IMAGE EVALUATION TEST TARGET (MT-3) k^ ^ 1.0 1.1 122 ^^ ■■I itt Itt 12.2 2f lift *■ ^ L& no 11.25 III 1.4 ■ 1.6 Hiotographic Sciences Corporatton ¥^ a>^ \ <^ as WBT MAIN STRUT VtfnSTIII,N.Y. MSM (716)t73-4»03 ^^ ^\ ^oN CIHM/ICMH Microfiche Series. CIHM/ICMH Collection de Canadian Institute for Historical IMicroraproductions / Institut Canadian da microraproductions hIttorfquM Tachnical and Bibliographic Notas/Notat tachniquaa at bibliographiqtiaa Tha Instltuta has attamptad to obtain tha bast original copy availabla for filming. Faaturas of this copy which may ba bibliographically uniqua, which may altar any of tha imagas in tha raproduction. or which may significantly changa tha usual mathod of filming, ara chackad balow. □ Colourad covara/ Couvartura da coulaur I I Covars damagad/ n D n n n Couvartura andommagia Covars rastorad and/or laminatad/ Couvartura raataurto at/ou palliculAa □ Covar titia missing/ La titra da couvartura manqua pn Colourad maps/ Cartas gAographiquas 9t% coulaur Colourad ink (i.a. othar than blua or black)/ Encra da coulaur (i.a. autra qua blaua ou noira) □ Colourad platas and/or illustrations/ Planchas at/ou illustrations an coulaur □ Bound with othar matarial/ RaiiA avac d'autrcs documants Tight binding may causa shadows or distortion along intarior margin/ La rs liura sarr^a paut causar da I'ombra ou da la distortion la long da la marga intAriaura Blank iaavas addad during rastoration may appaar within tha taxt. Whanavar possibia, thasa hava baan omittad from filming/ II sa paut qua cartainas pagas blanches ajoutias lors d'una rastauration apparaissant dans la takta. mais, lorsqua cala Mait possibia, cas pagas n'ont pas AtA filmAas. Additional commants:/ Commantairas 8uppl6mantalras; L'lnstKut a microfilm* la malliaur axamplaira qu'il lui a 4t4 possibia da sa proeurar. Las dAtdils da cat axamplaira qui sont paut-Atra uniquas du point da vua bibliographiqua, qui pauvant modif iar una imaga raproduita, ou qui pauvant axigar una modification dana la m4thoda normtth da f llmaga sont indiquAs ci-dassous. D D n Colourad pagas/ Pagaa da coulaur Pagas damagad/ Pagas andommagAas Pagas rastorad and/or laminatad/ Pagas rastaurAas at/ou paliiculAas r~yr Pagas discolourad. stainad or loMit/ D D Pagas dAcolorias. tachatias ou piquAas Pagas datachad/ Pagas dAtachias r~? Showthrough/ Transparanca Quality of prin QualitA inAgala da I'imprassion includes supplam«ntary matarii Comprand du material supplAmantaira Only adition availabia/ Saula Mition disponibia I I Quality of print varias/ pn includes supplam«ntary matarial/ I — I Only adition available/ Pagas wholly or partially obscured by errata slips, tissues, etc., have been ref limed to ensure the best possible image/ Les pages totalement ou partiallement obscurcies per un feuillet d'errata, una pelure, etc.. ont Att fiimAes i nouveau da fa^on A obtenir la meilleure image possible. This item is filmed at tha reduction ratio checked below/ Ce document est film* au taux da reduction indiquA ci-dassous. 10X 14X 18X 22X 26X 30X V 12X 16X 20X 24X 28X 32X ■^^^-ymw r« MdM modifiar •r una filmaga •• srrata to p«lure, m A Th« copy film«d htm has b««n raproduecd ttuinks to tho gonorosity of: LakahMd Unhrtnity ThundirBay Tho imogoo appooring horo aro tho boat quality poaaibia conaidoring tha condition and logibility of tho original copy and in icaoping with tho filming contract apooifieationa. Original eopioa in printad papor covara aro fllmod boginning with tho front eovor and ending on tifo laat pago with a printad or illuatratod improo- aion, or tho bock covor whon appropriato. All othor original copiaa aro filmod boginning on tho firat pogo with a printad or iHuatratad improa- aion, and anding on tho laat pago with a printod or IHuatratad impraaaion. Tho laat locordod frama on ooch micreficho ahoil contain tho aymboi — ^ (moaning "CON- TINUED"), or tho aymboi V (moaning "END"), whiehovor appliaa. IMapa. piataa, charta, ate., may k»o filmod at diffarant raduction ratioa. Thoao too larga to bo entirely included in one expoaure are filmod boginning in tho upper loft bond comer, loft to right and top to bottom, aa many framee aa rfHiuired. The following diagrama iiluatrate the method: 1 2 3 i L'oxomplaire film* f ut reproduit grice i la g4n*roait* da: LakaliMd UnivMiity Thunder Bay l.ea imagea auivantea ont *ti reproduitea avac la plua grand aoin. compto tenu do la condition at do la nattot* da I'axempioire film*, et en conformity avac lea conditiona du contrat da fllmago. Lea oxomplairoe originaux dont la couverture en papier eat imprim4e aont f ilm4a en commenpant par lo premier plat et en terminant aoit par la domlAro pago qui comporte uno empreinte d'impreaaion ou d'liluatratiofi, aoit par la aacond plat, aolon lo caa. Toua lea autrea exomplairoa originaux aont filmto en commonpant par la promMre pago qui comporte uno empreinte d'impreaaion ou d'illuatration et en torminsnt par la dernlAro pago qui comporte uno telle empreinte. Un doe aymboioa auivanta apparaltra aur la damlAre imago do cheque microfiche, aolon lo caa: la aymbolo -^ aignifie "A SUIVRE". la aymboio ▼ aignifie "FIN". Lea cartea, planchea, tableaux, etc., peuvent itra fiimto i dea taux da reduction diff Arenta. Loraque la document eat trop grand pour Atre reproduit en un aeui clich*. il eat film* * partir de I'angle aup4rieur gauche, do geuclM k droite, et do iMut en baa. an prenant la nombre d'imagea n*ceaaaire. ilea diagrammea auivanta illuatrent la m*thode. 32X 1 2 3 4 5 6 E] t FtTtl MAXm LOVKLL't teillBt OF •CHOOL JOOKt. ELEMENTS OF ALGEBRA; DMiomiD roR Tin usi or CAKADUK GRAHAR AID COHOX SCHOOLS, OOHTAIimiO PULL SOLUTIONS TO NBARLT ALL THE PROBLEMS, TOOttHtR WftB vmaaMwt izFUjrAi»nnr BmfAm. BT JOES RERBBRT SANGSTBR, M.A., M.D., KATBnCAIVOAL aCASTIft AVD I.Knn7RXR VX CmDaBTBT AXV irAT1TIU& rntLOsonnr is tos xosxai. bchoox. roa irEmt cavapa. .