. x' + Saj+lS. 
 
 7. y'+10y + 21. 
 
 9. y'+12y+20. 
 11. y'-6y + 8. 
 13. »n»-8m + 16. 
 15. m>-10»» + 2i. 
 17. m>-10»» + a6. 
 19. a«-14a + 24. 
 21. a«+14a+24. 
 
 2. a!«4.6a: + 6. 
 
 4. «*+7«+12. 
 
 6. «* + 6a: + 8. 
 
 8. y» + 8y+7. 
 10. y+14y + 24. 
 12. y'-4y + 3. 
 14. m»-10ar + 24. 
 16. m*-10w»+i6 
 18. m«-10»» + 9. 
 20. a«-10o + 24. 
 ^2' a"+10a+24. 
 
ELEMENTARY aLOEBIU. 
 
 95 
 
 23. m'n* + 6mn + 8. 
 25. a»»rt»-8am+12. 
 27. xyz^-20xj/z + 6l. 
 29. o»+4a-12. 
 31. m»-2m-63. 
 33. a:»-4r-21. 
 
 36. y» + 6y-:7. 
 
 37. a' -I- 4a -32. 
 39. a^-bx + 6. 
 41. 6' -76 + 10. 
 43. aV + 3aa:-70. 
 
 46. a' + 3a6+.26«. 
 
 47. c»-16crf-16rf>. 
 49. ar' - 1 7ay - 60y' 60 
 61. 4a:' + 12a; + 5. 52. 
 63. (a + by + 8(a + b)+l2. 64. 
 66. (a + by-5(a + b)e-lic\ 
 66. (a + 6)»-6c(a + 6) + 6c 
 
 2i. p'q^-4pq + 3. 
 26. aW+12am+ll. 
 28. a'6'-20rt6+19. 
 30. 6»-46-12. 
 32. m' + 2m-63. 
 34. «»+4a;-21. 
 36. y'-6y-27. 
 38. 6»-86-9. 
 40. a:» + 6a;-14. 
 42. y» + 9y-22. 
 44. a^x' ~ 9acc - 70. 
 46. oe' + bxij+Qy^ 
 48. n' + 4np + 4p'. 
 
 a:»+17ay-60y«. 
 
 9aJ + 36a;+36. 
 
 (w + n)»+ll (m + n)- 12. 
 
 To find the square of a trinomial. 
 
 Consider the expression a + b + c. We can put it in the 
 form of a binomial expression, if we place two of the terms 
 within a pair of brackets Hence we can find its square 
 thus : 
 
 {a + b + c}^ - {(a + b) + cy 
 
 - (a + 6)»+2(a + 6)c + c» 
 
 - a'^ + 2ab + b^ + 2ac + 2bc + <^ 
 
 - a' + b^ + c'+2ab + 2ac + 2bc 
 Similarly, 
 
 {«'-2aj+3} = {/ar»-2a;) + 3}' 
 
 = (»* - 2a;)» + 2 (»» - 2a;) 3 + 3> 
 
 - as* - 4a;» + 4{c» + 6a;2 - 1 2a! + 9 
 j== ar - 4ar -j- iOa;^ - i2a; + 9 
 
 fill 
 
 if 
 
96 
 
 ELEMENTABT ALGEBRA. 
 
 two term, ™' *"'' '™' *■>« P'^""' »« 've^r 
 
 To make ,„„ that we take twice the product of eyery „ir 
 
 of terms, we may perform the operation in tb^Z7^!Z 
 
 tw.ce the p^duct of each term and all tenn, thLhZorft 
 
 . EXE?ICI8B 31. 
 
 Write the square of : 
 
 1- '« + y + «. 2. :« + 2y + * 3. ^+2y + 3.. 
 
 *-«-y + .. 6.a,+ 2y-.. 6.x-2y:3. 
 
 nnd the expressions of which the following ^ the squares: 
 
 16. <''+2ab + !,'-2ac-2bc + c'. 
 
 17. »'-2a^+y>-2yj + 2aa!+«'. 
 
 18. 'f' + 4il»+V + 6M+12y« + 9j!. 
 
 19. <a!'-4a)s, + y>.I2aa + 63K + 9«». 
 
 20. a'-2ai + J»_2a + 26 + l. 
 
 To find the cube of a binomial : 
 
 3<J+"'"' '""'*'P"«'""' -« find that (« + 6>.,„U3«.» + 
 
 Since eyepr binomial may be put in the form « + 4 we a,» 
 able to wnte out its cuh« wifK^,,* *u i- 
 multiplication. ^'"* '^' "^^^'^^^ P~«««« of 
 
 Thus : 
 
 (a-6)» - a»+3a«(-6) + 3a(-6)2+(^5)». 
 = a'-3a% + 3a62_6s. 
 ^^A^d(2. + 3,)3= (2.), + 3 (2.,.(H.3y) + 3(H.2.)(+3yn 
 
ELEMENTARY ALGEBRA. 
 
 97 
 
 Al8o(2a;-3y)»- (2a:)» + 3 ( + 2a:)» ( - 3y) + 3 ( + 2a:) ( - 3y) 
 + (-3y)» 
 
 - 8a:'-36a;»y + 54jey»-27i/'. 
 EXEBCISB 32. 
 
 Expand : 
 1. ix + y)\ 
 4. (a;-2y)». 
 7. (a: + 4)». 
 
 10. (2a:+3)». 
 
 2. (a;-y)». 
 5. (3a; + 2y)». 
 8. (2a: +1)3. 
 
 3. (2a; + y)». 
 6. (3a-26)». 
 9. (1 - 2aj)». 
 
 12. (2-f)! 
 
 11. (2-3a;)». 
 
 13. {(a + 6) + c}». 14. [a+h + ef. 15. (a + 6_c)». 
 
 (aj + 2y + 3«)». 17. l2x-y + zf. 18. (1 -«: + «»)». , 
 Consider the following multiplications : 
 
 0) 
 
 a' + ab + h'' 
 a —h 
 
 ^^Ta^b + ab^ 
 
 a' -b\ 
 
 ' (2) 
 a'-ab + b^ 
 
 a +b 
 
 a' -a% + ab 
 
 We may write the results thus : 
 
 (a-6)(a»+a6+62) - a'-b\ 
 (a+b) {a'-ab + 6») = a»+6» 
 
 EXEBCISB 33. 
 Multiply : 
 
 1. (a: + y)(a^-ay+y»). 2. («-y) (««+ay + y»). 
 
 3. (c + d){o^-cd+d^). 4. (a-c)(a2 + ac + c«). 
 
 5. (2a: + a) (4ar» - 2a» + a«). 6. (2a - c) (4a'' + 2ac + c«). 
 
 7. (2a + 6) (4a» - 2a& + i«). 8. (2aj - 3y) (4ar» + 6a:y + 9f). 
 
 9. (2a + 36)(4a= - 6a6 + 96»). 10. (a: - 1) (a:» + -^- + 1). 
 
 o 
 
 11. (» + ^)(ar»--|- + i). 12. (a'-b^) (a* + a»6» + 6*). 
 
 
 if' 
 
 ,K4 
 
 in'^ 
 
 
 
 
98 
 
 ELEMENTARY ALGEBRA 
 
 15. (4a:+l)(16a:^-4.+ l). 16. (2+«:y) (4- 2^+4') 
 
 17. (2m + 3n)(4m='-6mn-f9n'»). "' 
 
 18. (a«- 1) (aHa2+ 1). 19. (l+«2) (1 _«»+««) 
 
 Complete the following statements by writing the necessary 
 
 ) = aj'+y'. 
 
 ) = aJ_y3^ 
 
 ) - a3 + 86». 
 ) = 8ar»-27y*. 
 )= l-8a«. 
 
 ) = V+i. 
 ) - 125w»-27w,» 
 
 28. (m»+n»n+«2)( ) = ^»_^8. 
 
 29. (a»-a6+J«)( ) - a»+63. 
 
 30. (4a;»+10ay4- 25y») ( ) = Sar* - 125y». 
 
 31. (9m2-3m+l)( ) . 27w»+l. * 
 
 32. (l+4m+16AA.'»)( ) = l-64m». ^ 
 
 If an expression can be written as the sum or difference of 
 the cubes of two numbers, it can readily be shown to be the 
 product of two factors. 
 
 Thus, 8a:» +27y» 
 
 - (2x)»+(3y)» 
 
 = (2a; + 3y) (4ar' - 6ay + 9y»). " 
 And, 126mV-l 
 
 -(5mn)«-l» 
 
 - (6mn- 1) (25mV+5mn + l). 
 
 21. {x-\-y) ( 
 
 22. {x-y){ 
 
 23. (a+ 26) ( 
 
 24. (2a:-3y)( 
 
 25. (1 - 2a2) ( 
 
 26. (2y+ J) ( 
 
 27. (5m -3n)^ 
 
 Find factors of : 
 1. p» + y». 
 4. |>»-89». 
 7. B-125*«. 
 
 Exercise 34. 
 
 2. />»-/. 
 
 5. 8p3 + 27^. 
 8. lOOOa^-y^. 
 
 3. 8p« + ^. 
 6. 27«i»~L 
 9. a^y» + «». 
 
ELEMENTARY ALGEBRA. 
 
 99 
 
 necessary 
 
 irence of 
 
 to be the 
 
 10. a» + 6V. 11. (a + 6)» + c». 12. a» + (fc + c)'. 
 
 13. (a + 6)»-c«. 14. a»-(6 + c)». 15. a» + (6 -c)'. 
 16. a'-(6-c)». 17. a'' + 3a26 + 3aA* + 6»-c'. 
 
 18. a»-6»-36»c-36c''-c«. 
 
 19. 27-a»-3a26-3a62-63. 
 
 20. Show that a; + y is a factor of (a: - 3)' + (y + 3)'. 
 
 21. Show that a; + y is a factor of (6a: + 5y)' + (4a; + 5y)^ 
 
 22. Show that a; - y is a factor of {{a+l)x + by\^- {{ax + 
 
 (1+%!'. 
 
 23. Show that a: + y is a factor of {(1 -w)a;+/)yl3^ 
 
 }ma; + (l-^)y}». /" 
 
 24. Express a:" -- 1 as the product of four factors. 
 26. Express 64 - y" as the product of four factors. 
 
 26. Show that a;+y is a factor of {(l-m) x+py + qzy + 
 {mx+{l-p)f/-qz}\ ' 
 
 We have shown how to find the factors of a trinomial of 
 the form a^ + mx + n, by arranging the expression as the 
 difference of two squares. 
 
 The same method may be applied in finding the factors of a 
 trinomial, such as 8a:'+ 22a;+ 15. 
 
 For, 
 
 Alao^ 
 
 8«2 + 22a; + 15 
 -8(«2 + -V-a; + J^) 
 
 -SUa^ + W-W + V-} 
 -8{(a: + W-|^} 
 
 -8(a: + V- + i)(a' + V-i) 
 -8(a; + f)(a;+») 
 
 -(2« + 3)(4a; + 5). 
 
 5a'- 12a + 4 
 
 =6}a«-Y« + j} 
 -5{(a-|r-fi + i} 
 
 - \" a/ \ ~ I 
 
 - (6a-2)(a-2> 
 
 I I 
 
 4 
 
 ml 
 
 mi 
 
 III 
 
 Hi J', S, 
 
 1 ^ 
 
 i 
 
 Ilia, 
 
 'r "^ ' 
 
100 
 
 ELEMENTARY ALGEBRA. 
 
 Factor : " 
 
 1. 6a:* + 5a;+l. 
 3. 6a^-x-2. 
 5. 5aP + 12xy + iyi, 
 7. 5c^ ^ 8cd - id\ 
 9. I2a^+a~20. 
 
 n. 12a'»-.31a + 20. 
 
 13. 12a'»- 43a -20. 
 
 16. 8ar»+22ay+15y2. 
 
 17. 8a»-2aJ-166» 
 19. 50ar»+61a:+l. 
 
 BZBRCISE 35. 
 
 2. ^3iP+7x+2. 
 
 4. 6«2 + a:_2. 
 
 6. 562-l2ic + 4c2 
 
 8. 6ar2^12a!y + 4y2 
 10. 12o2_a_20. 
 12. 12aH53a+20. 
 
 14. 12a2-53a5+206a. 
 16. 8a:2-22ay+15y2. 
 
 15. 8aH2ab-l5bK 
 20. 48«2+90a3^ + 27ya 
 
 the highest conunoa faTt r Tthe SoH °I1r ""^ 
 
 mon simple factors. P^oauct of all the corn- 
 
 Highest common factor is denoted by the Iett«r« TT m. 
 
 are2ando-i. **' *" ^« " * ) a^d 8 (a« - 2a4 + ft«) 
 
 Hence their H. C. F. is 2 (a - h). 
 
 Find the H. C. F. of : 
 1. aJ^anda^A. 
 3. m'n and m* 
 
 _ — .i« fc-Mt/ IT, 
 
 FXBHOISB 36. 
 
 2. aic» and a'Jc. 
 
 4. aiVfl?* and a*b'cU 
 
 6. 24ar^ and 36a:y. 
 
ELEMENTARY ALGEBRA. 
 
 101 
 
 7. ISaV*" and 25 y»«2 g 14^353^5 and 1V<?. 
 
 9. 4aft, 6a»ft«c and 126V. 10. a?x\ 20aV and lOas*. 
 
 11. ar'+.Saj + e and ar*+4aj + 3. 
 
 12. ar* + 5aj + 6 and 0!* + 3a; + 2. 
 
 13. 6(aj,-17a; + 70)and8(«»-3a;-28). 
 
 U. a2+6a6 + 8i»anda2 + 4oJ + 4^. 
 
 15. ai2+10ajy-24y2anda;2_4^2 
 
 16. JB* + 3ay + 2y2anda^ + 6a:y + 8y'. 
 
 17. 3a» - 4a6 + V and 4a2 - 5a6 + 1^, 
 
 18. (a + J)*-c«and(a + c)»-Jl 
 
 19. ar* + 5a; + 4^ 352 _ f and a? + 2a - 3. 
 
 20. a:» + y», a? + 2a:y + 2/2 and «» - y". 
 
 
 as. 
 
 A COnmon multiple of two or more algebraical expressions 
 is an expression which is exactly divisible by each of them, 
 and the lowest common multiple of the expressions is that 
 3ommon multiple which has the least number of simple 
 factors. 
 
 Lowest common multiple is denoted by the letters L. C. M. 
 The L.C.*M. of ^a^bc^ and Qab^<?d 
 
 isl2a26V(f. 
 Also, the L. C. M. of 
 
 6 {a? - J») aiid 8 (a» -2ab + 6"), 
 that is, of 6 (a-6) (a + J) and 8 {a-h) (a-6) 
 
 i8 24(a-6)2(a + 6). 
 
 J'! 
 
 EXRROIBB 37. 
 
 Find the L. CM. of: 
 
 1. €^hc and aV, 
 3. 
 
 u2^ » aiiu 
 
 
 5. a»-i»and(a + J)2 
 
 2. 3m'« and imM. 
 
 4. 5a-L-c- lOab'i^ and 15a%. 
 
 6. (a» + 6»)and(a + 6)'. 
 
 bii 
 
102 
 
 ELEMENTARY ALGEBRA, 
 
 9. ^'ft^-1) audi (6+1), 10. ^+y3and(x+y).. 
 
 11. a:» + 5:£ + 6anda:« + 7a;+12. 
 
 12. a:' + 9a; + 20and «»+a!-20. 
 
 13. a:'-3a:-4anda»-l. 
 
 14. 2ji?~x^SeLndix'^[2x+9. 
 
 15. a' + 6aj + 96»anda» + 7a6+'l26». 
 
 16. ^x' + x-Uandia-r^-ix-S. 
 
 17. 46»+ll^_3and*- + 66 + 9. 
 
 18. (a + 6 + c)» and (a + 6)»-.c«. 
 
 19. «« - 62 + e» - 2«c and «« - i» - e> - 26c. 
 ^0. « +4a; + 4.a:» + 5ar+6andar'+6a: + 9. 
 
ELEMENTARY ALGEBRA. 
 
 103 
 
 'fCll 
 
 t- 
 
 CHAPTER VI. 
 
 SIMPLE EQUATIONS. 
 
 An equation is a statement that two expressions are equal. 
 Thus 2a; + 3 = 3a:; or, aj (a; - 4) = a:» - 4aj. 
 
 The two expressions which are stated to be equal are called 
 the members or sides of the equation. /- 
 
 Equations are commonly divided into two classes: identical 
 equations and conditional equations. 
 
 An identical equation, or an identity, is one of which 
 
 the two sides are equal whatever numbers the letters stand 
 for. Thus, aj (a? - 4) =» «" - 4aj. 
 