\ PEINTBD AND PUBLISHED BY JOHN LOVELL, XVJ> SOLD BT BOBKBT XILLXR. ADAM iOLLEB, 61 KDSQ STSBET KAfTi 1864, |X\^3| QA Entered, ftceoiding to the Act of the Prorineial Parliament, io the jear one thoniand eight hundred and sixty-four, by Jon LoTiLii, in the Office of the Re^trar of the ProTinee of Canada. ,. .'» ; r^n 176474 PREFAGB. TbX following pages contain solutions to all, or nearly all the problems and exennses giren in the Author's Elements of Algebra. In many cases, two or more solutions of the same problem are offered, so as to afford the student additional illustrations of the best and neatest modes of working ; and of the application of artifices employed by the ezperienoed algebraist in order to obtain a required result. On this accounti also nearly every operation has been given at full length. The Author hopes that the Ket will prove serviceable to the many who are privately prosecuting the study of Algebra, or endeavouring, without the aid of a living teacher, to prepare themselves for entrance into our Univer* idties; and that it may likewise be of advantage to those teachers whose school duties are so many and varied as to render them unable to devote to the subject that time and study which long and intricate algebraic solutions in require. ToBONTO, Octobery.l$6li •♦ ^- ♦■ CORTIHTS. fixerclM 1 6 BnreiM IX 7 Exercise ZII 1 BxcdreCM ST^tlt.. 8 Exercise XIX 9 Exercise XX... 11 Exercise XXI. ...... 13 Exercise XXII 13 Exercise XXIII 17 Exercise XXIY....... 18 Exercise XXV 19 Exercise XXVI 20 Exercise XXVII 21 Exercise XXVIII 21 Exercise XXIX 23 Exercise XXJi 24 Exercise XXXI 25 Exercise XXXII 26 Exercise XXXUI 31 Exercise XXXIV 39 Exercise XXXV 43 Exercise XXXVI 46 Exercise XXXVIII 49 Exercise XXXIX....... 90 Exercise XLI.. 62 BMnitie XLII 63 Exercise ■xMciie Exercise Bxerdse Exercise Exercise Exercise Exercise Exercise Exercise Exei^oise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exsrcise Biei^ise Exercise fixericise Etereise Exercise Exercise xcm. XLtf. ... 65 ... 69 X£tL. ...'!.* ©5 XLTU....... 65 XLVlft 67 XLIX 69 L 71 LI 74 LII 76 LUI 7« LIV....... 88 LV 101 Ltt 108 LVII 109 LVm 113 LIX 115 LX 119 LXI 133 LXII 127 LXm. 139 LXIV J33 Lxv m LXVI. 138 LXVII 142 LXVIII I«3 MisdellMMoitt EtiroliM. .« 1(M> V >' U KEY • • • • 65 • • • • 5» • • • • 63 • • • • e« • • • • 65 • • • • CI • • • • 69 • • • • 11 • • • • 74 • • « • 76 • • • • ti • • • • 88 • • • • 101 • • • • 108 • • • • 109 • • • • 113 • • • • 115 • • • • 119 • • • « 133 • • • • 127 • • • * 199 • • • • ^32 • • • f 136 • • • • 138 • • • • 142 19 ELEMENTS OF ALGEBRA. EXIBOUI IV. 1. l»-l«l^l = 4. 3'-3x3s27-0a 18 3. 1 X 2 •!■ 3 X 4 « 2 + 12 a U 4. *l''x2='-(3^1)»lx4-2 = 4-t .a 6. va+T+i a V* p 3 6. •.' «p»ibO 7. 6 X (l^- ^) r, 6 X (9 - 1) ?: 6 X 8 s 48 8. (2»x 4»- 3 X 0)* « (4 X 16)' = ({^64>» « 4"» 16 9. (1+ 2) X (4 - 0)> s 3 X 4' » 3 X 16 =: 48 10. 4 {1 -- 14 - 3)}^ s 4 (I - 1)*^ = 4 X o'» a 4 X « lU M= 2x3x4^24 12. (4*- 2 X 3)»(3»- 2 X 3 X 4)»= (16 - 6)* (27 - 24)" W)»x3';=2700 13. 14-1 * + l"T 6+1 2+1 3 ''T = l"«; rri^iTT^TTT^T-T'^ i= 6, i+l J+1 i M. 14. 14 X 1 - (3 X 2 + 3) = 14 -9 = 5; 4»- 2 (2 + 3) = 16 - 10 •= e, ami 5 ' 6 KIT TO [EZ« !▼• 15. Isob ■ *.' M, oo« fliotor of Mcb, it eqaal ^(2 f 8) X (!« 5 17. 1x4x27-2x4 108-8 100 1-I-2 + 8 + 4 10 -jTT-lO, aad 2x(2 + 3) + 10 B 2 X 5 ■ 10 18. 1 X 8 -f - (4 - 3)» 8-1" ^2 (16 + 0)4- 2 (8 -1-4) ^2x25-1-2x7 |/84 « 8 -r ■ 2i i^id {4 X 3 - (4 -t- 3 -i- 2 -i- 1» « 12 - 10 » 2 ;8. (2 - a)(8 -I- 8 - 3) t {2 •(- (12 .- 6>| - 4 (• - 8) - (18 - (8 •)- 1)) ■»-|8-(3-K4)xl)2; ■0x8-l'(2-l-6)-4x0-(18-10)-»-(8-7)x2} »0-t-8-0-8-i-lx2- 1x2* 2 20. (0-l)<4-0)•l-0-^3{l•^3(4-3)J = 8x4■^0■^3(l-l-3) s 32 -I- 3 X 4 « 32 -i- 12 B 44. 21. {(1 - 2) -I- (3 -f A)Y -»■ {(3 -h 0) - (2 - l)j»- {(0 -I- 4) + (4 - 8)|» » ( - 1 -I- 7)»-»- (3 - !)•- (4 -f 1)»; a 6»•^ 2«- 6»= 36-I- 8-26« 18 22. V(l + 3)x4+^9x(l + 2).»-{2(4-l-6)»-l-(28-12))*-(24-H)* ■« V*^<^ + ^olTs -h (2 X io»+ 16)* - (26)^ • Vie + ^27 -f i^ite - (V26)' « 4 -f 3 -I- 6 - 5' « 13 - 125 => . 112 23. 7xV0-l-3V4.(8-H2) 36-28^^{64x(1^^3))' i ix6-f of +{(2-l)-H}{4.(2.»-0)j-V2*-16 O-t-6-20 36-28-fV250 „- 14 24 2-1-0 2x2 -^8s--5^•^-J-2a-7-^6-2»-3 24. i{2(l + 2)) - i {6(3 -hi)} -► i {(3 - 2)(2 -I- 3)J -f ^ f(4 -I- 3) (I + 6 • 6 + 4)»j « \C1 X 3) - J (6 X 4) + ^ (1 X 5) -I- } (7 X 25) « 2 - 6 -t- 1 -h 25 « 22 3(H-2-3)''-Hl((3+6)(2-2+2)| f(l-H2)^27-H0H3-V4)|* {(8-h2)-V4}(4 + 3+4-0) * ~~ 25. 0-1-^36-1 I (4-f 12-8)(4 + 3) 3x0»+ll(9x2) (169-37-7)' 8x7 7(4-»-4) (ll-2)(ll-0) 6-1 7x8 11x18 (Vl26)» 56 5« 9x11 r + •h7J7 = 2 + -r--H = 2 + 6 + l = 8 56 u M. O, U1.J ▲LaUBBA. Bxnoiw IX. a. 8a-4-6y-f«-Sa4-4-f0y-8a4-4>6««-6«-3 4. • + (-!- (-{-<»))))) -6-1- (-|-(«i)J)| - e + ( - { - (M) ) ) > 6 - { - (m) I ■ 6 + (») « 6 •!> M 5. aa-3«4-4(l-63«i-l>6«-i-4-l-tf-8a-l-4a-M-4 B lla-Sc-Sd-i-M 1. i-fi-i-M-fi-i-a 8. a^+3«-a^-l>a^4'2««-2M*-i-»*'i>a^-l'a«-l-«i*-f8<^-t-8x-»-dm> s8a»+7« + a** + 3»i». 9. tf^4-3c>-i>8o^-M-e-i-4a^-l'e-3c'-m«8aWe-aiii 10. Sa-aa-l-l-a-a-fa-l-l-a-aa-l-a-fa-l-I-a-t-l 11. -a-b>c4>aoc-e + a+aa-3fr-2c-8*<-a-^»r-o s a - 86 -. 6e 12. aM-e-Y-f5-7(M»4-c-l'3a+6aM*4iMi«-64'e«9-8tf*4a «.a-8a»«2e-17.- Eznom XIL 1. Sam - 3x 4 3y + Soar + lUttg + 2am - 2My + 4a» 4- 4« ■ Saiii4-Jtf3y4-9a«4c-5x-66c-3mBl0a-5x-56e4'26-4c-8M e 6 (2a - X - 6c) 4- 2 (i - 2tf) - 3m y. 8 KIT TO [Ex. xn, xfWCL 4. M-fMA-ScM^y-Scvy-f aay'-3cMv'-l-««4-ay'-<-e«jff««y ■ 2a« •»•««- S«au;y -> 2e«y + Say* * a«Niy'' •»> Ary «- ^1^ - /V* a 2«« 4- fuc - aati«y - 2c»y -t- a«y 4- Say* - Scmy" - fty* -/V K ($• ■»• ») X - (8«» + 2c - a) ^ 4- (8« - 2cM - A -/) y* 6. 8«y - 8Ay -»• 8cy - 2m9 4- ^« • 8«|ur - Somy 4- Smis - (8«iMi 4- 3«iiy 4- iamz 4- 2ex 4- 2cs 4- acy * ace) ■ 3ay 4' 8^.4> 8cy - 2m» 4- c« - aa«« - aomy 4- Sontc * 8mic *>8a«^ - Sonm - 2cx - 2cs - ocy 4- ac« ■ ^ - 86y 4- Scy - 2m« 4- cx - Qafix * Saoiy i- 2cie - ocy 4- acs as Say - 8^ 4- 3cy - 6aMy •'Oey- 2m* 4- cx - 6amx - 2cs 4- ocs ■ (3a - 36 4- 3c - 6am - ac) y - (9m - c 4' 6(Ni) x -> (2 --a) «z 6. lla«y 4- llbmy - 3€»y •¥ 3^y -Sexy - (2aeji 4* Socxy - 8cm + 6cxy - 3cy" - Bay - 3 3axy 4- 86xy •* Sexy - aocp * aocxy 4- 3cm - ecfry 4- 3cy» 4- 3«^ 4- 3ac B llamy.4> ll^my 4- 8cy> 4- Bay > 3«xy 4> 8ftxy - aacxy - 9cxy - 2acp 4- 3cm 4- 3ac » |U c| xy 4- 3 (m 4- a) c - 2acp Eonoui XYin. ^ 1. Ra - A) 4- €\{(,a - » .