 A conditional equation is one of which the two sides are 
 equal only when the letters stand for particular numbers. 
 Thus, 2a; + 3 = 3aj, is true only when x stands for 3. 
 
 The term equation when used without any qualifying word ' 
 usually means a conditional equation. 
 
 A letter which must have a particular value, in order that 
 the statement of equality may be true, is called the unknown 
 quantity. And the value of this unknown quantity is the 
 number which, when substituted foi* it, will satisfy the equa- 
 tion. This value is called a root of the equation. 
 
 To solve an equation is to find the root, that is, the value of 
 the unknown quantity. 
 In the equation 2a; + 3 = 9, 
 
 'JSfc + aia the left-hand member, or side, 
 9 is the right-hand member, or side^ 
 X is the unknown quantity, 
 
 3 is the value of x^ which satisfies the statement 
 and therefore 3 is the root of the equation. 
 
104 
 
 ELEMENTARY ALGEBRA. 
 
 A Simple equation is one which contains only the first 
 power of the unknown quantity. 
 
 The unknown quantity is usually denoted by the letter x, 
 although a«y other Ietu,r would do equally well 
 
 In solving equations, the following axioms are assumed : 
 
 If equals be added to equals the sums are equal. 
 
 If equals be taken from equals the remainders are equal. 
 
 If l'"?^r'5'P"'^ ^^ ^^"*^^ *^*» Pr«^-^« -e equal 
 If equals be divided by equals the quotients are equl 
 
 Consider the following equations : 
 (^\ 2a- = 6. 
 
 2 weThf •'^'*' ""^^ "^ '^" ^^'^"'^ "^'"^^''^ '^ *^^« «<i"-«on by 
 A we obtain equal quotients. ^ 
 
 If we add 6 to each of the equal members we obtain 
 
 ^»*i8> 5a;-3a; + 6. 
 
 If now we subtract 3x from each side, we obtain 
 
 6a;-3ar = 3a;-3a; + 6 ' ' 
 
 or, 6a: -3a;- 6, 
 
 or, 2a;a6. 
 
 •*• a; as 3. 
 
 ^^) 4a:-7 = 2a: + 5. 
 
 Jl Z tJ "^ ■"' '"■" ""■ ■"'" ="-''■*' - *"■» -* 
 
 4a!-7 + 7-2ar-2a:-2ar+7 + 6, 
 or, 4a:-2a; = 7 + 6, . 
 that is, 2a; = 12. 
 
ELEMENTARY ALGEBRA. 105 
 
 From these examples we see that any term may be trans- 
 ferred from one side of an equation to the other, provided we 
 change its sign; for this is equivalent to adding the comple- 
 mentary term to each side. 
 
 This transposition of terms enables us to place aU the terms 
 containing the unknown quantity on one side of the equation, 
 and all the other terms on the other side. 
 
 And by combining terms the equation is reduced to the 
 form 
 
 Dividing both sides by a, we have 
 
 b 
 a 
 
 (4) 4(a;-6) + 3(a;-2)=2(2a:-|.4)-16. 
 
 Removing the brackets, we have : 
 
 4a; - 20 -f. 3a; - 6 - 4a; + 8 - 16. 
 * Transposing terms, we have : 
 
 4a; -I- 3a; - 4a; = 20 -I- 6 -H 8 - 16. 
 That is, 3a;- 19. 
 
 a;=6|. 
 
 EXEBGISB 38. 
 Solve the equations : 
 
 1. 2r-8. 2. 2a;=-8. 3. -3a;«9. ' 
 
 4. 13a;= -39. 5. ma; = 4w. 6. axs _^*. 
 ^2a;4-5-17. 8. 3a;-7=:17. 9. 5a; + 3«l22. 
 10 lU- 45 -4a;. 11. 12a; -f- 5= 7a;- 30 
 
 12. 13a;-15.f.2a;-60. 13. 2«-2r- ;i><«4. 5 = 100 -3fli 
 14. 20a; -17- 18 -f. 13a;. 15. U-x^r-^t. 
 
 16. 3-4a;-5-f6a;s7-8a;. 
 
 17. 5 - 10a; - 8 - 3a;- 2 - 6a;.f. 30. 
 
 18. 2(a;-6) + 3(a;-i4)-d 
 
 r '1 
 
 f-} 1 2 
 
106 
 
 ELEMENTARY ALQEBRA. 
 
 10. 4«s-(a;+ 3) =12. 
 
 20. 2(a:-5) + 3(4-2a:)-6(2«-4). 
 
 21. a-4(2,-3) + 3(a:-.7). 
 
 22. 6(aJ-6) + 6(a:-6).0. 
 
 25. U-2f«-3(a:>4)-4}«24. 
 
 26. 5-3{2-x-4(5-a.)-3}-...60. 
 
 27. 3{«:-.2(3-2a:)}=lO-.2(«:-7). 
 
 28. 16 + 3(a:-7).4f2-3r2-«)+«,l 
 
 29. 2-[2-{2-(2-.2«,)}]-«^ 
 •30. (x-2) (a: + 3)=««+2aj+20. 
 
 31. (« + 4^-(«.-3)(a:4.7). 
 
 32. («-4)(«-5) + 17-«;(a; + 6)-100 
 
 33. 2(a;-6)4.(a.-4)«-a:(a;+10). 
 
 34. («'-*) («'-6) + (a:-6(a:-2)„(«+l)(«,^3^^^, . 
 
 35. 4 («-2)»4.3 («+3)»-.7 (»-l) («: +i 
 
 36. (2aj+3) (2aj-5) = 4 (a;-l)« 
 
 37. (a:+l)« +(a; + 2)»-(a:- l)»+(a.4.3).. 
 
 38. (2a:+3)(«:.6) + (a:+2)(«+7)-3 («-!)(«,. 4). 
 
 40. («+l)(aj + 3) («+6)-(a:-2)(x + 4) (« + 7)-100. 
 When some of the coefficients are fractions, if we multiply 
 
 each of the denominators, we get rid of the fractions. 
 
 .blfmtSe? "" ^- ''• ^' *'^ '^'^^"^^'^'^*^" -"^ ^ '^ -i^ 
 To solve : 
 
 Multiplying both sides by 1 2, we have 
 4aj+17O«l80 + .Sa; 
 .♦. 4aj-3aj -180 -17a 
 .*. a? -10. V 
 
ELEMENTARY ALOEBBA. 
 To solve : 
 
 Multiplying both sides by 6, we have 
 
 5(a;-2)-2(«-4)-60. 
 
 .-. 6aj-10-2a: + 8-60. 
 
 .•.6«- 2a; -60 +10 -8. 
 
 .'. Sx- 62. 
 
 /. a;-20§. 
 
 Note that ^{x-i) is the same as ^^. 
 
 o 
 
 107 
 
 Solve : 
 
 
 
 EXERCISK 39. 
 
 1. fe-8. 
 
 2x 
 44^ = 12. 
 
 
 
 2. |«=10. 
 0. -J-- 15. 
 
 7. fe+10- 
 ^' 4+T' 
 
 •|»+ 
 -lOJ 
 
 2i. 
 +». 
 
 
 » 
 
 9. 10-S|«-«+60. s 
 
 10. l(«-6)-|(«+12). 
 «+8 08 + 2 
 "4 T* 
 
 11. 
 
 3. |a:-3J. 
 
 - 2a; 
 6. f^ = 6. 
 
 
 ,„ « « 6a; 3 
 
 "• 4 •*• 6 +l2 - 2* 
 
 18. i(«-3)+i(2« + 4)«/^(a._8). 
 
 14. I (a;-2) (ar-6)-« (x-5) (3«-2) -^ 
 
 2a;-3 6-6* . 
 16. -J— -3 V. 
 
 16. |(6a:-9) + |(7a;+8)B23. 
 
 a^ 
 
 17. 
 
 3a;- 17 2a;- 9 a;+40 
 
 9 
 
 49 
 
 . "m 
 
 M^ir, 
 
 •li 
 
108 
 
 18. 
 
 19. 
 
 20. 
 
 ILEMKNTARY ALGEBRA. 
 
 2a:+l 
 
 1. 
 
 Rx + b 
 
 + 1. 
 
 6x+2 3x + 4 
 3aj+6 x+ia 
 
 2bx 
 
 -87. 
 
 - 6a; - 24. 
 
 6) -2. 
 
 21. f (a: + 2)-§(a. + 4)=:{(:c_ 
 
 23. l-^^10«-46 . 
 
 4 7 '• 
 
 24. TV^3a:-7,-^(2a;-4)-^ 
 
 (a: 4- 10). 
 
 25. 
 
 4aj 2a: + 3 3a: 
 
 (5aj + 8). 
 
 26. ^f- 
 
 5a: 6a;- 1 
 
 8 
 
 -12. 
 
 11 
 
 ■ 5aj - 9. 
 
 27 ^'^'^^ 2a; -3 a:-l 
 5 " 2 ~~l0"* 
 a:+2 6a: -3 
 
 29. — -^_ 
 
 a: 05-2 
 
 a: 
 
 30. 
 
 2a: + 9 
 
 10 "60* 
 
 4a; 
 
 1 |(a:-l)-3-^ 
 
 II 
 
 i; 
 
ELEMENTARY ALGEBRA. 
 
 109 
 
 CHAl lER VII. 
 
 SOLUTION OP PROBLEMS. 
 EXKROISB 40. 
 
 If :« 18 an algebraical number, state the meaning of .*ch of 
 the following expressions : 
 
 1. 2x. 2. -2x. 3. a: + 2. i x~2 
 
 6- « + 6. 6. 2x + b. 7. 2.^-6. 8 7 - '^a; 
 
 9. 2(.+ 3).10. (. + 6)1 II. (3a:-2)». 12. 14-V(, u6). 
 
 13. 2x=7. 14. (a:+2)' = a; x + 3). ' 
 
 15. 20-3(a; + 4) = 5. 
 
 EXERCISK 41 
 
 Using X to represent the unknown number, write an 
 algebraical expression which repr «ents : 
 
 1. Double the number. 
 
 2. Five times the number. 
 
 3. The sum of the number and 2. 
 
 4. The result of subtracting 4 f ror the number 
 6. The sum of double the number nd 20. 
 
 6. The square of the sum of the number and 10. 
 
 7. The product of the number and the sum of the number 
 
 and 7. 
 
 8. The amount by which the number exceeds 60. 
 
 9. The excess of the number over 60. 
 
 10. The excess of 100 over the number 
 
 11. The amount which must be added to the number to 
 
 make 40. ^ 
 
 II: i? 
 
 f?- 
 
 I Ki, 
 
 Hi 
 
 
 ! I' 
 
110 
 
 BMaraiNTAKV AMEBRA. 
 
 12. The ,^ount which must be subtracted from double th. 
 
 number to make 40. """"» me 
 
 13. That^the excess of double the number over 10 ia equal 
 U. That the square of th. sum of twice the number and 
 
 16. The number greater by one. 
 
 17/ The three roiiowing consecutive numbers 
 
 18. That the product of the two following consecutive num- 
 
 bers IS equal to 56. 
 
 19. That the square of the sum of the number and six is 
 
 greater by 2d than the square of the sum of ^e 
 number and five. 
 
 20. That the product of the next two consecutive numbem 
 
 IS greater than the product of the two preceding con- 
 secutive numbers by 42. 
 
 Consider the following problems : 
 
 (1) Find a number such that if 10 be added to double the 
 number the sum will be 50. "uoie ine 
 
 Let X represent the number. 
 Then 2a; represents double the number 
 
 ITZ^^ '"^"^T" *^" '""^ ^^ ^""*^^« *^« '^"'"ber and 10. 
 -But this sum is 50. 
 
 .'. 2a; + 10 = 50. 
 
 2a; = 40. 
 
 07=20. 
 
 Therefore, the required number is 20 
 
 (2) ^ has $100 and B has $40 how much must A give £ 
 m order thaf. « n}o»» u ^ ^ - ** 
 
double the 
 
 10 is equal 
 
 amber and 
 lumber. ) 
 sr the sum 
 
 tive num- 
 
 <nd six, is 
 
 m of the 
 
 • 
 
 I numbers 
 ding con- 
 
 the solu- 
 ouble the 
 
 r and 10. 
 
 I give B 
 
 ELEMENTARY ALGEBRA. m 
 
 ^ Let a; represent the number of dollars which A must give 
 
 Then 100 -a: represents the number of dollars which A 
 will have left. i 
 
 And 40 + a; represents the number of dollars which B will 
 then have. 
 
 .-. 100-a;=r40-fiB. 
 .-. -2a;- -60. 
 .*. a; =.30. 
 Therefore, A must give B $30. ^ 
 
 Hence we see that we can solve a problem by representing 
 the unknown number by ar, and then expressing the conditions 
 of the problem in the form of an equation. The solution of 
 this equation gives us the unknown number. 
 
 Exercise 42. 
 
 Solve the following problems : 
 
 1. The sum of two numbers is 45 ; if aj represents one of 
 
 the numbers, what will represent the other ? 
 
 2. If a; and 76 represent two numbers whose sum is 120 • 
 
 find K. * 
 
 3. A person has x dollars, a second person has 10 dollars 
 
 more than the first, together they have 40 dollars ; how 
 many dollars has each ? 
 
 4. The sum of two numbers is 73, and one of the numbers is 
 
 44 ; find the other. 
 
 5. The difference of the ages of two persons is 14 years, and 
 
 the age of the elder is 42 years; find the age of the 
 younger. %- 
 
 6. The sum of two numbnrR ih ?\d. and *^h^\r ''iff «« 
 
 find the numbers. 
 
 ^1:;^!: 
 
 if I ih I! 
 
 mi 
 
 
 rv'if 
 
 tiki 
 
112 
 
 ELEMENTARY ALGEBRA. 
 
 8. One boy has 22 maro.es more than another boy: if he 
 
 ^^h«6.o«,than double as many, find ho^ 'many' 
 
 9. The result of adding 5 to double a certain number is the 
 
 z^z rrr '' '- '"'■- *"- *« -'•«^- 
 
 10. One number is greater than another by 2 : their sum is 
 
 ov. j<ind the greater number 
 
 11. One number is less than anoth«,. by 3 ; their sum is 39. 
 
 i^ind the numbers. 
 
 ''■ ""olherb f ''f° "•■"•^™ '«««; oneisg,«.terthanthe 
 „ °™f ''ylO- What are the numbers? 
 3. Dmde 100 into two numbers whose difference is 10 
 
 Z^T^\ 'f ''"f ' "''°*"'' ' " *0 •» «dded to the les. 
 
 ,R _="'''« <J»»1>I« the latter. Find the numbem 
 B. Fmd the number which exceeds its sixth part by 35. 
 
 16. The sum of $93.05 was ™ised by A and I, t4ther, Jl 
 
 oontnbuted $8.79 ^o„ than A, how much did Lh o;,n- 
 
 17. Add 30 to a certain number, and the sum will be as 
 
 much above 31 as the original number is below 31. 
 What IS the number? 
 
 18. A drover bought a certain number of calves for $4.50 
 
 «»oh and 8 less cows at «30 each, and paid altogether, 
 »174. How many cows did he buy ? 
 
 19. How much chico,^at 10 cents a pound must be mixed 
 
 w^th 8 pounds of coffee at 40 cents a pound, to make a 
 mixture worth 22 cents a pound ! 
 20 The sum of «225 was raised by A, B Jnd C together ; n 
 contributed «19 more than A. and O. «.w „„■" f— » 
 ow much did each contribute ? 
 
'I 
 
 r I 
 
 ELEMENTARY ALGEBRA. ' 113 
 
 21. A father is 52 years old and his son is 4 years old : in 
 
 how many years will the father be exactly 7 times as 
 old as his son ? 
 
 22. If 42 be added to a certain number, the result is 4 times 
 
 that number ; find the number. 
 
 23. A is three times as old as £, and 7 years ago their united 
 
 ages amounted to as many years as now represent the 
 age of A ; find their ages. 
 
 24. A person ha^ $630 ; part of it he loans at the rate of 4 
 
 per cent., and the remainder at the rate of 5 per cent 
 and he received equal sums as interest- from the two 
 parts ; how much did he loan at each rate ? 
 
 25. John and Charles play a game of marbles ; John has 22 
 
 marbles, and Charles 13 before they begin, and at the 
 end of the game John has 4 times as many as Charles 
 How many did John win ? 
 
 26. Find a number such that its fifth part may exceed its 
 
 seventh part by 12. 
 