- *| « (a - 6)« - c» ■ kt, |a - (6 - c)}{tt 4- (6 - c)} « a» - (6 - c)» ■ *c. •a + (6 + c)}{o- (6 4- c)} « a" - (t 4- c)» * *o. 2. {4 f (3a - ac)}|4 - (3a - 2c)) « 16 - (3a - 2c)» ■ kc. |2tt - (X - 3ift')}{2a 4- (X - 3m')) « 4a» - (x - 3m")" ■ *o. |axy + (aa T- 3y)){2xy - (2a - 3y)) » 4xV ^ (2a - 8y)" « Ac. 3. {(2a - 3c) 4- (2x - 3y)) {(2a - 3c) - (2« - 8y» « (2a - 8c)» -(ax-3y)" = Ac. |(a 4- 3*04- (2c + 4m)}{(a 4- 3<0-<2c 4- 4m)) « (« 4- Srf)"- (Jc 4- 4m)' • Ac. Ex. ZTIU, XIX.] ▲LOSBRA. • - (2 - »»)" -Ac. KJo^ - 8x»> + (I 4- »»)M(»i^ - 8jb«) - (I ♦ 1^ - (liP - ««■)» -(l+y")»-*c. 5. (5a6 + te>- 66>) - (4a' - 16afr + 10^) ^ 4<» - «^ -> 4(4a* - 406 + 6>) «. Soft 4- ea"- 6fr>- 4a*-t- 1006 - 166>- 36 4' 4^- 16o> + I9ab - 46' - *o. «. (34o«y - 16o> - 9*y) + 8(4o» + 4o*3r + *y) - 7(«V- fo^) + 4(40^ - 13o«y 4- 9x*^) ^ 24axy - 16(^ - «xV 4- 130^4- lAury 4- Sx'y" - Tx'j" 4- tf 3oF 4- 16o^ - 48a«y 4- 8«jr»/ « fte. 7. (1 -*»)(! 4-af')(l 4-a*) 4- *c. to t tenns « (1 -jr^>(l 4-«*) ( 1 4- X*) 4- Ac. to 6 tormi - (1 - »•)(! 4- ««)(l 4- »»«) 4- 4e. to 5 terms = (I - **«)(l 4- *^«)(1 4- *■) 4- Ac. to 4 tormt - (1 - »") - xV; of f^nl three torBM = a*- x*ti^ ; of finit four tonus » «* - «*y*» abA so oo. Now the index of each term in the product of the flirst two factors^ 2s2»»2»-» ladcz of each term in the product of the first three factors = 4 » 2» = 2»-* Index of each term in the product of the first four factors ■ 8 - 2"= 2^-^ and soon Therefore llie index of each term m the product oi n such factors Si 2» - » .*. )» - (x»)' = (o» - i»)(«' 4- fl^x" 4- x«) 1. (a3 + »tt2x»)(tt« - wV) = Ac. Ae. 10 KEY TO (Ex. zix 8. (3a)» + *» » (3a + «)J(2o)* - (?»)»* + (2a)V - aw* + **J s Ac* ». 3* - (3c)* » {3» + (2c)>|{3» - (2c)"j « (9 + 4c»)(3 + 2c)(3 - 3c) 10. (3»)» - (2c)» « (3fli- 2c){(3»)* 4: (3»)»(ac) +^(3«)»(3c)» '^(3m)(3c)' + (2c)«} = *o. 1 1. (a')' + (*')* - («' + *')(«^* - o'x' + a»*) « *o. 12. («*)• + (»*)" s (0*+ i»*)(a»«- o"«*+ a'M" - a*«w+ m'«) 13. (c«)»+(»«)»s&c. 14. («»•)• + (»' »)• B («^« + m'") (»«« - «»»»»« + m««) a Ae 16. (a»* + e^)(a^ + cW)(a« + c«)(o» + c")(a^ - c^ - {(«•)" + (c»)^{(«*)' + (c*)'H(«V + (c*)1(<' + «")(«' - c») * Ac. 16. («^)" + (!»">»« Ac. II. (a»* + c»*)(a»' + c«»)(a«' - c"') « {(tt»«)» + (c'»)»j {(«»)» + (c9)«}{(o9)«- (c»)»} = (tt'« + e^*)(a** - a^U^* + c»«) (a* + c») (a»« - a»c» + c") (a» - c») (o»« + a'c* + c'«) X {(a«)»+ (c«)"H(fl^)" + (c')"M<«^* - («■)•}(«" - •*•«'• + «") (a' • - a^c* + c^ 'Xa^' + aM + c' •) « (a« + e^X^^ - "•«« + c»») (a«+ c")(o« - oV + c«)(a» - c»)(a6 + a'c« ■*- c«)(o»« - a^'c*' + c'«)(oV« - aH^ + c»«)(a»« + a'c* + c'^) « {(a^» + (c»)»} (o» + c») (o' - c») (a" - a^«c»» + c3«) (o'» - o»c« + c") (tt»« + a»c9 + c»«)((i^ - a«c« + c»^(o» - «"c» + c«)(o« + o»c» + c«) = Ac. 18. (»*•)» + (c*»)» = («)»j(l»l»« -»«•€«•+ C") » *C. 19. (o»)» + (m»)» = Ac. 30. (a«'»»«»)»-(p«'')»^ (a"»»*'-i»i»)(«"*»»"* + «"n»*V +j,»4) = {(a9m9)8_ (p9)»}(a»*m»* +oi»m«»j>«»+p»*) « (o'ltt* -.|»9) «' +j»««) « Ac. Ex. ZX.I ALQE9SA. 11 EUBOIBI XX. 1. a-X'l-x-a-a'fa + a-x4-a-«-a3a->2j( 2. 3(0^- ««) - 2(a«- iot + 4a:«) - (I2a» - Oa»- 4x») - i(0*»- o^ 1= 3tt»- 3x»- 3a»+ 84W - 8*»- 12a« + Oa«+ 4x» - 3e«» + 4o» U uV - 43x> - 4ax i+*)o*-«»(o»-*-o"-"*+o**»«"-Ac. (4) |a**"»+o*x»*«-«»x*-»-***« < 0) ll - 1) 1 (1 + 1 + 1 -I- 1| 4e. ii»-3,a rf»-»x*-l-a* *x* 1-1 + 1 1-1 1-1 1, *c. 8. (a9 + x9) (a» - x«) « |(a^)«+ (x»;^ |» - (x»)»j a (a»+ x») |[tt» - x')(a« - o^ x» + x«)(a» + o^ x* + 5;«V) a 4c. 9. x" m» (o* x» - 4a' xp + 4i>') » m» x» (o' x - 2j>)». 16. V4(6+3) . .,0 ^ {(2(8 + 4| - D" + g| - {12(3 + 4) 4 1} 12-0 + ^0 + V4 (12 + 1) - 3 - (2 + 3 + 1) jjQ _ 1(14 - 1)» + 6} - (84 + 1) ^ £ _ (169 4^6) - 86 12 V62 - 3 - 6 la V49 - 6 1 It5-8S 1 7-6 ~ - 00 a - 88J (11) 1 + 2434-2-1-1 1-2-H |l-{-2-i- 3 + 2-1-1 1 4 2 -I- 3 -I- 4. 1-2-3 i 4 2 4 3 4 4 -2-4-6-4-2 -2-4-C-8 IS unr TO [fix. 1 -t- 2 4- 3 -I- a + I -8-6-9-12 1+0 + 0-2 + + + 1 1 + 0-4-8-17-12 ««+09' +Ox«f- 2«"+0a:>+0«+ 1 a'»+0a«6-4+l -a«-4af'6»-8aV-nafr«-ia&' (13) a;4 + oz» - (a«i- 6 + c) a;> + (06 + oc) ;r - 6c ^ x^ + cue - c 1 -a + c 1 + - (a* - 6 + c) + ( - » w(2fl6 + 2flc + 26c - a' - 6' - c') « m(2a6 + 2ac + 26c - a' - 6' - c' - 4a6 + 4a6) = /n(2oc - 2a6 +^26c - a' - 6* - c* + 4a6) = m\(c-;a- b)(a + 6 - c) + 4a6J ■ m{(e - a -6)(a +6 + c - ac) + 4a6}| n »ij(c~o-6)(m- 2c) + 4a6)Bm{m(c-a-6)-2c(c-a-6) + 4aA|| a 8o6r + ih'O' -a- h) | 4, and sy<5 -I' 3a - 14 -I- 2x)(l - m), snd i»'¥2X* + 2 -ha); that is of «(x-f 2) [I - m), and (x + 2)(x + 2 -f- a). 5. That is of 3a*(a - x)(a 4- x), aid 4dFx*(a - x)' ; 6. That if -^f 3m"(a"-m»)(a+m); 4m»(a"-m»)", and4m'(a'- «•*) fa - m) ; that is of 3m' (efl - m*)(«^ + am + wF); 4m* (a* - m")', ^'4m*(a*-m>)(a-m) T. That is of (X - 7)(x •»• 3) ; (x ^ 7)(» - 6), aod (x - 7)(x + 12) 8. That is of a^(x - 1 )> and a*(x - l)(x - 2) 8. That is of (x •!• ^(ae •«• 1) ; (« - 1)», mtf (x - 1)(» -K 1) BaMmZXIL (1) (2) i+2ac-c»)- ^■-*-e)x»-8x-J4(l 2x^11*^ 21x-10)x*-8x*f21x^20x+4 x»-x-6 -4x-8 -4(x + 2) ■K2)x»-«-6(x-8 »»»2x -8x-e -8x»6 3x* - 16x» + 42x" - 40x + 8(x - 3 gX*-12x'-t-21x»-10x - 4x"+21x*-30x-t-8 " 4x»»24x«-42x-t-20 - 8x>12x-12 -8(«"- 4X + 4) 14 XBT TO tfex. xtu. j^. iX 4- 4)U^ - 13«> 4- 21« >• 10(27 - 4 -4x"+l3«-10 -4a;»'H6x-16 - 8« + 6 - 3(« - 2) jr-a)ap»-4x + 4(ar-2 ^-2« -2« + 4 -2« + 4 a(a . «) « t(» - «), ftnd i^(« - «) - 3(a - *) (a - tX* " *)» •■* C'' - ^)(* "• •) 4. «(«* f « - 18), ftnd J^(« -I- 4) -t- S(« -I- 4) »<« -f 4)(« - 8), And («* i- 6)(« + 4). 6. 41* . oft . 3^)0^ - 3afr •!• 26*(1 -a*(a-a6) a6-a6> a6-a6> 4(^6-a6"-3A" 4<»-ao(iy+iey t 1906* -196* 19ft*(a-6) ««ft)«^-5a5 406-1- 46" *4a»-l-4B* 'M. £x. xsit] ALGEBRA. 1^ T. Rejecting the faetor t from the first qnantitj .^ _ ^g 15a;«-9x«f47«'-ai«'l-a8)60«<-.86x>-i-48x«-45x^42x^45«-l-12 eOx* - 36g» + 188x* - 84*^ 4- 112x' - U0«« + Sd**- f0x'-45x-l- 12 . 8 - 420** + nix' - 210x« - 186x + 86 - 420a;* -f 2S2x» - 1316j^ + 588g - t84 - 136«*-f llOto*- 723x4-820 135X*- LlOes*^ 788«-820)16x« - 9x* 4- 4Tx*- 21x •!• a8(x + 20S 9 185x« - 81«* + 423X' - 189x •!