 27. A father's age is six times as great as that of his son, but 
 
 4 years ago it was 11 times as great. Find the age of 
 each. ° 
 
 28. How much tea worth 30 cents a pound, must be mixed 
 
 with 12 pounds at 50 cents a pound, to make a mixture 
 worth 36 cents a pound 1 
 
 29. Divide $300 among three persons, A, B and C, so that B 
 
 may receive twice as much and C three times as much as A 
 
 30. Dmde $480 into two parts, so that the first part put out 
 
 at interest for a year at 5 per cent., may exceed the 
 interest on the other part at 6 per cent., by $20 70 
 
 31. Find a number such that if 20 be added to it, the 'sum 
 
 will be three times the remainder when 20 is subtracted 
 from it. 
 
 32. A nan Anm ilfcO ^^A D dki ^er i -w-.- , 
 
 _ , ^^ „„cj x^ «,!. , u a aay. now long will it take 
 
 // to earn as much as 4 can earn in 21 days ? 
 
 i'n 
 
 i 
 
 I ; 
 
 An 
 
 1'ii 
 
114 
 
 33. 
 
 ELEMENTARY AtGEBRA. 
 
 At an election where 743 votes were polled, the sue 
 cessful candidate had a majority of Gh How rZj 
 ballots were cast for each ? ^ 
 
 he iMd twicer many left „^ then had. How 4™, 
 had each at first! "" umuij 
 
 36. If a man walks 49i mUe. in ploughing a fieia 40 rods 
 long and 20 rods wide, find the aver4e width of rh 
 fu^w, no aUowanoe being rn«ie. for'^tuming ^t Z. 
 
 '''\mml^ -Je disposed of .certain number of ho«es 
 Ih . r ' ^*"°«^'" P»y <=ows as horees at «30 
 
 rlr °r^^ *''P ^ '""^ *■"* ~*» together at 
 «5 each. The total proceeds of the sale amounted to 
 
 <.» J^ How many of each did he sell J 
 '>e»" a straight line with each other? 
 
 "■ ^ht*" *!^'''' f^-^ '™°°8 "^ f"*" «had«n. «. 
 that every tune the first receives one dollar the se«,nd 
 «ce.ves wo dollars ; and as often as the second re^s 
 
 ' '^'^°^"*^«"^™«'™ four dollar. Hown 
 does each child receive ? 
 
 first. 42 gallons are drawn; from the second, 6 gallons 
 
 tW rr ''.r^ " °°« ^'^"l » "0- thr^ tim" 
 that m the other. How much did each cade contain at 
 
 'raw 17, and then J has twice as many as ^ How 
 DMny had each at first > ' 
 
 42. ^ and £ have 128 apples between them- A »<™. » - 
 .uiBcient number to double his quantity. ^ t^y 
 
ELEMENTART ALGEBRA. 
 
 115 
 
 d, the suo^ 
 How many 
 
 > B 12, and 
 [low mani 
 
 la 40 rods 
 Ith of each 
 ag at each 
 
 r of horses 
 es at $30 
 ogether at 
 ounted to 
 
 clock first 
 
 >nce is 31. 
 
 ildren, so 
 he second 
 I receives 
 ow much 
 
 rrom the 
 » gallons, 
 fee times 
 mtain at 
 
 2kdAJB 
 How 
 
 t T VO ,&J Cm 
 
 id they 
 
 then have equal quantities. How many had each at 
 first? 
 
 43. A man has 5 times as many half dollars as he has dollars, 
 
 and 3 times as many quarters as he has halves. The 
 whole sum is $58. How many of each has he ? 
 
 44. As earnings for the past year are $75 less than twice 
 
 Fa ; Fs are $50 more than one-half C"«, and C"« are 
 $25 more than one third of A^s and Fa together. What 
 are the earnings of each ? 
 
 45. Two boys, John and James, have 74 marbles between them, 
 
 and John has 46 more than James. How many marbles 
 have each ? 
 
 46. A sixth part of the sum of two numbers is 17, and one- 
 
 quarter of their difference is 11. Find the numbers. 
 
 47. Two travellers, A and J?, agree to share their expenses 
 
 equally. At the end of the journey they find their total 
 outlay to be $48.50, aud that B must pay to A $6.25 to 
 settle according to agreement. Find the actual outlay 
 of each before settling. 
 
 48. In a family of three persons the average age is 21 years. 
 
 The father's age is five times one-quarter the combined 
 ages of mother and child, and the mother's age is 3 years 
 more than four times the child's age. Find the age of 
 each. 
 
 49. A rectangular field contains 2^ acres. If it is 26 rods in 
 
 length, what is its width ? 
 
 50. A cask contains a certain quantity of brandy. If half 
 
 the brandy be drawn off and 45 gallons of water poured 
 into the cask, and one-quarter of the mixture be brandy, 
 find the number of gallons of brandy originally in the 
 cask. 
 
 51. Find when, after 3 o'clock, the minute hand of a clock 
 
 first coincides with the hour hand. 
 
 .f: 
 
 m 
 
 'i if : ii 
 
 I I 
 
 ij 
 
 Ml 
 
 I, 4i 
 
116 
 
 ELEMENTARY ALGEBA.4. 
 
 52. Find the distance from A to 7V if a -i 
 
 for every day he w««, i^'« u u , , « '^^-eive {|pi.25, and 
 
 the end of tL T ""^^ ^^'^^'^ ^^ ^«^t«- At 
 
 «-ue ena ot the time ne received $28 75 FS„^ .i 
 
 number of days he worked. ^"^ *^'^ 
 
 54. A boy engaged with a farmer for one year for «128 .n^ 
 suit of clothes TT« u<j. j. t , '^ ^ *^^° *™ a 
 
 86. A „aa h« 11 hours at hi, disposal jCfo,^,.. 
 
 in a coach which fr»-.,«i. a V^ , ""^ "* "<*• 
 
 57. The rate of -. r^ ■ ^ °* ' °'"*» "» •><""• ' 
 
 =.iles down the L,^™^ r" f f""*™ **" "" " 
 68. ^ starts upon a wiTtt ^ "T^ °' *^" ''*™»^ 
 
 after ao'^uC^"!^ 't*,^ * "^'"^ •» "o-r. and 
 
 ^^ ^»--Henanrfe:r:^h:^:re°'-^*'^"'» 
 
 oZuZT^ It"*' °' '?'"*'■«'•• •»«' •- • bother 
 "■ """ "•" ""'«'• 9 «"»<« !««, the one wonld 
 
ELEMBNTARY ALGEBRA. 
 
 117 
 
 What sum had 
 
 have three times as much as the other, 
 each ? 
 
 62. A man seUs 50 acres more than one-half his farm, and 
 
 there yet remains 30 acres less than one-third of it. 
 How many acres were there in the farm ? 
 
 63. A bag contains a certain number of sovereigns, twice as 
 
 many shillings, and three times as many pence ; and the 
 whole sum is ^267. Find the number of sovereigns, 
 shillings and pence in it. 
 64: A huckster bought a certain number of apples at the rate 
 of 5 for 2 cents, and sold one-half of them at the rate of 
 3 for 1 cent^ and the other half at the rate of 2 for 1 
 cent, gaining altogether 4 cents. How many apples did 
 he buy? 
 
 65. A certain number of men and one-half as many women 
 
 were employed on a work. Each man received $1.25, 
 and each woman 75 cents; their total wages being 
 $45.50. How many of each were employed ? 
 
 66. A hare is 80 of her own leaps before a greyhound ; she 
 
 takes 3 leaps for every 2 he takes, but he covers as much 
 ground in 1 leap 4s she does in 2. How many leaps does 
 the hare take before she is caught? 
 
 
 w, 
 
 I 
 
 •'! r 
 
 miscellaneous review questions. 
 Exercise 43. 
 
 1. Divide So'+y* by 2x+y. 
 
 2. SimpUfy 2 L2-2{2-2 (2-a)} -al. 
 
 3. Factor ar» + 8a; - 560 and «» - 8a; - 560. 
 
 4. Solve 20a; - 13 - 565;- 3 (a; - 7). 
 
 5. Find a number which is as much greater than 20 a& thrs 
 
 times the number is greater than 70. 
 
 I. ii 
 
 '^- 
 
 j" ^1 
 
118 
 
 ELBMBNTART ALQBBRA. 
 
 6. Divide 256«* + 1 6a!» + 1 by l-4« + 16«» 
 
 7. Expand (a ~ 2b ■\- c - 3dy. 
 
 8. Find the product of (a - 26 + c - 3d) and (a + 2ft + c + 3rf) 
 
 9. Solve 4 (a; - 2) - 5 (« - 3)r= 6 (2 - a;). 
 
 10. Divide 16 into two such parts that if four times one part 
 be added to five times the other part, the result may b« 
 76. ^ 
 
 11. Dividea:5 + y6bya; + yandaJ»-y8bya;-y. 
 
 12. Divide the diflference of the squares of as* + a: + 1 and «» - « 
 
 + 1 by their sum. 
 
 13. Simplifya-2[a-2{a-2(2-a)}]. 
 
 14. Solve 3aj - 2 (2a; - 4) - a;. 
 
 15. Express a;" - y>« as 'the product of five factors. 
 
 16. Divide «* - y« by a; - y. 
 
 17. Solve (2aj - 3) (3a: - 2) - 6 (a: - 1)». 
 Y. Find the factors of 64«»-«» 
 
 19. Find a number such that double the number is as much 
 
 greater than 40 as three times the number is less than 
 85. 
 
 20. Show that (a + b) (b + c) (e + a)^ a^b + c) + bUc + a) + e^ 
 
 (a + b)+2abc. 
 
 21. Divide as* - y« by ar +y. 
 
 22. Find factors of a* - 17a; - 60 and sb» + a: - 20. 
 
 23. State in words the meaning of (a; + y)» = a;» + y» + 3ay (a; + y) 
 
 and prove the equality. 
 
 24. Solve 2a; - (3 - 4 (a: - 2j+a;} = 60. 
 
 25. Show that (a-ft) (ft-c) ((;-a)=-a» (c-ft) + J= (a-c) + c» 
 
 (b-a). 
 
m 
 
 ELEMENTARY ALGEBRA. 
 
 119 
 
 26. Multiply b* + bc + c» by 5» - 6c + c\ 
 
 27. Find the factors of Iboc^ + 34ajy + 16y» and of 15aj» - IGajy 
 
 - 15yl 
 
 28. Prove that the product of the sum of the squares of two 
 
 numbers, increased by their product, and the difference 
 of the numbers, is equal to the difference of the cubes 
 
 of the numbers. 
 
 29. Solve (aj + 2)(aj-9)-(«-6)(a;-3)-20- 13a;. 
 
 30. Prove that (a + h + c) (a^ + b^ + c^ -be - ca- ab)~a' + b' 
 
 + c"-3a6c. ^- 
 
 31. Multiply «*"«' + aJ'-aJ+l by a!* + ac' + a!* + ae + l. 
 
 32. Prove that {x-hy + z) {a?+y^ + z^-yz-zx-xi/) = at? + y^ 
 
 + «* - Sxya. 
 
 33. From the preceding write the product of {x+2y + 3z) and 
 
 (a.2 + 4y» + 9«9 - 6y« - 3zx - 2xy). 
 
 34. Find two numbers whose sum is 156 and whose difference 
 
 is 40. 
 
 35. Find the value of x^ ■i-y' + 9?-Sxyz when a; = a+l, y- 
 
 a - 1 and « as - 2a. 
 
 36. Find the product of f* - ^m + m' and P + lm + m. 
 
 37. Write the quotient oia^ + y^ + z^- Sxyz hj x + y + z. 
 
 38. Write the quotient of ar* + y* + Bar* - 6a!y» by a; + y + 2«. 
 
 39. The difference of two numbers is 6, and half their sum is 
 
 12, what are the numbers Jf 
 
 40. Solve {x + 2f«'(x+iy{x + 4). 
 
 41. Show that (6-c)« + (c-a)>+(a-6)»=2 (a« + 6« + c»-fcc 
 
 -ca~ab). 
 
 42. Write the quotient of a* - y* + z* + Bxyz by « - y + «. 
 
 43. Simplify2[a;-3{a;-4(aj--a)-a}-a]-a. 
 
 -ii 
 
 I 
 
 
 :i;| 
 
 n 
 
 ^i 
 
 :« 
 
120 
 
 ELEMENTARY ALGEBRA, 
 
 44. Two numoers differ by 1 ; show the difterence of their 
 
 squares is equal to the sum of the numbers. 
 
 45. Solve(a; + 7)«+(5-a:)(^ + 6)-36aj. 
 
 4G. Write the square of 3x-y-z. 
 
 47. Find two factors of a" - J» + c» - 2ae. 
 
 48. Show that (x + a) (x + b) (x + c)^'^ + (a + b + c) a?Uhc 
 
 + ca + ab)x + abc. From this expand the product (x +2) 
 {x + 3)(x + i). ^ ', 
 
 49. Divide n^^-f- 27«» - ^^yr hjx-y-Sz. 
 
 60. If a is 2, find the value of x which will make (x-aY^ 
 
 a;(a:-14). ' ' 
 
 61. Find the continued prknluct of (x+l), («._!), (x^ + x+ I) 
 
 and («'-«+ 1). ' 
 
 52. Show that (a^,b^) (c»+e/») = (ac+bd)^ + (ad-bcy. State 
 
 m words the meaning of thia identity. 
 
 53. Factor j:» + 3a;y + 2y' + a- + y. 
 
 64. Divide 72 into two parts, such that three times one part is 
 
 equal to five times the other part. 
 55. Solve 2(a;+l)» + 5(a:-2)»==7a:» - 2i (a:+3) (a:-9) 
 
 66. Show that (6-c)» + (e-ay + (a-by -.3 (6-c) (c-«) 
 
 67. It x.2a^b-c, y^2b^o--a, «-2c-a-6, show that 
 
 ar' + 2/' + «» = 3ay«. 
 
 68. Factor 1 2a' + ab- 20b\ 
 
 59. Simplify a:« - [(a- - «)« _ {.2 _ (^ + ^yXl 
 
 60. Solve (•3a; + 2) - (7 - •4a:) == 1 4-6. 
 
 61. Show that {x + y + zy~(y + zy + (z + xf+(x+yy^aa-^ 
 
ELEMENTARY ALGEBRA. 
 
 121 
 
 62. Divulo a:" - 3a;' + 3a; + y« - 1 by a; + y - 1. 
 
 63. Find the value of (99a; + y/+ (x'-»-99y)', when a; -49 and 
 
 y--49. 
 
 64. What is the price of bread per loaf if ah increase of 25 
 
 per cent, in the price would reduce the number of loaves 
 that could 1)6 purchased for one dollar by 2 1 
 
 65. Solve|(2x+3)-i(3a;+4; = 12. 
 
 66. Divide {a? + 5a; - 14) (x' + 8^-9) by a;» + 6a; - 7. 
 
 67. Solve(a;-6)(x + 4)-(x-8)(a;-f3) = 0. 
 
 68. A father has three sons : the father's age is 36, and the 
 
 joint ages of the sous is 30. In how many years will 
 the joint ages of the sons be equal to that of the father? 
 
 69. Expand (m + 2n -;> + 2q)\ ^ 
 
 70. Find the value of il' + ^, when il= (2a+2^-c), B* 
 
 (2c - 6 - a) and a + 6 + c = 0. 
 
 Mil' 
 
 jl 
 
 
 •1 
 
 1 
 
 :;l 
 
 |B 
 
 =i 
 
 ij^H 
 
 ■:h 
 
 
 ' "■ ii 
 '.'if 
 
 H 
 
 ., i 
 
 1 
 
 I 1 
 
 T ! 
 
 11 
 
 1 
 
 
 1 
 
 ■ 
 
 ■ii 
 
 1 
 
122 
 
 ILEMENTAHY ALGEBRA. 
 
 CHAPTER VIIL 
 
 Fractions. 
 
 «nllTT'^ '"'^""^ *'"' *^'™" °* "»» «P"»»ion by 
 
 sZraL f '' !T "■" «'^"'^P'^-''« "bove the other aod 
 separatod from it by a bar or stroke. 
 
 IW, lis the quotient obtained by dividing a by J. 
 
 Such an indicated quotient is called a fraction 
 
 f J ?r™i"? "'"""^ *'"™ "•« ""o i« <»"«J the nnmera- 
 
 tor and that below the line is called the denominator 
 
 the'tc«„~' '"' '"""^'-'^' - -""^ »"« teJnu o£ 
 
 and'rf "JTT '^ ■"" "*"^'' « '"' ""•'tiply both divisor 
 o7a f^f • ^ Tr" "'"'"""^' i'-evidenuirthe vX 
 
 That is, " « ^ 
 
 • And since tlio quotient is not altered, if we divide K^fi 
 
 Jjby the'sa^::::::,;-' ""•™'' •.ydividingoachon. 
 Thus, 
 
 «Dd 
 
 a' 
 
 a 
 
 ab 
 
 a^~a? a-x 
 
 {a + xf a + x 
 A fractioa is said to be in its lowest terms when the 
 rr"i "f d-o-inator have no common factor. We 
 
 ZI^T " *"l*" '"^ '°"^^* *^^°^« ^y dividing nunJa 
 tor and denominator by their H. C= F. "umera 
 
ELEMENTARY ALCJEBRA. 
 