• 252 185x* - ll06x» + 728x« - 820x 1025X* - 300x' + 631x 4- 252 27 a7676x" - 8100X' 4- 1 7037x 4- 6804 a767Sx» - 2267S0x« 4- I482l5x - 168100 SISOSOx'- 131178X 4- 174904 43726(6x>-3x4-4) &r*- 8.'; + 4)186x* - llOOs" 4- 728x - 820(27x - 205 I35x»~ 81x«4-108x -1026x«4-615x-820 ^ "^0^Sx* + 81S^~S20 8. That 18 of 26(30" - 3a^ > y* 4- ay>), and 36(4a' 4- y* - Say) That ii of 2h{(Z- y^}, and 3&{(4a>- 4ay) - (ay - y>) - 26{3o»(a - y) 4- y'(a - y)J, and 36{4a(a - y) - y(a - y)} 2b(a - y)(3a* 4- y*), and 3fr(a - y)(4a - y) ; Otherwiae, 4fl^ - 6,«idl2«i^(a«-2ab-|-5*> Tb«t it of 8ft* (• -ft)*, and 12dF («-»)« 11. Renting the foetor 2 from tbe first quantity and nral* tiplyiag th* MCMid b7 8 (a + 2 8a*'t> 10«^8i^-24fl^ lUi-f e)ad^'l- 12a*-9ii^48a^H-33a^+36a-.27 1 8b« -t- lOtt* - ea* - 24if -f lltt»+ 6a aa*-8«*-24(i"4-324^-f30a-.27 8 - 6a»- 9a*-T24i^+ 6«|i^ + 90a-8l 6g*»20a»^W- 48a^-l-22atl2 - 29«* - 60a^-hlI4'4-68a^93 |9a^60a^-114(i'-68a4-93)3a>-l-10a«*6ii^-24a> -I- 11a 4-6(30+1101 29 8Ta^ -t- 2900* - 174a^ - 996a* -i- 3l9a -i- l7f 87»» + 180«« - 3420^ - aOia«> 2 t9a -^ UOa*;ma^- 4920*4- 4->-81a^-188a»»8 12. Bejeoting the fikotor 3 from the fint, and 8c from the eeooiid - af« - 806* •!• 66^ 0* - 3an» - 8a%> 4- 18a6> - 8»« (a - 26 0*- «^6-8aW+ eaft* '2tn + 1206*- 86« - 3«^ + 2aV + 16afr»- 13ft* -3<^- 40^4- 4^ -8PCa?>-l-aafr-aft>) iF 4- a«ft - 86^ 0^ - 0% - 8a6> 4^ 6ft* (a - Sfr al'-t-3<^ft-2aft» -8a«-«aft»-l-««P -StM-eoft'^'Cl^ ^S iaP-f30a-37 Isnoim Um. 1. 4 X - 3 X i^MeV " - ISdVV 2. 4 X 3 X o»*^« s 12a^«y«» 3. («-»)»(«»- y)«. {(x-y)(«»-y))« = (*»-««y-«V + j^ 4. (x*+ ay ■!-»») (»*- j*)»««+«"i + *V-«V-«»*-»* 5. x»(l -«>»;(«- l)(« + 1), and 4*(l + «) that is »»(1 - »)» ; |(1 - x)(l + *), and 4x(l + x) a 4««(i - »)« (i + *) s 4*» - 4x* 4x»*4si» lb ifj|r TQ [Ex. ZZII, XXIY« 6. «*-4*«Ottteiiit - 6') a 36 (a^ - a*b - a6* 4- fr') c 86a^ - 36a*b • 86a»* + 866' 7. x(«~8); («<-3)(«-f); aii(l«(«-7) .-. i. c. »i. ■ * (*• - 10* + 21) » *• - 10** + »!• 8. (o" - «•), ud (a"- at) - (o - *) al» - *•, and a (a - x) - (a - «) tf» - *•, and (tt - «) (a - 1) .•• /. , and { 4- aft 4- ft') } 4(a - ft)» ; 6(a - ft)(a 4- 6)(a' 4- ft") | 6(a-ft)",and(afft)«(a-ft)« , .-. i.c.m. "3x4x6Ca-ft)"(« + ft>»(a«4-aft 4-ft") (o"4- ft") ■ 60(oi» + a»ft - o"ft«- 2a''ft»- 2o«ft*4- 2a*ft« 4- 2««» 4- tt%" - oft* «ft»o) ♦ BXIB0I8I XXIY. 1. 3. 6. T. 9. a(x + y) c(14-tt) »Ki4-«) aftc* ft(a4>e) 7x"y"(3->Sx) ' 14x^~ (g 4- ft)(o" » oft 4- ft") (a4-ft)(a-ft) 2. »(^o4-«x-m*) m(3o" 4- m) a"ft(14-ft4-»i) * x(l4-ft4-m) oxV *• x(o^xfli4-oy4-xV«') o-w 8. 10. (a-}n)(o4-m) (o>ft)(o>ft) (o - ft)(o" 4- oft 4- ft") £X. XXIU, XXXf.] AliOBBBA. T$ u. 13. 15. 17. 19. 20. (a4-d)(a^-a>->-y) (a-*)(<^-t-afr4-»') (g-7)(*~4) (X -»)(« + 8) ?(2x"-3xy-6^ 13. 14. 16. 18. 7(««-8g4-6) ll(«>-3«-i-6)' (2g •>> 8)(ax •¥ 8) <2Jf + 8)(«-4) (a^-a6 + y)C~*) (am + bm) + {Up + 2l!p) m(a + b) + Zp(a-t-b} (m 4- 2p)(a + b) 22. (g-l)(2g»+3J:'-ft) (*-l)(7«-6) 21. 23. (X + a)(ag + by (x + eX*'*'b) (g + «t)(a?' -I- aom -Hw^ - **) (a4-m)('^'t-24- 2am + «i»-x') (x*- a^i^ 2a«i- M^) (tt* + X*) (g' - g*g* + »*) 24. («* + **)(tt*«-a"jp* + a»x» -««»*•+»»•> Bznoui ZXT. aF + a^-l'a-'a*>g-l-l-2 a^+ 1 g-1 g-l 3. 3ax + 9a~yx-8y-(3a*-»30) 3ax-(-9a-xy->3y»3a'-('30 X4-3 x-|*3 Sax » 3ay + xy ^ y' <- aa - xy 3ax > Sa^ - 2*- y* x-y «-y 3a'x + 3ax»-ay»~xy» + am4-mx-8ax»"Xy> 5. a-i-x 3aV - «^ - axy' -1- am •!■ mx g-fx [tx. xt% ixtt « 4- 9» s -fSn s-t-aw A-i-b aTb a + » '*tf4-a< a*+c*^df4'a4ix-s' aox ». arra -jTfp 9. •-»)i|^ + «*(a + « + ^ BnBGin ZZYtt ifi *' ttas •-« ax4>s' 8. «-)'y)«*-l'axy-i-y*-f«*-htf*(Aiyf4^«4(y'f/«- «4-y -W»-y* «y*-^ Jcy» + y* «i«|l 6A*p-6«»j8* X Smp* - ftp* + 8 I A . zsv, kzti lx|f«»n - a + * ^. JiXTP, x»Tin.l • •p>l 6.«ft.ft)g.-6-«+l<4i.jp-p-#-1.- -«+l 6. •I4>ft)fii-t-d6 + 64im(l f 5a mf6 ai4'l 6aai-f Oofr •4«i#-* ExiBOiii XXYIt *• 2(x»-^) ' 2(jt«-y*)' 2(**-V) ' 0^4* ae'-H . 3f + 2 3a(«'"l) 4a;(«'-l) Emoira XXVm. 1. 3. 4 :: ^8 *• (^+f>^ ■ (. + »)• ^ «. c ((>»») -m (6 .1.1?) » 6 (g 1.C) p l^c obc S2 ' tir 10 IBs. %xrm. % ■■'< 8(3«-l)-4(l-6a)-T(l«-f 1) 1141-14 14-lla 9. MaltiplTiog both nam. and den. of lit flwet. b^ ordor to ehftngo tbo ilgu of the den. we get x(x - 16) •!-(« + 7)(U -f 8) - (S - 8c)(3 - «) iT? «*-ie« + (a«»+ T« -f 6)-(4 - 1» •!• 8ar<<) TriT *o. 10, 11. « + y « + y « + y «-y « + y «»y g»y-ap»y "TT+'ft — sr+nr""*; — r""~n5 — " = (■n- |l)(ln■.p)4'(|»^^g)(J» -•)•!• (en- »Kg-m) "T (p-«)(«-ll)(M-Jl) fte. (HI* -f^ + (p« -«•) + (««- m") 12. (« - ft)(6 + c) + (* - c)(a + ») 806 -2m (a-i-»)(»-fc> •6 + 6e •!• oe •!• 6e - 6c -f 6' 806 - afrc - 2e6 4- 2ae ■ *c. 13. l^.a;.(l.«) 8(H-2x)-8(l-ajr) 2« TTJT l-4«" 12x l-4i« 2g ~ Sjc* -Hax - 12«» i4g-ao«* = (l-x")(l-4«>) ■ l-6T»f4«« 14. MnltiplTing both teon^ cf t^^* 't 9f the 1 ' .iro Orec. bj - 1 we get la fM ' . in a(aTF)15rr5 " fr(a-6)(ft-e) " e(a-e)(c-6) Ml (ft - (2fty-yc-6c»»(a - c) "■ fr(a - 6)(6 -«)"** «(• - «)(* - O wbenee we hare I. e. m. of the den. « abc(^» - ft)(6 - e)(a - c) ftgw(6 ~ c) - omCa » e) ■► o>wC - 1) ... the flren frMtiom o^a - t)(o - e)(6 , c) Wcw - ><% - a*em ■¥ aAn ■¥ a*bm ■» mb^ abe{a - 6)(a - c)(h - e) a»c(a - 6)(a - e)(6 - m a6e(a - 6)(a • c)(fr - c) * o^c(« - e)(6 • e) i n(c* - flc -» 6c -Kit) m{(tt> - >c) - (Qg - <*)\ abc(a - c)(6 - c) " aie(a - c)(* - c) iii{6(a~c)-«(o«c)} in(6 - c)(g « c) m ofre(a-c)(6-e) " a»e(^-«)(»->c) * oSe* Entoiu XXIX. 2gx Sac 3x* ^' 5 X 2a * 6a 2MXX*X|f* 2. — - ■ t xyxmjfxs 2(a + 6) x(a-.6) 2a->2& 3. — r- — X xy 3(0 + 6) ~ 3y ** 1 ** 2a '^0+6* 2(a + 6) " aa + 26 (g -. ag)(a + x) (g + 6)(g-6) g «(tt - ft) g + 6 ** g + * '*«(g^"x)" x my a-m my my r (** "*)(*•*• ») 4ax« ig(g - x) 4gx - 4x* 24 EBY TO [Ex. XXIX, XXX. (ag-T)(x~6) («~6)(»-»4) (»-t)(af.4) x«- 11a + 28 *• x(af-6) ^ «(x-6) "«»"«» ahedm am 9. 10. 11. 18 bet^fY" f*y (tt-2)(g + 2) t?'l g- 2 (g-2)(a»a) (a-2)» (flF-1) ** 2a '*a+2* 2a " 2a (X - a)(x + g) x(x + &) + c (x + 6) x(x + 6) - g(x + 6) ** x(x + c) + d(x + c> x-f g x+H (x-g)(x + a) (x + 6)(x + g) (X + 6)(x - g) ^ (X + c)(x + «0 (x + 4)(x-8) (x-6)(x+Y) (x-3)(x + Y) -4x-2I '^- (x-8)(x-6)'*(x+4Kx-ll)*(X-8Xx-ll)*x^l9x+88 13. 1 " g + g» g* 4- g -H _ {(g« + 1) - g}((gy -H) •!■ g} g" (g'+l)»-a^ g* + g»+l (2g+4>tt)(2g-4ro) ^*' g-2Bi ^ 2(g - 2in)(a -1-2111) 6g a 4- 2m B(2a-f4ii)(2a-»'4») 2(a-.ajR)(a-f2m) 4-> g+ X g-f 5 ~ » j? 1 X g- X g (g-6)« SxiBoisi XXZ. y y \ ^- g-6 ** (oTftp g-f X g a '*g-« g-6 ■ g+ 6 (g«+x«)(g4-f )(g~x) (y4.2Xy"2) y + 2 g-x x-3 (g » 9) (X - 8) x-3 ***-« ** (;t-8)(x-7) '*-T adaCtffxXy^-a) r.ii- BjC. XXX, ZXZI.] ALQUBRA. 26 o» + 6» a« + 6" + 6« a» - 6« T» + a" - 6« " o» - 6« ■ a' I"*"** «?TP "T- I a-x (a - X) a+ X a' + ax + jc* a*-ux + x* (g - xXo* 4- 0x + ar*)(a + g)((^ ■» ox + ac») 1 a-x (a-x)(a-x) + x a^ + ax + a^-ax + xr 8. 