 123 
 
 Exercise 
 Reduce to their lowest tonus . 
 
 44. 
 
 1. 
 
 10. 
 
 13. 
 
 16. 
 
 18. 
 
 20. 
 
 22. 
 
 24. 
 
 4x^ 
 3a>»* 
 
 20a*6'' 
 
 bx + bt/ 
 
 2. ^ 
 10 
 
 a 
 
 11 
 
 24sc»y»» 
 SOa^'a' 
 
 a^ + xtj 
 
 3. 
 
 6. 
 
 12. 
 
 6a« 
 9a*' 
 25Pmy 
 
 soimy 
 
 a?-b^ 
 
 (a + bf 
 
 a'-b' 
 
 c' + 2crf+rf' £B»-2ajy + y« " (a-6)» 
 
 ^ + 3ay+2y» ,. a^ + 2xi/ + y^ 
 
 {x + yf 
 a'+5a + 6 
 a' + a - 6 ' 
 (^-1)» 
 
 a:»-l 
 6'+6+l 
 
 6»-l 
 
 o' + 3a'4-2a 
 a» + 3a« + 3a+l 
 
 aJ»-l 
 
 (aJ + y)* 
 
 15. 
 
 m' - 2mn + n ' 
 m'-3mw+2w' 
 
 17. p'-^Qpy-24g' 
 
 19. 
 
 21. 
 
 a» + l 
 
 ar' + 3a;«4-3aj+l' 
 
 a?^y^ 
 
 C^ — 1J* 
 
 r 
 
 23. ^^+13a;y + 6 .V 
 
 a:" + 2a;'+l 
 
 26. S^ 
 
 (x+yy-z" 
 
 a^-{y+zy 
 
 26. 
 
 27. 
 
 a« - 6» - 26c - c' 
 
 a 
 
 {h-c-df 
 
 Exercise 45. 
 Complete the following identities by writing the required 
 
 term : 
 
 1 ^ - 
 
 X 
 
 m 
 n 
 
 mp 
 
 2. JL 
 
 y yz 
 
 6. * -. ^. 
 
 6 
 
 p- 
 
 11 
 
124 
 
 ELEMENTARY ALGEBRA. 
 
 10. ^=__. 
 ic bx 
 
 1 1 mx a7K. 
 no; 
 
 a + aj 
 
 15. ^Jl^^ «'-2a;- 35 
 x~^ 
 
 22. J^„ " 
 12a6 ~ 
 
 14. ^., ^ 
 
 16. '^^^^ 
 
 17. a!::ja-2o .^ ^""^^ «'+^' 
 
 18. g+^86+15^ ^ J 
 
 20. "IZl^ ^, , 
 
 1 x-y ^ ^^' 0-c^. 
 
 M Az ^-fc 
 
 22. 6-0= *' + 46c-63' 
 
 23. 1 = 
 
 ^^— 24. a« 
 
 ' Exercise 46. 
 
 Cfhaiige into equivalent fractions havinir th. « 
 
 denominators : 
 
 1 a b 
 
 X 2x 
 3 _f X X 
 
 be ca * ah 
 5 2 3 4 
 
 T* -r» — 
 
 7. " * 
 
 a + 6 o^T* 
 
 9. -J L 
 
 « + 3 a; + 4* 
 
 tile 
 
 2. -^, i. 
 
 4. — _*. <^ 
 6c ca oA * 
 
 6 « _*_ 1 
 * ba?* a^x* a»F* 
 
 a -A--, J-. 
 10. .^^+3 4 
 
 al 1 "ie _ . o* : — ■ • 
 
ELEMENTARY ALGEBRA. 
 
 1 
 
 125 
 
 11. « <t±}. ^. 12. 
 
 '^ a-l a+l 
 
 2 2a? 2a;* 
 
 x+l* {x+lf (x+l/ 
 
 1 1 ft 
 
 1 
 
 a?+6a:+5* a^ + 7x+10' 
 
 13. 
 U. 
 
 6-c' 6 + c* b^-e^ (6 + c»)' 
 
 x-y x+y x'-y* 6' d* h-d 
 
 17 _J L_ 1 
 
 18. 
 
 a + 6 
 
 6 + c 
 
 c + a 
 
 ^ {h-c){c-af {e^a){a~h)' {a-h){h-cj 
 
 The sum of the quotient obtained by dividing several 
 numbers by the same divisor is equal to the quotient obtained 
 by dividing the sum of the several numbers by the same 
 divisor. 
 
 That is, 
 
 a h c a+b+o 
 
 XXX X 
 
 Also the di£ference of the quotients obtained by dividing 
 two numbers by the same divisor is equal to the quotient 
 obtained by dividing the difference of the numbers by the 
 same divisor. 
 
 That is, 
 
 d' d" d 
 
 Hence we see that any number of fractions which have a 
 common denominator may be combined into a single frac- 
 tion. 
 
 Thus, 
 
 _« y « x-y+z 
 
 a a a a 
 
 and 
 
 at 
 
 tf 
 
 x-v-z 
 
 6-0 6-0 6-0 
 
 x-y 
 
 /•I 
 
126 
 
 ELEMENTARY ALGEBRA. 
 
 To combine ^ - * 4. 
 
 2a;- 1 2a;+l 4a'2-l 
 The L. 0. M. of the denominators is i**- 1. 
 
 into a single fraction. 
 
 5 
 
 7-+- 
 
 2x~l 2a;fl " 4ar'-l 
 ^ 6_(2£+l)_4(2«-l) 5 
 
 ^ 10a! + 5-8a; + 4 + 5 
 4aj»-l 
 
 = ^^ + 14 
 4aj»-l' 
 
 To combine iZ* + *rf + £:2i* 
 oh be ah ' 
 The L. C. M. of th^ denominators is abc. 
 
 ab 
 ae-hc 
 
 be 
 
 ca 
 
 ^ ab-ae bc-qff 
 abe ~~abc~ 
 
 abe 
 
 *»e~be + ab ac + bo-ah 
 
 ahe 
 
 — a 
 
 abe 
 
 ExiBoiSB 47. 
 Combine into a single fraction : 
 
 1 ^x * 
 3 1 1 
 
 7. ^^ ^ 
 «+y «-y' 
 
 o 2aj 3aj 
 
 G. 
 
 « 
 
 «-y «-y' 
 « + V as - «* 
 
jle fraction. 
 
 ELEMENTARY ALGEBRA. 
 
 9. _L--1. 
 
 * x-2 x + 2' 
 
 11. -J_^ 1 
 
 10. -,.l„+_l.. 
 
 a-4 a+4 
 
 ^. TO ^ + J',a;-y 
 
 13. JL4 1 
 
 a 
 
 15. 
 
 a + 2 a- 2 a'-i' 
 
 I , 1 
 
 «' + 2a;+l x' + ^iK+T 
 
 14. ...L - 
 
 «+y x-y ai?-y* 
 
 16. 
 
 2 
 
 3 
 
 ar'-3a: + 2 a:»-r 
 
 a;+2 
 
 a; + 4 
 
 17. , 
 
 o' + Saj + e a;* + 4a; + 3* 
 
 « + * « + i a:-y x^y 
 
 20. _^ + _X 
 
 «-y y-aj 
 
 22. 
 
 21. J- + -fL+_^ 
 rt-6 a+6 6'-a'* 
 
 1 
 
 23. 
 
 (a-6)(a-c)'*'(6-a)(6-.c)"*'(c-«)/o-6)' 
 
 J _1_+_1_-JL 
 
 a-l a-2 a + 2 a+l' 
 
 24. 
 
 «+! aj» 
 
 «-l aj-l «*-! 
 
 + 1. 
 
 127 
 
 '*; 
 
 \y 
 
 To multiply - and -, . 
 rt 
 
 Since the product of quotient and divisor is the dividend, 
 
 a 
 
 .*. -^ xhmta. 
 
 Also -7 xd^c, 
 a 
 
 ■••V X— xbd^ac. 
 
 a 
 
 ao 
 
 h "* d'Td 
 
 fii a 
 
128 
 
 ELEMENTARY ALGEBRA. 
 
 Hence the product of two fractions may ^ be written as one 
 fraction, whoso rium.'3rator is (.he product of the numerators of 
 the two fractions, and whose denominator is the product of 
 the denominators. 
 
 Evidently we may find the product of any number of frac- 
 tions, by multiplying together the numerators for a new 
 numerator, and the denominators for a new denominator. 
 
 To multiply ^, 
 
 iad 
 
 and 
 
 Taa^ 
 
 '2aX3ba^xl0c^d 
 5c X iad X 7aar» * 
 
 The product is 
 
 that is, .. 
 
 This result, whea reduced to lowest terms, is 
 
 3bc 
 *i ax 
 We may shorten the work by cancelling like factors in 
 numerator and denominator, just as is done in arithmetic 
 Thus: 
 
 X ?-x 
 
 a?~y^^ a-b 
 ^ (g - 5) (g + h) 
 
 {x-y){xTy) 
 
 e 
 {rs-i/){a + b) 
 
 (a + bf 
 
 x + y 
 
 X ^f X 
 
 a-b 
 
 («+6); 
 
 13. '^ 
 
 SXBROISB 48. 
 
 Find the following products in their lowest terms : 
 ^ „ Say „ 2a= .. 36'c 
 
 5y 
 
 . 2x 
 1. -—— X 
 
 4aj» 
 
 2.21= 
 6be 
 
 o a 
 3. -r X 
 
 V 
 
 — X 
 
 
 
 iad 
 I 
 
 XX 
 
 i. — X JL X -1., 
 y« 
 
 
itten as one 
 imerators of 
 9 product of 
 
 ber of f rac- 
 for a new 
 inator* 
 
 factors in 
 imetic. 
 
 9. 
 
 11. 
 
 12. 
 
 13. 
 
 U. 
 
 15. 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 20. 
 
 ELEMENTARY- ALGEBRA. 
 
 129 
 
 5xw 
 
 4yw 3yz 
 
 a- 
 
 -6' 
 
 5zw 
 
 a + b 
 (^^» 
 
 a + b 
 a^-ab ab + b* 
 
 x^-l x+2 
 
 aP-i x-l x+l 
 
 8. 
 
 a" 
 
 b^ ^ 
 (x + yf 
 
 6» 
 
 a 
 a 
 
 x"- 
 
 -b 
 
 r 
 
 10. 
 
 X 
 
 a 
 
 x-y 
 
 X 
 
 x+l x+2 
 
 x+l x+2 
 
 X 
 
 X 
 
 X' 
 
 -f 
 
 (x-yf a? + 2xy + y' 
 ar»+3a; + 2 a^+bx + Q 
 x^ + 4a; + 4 x^ + ix + 3' 
 
 a' 
 
 -h" 
 
 a 
 
 w 
 
 + ab-2¥ 
 
 a 
 
 + 26 
 Tb' 
 
 m 
 
 m^ + 4 WW + 3w' 
 
 m' — imii - 5n^ m +Sn 
 
 ar»-l 
 
 X 
 
 a^-l 
 
 {x+iy a^ + x+l 
 a?-(b + cf {a + cf-b^ 
 
 a' 
 
 X T- 
 
 -{b-cf {a-cf-b 
 
 a^ + 5x + Q !x?+5x + 4: x+l 
 
 (x+lf ^ (x + 2f ^ ST4* 
 
 a + b a^-V a^-b* 
 
 (a-bf ^ a^ + b" ^ {a + bf 
 (a-bf-c" c^-(a + b)^ c'i-{t>-cy a + b + c 
 
 {a + bf-G' ic-bf-d 
 
 - .r X 
 
 i^-ib + cf a-b + c 
 
 III 
 
 
ISO 
 
 ELEMENTARY ALGEBRA. 
 
 To divide 
 
 a 
 
 T 
 
 1 
 
 Since. "^ X i. = « 
 be d T' 
 
 . a . « ad 
 
 "T -^ d ~ bT' 
 
 But "^ - « X ^^ 
 
 
 
 
 . a . c a 
 
 •• 6 ■" rf " -T ^ 
 
 c 
 
 Hence we see that to divide by a fraction we invert the 
 fraction and multiply by it. 
 
 Thus' ^ ^ S!L ~ V 
 
 *y ay 
 
 by ab 
 
 " IT- 
 
 A^d "^-y ^ ^ty 
 
 ic» a? 
 
 07 
 
 -y' 
 
 SB* 
 
 X 
 
 X 
 
 X + 'i 
 
 Exercise 49. 
 
 Simplify : 
 
 I. 
 
 ax 
 
 . bx 
 
 
 by 
 
 ' ay" 
 
 
 3. 
 
 4ab 
 
 . 2ac 
 
 
 
 w_ 
 
 
 
 5c» 
 
 b'' 
 
 
 5. 
 
 a». 
 
 ■b^ ,a 
 
 + b 
 
 
 («- 
 
 6)> ' a 
 
 -b' 
 
 2v ax 
 
 y' fry 
 A mnp _^np* 
 
 * aJ«+7ajy+10^2 ' (a: + 5y)«* 
 
ELEMENTAllY ALGEBRA. 
 
 131 
 
 invert the 
 
 7 «'~'* ^ (a?+2)' 
 ' sx»-9 ' {x + Sf 
 
 c-2rf 
 
 '-4rf» 
 
 9. 
 
 c + 3d' e' + bd+Q<P 
 
 12 o'-^ ^ a'-8o+15 ^a-2 
 
 * a' - 25 o* - o - 6 * + 5* 
 
 13. ^-y^A^+y^f y^ a^+y' 
 
 os' + y' ' X* -y* {x- yf 
 14 <»*-6* ^g' + q^ a-6 
 
 * a2-2a6 + 6' ' a-h ^ 6(.t + 6)' 
 16. / a? + 2 _ ag-l \ a; 
 
 * Va;-2 a; + 2/ * x + f 
 
 16. 1 -^(«LZ.J + «JL^. 
 >a + 6 a-b' 
 
 8 »* + €a? + 5 ^ a; + 5 
 ' «'H:7a:+12 * «+T' 
 
 10 ^*~^ ^ °' + <'+^ 
 
 
 I 
 
 J: 
 
 
 I 
 
13^ 
 
 . ELEMENTARY ALGEBRA. 
 
 Cti^Pl'ER IX. 
 ICQUATIONS AND PROBLEMS INVOLVINa PRAOTIOKB 
 
 equaL S the L^r^^^ ^th sides of the 
 
 / ne i..i..M. of the denominators of the fractions. 
 
 To solve 3 _ 4 
 
 Mnltipiying both sides by"(f - 1^, .. i^ ^, ^^^ 
 
 3(a:,-l) = 4(a:-2). 
 That is, 3x~3 = 4x-8. 
 
 3x-ix=S~'8 
 and /i._K 
 
 > y — O, 
 
 To solve ^j^l _ u. + 4 
 
 x~ 2 x+2' 
 Multiplying both sides by (. - 2) (. ^. 2) we get 
 
 (*-l)(«r+2) = (a:_2)(a: + 4), 
 or «»+a:-2 = ar' + 2a:-8 
 ••• -« = _6, 
 
 and « = 6. 
 
 
 
 
 
 
 
 EZKRCISK 
 
 fiO. 
 
 
 
 
 
 Solve the 
 
 > equations 
 
 * 
 • 
 
 
 
 
 
 
 
 
 1. 
 3. 
 
 a; 
 12 
 
 X 
 
 3 
 
 4x 
 
 f » 
 
 X 
 
 11 
 
 4 
 "3^ 
 
 2. 
 
 4 + 
 
 — • 
 
 5 
 Qx 
 
 3 
 "2i- 
 
 
 
 
 4. 
 
 1 
 
 2 
 
 
 
 6. 
 
 20 
 
 
 39 * 
 
 -■ 
 
 
 
 
 1 +^ 
 
 3s; 
 
 « 
 
 
 — - 
 
 - - 
 
 
 0. 
 
 X x + 2 
 
ELEMENTARY ALGEBRA. 
 
 138 
 
 6. 
 8. 
 9. 
 
 11. 
 13. 
 15. 
 17. 
 18. 
 19. 
 20. 
 21. 
 22. 
 22. 
 24. 
 
 25. 
 
 26. 
 
 27. 
 