9. 3(a'-l) 2a(a + 6) 3a(fl«-l) afl^^aa 2(<»+6)^ x*-l " x«-l " x«-l (xy + y«) + y» + x(x + y) (2xy+2^ + x (x + y) - xy xy+y xy + y" 2y'+2xy + x' xy + y« 2y' + 2xy + x" 10. xy + y* 4ay 0*- 6« * rf"-** * (o«-6«)(a»+ 6») ^ 40b " a« + 6» 4a6 4a%> i^-ft« a6 SSIBOIM ZZXt. a-( 9a-2x 3 _ 6(a - 6) Oa+96 lOo •». 96 6. 15 X I OJt a + 2x a-i-2i a 15-6x4- 6a 10 10a -i- lOx -6 16 8a l-4o« T7W 4a If 4a* 21 - 12x 30 7a -Ix 21 3(21 «12x) 10(a«-l) 6 i-4o« 3(16-6x + 6a) 2(10a + lOx - 6) -2a l-a> -ll 26 KIT TO (Ex. XXXI, xxxu. a' + fr'-oft 8. 9. ab xy - 1 - «y a-T"! (a+6)(aF-o5+6») - 1 -1 1 — I — =f 1- «y - 1 l- «y 1- icy ~ I ary-l-gy «y-l *y -I -1 *1 ^1 l- xy - 1 -1 l- - xy -T+«y xy i9 BnBoira XXXII. 1. 12x -If 4x » 84 - Zx, or 19x ■ 84, or x ■ 4^ 2. lOx - X « 6x + 20, or 4x <■ 20, or X ■ S 3. 168x - 28x •M2x ■ 63x - 231 4- 84x 4> 766, or 5x or X s lod 526, Ex. xxttt.] ALGBBRA. 27 4. 30dr - 105 + 9j; - 3 s Sx + 40 - 30ar, or 64« a 148, or x at 2fe 5. 46 - 4x + 20 s 84 « 7x + 49, or 3x s S7, or « s lo 6. 56x - 8x = 21x + 7 + 14ar + 84, or 13x = 91, or x a Y 7. 3x — 65 = 35 + 2x, or 6x a 100, or x = 16| 8. 15x + 45 - 12x - 48 - 960 = - 20x - 20, or 23x s 943, or x = 41 9. SOt - 8x - 76 a 300 - 35x - 55, or 107x s 321, or x « 3 10. 112x + 480 a 3024 ~ 39x + 84, or 151x a 2628, or x a llf^i^. 11. 208x - 442 + 308x + 374 a 858x - 4433 + 143x, or - 485x a - 4365, or X a 9 12. 4x + 4 - 3x a 6 + 14 - 3x, or 4x a 16, or x a 4 13. 360x - 160x + 200 + 48x a 2040 + 60 - 180x + 45x + IS, or 383x a 1915, or x a 5 14. Haltiplying bj 12 we get x -H 40x - 60 34x - 108 = 12- 39x -f 12 36 - 23x This X 4 and redaced gives 160x - 240 136X > 432 a 3x - 36, or 800x - 1200 - 952x + 3024 » 105x - 1260, or 257x « 3084, orxa 13. 15. 60x + 30x -f 15x -> 36x + 252 s 120x -> 166, or 51x a 408, or X a 8 . 16. 336 - lOx + 10 - 776 + 66x a 16x - 3x ■!• 11 - 144, or 33x -. 297, or X = 9 17. 30X + 20X-I- 15x4* 12x + lOx a 60x + 25x + 240, or 2x =^ 240, or X a 120 18. 12x - 20 + X + 60 a- 9x- or 4x = - 40, or x a - lo 19. 36 4- 20x - 20x a 86 - 125x -f 600 125x 4- 600 9X-16 ' or 9x~16 « 50, or 125x ^ 600 a 450x - 800, or 326x ■ 1300, or x ■ 4 KBT TO ^4M^^# ^^^^^H^|V «A ..t •« HO't-yO* . ^ . 15a; « 6 5 5»p ^ gS , ^ ao. 8S1 - 80« - •— ^ -f 9 -h 6jr « g j (i) ■ tlM f ires tqiMt, X 10 awo - aoo« - g nduMd ftftd X 946« ^^ 900 -I' 60O« > a9535 (in) ■ n reduced tiid k 9 1605« M 2307S, or X » 16 ■ 16c - 06 - 110* + ItO (») s (I) 21. 9C'f 30 144«-432 •Mq ScTl — +' ^* (0 * gi^en eqnat. x 36 ^ 1000 - 00 « 1444r - 433 (n) - i redttcsd and x (6x *- 4) 44j; - 363f or c ■ 8 32. SOc f 20» •(• 00 - 16« •!• 00 - 13r 4- 60 -I' 1900, or 2dxmlBi^ or««60 23. 90c - 36c - to ■ 76 -(• 20c -i- 10 - 61 •(- 9c, or 20c » 104, or c ■ 4 12c' + 36c + 27 24. 16c+10»»-10«'+l8-aT + 18c. j^r^ — - (i) •" ffirtn tquftt. x (3 -f 3c) 13c» + 80c^27 , , ' . . 3c + 9 ■ z^ ix ^"^ ■ ^*^ ♦'•Mp. and coUectod 9c 27 •(• 13«« 'I' 80c ■ ISC' •!• 80c -f 27 ^ «'*-irr0T8a £ir. HKxa.] Ji^i^lSA, 29. 4a%« - 6a*+ 2ax s Sofrx -abx + i^x(i)» girth eqo*. x a«6 (4tn + 2a-db^ V^x « 6a' (ii) » (i) transp. and bracketed eg" 30. ISo&c - lOcx - Sac 31 20a& - 166x - obex + ft^ (i) s giren eqnat. x 5&c (156 + ofrc - I0c)xs 20ab + V^c^ Hoc - Ifiabc (u) s (i) trans- posed and bracketed • X 20ab + 6^6 + 6ae - I5abc 15b + a6e - 10c 3l.ldii + adx + bcx & fr(^, 6t (bd + ad + be)x s bd/^ kt. 32. dft« 4- 4a^ •^•la' + ia6« - 4abx n 4a%* - I0a> -I- 126« 4- 4a'«, by mnltiplying the given equation by 4a ; and this reduced and T by a gires 3bx + 4axs lOa - Aab*, or (36 + 4a)x[3 10a - 4a6> m-4a6* • * - 36 + 4a 33. a6« - a% - 6'c 4- a6e « 6'x, or (a6 -^>^- 6^ a 6^ - a6c, 6c(6-a) *>'*'a6-a*-6» 34. 11a* - 3ax > llafr + 36x - (6a> ^ eab ^ 6ax '• 56x) -. (a + by + 2x (i) ■ given cquat. x (o^ - 6») 2ox -f 86x - 2« s 6' + 19a6 - 4a' (u) » (i) reduced and transp. 6* + 19a6 - 4a' (2a + 86»2)x8 6'-nea6^4a'', orxs 2a + 86 ' -2 a 36. o* + 2ax + x* - 4a6x » x», or (46 - 2)x -a, or x a • \. ^- 3a6e 6x 2a6 + 6' \ a%' 3«- JTT - T <^ - ^?256TP; + (5+6? ' 3CX (1) s giren equa. with num. and den. of Ist term x 3, and 2nd and 5th terms factored 3«6c 6x f o« "I o^ iTTS - T \(J76p/ + (^+6)« ■ 3<» (n) » (0 with 2nd term red 80 jonr xo [ES. XZ2i£ ^+6 " (HTftp + (a + 6y'^^ (ni)«(ii>with2dtemfurtlierred. 3rd| and and and 4th terms factortd oft ^ ^ , a6 ) 37. 3000 + n20x - 2210« ~ 203a; (i) s given equa.' x 1000 693« B 3000, or X s 4/^ „« 3« ^ 28* 38. -g- + 6Jf - ojj « 3a - -^, or 33* + 694a; - 99aar » 287a - 23x; or OSOo; > 99ax > 297a, or (650 - 99a)» s 297a, 297a 106x + 30a;, *^V* * 660-99a 39. 42(a; - J) + 35(1 -a; - J) - 30(« - 1 - -f") by multiplying tlie giren equation by 105 ; and removing the .brackets from tliis we get 42a; > 14 + 35 - 35a; > 14 - SOa; 4- 30 + 10» » 135a; ; or 14da; a 37 .*. a: = i 40. 72aa; - 9fr - 756 ■ 180 -456- 35c, or 72aa: s 180 -I- 396 * 35e ' 180 •!• 896 - 35e .'. a; a 72a 41. a%' + d^x - 6'a; - a;^ - 3a6 •(■ 3a6a; s ex - ac + oa; » «> tfix "Vx-i- Zabx - ex o ax B 3a6 - ae *- cW (a> > 6> 4- 3a6 - c - a)x » 3a6 - ac - flA* 3a6 -cu:- a'6* •*'*'a^ + 8a6-6»-c-a Kcuaun.] ALOEBBA. Exnoiu XXXIU. 81 1. Let * « greater, then 47 •» x s the lesB, ud « • (4t - x) 3 13, or 2x - 4t K 13 2x 4* 21 2. Let X a the less, then x -f 21 > the greater', — ~ — ■ 3, or 2x + 21 s 3x • ^ 2x 3. Let X B money*, Y *** y " P*'^ P^^ away; then x 2x 2x ^^ ^^ « -J + Y + $2-60 4. Let X B the number; then X - 21 6. Let X s the quotient, then 2x •!• 3x •!• 4x ■ 64 2x 8x 6. Let X s debts ; then y ■ l8t payment, and -g* ■ remainder } •'• T •**^ T " 35 * ^''^ payment; then -g- + 35 + W3 = x T. Let X s the number of cattle in the drorei then y + -g- + -J + 9 = X 8. Let X B the number of sheep in eaeh floek ; x • 19 is twice as great as x - 91, that is x •- 19 > 2x - 182 * » 9. Let X B the number; then 'T " "7 " ^ '3 X n-g 10. Let X B the number ; then 3x - •?• of -?- s 25, or 2x - tt s 25 11. Let X s the number; then x + -^ ^ 39 / X x\ 12. Letx B the number ; then *"('2* + T")*'l''i X X 2x — 15 3x 13. Let X s the number ; then — g — + t « -j- + 3 6(x + ll) 14. Let X « the number ; then 5 s 85 32 X£Y TO lEz. xxuii /36 ; (7-5). 