 «-4 
 
 x-2 
 
 
 2 
 
 3 
 
 1+ 2 
 
 X- 
 
 x-2 
 x-i 
 
 -4 - ^ * 
 a;+l 
 
 x-2' 
 
 X 
 
 4x 
 
 -2 
 -16 
 
 x+l 
 
 ix-8 
 
 2x 
 3x 
 
 -6 
 
 1 
 
 -8 
 
 2a!- 5 
 3* -7' 
 
 Sx 
 
 -5_ 
 
 2a;- 7 j 
 
 a; - 4 
 
 X 
 
 -2 
 
 8 
 
 x-l 
 3r-^l 
 2i-l 
 -4 
 
 X 
 
 -3 
 4?. -2 
 3/'- 2 
 
 10. 
 
 12. 
 
 14. 
 
 16. 
 
 X 
 
 3a;- 6 3a; + 3 
 
 ITTi "^ -• 
 
 ea!-1 8 
 3a; -8 
 a;-3 
 a;-4 
 
 6x-15 
 3a;- 7' 
 -6 ^ 
 
 aj+1 
 
 18a;- 6 _ 24a;- 12 
 2a;- 1 3a;- 2 
 
 «1. 
 
 a; 
 
 a; 
 
 2a; 
 
 a; - 6 a; - 6 (.r - 5) (a; - 6) 
 
 a:-7 a;- 8 
 
 05-1 
 
 a;-8 a;-9 
 a; - 4 Z"5 
 
 a;*-17a;+72 
 x-7 x-S 
 
 x-5 x-Q 
 
 x-S 
 
 x+l 
 
 a;-8 a; + 2 
 
 x-9 
 
 x-7 
 
 X 
 
 X 
 
 ^ 
 
 X 
 
 S+x 2+x l+x 
 
 Z-x 
 
 X 
 
 l-x 
 
 0^-x+l , ar'+a+l 
 
 x-l 
 
 x+l 
 
 2x. 
 
 4«+17 . 3«-10 
 
 z + 3 
 
 + 
 
 -4 
 
 -7. 
 
 37 
 
 z + 2 
 
 z + S *» + 5« + 6 
 
 i(«'-i)-i(*-i^ + i(«-i)-o 
 
 1 
 
 6 
 
 2y-5 y-3 3y-l 
 
 - «! 
 
 '■ i^ 
 
 .:llH 
 
 
 {■■ 
 
 - »v- 1 
 
 ^^B 
 
 .;,,-, i 
 
 j^^^H 
 
 
 ■ 
 
 1 
 
 1 
 
 ,11 
 
 H 
 
134 
 
 ELEMENTARY ALGEBRA. 
 
 28. 
 
 29. 
 
 6a; +1 
 15 
 
 2a;- 4 
 
 7a; -16 
 
 2a; -1 
 
 «+2 3a;- 1 1 6a;+ 13 
 
 ^8 
 
 30. -1. - -L _ 
 
 a 
 
 3z 
 
 1 - 
 
 8 
 
 2« 
 
 31. 
 
 32. 
 
 33. 
 34. 
 35. 
 
 36. 
 
 37. 
 
 y. _ y'-5y 
 
 X 
 
 as- 1 a;- 1 
 
 3y-7 
 
 T' 
 
 6-a; 
 
 3 
 
 6 
 
 a:-l *" 
 
 a;+l 
 
 a;" 
 
 a;-l 
 
 1-a;* 
 
 X 
 
 {x-i) (a;- 6) (a;-8) = (a;-l) (a;-3) (a;-5) (a;-9). 
 
 38. 
 
 39. 
 
 (3a; - 2) (2a; - 5) (5a; -9) (2a; - 3)' 
 2^'-3y+l 2y-3 
 
 ^"-2^ + 2 " y-2* 
 
 ~ + (a;+2)»- (x+iy. 
 
 X x-i 13 -2a; 8-a; 
 
 32 
 
 40 
 
 8 
 
 t(^4)-I(t-|) = *|- 
 
 Exercise 51. 
 
 1. Divide 108 into two parts, so that 25% of one part may 
 
 equal 20% of the other part. 
 
 2. What number subtracted from the denominator of f will 
 
 make the fraction equal to f ? 
 
 3. What number must be subtracted from the denominator 
 
 of yV and added to the denominator of i to make the 
 resulting fractions equaU 
 
ELEMENTARY ALGEBRA. 
 
 185 
 
 4. What number must I add to the numerator and subtract 
 from the denominator of y\ to make the fraction equal 
 
 to|? 
 6. Eight times a number, consisting of two digits, is equal to 
 three times the number composed of the digits reversed. 
 If the units digit is 5 greater than the tens digit in the 
 former number, find the number. 
 
 6. Find the number whose one-half and one-fifth exceeds its 
 
 one-third and two-sevenths by 34. 
 
 7. What number subtracted from both numerator and de- 
 
 nominator of f will reduce the fraction to ^ ? 
 
 8. What number added to the numerator and denominator 
 
 of Y- will make the fraction equal to 2 ? 
 
 9. What number added to the numerator and subtracted from 
 
 the denominator of f will make a fraction equal to ■j\'i 
 
 10. What number added to the numerator and subtracted 
 
 from the denominator of ^ will make the same fraction 
 as when twice the number is added to the numerator 
 and also subtracted from the denominator of 1 1 
 
 11. A company took a risk at 4%, and reinsured | of it at 3%. 
 
 The premium received exceeded the premium paid by 
 $27. Find the amount of the risk. 
 
 12. A man lends $375 at a certain rate of interest, and $412 
 
 at a rate 2% higher. If the interest for one year from 
 both investments is $47.59, find the rate at which each 
 
 was lent. 
 
 1 3. A man sold a horse at 20% profit. If the horse had cost 
 
 him $40 more, and had sold for the same amount as be. 
 fore, he would have lost 57„. What was the cost 1 
 
 14. A man divided a farm among three sons. To the first he 
 
 gave 110 acres ; to the second | of the whole, and to 
 the third If as much as to both the others. How many 
 acres did the farm contain t 
 
 - ii = 
 
186 
 
 ELEMENTAllY ALOEBRA. 
 
 16. 
 
 16. A man has a certain sura of money invested at 4°/ and 
 3 times that amount at G"/,. From botli investinei'iL he 
 obtains an income of $382. 1 4. What is the total an,.»u«t 
 invested ? 
 
 A quantity of goods was sold at 25°/, gain ; but, had 
 they cost $40 less, the gain at the same sellinir price 
 would have been 35%. What did the goods cost ? 
 
 17. A person gave 5 cents each to a numl)er of beggars, and 
 
 had 1 4 cents left. He found that he would have requir- 
 ed 22 cents more ta enable him to give the beggars 8 
 cents each. How many beggars were there ? 
 
 18. Divide 78 intj two parts, so that 9% of one part may 
 
 equal 17% of the other. 
 
 19. In building a house the oWner pays twice as much for ma 
 
 terial as for labor. Had he paid 5% more for material 
 and 7% more for labor, the house would have cost 
 $10,144. What was the cost ? 
 
 20. A merchant bought 100 barrels of flour, part at $7 per 
 
 barrel, and the remainder at $5 a barrel. By sellin- 
 the former at 15% ^ain, and the latter at 14% loss, he 
 just cleared himself on the transaction. How many 
 barrels of each did he buy ? 
 
 21. A grocer spent qual sums in tea, coffee and sugar, making 
 
 12% on the tea, 8% on the coffee, and losing 15% 
 on the sugar. His total gain being $63.50, find the 
 cost of each commodity. 
 
 22. Divide $500 into two parts, such that the simple interest 
 
 on one part for 4 years, at 6°/^ per annum, may be $12 
 more than that on the other part for 6 years, at 5°/ 
 per annum. 
 
 23. Find a number consisting of two digits, whose units digit 
 
 exceeds its tens digit by 5, and when the number is di- 
 
 vifJftfl \iv f.Iio Hill)) ei.f +l,i» ,1;™:*,, iU_ I.- , . n 
 
 .-i_ .y iiigii;:^ Lhc 4uuoicac 15 o. 
 
ELEMENTARY ALGEBRA. 
 
 137 
 
 24 What number divided into 367 will give a quotient 21 
 , and a remainder 10 ? 
 
 25. A can do a piece of work in m days, and B \n n days. In 
 
 what time can they do it working together 1 
 
 26. A and B working together can do a piece of work iii p 
 
 days, and A can do it alone in q days. In what time 
 can B do the work himself ? 
 
 27. A can do a piece of work in x days, and B in y days. 
 
 How long will it take B to finish the work if A has 
 worked at it z days ? 
 
 28. A can do a piece of work in 10 days, and B in^m days. 
 
 If both working together could do it in 6 days, find m. 
 
 29. A can earn %m a day, and B %n 9, day. If they work to- 
 
 gether, how much can they earn in t days ? 
 
 30. If they earn together %p; how many days do they work, 
 
 and how should the money be divided? 
 
 MISCELLANEOUS:) REVIEW QUESTIONS 
 Exercise 52. 
 
 1 . Multiply a-^-h + chy a + h-c. 
 
 2. Show that (a + if +2 (a + 6) <; + c'-(a + 6 + c)». 
 
 3. Factor 1 - 2a; + a;' - if - «' + 2y«. 
 
 4. A man receives $5 a day for his work, and forfeits $3 a 
 
 day f(»r each working day he is idle ; at the end of 20 
 days he ^ ceives |28. How many days has he worked ? 
 6. Divide {x" + Zxf - 7 {x" + 3a;) - 1 8 by a:^ ^ 3^. ^. 2. 
 
 6. Show that (a; + ^)2 =- a; (a; + 1 ) + \. 
 
 Infer an easy rule for squaring such a number as iO\. 
 
 7. Find the continued product {a->rh-\-c) (h + c-a) {c + a-b) 
 
 (a + 6 - c). 
 
 8. Divide a'-ft'-c' + cT f2fto-2arf by a + ft-c-rrf^ 
 
 9. Solve (a; - 4)' - (aj - 6)' = 4 (» - 5). 
 
 ^ J 
 
138 
 
 ELEMENTARY ALGEBRA. 
 
 10. Ten apples and six pears cost 22 cents, and one apple cost 
 half as much as one pear. Find the cost of each. 
 
 11. Find the continued product of (x + 2\ (x-~2\ (se'4'2x 
 
 12. There are two numbers whose sum is 10 ; then- product 
 
 w 24|. Find the sum of the cubes of the numbers. 
 
 13. Find factors of 50a:'- 151a; +3. 
 
 U. A, B and C are three houses, in order, along a road. 
 The distance from ^ to C is one mile, and the number 
 of feet in the distance from il to ^ is the same m the 
 number of yards in the distance from BtoC. Find 
 the distance from BtoC. 
 
 16. Solve2(aj-3)(a;-ll)^(2a;-l)(a;-7)=10a 
 
 16. Find the value of (a-6)» + (6-c)» + (c-a)3-3 (6-c) 
 
 (c-a)(a-i); when a»l, 6-2 and c = 3. 
 
 17. Find the continued product of (a + b), (a + c) and (a + d). 
 
 From the result infer the product of («+!),(»- 2) and 
 (« + 3). 
 
 18. The sum of two numbers is 20 -, the difference of their 
 
 squares, 20. What is the difference of the numbers 1 
 
 19. Find two numbers whose difference is 10, and whose sum 
 
 diminished by 10 is 10. 
 
 20. Solvea;(a;-3)(a;-7)«(a;-l)(a;-4)(.«-5). 
 
 21. Fmd factors of (ar^ + ixf - 9 («» + 4a;) - .36. 
 
 22. Find the expression which multiplied by l-y-^« will 
 
 give 1 - y" - 3y« - 2«. 
 
 23. Simplify a- [26+ {3c-3a-(a + 6)}] + 2a-(J+c). 
 
 24. Divide (a+ 6 + c) (6c + ca + oft)- oic by a + 6. ' 
 
 26. Solve (a;-6) (aJ + 2) + (5 + a:)(aj-5)-(7 + a;) (2x-3)-a 
 
ELEMENTABT ALGEBRA. 
 
 139 
 
 26. If 2««!a + 6 + c,showthatl6«(«-a)(»-5)(9-e)-2&'c' 
 
 -f2c»a»-5-2a«ft«-rt«-6*-c*. 
 
 27. Put a + 3 for fls in the expression {c* - 6a^ + 12a; - 8, and 
 
 arrange the result in descending powers of a. 
 
 28. Show that the sum of the squares of two consecutive 
 
 numbers is greater than twice their pr( duct by 1 . 
 
 29. What value of x will make (a; + a) (a; + h) greater than 
 
 (a; - 2a) (a; - 35) by 32, where a « 1 and 6 » 2 ) 
 
 30. Show that the sum of the cubes of any three consecutive 
 
 whole numbers is divisible by three times the middle 
 number. ^ 
 
 31. Factor 30aj» - 73ay - 5y«. 
 
 32. Show that 2 (a» + 6' + c» - 3a6c)- {(a-6)?+ (6-c)»+ 
 
 Z6, Expand (a + ^ + e)'; from the result infer the expansion 
 of (a;+2y-«)'. 
 
 34. Add a term to 16a'&' + 2a5c^ which will make the expres- 
 sion a square, and state of what expression it is then the 
 square. 
 
 36. Solve(«-6)» + (2a;-4)«-(3a;-7)«-(2aj-3)*. 
 
 lii 
 
 36 Fmd the L. 0. M. of a?- 1, a?+l and a^-aj+l. 
 
 37. Simplify ^ + ^Lj^ 
 
 38. If a;»2, ^=^3 and s-iS, find the value ox 
 
 X y z 
 
 39. Show that the product ot two consecutive even integers 
 
 is 1 less than a square integer. 
 
 40. Solve Z. 
 
 5^ 
 2» 
 
 2ie 
 
140 
 
 ELEMENTARY ALGEBRA. 
 
 41. Prove that x(x+l) {z+ 2) (a! + 3) + 1 ^(a?+Sx+ !)• 
 
 42. Infer from Ex. 41, the square root of 12 x 13 y 14 x 15 + 1. 
 
 43. Solve ^ - ?^± 
 
 X- S 2a: - 8 
 
 4t. The sum of two numbers is 15. The fraction formed by 
 dividing the less by the greater is equal to §. Find the 
 numbers. 
 45. Show that 
 
 q' + y-c' _ (a + b + c){ a + b-c) 
 2ab " . 2ab 
 
 If 
 
 46. Sunplify 
 
 (a + b)^-(c + d)^ X (a-hy-{ c-dy 
 {a + cf -{b4 df {a -cf - (6 - df 
 
 47. A sum of money is to be divided among a number of boys. 
 
 If 8 cents is given to each there will be 5 cents over; if 
 9 cents is given to each there will be 5 cents short. 
 Find the number of boys. 
 
 48. Solve 1^3)J?^t^)- 2-0. 
 
 (ar-l) ^aj + 8) 
 
 49. Prove that 
 
 l+» + Jn(n + l) + ^(n + l)(n + 2) 
 •=i(n+l)(n + 2)(n + 3). 
 
 50. Show that the square of the sum of two numbers is 
 
 greater than the sum of their squares by just as much 
 as the sum of their squares is greater than the square of 
 their differ<mce. 
 
 I! r 
 
-A.3Sr£3"V7 
 
 1.(1) +30. (2) -30. (3) -30. (4) +30. (5) +5. 
 (6) - 4. (7) - 1 76. (8) A debt of 3 dollars ; an asset of 4 
 dollars. (9) A debt of 5 dollars. (10) An expansion of 
 1^ inches. ^- 
 
 Ex. 2. (1) +10. (2) -9. (3) -3. (4) +21. (5) -f 
 
 (6) +2,^. (7) +42. (8) 198-2813. 
 Ex.3. (1)+ 2. (2) +8. (3) -8. (4) -2. (5) -46. 
 
 (6) -30. (7) +20. (8) -9.' (9) +20. (10) -61. 
 Ex. 4. (1) + 8. (2) - 8. (3) -8. (4) +8. (6) +28. 
 
 (6) -334. (7) +162. (8) +, (9) +. (10) -. 
 
 Ex.6. (1) +3. (2>~3. (3)-3. (4) + 3. (5)-6. (6)+70. 
 ^7) +. (8) -. (9) -. (10) +. 
 
 Ex.6. (1)- 2. (2)0. (3)-36. (4)-l. (6)-28. (6)+9. 
 
 (7) +1. (8) -25. (9) -9. (10) +31. 
 
 Ex.7. (2) -18; +18; -12; +18; +72. (4) -24; 
 
 + 2; -10; -14; -10; +24. (5)-+l; -23; -46; 
 
 -16; +62; -4; +2400. (6) + 10. (7)-.10. (8) + 10. 
 
 (9) -4. (lO)-lO. (11) -10. (12) +10. (13) +4. 
 
 (14) +60. (15) +14. (16) +60. (17) -10. (18)+7. 
 
 (19) +36. (20) -10. 
 
 Ex. 8. (1) +5a. (2) +5abc, (3) +&r- 8y. (4)'-y- 14a5. 
 (5) +|8ar. (6) +^5c. (7) +aj+y+x (8)'+3^+3p 
 + 2q. {9)ap + bq-27pq. (10) + 7a;-2«y. (ll)+18a*crf. 
 (12)15a:+12y+«. (13) -21a;-23«+l()y. (14) -5mn 
 + 6n/+40^. (15) +12mn. (16)-12a6c (17) 43aa^« 
 + 96ay«. (18)22X-7r 
 
142 
 
 ANSWERS.— EX. 9-12. 
 
 Ex. 9. (1) 12a + 46. (2) 5a. (3) 7a + 226 + c. (4)8a-46 
 
 + 23c. (5) 49a; + 36y-24«. (6) a; + y + «. (7) 13a; + 8y 
 
 + 3«. (8) 17a -196 + 26c. (9) 2c + 4rf. (10) 50xy + 
 
 S5yz-I9zx, (11) 3-2a6 + 66c. (12) - 16/>yr + 46a6c 
 
 -83. (13)^b+ibc-^d. (14)--15a6--136c + l-65crf. 
 
 (15) Y« - y + ^xyz. (16) 25a6crf - abde - 25b^de. 
 
 (17) 43^ + 46^r-69jt)r. (18) - 29a;y- n29a:y» + 1403 
 
 xyzw. (19) -2aj-9y-2M;+4a-26. (20) - 16 - f c - ^a. 
 
 Ex. 10 (1) 4a + 26 + 2c. (2) a - 26. (3) 26 + 2a (4) Sx+ 
 2y-5«-16M;. (5) -2a-6 + 4c + 3rf. (6)23a;+13y- 
 21«-w. (7)m-n-p. (8) Sm-3n + 3p. (9) 2a^ + 6c 
 ^ca. (10) - lOa^a - 1 7y«M; + 1 3zwx. (11) 2ajy - '2yz - 
 izx. {12) 5l-7m + 8n. (13) -a:-y + 2«. (14)-26» 
 + 6c«. (15) -aaji+5a + 6. 
 
 Ex. 11. (1) x + y-z. (2) «-y + «. (3) 9a-5y. (4) o. 
 (5)« + y + «. (6)aj. (7) 5aj - 2y. (8) 38 - 3a. (9) z-x. 
 (10) p + q + r. (11) a. (12) 5a + 66 - 8c. (13) 26 + 2rf. 
 (14)-2«-2y. (15) a. (16) 3a;. (17)26-c. (18)a;- 
 y + 7« + 7M;. (19) 3a -36 + 3c. (20) 2-5«. ^ 
 
 Ex. 12. (1) 2a-6 + (-3c + 4rf-«); 2a-6-(3c-4rf+c). 
 (2) a-6 + (-c-c?-c)+/; a-b-{c + d+e)+J. (3) 2x 
 -3y + (4«-3a;-3y) + 3«; 2a;- 3y-( - 4« + 3x+3y) + 3«. 
 
 (4) 2af-y + (-3a;-4y-4a;) + 
 
 y; 2a;-y-(3a; + 4y + 4a;) 
 
 +y. (5) a + 6-r(-c-a + 6)-c; a + 6-(c + a-6)-c. 
 
 (6) 2a;-3y + (4«-5w; + 6M); 2a; - 3y - ( - 4« + 5m; - 6i!*). 
 
 (7) 3a-36 + (-4c-4rf-e)i 3a-36-(4c + 4c^+e). (8) 
 26-4 + (-2c-y + 32^);26-4-(25+y-3«). (9;-a; + y 
 + (-«-m; + 6w); -x + i/-(z + w - 6u). (lO)-^x + y + 
 
 {-h + if^-h)i -|«+Jy-(i«-iw+§w). (11) a- 
 b + {c-d)-e\ (12) 2a;-{3y + (4«-5M>)}. (IS) a;- 
 
 i 
 
 .y + (« + 3)}. .(14) a6-{-6c + (-ca + a6c)}. (15) 
 -a;-{-y + («-M,)|. (16) -^a- { -|6 + (^c-i|rf)}. 
 
 /17)a- 6x^/._^\_« jI /iq\ 
 2a: + a;-2y-M;}. 
 
 Ktfr 
 
 ( 
 
 ^iZfZ 
 
 0- 
 
ANSWERS.— EX. 13-16. 
 
 143 
 
 2»2 
 
 Ex. 13.(1) -12. (2) +15. (3) +10. (4) -12aJ. (6) 
 ^Ibxtj. (6) +10wn. (7) -126*. (8) +15a^y. (9) 
 + 10mV. (10) -Uh\ (11) +15a;^ (12) +10m»«. 
 (13)+20a*6H (14) -i««6V'rf«e. (15) +10a:y2;^ (16) 
 -6mV/>V. (17)-ar». (18) +ar*. (19) - 8a»6». (20) 
 -8a«63. (21) -243. (22) +16a«6*. (23) +a26*c«. 
 (24) -36a»yV. (25) +a^»66c». 
 
 Ex. 14. (1) ax + hx. (2) cm + c?m. (3) 2ax + Zhx. (4) 3cm 
 
 + 5rfm. (5) 8aa;+126a;. (6) 30cm + 50rfw. (7) - 4a« 
 
 -66a;. (8) -9cm-15c?m. (9) -6ac + 8Jc. (10) -8c 
 
 ■i- 16c?. (11) a'+a6. {\2) cd + d\ (13) 2a:»+ 2ar». 
 
 (14) ^^-^x\ (15) a?+2x^ + x. (16) 3aJ* - 6a;» - 6.r». 
 
 (1 7) x* + a?y + ^y\ (18) - ar^ - ar'y' - a:y». (19) «»y + 
 
 ary» + y*. (20) -4a26c -.6a6«c + 8a6c\ (21) 7^-^' 
 
 (22) a;*^' - y*. (23) - 2a;*y2 - 'lo^y'z - 2a?yz'' - 2a;V 
 
 (24) a« + 6' + <:^-3a6c. (25) a^ + x^y^ + y*. 
 
 Ex. 15. (1) 2a;» + 7ay + 6/. (?) W'+lUh + W. (3) 2a» + 
 7a6 + 66«. (4) Qx''^■\1xy^-by\ (5) 2m2 + 7wm + 6w''. 
 (6) 6jo«+17^y + 59'«. (7) 2aN3ftA_26'. (8) ear* - 13a;3/ 
 + 6y2. (9) 2m2 + 3mn - 2/1*. (10) ^rr^ - 15mw + 6n«. 
 (11) 2a2+17a + 30. (12) 21a;'' + 23aj-20. (13) a;' + 
 17a: + 60. (14) x^ + 16aj + 60. (15) x" + 19* + 60. 
 (16)ar» + 23a; + 60. (17) a;2 + 32a; + 60. {U) si?+%\x + ^{i. 
 (19)a;'-t7a; + 60. (20) a;*. j^^^gQ (21) ar»- 19a; + 60. 
 (22) a;''- 23a; + 60. ( 23) a;* _ 333. + 60. (24) ar*- 61a; + 60. 
 
 (25) a;'' -7a; -60. {2%) a? + lx- QO. (."7) ar*+ lla;-60. 
 (28) ar^ - 1 la; - 60. (29) a? + ix- 60. (30) ar* - 4a; - 60. 
 (31)a;2+17B-60. (32) a;"- 17a; -60. (33) a:^ + 28a; _ 60. 
 (34)a;2-28a;-60. (35) ar» + f'>a.-60. (36) ar» - 59a; - 60. 
 (37) 9a;2^. 75^+24, (38) 9«2- 69a;- 24. (39) 9a;2^.42a. 
 + 24. (40)9ar' + 30a;-24. (41 ) 9a;' + 33a; +24. (42)9a;2^ 
 15a; - 24. (43) ^x" - 33a; + 24. (44> %x^ - 36a; + 20. 
 (45) c(? - y\ (46) a? - h\ (47) w? - n\ (48) p« - y». 
 
144 
 
 ANSWERS.— EX. 16-19. 
 
 (49) 4a5» - y-. (50) 4a:» - 9y«. (51 ) a»6» - c*. (52) />y - 
 6W. (53) a* - 6*. (54) a* - ¥c\ (55) 4a* - 96*. 
 (56) J?m^'n?-'pYr'^. 
 
 Ex. 1 6. ( 1 ) Car' + Stcy + 9a;«. (2) 4ay + ly^ + 6ya. (3) 6a:» + 
 "Ixy + 9a;2; + 2y2 + 6y». (4) a«6 + a'c + rt6=» + ac^ + 6'c + 6c=* + 
 3a6c. (5) a?' + y2 + «''+2a^+2ya +2»a;. (6) a' + 62 + c' + 
 2a6 + 26c + 2ca. (7) 4aj2 + 9y''+16a» + 12a.-y + 242/»+16«r. 
 (8) «3^y3^^_3^y^ (9) ar'-y» + »»4.3a^,«. (10) «»- 
 y' - 2* - ^xyz. (11) 6a* + 13a» + 18a« + 10a - 12. 
 (12) 2a^-lla^+18a;2-21a;4-18. (13) a;*-y*. (14) 5m» 
 + 8m*-5m»-8. (15) 12a«+ 17a*+16ar'+ 10«'+42! + l. 
 Ex. 17. (1) 2aJ*-V-8a'y+nay'-4y*. (2) a:» + a^ + a;7 + ' 
 a:6-a;*-aj3-«2_l. (3) 8a« + 276» + c3- 18a6c. (4) «> + 
 y3 + 3a:y - 1. (5j a« + a*i - 2a'62 - 2a26' + ah*' + J«. (6) 1%^ 
 + .«-a.''-8?^ + 2af'-3a^ + 4.^-l. (7) a:« - 3a^/ + 3x>y* 
 - y'- (8) u^ y + ar** + a;y' + xz^ + y'« + yz^ + 2.r;y». (9) a'^J f 
 a'c + a6» + ac^ + hH + 6c«+ i'aJc. (10) ar» + 9rc2 + 26a; + 24. 
 (11) a'-9aj'+26a;-24. (12) ar* + 9ar»y + 26a;y + 24^^. 
 (I3)ar'-9a:2y + 26«y2_24ys. (U) a:< + ar»+ 1. (15)»* + 
 a^r' + y*. (16)a:^-l. (17) a;* + .r»y'* + y*. (18)12a»-8a. 
 (19; 15a2+10a6. (20) lOa + 92. (21)0. (22) -Sa" 
 + 2a. (23) a - 46 + 9c. (24) (4a - 36) aj + (1 16 - 6c) v 
 (25) -12i.a; + 4ay. 
 
 Ex. 18. (1) (4m + 5w)a:». (2) (2a + 56 + c)a;. (3) (af6)aj 
 ^{c-\-dy. (4) (2 + 5c)a; + (3+4w)a;». (5) {m-{.n)a? 
 + (4mn+;)3r)a;. (6) (2 + c?) a; + (e + 4) ar* + (c -r/) ar* (7) 
 {a^h + a*62)a: + (a''6 + a6 V + a^^, (8) (2 - a)a; + ( - c - 3)ar' 
 + (6-c)ar'. (9) (-a-6-c) a; + (a-6) ar' + (a-c) a' 
 (10) (3-6)a;-2aa;2+.6-3a)a». (ll)^a; + y)«. (12) 
 (« + y) (a + 6). (13) a; + (a + 6)(c + <^). (14) m^-.{m- 
 ' n)(m + «). (15) (a + 6)''. (16) (a + 6) (a-6)-a2-62. 
 Ex. 19. (1) 6. (2). a. (3) 1. (4)36. (5) +4a. (6) -5. 
 (7)a^. (S)*!^. (9)a^V. (10) 7a^ (li)-4a;y2, ^jg) 
 
ANSWERS.—EX. 20-23. 
 
 145 
 
 10ay»««. (13)26. (U)-xff. (15)-2j9V". (16)-8aW. 
 
 (17) -bo. (18) a^bod. (19)- 72. (20) 5«. 
 Ex. 20. (1) a+b + o. (2) a + b + c. (3) 2a + 36 + 4c. (4) 
 
 a + 26 + 30. (ti) 3a -66 + 46c. (6) 2a -46 + 3c. (7) 
 
 26y-a6 + 36Vy». (8) 4ay - Baa^y" + 5aV. (9) ac + 
 
 2ad+3cd. (10) 4a»6-a6»c-2c''rf2+l. 
 Ex. 21. (l)a;+2. (2)aj + 2. (3)a; + 3. (4)a+l. (5)a: + 2. 
 
 (6)a; + 6. (7)a-2. (8) a: -7. (9) a -4. (10) a- + 2. 
 2. (12) c - 6. (13) a; + y. (14) a; +>. 
 
 (11) 
 
 (16) a: + 2y. (16)a:+2y. (17)a-126. (18) ^-7w 
 (19) 3a; + 2. (20) a; + 6. (21) a^ + Za^y + Zxf + y». 
 (22)w*-3m» + 4m+l. (23)«2-a;-2. (24) 2y*- 4y2+l. 
 (25) ar'-xy + y». (26) a:»y«-3a:y+l. (27) 'l^y'-^z 
 - 3»». (28) 2 + a: - a:". (29) a* - a»6c + a'»6»c' - a6V + 6V. 
 (30)m'+2w-3. (31)a: + y+l. (32)a-6+l. (33) 
 a + 6 + c. (34)a; + 3y. (35)^+2m. (36)a:-y-«. 
 
 Ex. 22. (1)- 7. (2)14. (3)-5a + 6c. (4)1. (5) a;«-a;^- 
 a;+l. (6) a + 26 + c. (7) 2a*-26*. (8) ar'-2a: + 4. 
 (9)17. (10)0. (ll)-2-a;-3a;». (12) 7y*+6a3/-a:». 
 (13)l+a; + ar' + a;' + a;*. (14)-6. (15)8. (16; - 26c + 2a<?. 
 
 (17) aj*. 5a;2y2 + 4y4, ^jg^ 3a-24a6. (19) «» + 2a:2y + 3 
 ajy» + 4y3. (20) 15. (21) i>? + 3a;V + 3a:y» + y». 
 (22) 3a2-14a6 + 86». (23)21. (24) -2^. (25) a = 4, 
 6 = 6, c =a 6. 
 
 Ex. 23. (I)ar^+2a:y + y2. (2)y2 + 2y« 
 
 + «'. (3)m»+2mn + n'. 
 
 (4) 4ar» + 4a^ + y'-'. (5) 4y2+12y« + 9*. (6)9m»+24mn 
 + 16«l (7)ar'-2«y + y». (8)y»-2y2 + »l (9)»n*-2m 
 n + »i». (10) 4ar» - 4ay + y». (11) 4y» - 12y» + 9«l 
 (12)9m»-24mw+16n«. (13) 4y + 4jtfj + <^2. (14)0*62 + 
 2a6c + c». (15) a'»6'» + 4a6c + 4c». (16) 49a''6» + 42a6 + 9. 
 (17) 4^y2 + 20a;y + 25. (18) 9mV - 24m7ia + 16a'. 
 
 ^x*,/* +»,«; + 2 J. ^ij0)a;- + _; + ^. (21) 4ar'+ _ +-1. 
 
 3 9 5 62* 
 
146 
 
 ANSWERS. — EX. 24-25. 
 
 Ilf 
 
 (23) 4a' + 6ab + 
 
 96> 
 
 ^ + ^ {26)U^j/^ + ixyz + Z^. (26)49ar'-flf+^. 
 
 (24) 9m«+ 
 21a;. 9 
 
 1 
 
 (27) 4«*-22K>4--i.. (28)4-2a^»+^.(29)20« + 2x20x3 
 
 + 3» = 529. (30) 1024. (31) 3721. (32) 10609. 
 
 (33) 10404. (34) 11025. (36) 10100^, (36) 9900^. 
 
 , (37) 1006009. (38) 4002000^. (39) 2a!y. (40) -2xy. 
 (41) y\ (42) 4y\ (43) 25. (44) 9. 
 
 Ex. 24. (1) (a; + y)'. (2) (cc-yf. (S){a + bY. (4) (c-rf)« 
 (5) {x + 2yy. {6){x- 2yf. (1) {2x+\ ,\ (8) (a + 2h)\ 
 (9) (a + Jc)'. {\Q){Zp-2q)\ (ll)(4y-l)«. (12) (a: + 
 
 r. (13)(^- + 1)V (14) (?^+l)«. (15) (2«6-cA 
 
 <^^>(t-^)- <^^>(7^^)' <^«)<^--)Ml9) 
 (5a-|)'. (20) {(a + 6) + c}«. (21) {(a; + y) + «}> (22) 
 {{a + h) + {c + d)]\ 
 £z. 25. (1) (m + n)>. (2) (j> + q)\ (3) (2a: + y)». (4) 
 (m + 3)2. (5)(;> + 4)'. (6) (2a;+3)». (7) (w-w/- (8) 
 (p-qf. (9) (2a;-y)«. (10) (m-3A (11) {p-if. 
 (12)(2a;-3)l (13)(a;-8)2. (14)(y+l)». (15)(«-2)». 
 