16. Let * s the number ; then "a" + T + T " ~8~ "*" '^ 16. Let X ■ prioe per barrel; then •-- » nomber of bwreli; 3« and 6 s nomber of barrels sold the seeond load X = 21, or 36 - 6x s 21 lY. Let X s distance in mileS) then Ix » half distance ; 1« ■{■ 1 = -=- " times in hoars Ji travels ; ix -i- 4 s — ■ times in hours B 281 trayels; then |x - J« « -^ = JJ, or x s 26^ 18. Letx = the time in hours, and since the three runs of stones seyerally require t2, 84 and 90 hours to empty the granary, they will in 1 hour empty respectively ^, -^f and ^ XX X of it, and in x honrs they will empty =^, -^ and g7; similarly X X the teams will respectively fill in x hours ^ uid -^ \ then 60 78 «K •» ' «V «P M? H ■** 84 "*" 90 " 60 "" 78 ■ ^ 19. Let X = date of abolition of slavery in Canada; then 3(x - 1780) -I* 1620 s year of massacre of Lachine. Therefore X + 3(x - 1780) + 1620 ,„.^ ^^ 2~^ + 1^6 « 1862 20. Let X u J's share, then x « 120 s IPs, and x - 106 * Cg. Therefore x -i- x - 120 + x - 106 a 7400 24^ - 800 21. Let X -~ price in cents of a music lesson, then — rr — /24x-300\ s price of a^drawing lessson ; therefore 32x s 24 ( • ^^ — 14- 1000 22. Let X s the number of volumes on science ; then 8x = number on travels, 3x s number on biogfaphy ; 4ix s number on history, and 9x = number on general literature. Therefore X + 3a: + 3ar + 2x + 9x = 14i»6; whence x » 70 £z. ZZZtll.] ALQBBEA. 88 jr-106» Cb. 23. Let X M length of Niagara rirer, wherefore 4c - 6 » length of Ridean eanal ; then 2(6« - 6) - 100 » 230 24. Let X ■ dajB required to finieh the work. Then since J does 1^, S, iV, and C, ^ of the work in 1 daj, Ji and B work- ing 1 daj, and B and C working 2 days will finish it + ^ + •?£ a ^ViT of it, and the part remaining to be done ^)^%\ in x days XX X wf does r^h'' i ^» Tr'^* '^^^ ^' li'^' ^' ^^^ work, therefore X X St 12 "•■ 16 ■*■ 18 ■ *^*' **' ^^' ^ ^^* ■*" ^^* * *°* 25. Let X B greater part ; then n <- x the less ; and x - (n - x) 26. (i) Let X = minute divisions the hour hand passes orer; then since the minute hand travels 12 times as fast as the howir band it will pass over 12x; but the minute hand also passes completely round the circle (60 minutes), and then in addition over the x minutes. Therefore 60 + x is also equal to the number of minute divisions passed over by the minute band ; then 12x = 60 + x, or llx = 60, or x = 5-^; hence the hour « 6j»/ X 12 » 1 h. 6^6,- m. (n) To be opposite the hands must be 30 minutes apart ; then letting X s space in minutes passed over by hour hand, and remembering that the minute hand travels 12 times as fast, and also goes over 30 -I- x minutes, we have 12x s 30 -f x, or llx = 30, or X = 2i8f and 2 1*- x 12 = 32-jar' past 12 (ill) By similar reasoning to the above 12x « 15 4- x, or llx « 15, or X s 1^-, and 1,^ m. x 12 s 16-|*f m. 27. Let X = price in dollars of first field ; then x •(- 90 > 23 = price of second field, wherefore (x + 90 - 25) -f 90 = 2x 28. Let X » days required by A and C to finish the remainder ; then ^ - (a'ff + ^y) * tJ^it « P«ft C does in 1 day, .•. in 11 days C does f ^o^f , and B and C together in 5 days do Hs^ii* iis u KBT'tO t£x. ix^itt. ■ Tff. H«nc« part remftlalng to bo dono ■ 1 - (^VVtf * /o) ■ HU " iVW " HI f tben In x aayi ^ and C will do g^ + c 19x •'• Ao **■ Taoo ■ II Ji or 26x + 19x « 896, or « ■ 19 J^ days 19x 3a 29. Lot « ■ Ca snare in cents ,then y - 2540 > D's share ; t/ « 4x . 8x ^C-jl^ - 2640) + 4000 = y + 2984 = JB*!* share, and (« + y - 2640 A» 8« 4t + y + 2984) « 2x + 444 » wtf's. Then £ + y - 2840 + y + 2984 •f 2« + 444 » 718900, or 4a; + 888 » YldOOO, or 4x = 718012 .*. x 3x 538609 « $1796;03 » Cs share ; y - 2640 » — ^ 2640 » $743-89?, Ice. 80. Let X ■ the number of days required ; then since 4 men can do it in 9 days, 1 ma 21000 16X - 21000 . 1 /16x > 21000\ year 9 *TV ^ — ; - jooo )r 131« » 1260 Ex. XXXIII.] 64« - 84000 ■ 27 64a; - 84000 ALaSBSA. 86 - 1000 ■ capitel at wA of 8r4 jut» Htnet 27 - 1000 *. 2» 86. Let X ■ the distance In feet, then —' ■ namber of rerol- ations of the fore-wheel, and -^ « reTolntioni of the hind-wheel. X X Hence ~ s -yh n .\ or ■ ax -f o6n, whence to • ax ■ aftn .*. ofrn * TTa 36. Let X 3 namber of minute dlTilions the hour hand passei over before the minute hand orertakei it; then the minnte band must pass from XII to ZII, i.e. 60 minutes plus x minntei in order to overtake the hour hand, that is while the hour hand paeies oyer x minute divisione the minnte hand pasiei over 60 + x minute dlTisions, but the minute hand moves through twthi timet the space the hour hand travels in a given time. Hence 12x s the space travelled over by the minute hand, while the hour hand goes over x minutes. Hence 12x a 60 -h x .*. llx a 60 and consequently x s 6^^ • that is the hands will be together for the first time after XII when the hour hand has passed over 5]', of the minute divisions, i. e. in 6-^- x 12 s l h. S^f m., and similarly they will be together again 1 h. 5^f m. afterwards, and so on. Hfcnce they will be together at 1 h. 6^c m., 2 h. lO^f m., 3 b. 16-t^f m., 4 b. 21-^^ m., &c., and they will be together as often as 1 h. H-ff m. is contained times in 12 h., i.e. 11 times. 37. Let X s the greater, part, then 96 - x a the less. Hence y -h 3(96 - x) = 30 ; clearing of fractions we have x -f 2016 - 21x ± 210, whence x s 90^'^ « the greater, and 96 - x > 96 •* 90tV « 6/^ = the less. y-^^ ■^*'vJt. 80 KfiT to [Ex. xxxiti. 3x 8s Ox 88. Let X » £*• ibare, then "r- " •^^ share, i of ^ ■ -r- 3x Ox C« share, consequently * + "T "^ T " ^'^^^^ whence hj clearing of fractions 4x H- 6x + Ox » 10240 ; that is lOx ■ 10240, whence x e ${(38-04] i = B's share, /. Si share * J of J's « $808-42^, and Cb share » ; of ^'s := $1212-63,V 30. Let X = rate down .*. 28x ■ distance, and x - 6 « rate np the river, and x - 3 « rate up the lake ; length of river ■ f of 28x > 12x, .*. length of lake « 16x I ^°®° x-5 ■*'x-3 ' 2lVx-B/ -''x-S ■''x-8 " 3(x-B) 10 12x - 60 ■ lOx - 80, X K IS, X - 5 B 10, (x-3) 3(x-6) • X - 3 a 12, and 28x s 420 40. Let X s the whole property, then $1800 4- ^(x - 1800) e $1800 + -^ - $300 " -^ + $1600 a share of the eldest ; also 6 6 (X 6x -J 4- $1500) "» g- - $1600 3 part remaining, and $3600 ^ i\J - $1500 - $3600) « $3600 + g^ - $860 » g^ + $2750 e share of the second, but these shares are equal. Therefore •p fix X -— + $1600 = ^ + $2750, whence x » $45000 and — + $1600 o ou o e $9000 s share of each ; also $45000 f $9000 = 5 = number of ebildren. 