 (16)(x + 4> a7)(y-|.)«. (18)(2« + ^y 
 
 5 
 
 (a;y + 2)«. (20) (2a6 + 2)«. (21) (j^-^)'- 
 
 (^^-\Y (23) (-|.+iy. (24) (i+iy. 
 
 (m + w)» or {w. - nf. (26) (p + qf or (j9 - q^. 
 (1 +a)» or (1 -a)» (28) {\-a)\ (29) (3 + m)». 
 (8a+l)». (31) (2w« + l)l (32) (3m + mw)l 
 
 (19) 
 (22) 
 (25) 
 
 (27) 
 (30) 
 (33) 
 
 (a: +!-)«. (34) (x^tW 
 
 %) 
 
 (35) (aj + c)'. (36) (2aj + 
 
ANSWKRa— EX. 26-27. 
 
 147 
 
 ■^y. (37)(2«+|)«. (38) (2^+1)1 (3»)(l + «:)'. 
 (40) (l + 2x)« or (i-2a.)«. (41) (^-xV (42) 
 
 (4-4)'- 
 
 Ex. 26. (1) x'^f. (2) m»-n«. (3) 6«-c«. (4) 6»-4c». 
 (5) 4m»-n\ (6)4m»-9n». (7)«»-49. (8)a:»- 
 
 (9) 
 
 1. (10) 1 
 
 16. 
 (11) l-4a^. (12) 9-4«». 
 
 (13) ia'b' - c». (14) aW - 4c\ (15) 16A'm' -/25> 
 (16) 4a«6*-i. (17) a*-6*. (18) 4a*-6V. (19) a« + 
 2ab + 6» - c». (20) a» + 2ai + i^ - c». (21) 4a» + 4ai + 6' 
 -4r». (22)4a2+12a6 + 96»-4c«. (23) P + 2ln + n' - tn\ 
 (24) 4a»+12ac + 9cV6». (25) a'-b'+2bc-c^ (26) 
 4a»-J» + 66c-9c». (27) a:* + ar'+l. (28)aV + aV+l. 
 {'29) ^ + xy + y\ (30) l+a;* + x«. 
 
 Ex. 27. (1) {c + d)(c-d). (2) (m + n)(m-n). (3) (q + r) 
 (g-r). (4) (aa; + y) (ax - y). (5) (x + yz) {x-yx). 
 (6) (ajy + zw) {xy - zw). (7) (2m + n) (2wi - n). 
 (8)(3^ + 2y):3;>-2y). (9) (4« + 3y)(4a:-3y). (10)(2 + a:) 
 (2 - X). (11) (3 + 2y) (3 - 2y). (12) (l + -|-) (l - ^). 
 
 (13) (2ai + xy) {2ab - xy). (14) {sx? + 3) (a^ - 3) 
 (15) (4 + y») (4 - y3^. (16) (Ubc + 3) (7a6c - 3). 
 (17) (a¥c + rf') (abh - d% (18) (1 + idbc) (1 - 4aic) 
 (19) (5 + W) (5 - ipq^). (20) (1 + bpq) (1 _ 5^>. 
 (21) (9a + 56) (9a - 56). (22) (a» + 56) (a^ - 56). 
 (23) (a» + 6«) (a + 6) (a - 6). (24) (a* + 6*) (a» + 6«) (a + 6) 
 (a-6). (25) (4 + a») (2 + a) (2-a). (26) (25a:« + y'; 
 
 (5« + y)(5«-y). (27) (I 4 «yj (i.- a:y). (28)(6V+16) 
 
 (6»c»+4) (6c + 2) (6c -2). (29) (aj* + y*) (ar» + y«) (ar + y) 
 («-y). (30) (tt«6* + c*) (a'6^ + c') (a6 + c) {ab ^ e). 
 
148 
 
 ANSWERS.— EX. 28-29. 
 
 (31) (27 + 23) (27 - 23) « 60 x 4 - 2*5\ (32) 2*-6«-3. 
 (33) 5''3»-43. (34) (« + 6 + c)(a+i-c). (35) (a+6 + c) 
 (a + b-c). m)(2a + b + c){2a + b-c), (37) (a- 26 + 2c) 
 (a-26-2c). {38) (a + b + c){a-b-c). (30) (a + 6 + c, 
 (a-b-c). (40) ^a + 6-c) (a-6 + c). (41) (3a;- 2y 
 + 5z) {3x-2y-5z). (42) ixy. (43) (a + 6 + c + e^ 
 (a + 6 - c - fl?). (44) {a-b + o + d) {a - b - c - d . 
 (46) (a-26 + 2c-rf) (a-26-2c + rf. (46) (3 + 2a;-y, 
 (3 - 2« + y). (47) {a + b-c + d){a-b~c~d. 
 (48) (2a + 36 + c-5rf) (2a-36 + c + rrf). (49) (a:» + ay 
 +5^) (a^-rry + y^ (50) (x' + aiy + y) (ar«-ay + y2 
 (61) (a? + x+l)(x'-x+l). 
 
 Ex. 28. (I) a!» + 6a: + 6. (2) x^ + Sx^-W. (3) x»+18x-|-77. 
 (4) a?-5x + 6. (5) a;»-8a;+15. (6) ic* - 18a; + 77. 
 (7) «« - a: - 6. (8) ar* - 2a; - 15. (9) a^ -%x - 77. 
 (10) ar» + a!-6. (11) a;»+2a;-15. (12) a;»+4a;-77. 
 (13) m2+2w-63. (14)^^+2^-63. (16) y»+2y-63. 
 (16) mW+ 2mn - 63. (17) rfV - 2de - 63. (18) yV + 
 2y« - 63. (19) 4a;» - 4a; - 15. (20) 9y« + 12y - 77. 
 (21) 9«» - 42« + 48. (22) 4mV + 24mn - 72. (23) 9a«6V 
 -6a6c-63. (24) 16a;V+32a;'y- 65. (25) 9aV + 
 72a»y+119. (26) a^' + 6a; + 9. (27) m2+22m+121. 
 (28) 4a«'» + 20a; + 25. (29) 4m»+28m + 49. 
 
 Ex. 29. (1) a;* + 3oa; + 2a». (2) a;* + 76a; + 1 26'. (3) x'^2ax 
 -8a». (4)ar» + 6a;-126». (5) 4a;2 + g^^, ^ 3^, (6) 9a;» + 
 96a; - 286». (7) 16mV - imnp - 2p\ (8) o»6V + 3a6ca; 
 - 10a;». (9) a;« - 3a^ - 10y«. (10) 9«» - 9a;y - 10y». 
 (11) 4a;»-32aa;» + 63a2. (12) 9a«- 1006'. (13) lOO^WV 
 -40/jnna;-45a?. {U) a?if + xyz -{■ xyw + ^. (15)(a + 6)» 
 + 5(a + 6) + 6 = a2 + 2a6 + 6' + 5a + 56 + 6. (16)m» + 2»nn 
 + n«-m-w-30. (17) 4m« + 4ww + n2- 10m-5w- 150. 
 (18)a« + 2a6 + 6« + 7a + 76 + 12. (19) m* + 2mw + n« + 3to 
 
 + .^«l — 7f) ^^^^ /».2j_0«._ f{Q /01\ 1C_S„2 Oi — „ 
 
 -J- — s _, V""/ ~ ! "tw ~ vif. y^*-/ *«"*■/* ~" <s«7/ti» ■ 
 
 
ANSWERS.— EX. 30—81. 
 
 IM 
 
 (22) a»-.2o6 + 6»+7a- 76-30. (23) «» + 4try + 4y» - 5a;« 
 - 10y*-24«». (24) 4a:»- 12ay+9y»- 6ar2 + 9y«- 10«». 
 
 (25) 4m V + 1 2mwp + 9/>' - 6mn - 9p - 28. 
 
 Ex. 30. (1) ix + 3)(x+l). (2) (x + 5) (x+l). (3) (a;+8) 
 ix + 2). (4) (x + 4) (a; + 3). (5) {x + 5) {x-\-S). (6) 
 (x+4) (a;+2). (7) (y + 7) (y + 3). (8) (y+7)(y+l). 
 (9) (y+10) (y + 2). (10) (y+12)(y + 2). (11) (y-4) 
 (y-2). (12) (y-3)(y-l). (13) (m-4) (m-.4). (14) 
 (m-6) (m-4). (15) (m-7) (m-3). (16) (m - 8) 
 (m-2). (17) (m-6) (m-5). (18) (/»-9)(m-l). 
 (19) (a-12)(a-2. (20) (a - 6) (a - 4). (21)(a+12) 
 (a + 2). (22) (a + 6) (a + 4). (23) (mw + 4) (wn + 2j. 
 (24) (;,<y - 3) (;>y _ 1). (25) {am - 6) (am - 2). 
 
 (26) (am + 11) {am + 1). (27) (a^» - 17) {xyz - 3). 
 (28)(a6-19)(a6-l). (29) (a + 6) (a - 2). (30) (6-6) 
 (6 + 2). (31) (m-9) (m + 7). (32) (m + 9) (m-7). 
 (33) (a; -7) (a: + 3). (34) (a; +7) (a; -3). (35) (y4-9) 
 (y-3). (36) (y_9) (y + 3). (37) (a + 8) (a -4). (38) 
 (6-9)(6 + l). (39)(x-2)(aj-3). (40) (a: + 7) (ar-2) 
 
 (41) (6 -2) (6 -5). (42) (y+ 11) (y-2). (43) 
 
 (a.r+10) 
 
 (aa;-7). (44) {ax-li) {ax + 5). (45) (a + 26; (a + 6). 
 .(46) (a; + 3y) {x+2y). (47) {c-Ud) {c + d). (48) 
 {n + 2p){n + 2p). (49) (aj-20y)(a:+3y). (50)(a;+20y) 
 (x - 3y). (51 ) (2a: + 5) (2a; + 1 ). (52) (3a; + 5) (3a; + 7). 
 (53) (a + 6 + 6)(a + 6 + 2). (54) (m + n + 12) (m + n- 1). 
 (55) (a+ 6 - 7c) (a + 6 + 2c). (56) (a + 6 - 3c) (a + 6 - 2c). 
 
 Ex. 31. (1) a^ + y^+z^ + 2xy + 2yz-j-2zx. (2) a^+ 
 
 iy^ + z'' + 
 
 ^xy + iyz + 2zx. (3)ar» + 4y2 + 9«2 + 4a;y+12y« + 6«a;. (4> 
 ^ + y'' + z'-2xy-2yz + 2zx. {b) si? ■¥ iy"" + z^ + Axy - ^yz - 
 2zx. (&)a? + 4y^ + 9z''-ixy+l2yz-6zx. {7)a^ + b-' + c^ + 
 d^ + 2ab + 2ac + 2ad+2bc + 2bd + 2cd. (8) a^ + 6^ + 40^ + 
 9flP - 2a6 -r 4ac - Gad - 46c + G6c? - 1 2cd. ( 9 ) in?+ 9^/2 + 
 ior + 2om;2 - 12xy + 16a;« - 20a;M; - 24y» + 30yw - 40««;. 
 
150 
 
 ANSWERS.— EX. 32-84. 
 
 tf\ (12)aJ* + 4ar'y-»-6xV + 4ajy3+/. (13) a:« + 2tB» + 3aJ* + 
 4ar» + 3^+2a;+l. (14) jl^ - 2x^ + 3x* - i:^ + 3x' -2x+ \ 
 (16) a«+ 6a»6 + 15a*62 + 20a»6«+ 15a«6« + 6a6» + A« (16) 
 {a + b-c). (17)(x-y4-4 (18)(a: + 2y + 3«). (19) (2 
 
 -y-3«). (20)(a-/>-lj 
 Ex. 32. (1) af + 3x^y+3xy^ + i/> 
 
 \x 
 
 (2) ar'-3a,«2y + 3xy»-y3. 
 
 (3) ^^ + \'2x^y + (Sry'^ ^ ,f, (4) a^ - 6c«V+ 12a^»- 8/ 
 (6) 27a;» + 54ar'y4-36a,y4-8y'. (6) 27a^' - 54a^6 + 36a6» - 
 
 86«. (7) «3+] 'c'^+48 
 
 (9) 1 - 6a;+ 12ar - Sar*. (10) 8ar' 
 
 te + 64. (8) 8ar»+ ]2a;'»+ 6* +1. 
 
 + 36a;»+r)4a;+27. (11) 
 
 8 - 36a; + 54a;» - 27a,-«. (12) 8 - 6m + 1,^2 _ ^' (13) 
 
 a» + 6» + c»+ 3a*6 + 8a«c + 3a6^ + Sac^ + 36»c + 36c'' + Qahc, 
 (14) See 13. (15) a' + h' - c' + ^a^b - 3a?c + 3a6»+3ac'^- 
 m + 36c' - 6a6c. (1 6) ar* + 8y + 27r' + 6a:»y + 9ar'^ + 
 12ajy2 + 27a:»« + 36y='« + 54y2' + 36a;y«. (17) Sar' -y» + «3- 
 12a:V + 12x-"^ + 6ay+6a:«''+ 3y*« - 3y*»- 12a!y«. (18) 
 1 - 3a: + 6a;='-7ar'+6x*-3x-« + «,-«. 
 
 Ex. 33. (1) 3? + f, (2) 
 
 a;^ 
 
 -/A (3) c' + d^ (4) a»-c'. 
 
 (5)8ar' + a». (6) 8a''- c«. (7) ha^ + b\ (8) 8a^- 
 
 (9) 8a3 + 276» (10) ar'-^V (H) 
 
 27y». 
 
 af' + f (12) a'-ft" 
 
 (13) aV + 6«. (14) a^ + 3^^.'. (15) 64ar3+l.' (16) 8 + 
 
 (17)8m'+27w''. (18) ««-!. (19) 1 
 (21) a?-xy + y\ (22) ar»4- 
 
 (16) 8 + ar'y'. 
 
 + a\ (20) 8a:3 + |.. 
 
 (^ + y\ (23) a2-2a6 + 46«. 
 
 (24)4a:2^6^^^9yj (25) 1 + 2a*+4a* (26)42r'-y + i 
 (27) 25m«+15nm + 9/i». (28)m-w. (29) a + 6. (30) 
 2«-6y. (31) 3m +1. (32) l-4m. 
 
 Ex.34. (1) {p-\-q){p^-pq + q^) 
 
 (2) (;>-y) (y+^4.^2). 
 
 m/o^ ^^lltr^^'"^^- <*> (/'-27)(i>^+2;,y + 4^). 
 5 (2p + 3q) (4^ _ 6^ + 9^2,. (6) (3m - 1 ) (9m^ + 3m +1 . 
 
 (7) (2-5.)(4 + 10. + 25.«). (8) (10ar»-y)(100ar'+ lOar^y 
 
 + .v'). ^9Wa»/-».«\/'.J-»2 
 
 -/ V 
 
 ,-, I Ji\ 
 
 
 /1/\V / 
 
 /•• \ i v; ^« + oc) (a- - wo 
 
ANSWERS— EX. 85-38. 
 
 161 
 
 + 6V). (11) (a ■{■ b + c) {a^ + '2ab + b^-ac-bc + c'). 
 (12)(a + i + (;)(a'-a6-ac-h6» + 26c+'''0. (13) (a + 6-c) 
 {d^ + 2ab + b'^ + ac + bc + c^). (U) {a-h -c){a' + ab + ac + 
 6' + 26c + c'). (15) (a + 6 - c) (a^ - a6 + ac ■{• b^ - 2bc + c^j. 
 (U)(a-b + c){a' + ab-ac + b^-2bc + c\) (17)(a + 6-c) 
 (a» + 2ai+6' + ac4-6c + c»;. (18) (a-b -c)(a'' + ab-\-ac + 
 b^ + 26c + c^). (19) (3 - a - 6) (9 + 3a + 36 + a" + 2a6 + 6«). 
 (24; (x-l) (x+l) (x'-x+l) (x'-' + .r+l). (25} (2-y) 
 
 (2+y)(4-2y + y»)(4+2y + y0- 
 Fx36. (l)(3a;+l)(2a;+l). (2) (3u;+ 2)(2x'+ 1). (3>(3a;-2) 
 (2«+n). (4) (3a; + 2) (2a;- 1). (5) (5aj + 2y) (a;+2i/). 
 (6) (5^ - 2c) (6 - 2c). (7) (5c + 2d) (c - 2^/). (8) (5a; + 2y) 
 (a;+2y). (9) (4a -5) (3a + 4). (10) (3a- 4) (4a + 5). 
 (11) (3a-4)(4a-5). (12) (12a + 5) (a + 4). (13) (12a 
 + 5)(a-4). (14) (12a -56) (a -46). (15) (2a;+3y) 
 (4a: + 5y). (16) (2a; - 3y) (4a; - 5y). (17) (2a -36) 
 (4a + 56). (18) (2a + 36) (4a - 56). (19) (60a; + 1) 
 (a;4-l). (20) (8a; + 3y)(6a;+9y). 
 
 Ex. 36. (1) a6. (2) a6f. (3) m\ (4) a6Vrf. (5) 3a6»c. 
 
 (6) 123.^ (7) 5yV. (8) 76V. (9) 26. (10) ar». 
 (11) a; + 3. (12) a; +2. (13)2(a;-7). (14)a + 26. 
 ^15)a;-2y. (16)a; + 2y. (17)a-6. (18) a + 6 + c. 
 (19) a;+l. (20). x + ij. 
 
 Ex. 37. (1) aWc. (2) 12mV. (3) 24xj/^i^vr'. (4) 30a«6V. 
 (5) (a - 6) (a -f bf. (6) (a + 6) (a" + 6"). (7) (2a; - 1) (2a; 
 + iy. (8) (a;-2)(ar« + 8). (9)6(6^-1). (10) (ar' + y) 
 {x + yf. (11) (a; + 2)(a; + 3)(a; + 4). (12) (a; + 4) (a; -4) 
 (a; + 5). (13)(a;+l)(a;-l)(a;-4). (14) (a;+ l)(2a;-3)». 
 (15) (a + 46) (a + 36)^ (16) (2a;-l) (2a;+3) (3a;-4). 
 (17) (46-1) (6 + 3)1 (18) (a + 6-c)(a + 6 + c)^ (19) 
 (a + 6 + c)(a + 6-c)(a-6-c). (20) {x+2y{x + 3f, 
 
 Ex. 38, (1)4. (2) -4, (3)-3. (4)-3, (5)4, (6) -a. 
 
 (7) 6. ' (8) 8. (9) -5. (10)' 3. (11) 1 7. (12)6. 
 
^f 1 
 
 162 
 
 ANSWERS. — EX. 39-42. 
 
 (13) -13|. (14)5. _(16) 10. (16) A. (m -6 
 
 ^t■ (ftv to)iii^*;3if!uV)"o^%ri 
 
 Ex.39. (1)12. (2)13i. (3)17i. (4)18. (6)20. (6) 
 16. (7)66i. (8)70. (9; -12. 10)-60. (in-20 
 
 ». (18) 10. (19) 8. (20) 5. ^21) 6^2 ^99\ is 
 (23) 5. (24) 5f. (25) 56. 26) ^\k'^ fag f 
 (29) -f (30) 3^. ^ ^ ^ ^ ^* ^^®^^- 
 
 Ex. 41. (1) 2a;. (2)5a:. (3)a:+2. (4) a:- 4. (5) 2^;+ 20 
 (6) (.+ 10/. (7):»(. + 7). (8 ;_60. (9)t!50 
 
 : ^(lf)(2«' + 3)»=4a^. (15)3a. + 20 = 50. (16) 
 a:+l. (17)a;+l,a;+2,a;+3. (18)(aj+l) (x + 2U56 
 
 (?i2tV2!"''^^^"'^- (20; (.Vi) (i;2)t.'i.'f^ 
 
 42. (1) 45-a:. ' (2) 44. (3) $15. (4) 29 (5) og 
 (6)38,16. (7)16,38. (8)16. (9)10. lb) 26 (11) 8 
 (12)20. (13)45. (14)14. 15) 42.\'lV/$42 13 
 (17)16. (18)4. (19)12. (20)^143.^21)4 m) 
 14. (23) A 42. (24) $350. (25) 6. (26) 210! 27 
 Son 8. (28)28. (29) ^ $50. (30) $450. (31)40 
 
 (37) lOjo niinutes. (38)29. (39) $9. (40)60. (41) 
 
 ^54, (42)^96. (43) 8 dollars. (44) ^ $125. 45 
 
 James 14. (46)29. (47) ^ $18. (48) Child 6. (49 
 
 16. (50) 30. (51) 16A. (52) 56. (53) 25. (64 
 
 $16. (55) $20. (56) 24. (57) 2^. (58) 2| hrs (69) 
 49. (60) Father 54. (61)21. (62)120. (63) 240 sov 
 (64) 240. (65) 28 men. (66) 320. 
 
 Ex 
 
ANSWERS.— EX. 43-44. 
 
 153 
 
 Ex. 43. (I) ix' - 2xy + y\ (2) 12 - 10a. (3) {x - 20). 
 (a; + 28), (a; + 20) (a; -28). (4) ^V (5)25. (6) 16ar» + 
 4a; +1. (7) a'+463 + c2 + 9<i2_4a6-46c + 2ac-6arf+ 
 nhd- 6cct (8) a«-46'» + <r«-9rf=' + 2ac-126rf. (9) 1. 
 (10) 12, 4. (11) a*-x'y + xy-xf-\.y\x' + a?y^^yi + 
 xy' + y'. a2)2x. (13)lla-16. (14)4. (15) (a;- y) 
 (a' + y) i^^fYix' + y') (a^ + y8). ^16) a^ ^ a*y + a?f + x' 
 y' + x^ + y', (17) 0. (18) (8a; + «)(8aj-4 (19) 25. 
 (21) a^^-a^y + a.V-a^y' + ay-yS (22) (a;-20) (a; + 3), 
 {X + 5) {X - 4). (24) 14f (26) b* + 6V + c*. 
 (27) (3^ + 5y) (5x + 3y), (3a; -5y) (5.c + 3y). (29) 3|i. 
 (31) a:« + a;« + a:* + a3«+l. (33) ar' + 8y3 + 27«» - 18a:y«. 
 (34)58,98. (35)0. (36) l* + Pni' + m\ (S7)x' + y' + 
 z" - xy ~yz - zx. (38) «' + ^ + 4«2 - ay - 2y« - 2«a;. 
 (39) 9, 15. (40) - 1^. (42) x^ + f + z' + xy + yz-zx. 
 (43) 20a; - 21a. (45) 3j\. (46) 9aP + y^ + z'-6xy - Qxz 
 + 2yz. (47) (a-b-c) (a + b-c). (48) x' + 9x'+26x 
 + 24, (49) a? + y"" + 9z^ + xy + 3xz - 3yz. (50) -|. 
 (51) a;« - 1. (53) {x + y) {x + 2y + 1). (54) 45, 27. 
 (55) -8f|. (58) (3a + 46) (4a -56). (59) -{a^ + z'). 
 (60)28. l62)a;2-a:y + y2_2a; + y+l. (63)0. (64)10. 
 (65) -120|. (66)ar» + 7a;-18. (67)0." (68)3. (69) 
 m? + 4m2 +;?2 ^ 4^2 ^ 4^^ 2wp + 4wk7 - 4n» + 8n<7 - 42w 
 (70) 0. /- -^ /«/ 
 
 Ex. 44. (1) ^. (2) « . (3). 
 
 5?wi 
 
 (12) 
 (16) 
 
 /OA\ 
 
 (7)^ 
 
 a + 6 
 
 (« - */ 
 a + 2 
 
 a -2* 
 
 1 
 
 6^1* 
 
 (9) 
 
 a; 
 
 a; + 2y 
 
 x + y' 
 
 (8) 
 (13) 
 
 3a 
 a-b 
 a + b 
 
 (14) 
 
 (4) 
 
 4a6" 
 
 (5)i^. 
 5y 
 
 (10) 
 1 
 
 d 
 
 c + d 
 
 (11) 
 
 (6) 
 
 x + y 
 
 (21) 
 
 p + 2q 
 1 
 
 8;^ + / 
 
 (18) 
 
 (22) 
 
 »+y 
 
 a;' + a;+r 
 a'' + 2a 
 a' + 2a+l* 
 
 (15) 
 
 (19) 
 
 x-y 
 m-n 
 m-2n 
 a^-x+l 
 
 ^+2x+l' 
 
 (23) 
 
 2x + 3y 
 2x+y' 
 
154 
 
 ANSWERS.— EX. 45-47. 
 
 (24) 
 
 
 (25) 
 
 a + b + c 
 d — h — c 
 
 (26) 
 
 Ex. 45. (1) ac. (2) xz. (3) ab. (4) 
 
 x + y-z 
 x-y-z' 
 
 (27) 
 
 a—h~G-d 
 a+b -c-d 
 
 np. (5) yp. (6) ab. 
 
 (7)8xyz. (8)12abc\ (9) 4aa^. ()0) ax. (U) an. (12) 
 06. (13)(rt + a;)2. (14) £c2 + 5a: + 6. (15)(a:-7)l 16 
 a'+2ab. (17)a2-a-2. (18) 6'' + 26- 15. (19)a-6 
 (20)(a:-y)''. {21)b^-c\ (22)6 + 5c. m)a: + y. (24) 
 
 + a6 
 Ex. 46. (1) 
 
 (5) -^^ 
 aoc 
 
 f }-xy 
 
 2« b 
 2^' 2x 
 
 (2^ 
 
 aaj 
 
 (3) 
 
 a^c 
 
 (4) 
 
 a' 
 a6c' 
 
 (6) 
 
 a^ 
 ^ 
 
 'CC^ 
 
 (7) 
 
 rt 
 
 -ab 
 
 a 
 
 -^2 
 
 (8) 
 
 (11) 
 
 (13) 
 (15) 
 
 y 
 
 ,2^ 
 
 (9) 
 
 3a; + 12 
 
 o^47n+l2' 
 
 (10) 
 
 4a + 8 
 
 a'+6a + 8 
 
 a* - rt 
 
 2(a''« - 1)' 
 2^_x_+JY 
 {x + If 
 
 a + b 
 
 (12). 
 
 aj+2 
 
 (aj+l)(a;+2)(a; + 5)* 
 
 (14) 
 
 (6 + c)!" 
 
 i6i ^^{^ - ci) 
 
 (6 - c)(6 + c/ 
 
 (16) 
 
 bd{b - d) 
 
 (17) 
 
 {a + b)(b + c){c + a)' 
 
 (18) 
 
 a 
 
 i« - 62 
 
 17a; 
 
 Ex. 47. (l)f . (2) Ijl. (.•]) 
 
 1. (7) 
 
 x 
 
 7y 
 
 a;2- 
 
 2a 
 
 y^ 
 
 12 
 
 (8) 
 
 6a; 
 
 (a-6) (6-c)(c-a)' 
 
 (4) 
 
 ^-2xy~y^ 
 
 a2-16 
 
 (11) 
 
 «2- 
 
 y^ 
 
 (14) 
 (17) 
 
 -y 
 
 x'-f 
 
 (15) 
 
 a;2- 
 (12) 
 
 2.x- + 4 
 
 y 
 
 il 
 12a;' 
 
 (9) 
 
 (5) 1. (6) 
 
 (10) 
 
 X 
 
 (x+l) (a;+2)(a; + 3)' 
 
 (a;+l)»(a;+3) 
 
 (18) 
 
 2Ja^3^) 
 
 (16) 
 
 2a + 6 
 a + b' 
 
 (13) 
 8^ 
 
 a 
 
 X 
 
 (19) 
 
 X 
 
 (2a; -y) 
 
 (20) 1. (21) 1. (22) 0. (23) 
 
 a;-; 
 -6a= 
 
 (a=^-l) (a?~i) 
 
ANSWEB8.— EX. 48-51. 
 
 156 
 
 (2i) li:^-i^'. 
 
 Ex. 48. 
 
 nw 
 
 (1) 1. 
 
 ^ ' 10 
 
 (2) 
 
 ah 
 Td 
 
 (b) 1. (4)'l. 
 
 a' 
 
 (7) 
 
 (a + bf 
 
 a; - 2 
 
 
 (8) 
 
 x + y 
 
 X 
 
 (9) -1- 
 ab 
 
 yzw. 
 (10) 1. 
 
 (13) 1. (14) 1. (15) 
 
 m- 5n 
 
 (16) (^-1)'. (17) («_+* + '')'. ns) ^^ + 3 , 
 
 (20) 1 
 Ex. 49. (1) -^. 
 
 (2) 
 
 ix 
 
 (3) 
 
 2b' 
 
 (6) 
 (10) 
 
 a; + 5y 
 
 a;-3y' 
 
 a-l 
 
 ay ' 5c^ 
 
 (7) («^-2)' (a; + o) 
 (a; + 2) (a; -3)' 
 
 (8) 
 
 (4) -1. 
 pq 
 
 a;+l 
 
 a;+3* 
 
 (5) 1. 
 (9) 1. 
 
 a+\ 
 (a' + b') 
 {a + by 
 
 (11) az. (12)1. (13) (^m'. (14) i^) 
 
 (15) 
 
 7a;+2 
 
 (16) 
 
 X^ + y^ 
 a'-b' (17)«' + *' 
 
 ah' 
 
 x(x^2) ^~'' 2(a« + 6V ^ •' a + b 
 
 Ex. 50. (1) 2. (2) f. (3) 3. (4) 2. (5) 2,?^. (6) 8. (7) 
 8. (8) 8. (9) 8. (10) 8. (11) 8. (12) 2. (13) 2. 
 (14) 3f. (15)5. (16)2. (17)2. (18) -i. (19)0. 
 (20) 7. (21) 3. (22) II (23) 0. (24) 2. (25) 1. 
 (26) ^V (27) 2|f (28) -2. (29)2. (30) f (31) 
 -7. (32) 3^. (33) -2. (34) 4|. (35) 7f. (36) H. 
 (37) -If (38)4. (39) iJ. 
 
 Ex. 51. (1)48. (2)4f. (3)3|. (4)3|. (5)27. (0)420. 
 (7) 2. (8) 1. (9) 1/^. (10) ^V (11) $1542«. (12) 
 5, 7. (13) $152. (14) 720. (15) $6948. (16) $540. 
 (17) 12. (18) 51, 27. (19) $9600. (20) 40, 60. (21) 
 
 9>i-.-v. i^z^f iipouu. (23) U7. (24) 17. 
 
 (25) 
 
 mn 
 
 m + n 
 
156 
 
 ANSWERS.— EX. 62. 
 
 (26)-^ 
 p mp 
 
 (27) 
 
 {x-z)y 
 
 X 
 
 (28)15. (29)(m + w)<. (30) 
 
 np 
 
 m + n m + n wr -fn 
 
 Ex. 52.{l)a?+2ab + b^-c\ (S) (x-y + z-l) (x + y-z-l). 
 (4)11. (5) ar» + 3a;-9. (7) 2a^62 + 26V + 2c'a -a*- 
 6*-c*. (8) a-b + c-d. (9) Identity. (10)1; 2. 
 (11) a*-U. (12) 257f (13) (50a;- 1) (x-3). (14) 
 1320. (15) -3VV. (16)0. (17) a' + a^b + a''c + a'd+ 
 ahc + abd + acd + bcd,a!' + 2a^-5a~6. (18)1. (19)5, 
 15. (20)2^. (21) {x-2) (x+1) (x + S) (x + 6). 
 {22)l+y + z + f-yz + z\ (23) 7a - 26 -4c. (24) a6 + 
 be + ca-yc\ (25) - 1^^. (27) a» + 3a'»4-3a+ 1, (29) 3^9^. 
 (31) (15a; + y) (2x - hy). (33) a» + 6» + c" + 3a^b + 3a?c + 
 Sab^ + Zac^ + 3¥c -f Zbc" + 6a6c, a? + Sy' + ^^ + 6a;2y _ 33^3, ^ 
 
 l2xy^ + Zxz'-l2fz + 6yz^-l2xyz. (34) ^ 4a6 + i!-. 
 
 16 4 
 
 (35) -f (36)(x-l)(a;»+l). (37)^^f + f). (38) 
 
 -i. (40) 2. (42) 181. (43) 4^. (44) 6, 9, (46) 
 1. (47) 10. (48) 7. 
 
 I 
 
\-n)t. (30) 
 
 10) 1 ; 2. 
 -3). (14) 
 
 ■ (19)5, 
 ') (a; + 6). 
 (24) ab + 
 (29) 3^9,. 
 ?b + 3a'c + 
 ^y-Za?z + 
 
 4a6 + iL. 
 
 P- (3*8) 
 i, 9. <46>