41. Let X s the left hand digit, then x + 7 b the right hand iij^t ; also lOx + x + 7 » the number, and x 4- x+7s2x + 7 . , - lOx + x + 7 7 « sum of the digits. Then — ox*! ** ^ "*" 2x + 7 '^^''"®® llx -f 7 s 4x + 14 + t, and *. x « 2, x -t* 7 :* 9, consequently the number is 29. £x. xzziii.) ALOBBBA. 87 lal. Therefore : 6 = number of 42. Let *m Ft ihare, x - 30 ■ Cf , and }(a« - 20) •»• 80 > J't. 2(2x - 20) Then x + x - 20 + -I- 80 ■ 2100; whence lOx - 100 > 4x - 40 + 400 ' 10500 .-. 14x « 10240, and x a $731-429 • Bt sliAre; also $731i2? - $20 * $711-42} « Ci share, and ^($731-429 + $711-42?) + $80 - $657l4f - A'* ihare. 43. Let X « the number ot rows, then x'+ 75 « number of trees also X + 6 rows each containing x - 5 trees ■ x* -f x - 30 4- 5 > the number of trees. Then x' + x^ 30 + 5 ^ x* -f 75 ; whence x a 100 .-. x' + 75 a 10000 + 75 s 10075 ■ number of trees. 44. Let X » one part, then a - x 3> the other ; and x ■ —(a - x) .-. mx s na - nx, or tax -f nx a fia .*. X a m + n ; also a - X na ma^na-na - a- ma TO + n TO + n TO + n 45. Let X and 60 - x « the two parts, x being the less ; then z(60 - x) a 3x' .•. eo - X a 3x, and x a 15 a the less ; whence 60 - X s 45 s the greater. 46. Let X s the growth in acres of one acre of grass for one week. Then the growth of 3} acres for 4 weeics a x x Y >< 4 40x a -o~ ; and the growth of 10 acres for 9 weeks a x x.lO x 9 a 90x. Therefore the whole quantity of grass eaten in the first case 40x -3-+3i = 40x ■(- 10 acres, and the quantity eaten in the second case = 90x +10 Hence in the first case the quantity of grass eaten by one ox 40x +10 20x + 5 - ^ X 1 X -jif a — ^^^ — , and in the second case thequan. 90x + 10 tity of grass eaten by one ox = (90x + 10) x ^ x ^^ a — -— — 188 But by the question an ox in the first case eats as much as an oj( iu the second case 38 KEY TO [Ex. zxxui. Therefore 30X-I-S 00X4-1O Hence 72 189 20x •(- 5 ^ ■¥ ti ; whence x « -jiy of an acre 80 12 ta * 12x72 ■ ^* ' fractional part of an acre of grass eaten by one ox in one week, .*. one ox in 18 weeks will eat /{ X 18 B } acres. Now since each acre increases at the r&te of -^ of an acre per week 24 acres will increase 2 acres per week, and in 18 weeks the 24 acres increase by 36 acres, and therefore become equiv- alent to 60 acres Then 60 acres * f acres - 36 oxen. 47. Let X s the first, then nx « the second, and nue s the third. Therefore x + nx + rnxsa. whence x a - . ^^^ b fipst; second ' 1 T » + III ' na .... ma \ s nza and third s mx l + M + n' "•""'""'* '"" l + m + n mx px 48. Letar s thit first, then — - s the second, and — a the third. ' n 9 mx px Therefore x + -jj— + —•*«, whence nqx + mgv + npx a nqOf and anq Second part nq + mq + np mx n the first part anq m — X - n nq + mq + np anq amq nq + mq'irnp anp px p Third part = t ■ T *< «« j. -.„ j. «« «« j. «./. j. «. 9 9 ^"^ "^ + ftp nq + mq + np 49. Let x a the number thrown by the first after the second commences ; then x -f 36 a whole nnmber thrown by the first ; 7x —• s the number thrown by the second. But every 4 charges of the first consume as much powder as every 3 charges of the second, and they are to consume equal amounts of powder, X -f 36 7x .'. — 2 — a rr; whence x ■ 216 ■ balls thrown by the first hx. XXZIV.] ALQEBSA. after tho second commences. Therefore 2 of 216 " 189 a balls thrown by second. ExiBoui XXXIT. 10. 4x 15x + 6y ■ 2a I - 6y = 36 ) 19z ■ 2a + 86 2a + 36 19 32^ s a - 2« 4a 4- 66 I9a . 4a - 66 3y = 3» = 19 15a - 66 Id 6 0-26 19 12. 2a6x - 4a6y s 26* 2a6x - a6y - ae 3aby s oc - 26' a4;-26> 2ae- ox r- 6 •(- 2ay « 6 4- — gj 11. 12x-l-4ays4M 12x + 36y '. > 3n 4ay - 36y > B 4ffi- -3i» (4a-36)y « 4m -3» y 4m "4a -3fi -36 8« ■ m - ay = m- 4am • 4«- - 3an • 36 4am - 36m- 4am + ^n ji* — "■■■ ■*■ 4a- 36 3an-j )6m 8xs «K 4a-36 on-6m 4a-36 13. X- . X - y « al *«-y»»6J (x + y) = 6'l (x + y) = 6j 46* ax ax B 36' + 2ac - 46» 36 2ac~6» 86 2ac-6« 306 (x-y)(x + y) = 6'] ax -I- oy ■ 6 ox -oyso* 2ax 3 a^+ 6 a^-f6 *--2r 2ay mh-tfl 40 KEY TO [Ex. zzziv. U. cX'-'Oym aem (m - c)x + (» + c)y ■ aem, } c(a» - «)x - a(m - e)y ■ acm(0» - e) e(m - c)g + e(m + e)y ■ flc^m mcy + c'y -f amy -aey* acHn - acm* -f oc'm flcm(2c-m) ^'mc + e^ + am-ae ex s ocfA + ay s ocm + 2a^e*m^ a*m* am - 00 «s «s am(wc 4- c* 4- am - oc) -H 2a'ciit - o^ mc + (T* ■(- am • ac am? e 4- aiwc' + a'm' -» a'cm ■»■ 2a^cm - ah/fi mc + c*-^am-ac am*c + amc' + g'gm ^ir+c^TamTirac 15. bm bn + — m ab y 19, («-y)(af + y)a65 « + ys 11 6m 6n — + mq bn mq * ab " bm (ab - bm)y »bn-¥mq bn + mq ll(x - y) ■ 6ft x~ya& « + yall 2x * 16 «a8 } — sa m — = a m y ' db^bm n n --.a ~ 6n4'«g ab-bm a6n - bmn ~ 6n + mq abn f am? - a4« 4- bmt } 6n4-m(f 1 aq +bn X " mq-\-bn mq^bn * " aq 4-6n 3y«6 ysS 17. 459x - 463y » - 495 - ftx 4- 19y = 13 1 2296X - 231Sy =: - 2475 » 2295x4- 872 ly = 6012 9 640dy ' 57654 . y ■ 9 (krBl9y-131 * 171-131 ftr»40 «*8 Ex. XXXIV.] AL6EBBA. 41 18. ex -ay s acp ax " cy s a^ + 1^ acx - d-'y = a*cp acx - c' y = a^c + d* c'y -aJ'ys a'cp -t?e~^^ o?cp "a^c " <^ y~ c* - a' ah^ - a V - c* m: = o' + c* + cu = a' + c^ + — -jt C*^a* + a^c'p - aV - c* oA» - »^ - ae* ax = s — rs .'. * = «a-a" c»-o» (19) 35x - 126 + 28xy + 49y=s 28xy + iQy 72x3- 90j.y . Yo'« + 42y - U = t2j:»- DOzy +930y- 1002*+ 2666 35x- 27y= 1261 8293X - 2960y = 8917J 115255.«- 88911y = 414918 115255g - lOaeOOy = 312095 I4689y = 102823 '*• ^ * ^ 35a; - 126 + 27y - 126 + 189 = 3l6 /. X <« 9 } 3a6^ 3a6'e (20) 3(a» - b^x + 6(o» - b^y = 80*6 - 206" SCa^* - 6=»)x + 3(a + 6 + r)6y = Za^b + 6a6" + ~"^^ 5(0=* - 6»)y - 3(06 + 6*+ frc)y = Sa^ii - 206" - 3a='6 - 606*- ^. ^ 50^6 - 3o36a - 806" - Sab^'c (5a'-86»-3o6-36c)y = (5a»-86'-3a6-.36c)y = a + b (6a» - Sft'-* - 3ab - 3bc)ab 8a*ft-2o6» 3x = — -:5~ £5 6y a y-TTb 80*6-206" (o + 6) 3x = a»-6» 3t^+_3a6» o» - 6'' a" -6"' 3a6(a + 6) "o" - 6» 5 5a6 a +"6 8a>6-2a6'-6a*6+5a6> 3x = 3o6 a-6 X s 42 KBY TO Bmcm XXXY. lEs. V. (•) 1 T + 7 (») *+ y «y X ■¥ 2 a 2X2 ay + 22 s 3y« 116 8 1 y "7"7" 4 "la 1 1 : » 7*T "la a 8 7" ijj 8y a*j»-8 1 1 ji^ 1 * " y " * 2 2 1 1 7 8 ^ # ■ la 1 .7 I I 7"Ta-7"T'*''"* T"7 ■ 2 2 T + 7" 4 7 ■ 6 ,*. X 116 « ^ 8 " 6 16 11 7"7-7-7 •■» rr + — « 1 (•) « •i' 8y 4- a« ■ ft ftr •»• 6y - a« « m 8« - 2y •f a« ■ 2n\ — + -r» 1 *="¥ } Ax-¥iy»b ^ m llg'f 8yffl » an ia« •I' a4y " 86 -t- 8» 88« 4* a4y ■ 8m •!• 18n f«7 (10) Add all four eqaatious to- gether and then 4- 3 » + * + y + 2 » 28 X ■ 6m'l-10»'86 6m^ien-86 ' 6/» » 16» » 86 19 76 6+fH • + x + y 13 «=10 v-l-x + 2*17 y •« 6 » + y •|-2»18 * * = 8 x + y + *3 21 Ex. XXXV.] ALGEBBA. 48 4' 8 latious to- ■y + a-a !28 y .13 z= 10 + «*17 y •« 6 + Z'19 s S 8y ■ 6 + m - 6w -I- 16» - 86 19 8y 226 + 14m - 16n 19 < ll6 4-ti»-8>i 76 • ■fi-4x+y; «»n- Sm + 16n - 36 116 -I- Ym - 8n 19 IT ten - 20m - 64n + 126 -t- 116 -I- tm - 8fi z * Zs 236_+4tt-18m 76 76 (9) abx + 6*^ = 6c a6x + acz = o" b^y">aczs bif" tfi c^ + acz = be 26c -1^ (6» + c»)y = 26c - o>' J'" 6' + c» a«s fr.cys 5 - Z»' 6 * - 60^ + tt' c a6" + ac" 26c* -g^ 6» + c» ' «r«i oo; B c - 6y <■ c ~ c '-6^ + a% a6' + ac* 26'c ~ 6a» 6» + c« » ax< 6»4-6c»-26c' + a^ 6> + c" c6'-l-c>-.26»c-l-a% >¥?" «s (11) ax H: ay -f dz s a' -f a6 + oe ftx + c y + gg « a' + 6* + c* (a - 6)« + (a-c)y»a6 + gc-6'-<^ 6x + 6y + 6z s o6 4- 6' + 6c ex 4- ay + bz = a* + 6' + c* (6 - c)x + (6 - a)y 8 a6 4- 6c - 1^ - ^ (a - 6)x + (a - c)y s g6 4- ac - 6* - (6 - c)x 4- (6 > a)y 3 g6 4- 6f (2a6-6'-(^)x4-(a'-c)(6-a)y « 2a6*4-a6c-6*-6c'-a%-a%4-ac> (ab - oc - 6c'4- c*)x 4- (a -- c)(6 ■i'g)y « a'6 - <^» oc* - 6c' 4- o"c 4- c* -6«-en -g»-c»J 44 KEY TO [£z. zzzv. sb+c-a (62+ o»+c»-a6- - c* - (a - ft)(6 + c - o) (a - c)y s a' - a6 - c^ + frc .'. y a a + c - 6 x + y + ««a + 6 + c.', ««ia + 6 + c-(6 + c-a)-(a + c-6) ars a + 6 + c-6-c + a-a-c + 6j X3a-l>6«>c (12) ax \- tC'hf ■¥cfiz*am ax+ y +ax mn otc + ay + z -p ahi-y-{-c^z~az»«m~n ay -y-\- z-az^p (tt' - l)y + (a» - o)« K am^ii (o -l)y-(a - l)«*p ::} (a^ - l)y + (o* - a)« b am - (a* - a)y - (a' - a)« * op (2a' - a - l)y s ai» - n + ap - an .*. y ^ -a«J am - n f op - on 2a' - a - 1 (a - 1)« ■ (a - l)y - p + tt (a - l)z m am " n ■¥ ap - an 2a +1 -p + n (a - 1) «■ am - n •{■ ap ~ an - 2ap 4- 2an •'p + n 2a + 1 (a - 1)* - am "Op + an — p 2a +1 Z s am — ap 4- an — p 2a»-a-l i- = m - a(y + «) ■ m -( am -n-^ap-an + ain'~ap + tm — f 2a» - a - 1 ^ ) X = m-a /2am -n- { 2a> -n-p\ -a-l ) »«m> 2a*m - an - ap 2a«.a.l X 5 2a' /n - am - m - 2chn, '\ren'¥ap 2a»-a-l op — am 4- on •■ m 2a? - a - I £x. xxsvi. ALQ^KA* EXBROISB XXXVI. 45 1. 7(x + y) + 4y = 60, and 2(x - y) + 3x = 16 2. X + y = a, and bx-cy = 3. Let X 3 price in cents of hay per ton, and y = price of 4y oats per bushel : 2x -i- 35y > 4400, also -r- = reduced price of oats and -r- = increased price of hay; then y + 28y = 6120. 8x Hence the equations are 2x + 35y - 4400, and -j + 28y » 6120 4. Let X 3 length, and y = breadth ; then xy = area. Then (X + 20)(y + 24) = xy + 4180, and (x + 24)(y + 20) = xy + 3860, which tvo equations when reduced gire 6x -f 5y = 926, and 6x + 6y - 845 5. Jx + Jy=ll; Jx-1 = Jy 6. X + y = 144, and fx - ^y = 1 J 7. X + y = 48, and -j = — , or xy « 16x, or dividing by x we get y = 16 8. X = 5(y + s) ; y = 1 (x + «) + 6 ; 2 = i(x + y) -3] whence by reduction we get 4x - 5y - 62 * ; - x + 2y - s =» 12, and V -x-y + 3« = -9 9. Let X = sulphur, y -■ saltpetre, and z = charcoal X -i-y + z^ 4000 ; y + c - x = 3240 ; x + y - 2 = 2760 10. X + y + « = 72 ; Jx = Jy ; |x = {z n. Let x = space occupied by one shilling, and y » space filled by a ten cent piece; then 16x 4- 27y » 1, and llx + 13y = -?g, whence x = tHti ^^^ If ~ «V > wherefore the pnrse would hold 1^1^ = 40!iJ shillings, or ^ 7 44) ten cent pieces. 46 ItBY to [Ex. xxxtu 12. Let X » number of lioeSi and y » nnmber of letters in a line ; tlien xy = number of letters on a page. Then (x -t- 3)(y -f 4) s zy 4- 224, and (x - 2)(y - 3) « «y - 146 ; or 4x -l- 3y = 212, and 3x -I- 2y « 151 13. Let X « left hand digit, and y s right hand one ; then the 10* + y number will be represented by lOx + y. Whence 2x -f 2v - 4 lOx + y = 3, and ^^^^ = 13 ; or 4x - 6y = - 12, and 23x - 12y = 66 14. Let X = number of ten cent pieces, and y = number of twenty-five cent pieces ; then lOx -l- 26y » 8160, and y - 6x « 4 ; or 6y • 12x a 8, and Cy -t- 2x = 1632 X xfa - 1) 16. Let * a rate before the accident .♦. x - -- s -^ a a B rate after the accident Let * s the number of miles from Kingston at which the ac- .cident occurred. n n , an n ••• x(a.lV *T + * ••• ^iTTiy - T + * <»> a And n " e n^e , acn - c) n-e an i From (I) . ^ d(n-c) n-c From(ii) .^^.. .—- x(a > 1) X x(a - 1) x(« - 1) n 6(in) «I(IT) x{a - 1) « e x(a — 1} ' 0*1 Whence x(6 - d) a- 1 .♦. x« 490 ; whence x » 5 = inside passengers 17. Let X and y ~ the digits ; then the number will be lOx -^ y Then lOx -f y » 2xy (i) and lOx ■!■ y s 4x -i- 4y ; or 6x - 3y = ; or 2x - y » (n) Adding equations (i) and (ii) we hare 12x « 2xy » 0, and dividing this hj 2x, and transposing, we have y » 6; whence x,A 3, ttnd the number s 36 18. Lot X, y and x be the digits, then the number will be lOOx -f lOy -i- 8 lOOx+lOy + x. Then y s |(x + x) ; 48, and « + y + « lOOx 4- lOy -I- « - 198 s lOOz -I- lOy 4- x. These reduced give the equations - x f 2y - « = 0, 52x - 38y - 47s - 0, and x - x = 2, &c. 19. Let X = the oz. of ^, and yoa.otB; then « -f y » p (i) b since j9 oz. of Jt loie b oz. in water, 1 oz. will lose — , and .*. x bx cy bx ey oz. lose — ; similarly y oz. of B lose — oz. in water .*. — + — so(u) From (ii) Ax 4- cy a op (lu) and multiplying (i) by b we get bx-i-by*bp (iv); then (in) - (iv) gives us cy - fry « ajp - Ap (« - ft)P . ., , ^ (g - tt)p 48 SET TO (Ex. ZZZTt ■4 ^ ^ > -^ 2 « I I 8 « e .S (* 1 1 N N 1 I I I I I I I « N B ^ 2 » II II 9 »» n s^ O O ^-^ ^ ^ to M ^ 00 to .. „ "^ N 00 « « N ^ 1-4 I I I s H .► 9> I I 8 s ft - CO »a 2 « C9 d N M CO CO CO CO ,^ 00 M 9 ft CO CO 00 *♦ I i I I I i «« I I e«i 00 8 8 w 2 I (M 00 lO C4 a> H t)0 •& -^ d «» + II -^ 00 X H >> + a % - H + I I I I S 9 S S » + ^ I I I I I e* 0) 00 CO Sa a a P 3 3 M be t« ea If '2 *? •*> >" en H »4 »-• C^ CO I « [Ex. ZZZTI Ex. ZXXTUI.] ALQBBBA. 49 ExiROiM XXXYIII. 11 9» II II 9 to II o 00 a + I + O •» lO I o CO »^ II o to 5 a % ^ p /-s II + + + + + II <^ II 00 » H II . • • «* CI 00 + 9 ■4 faO + •a T. (2« + 3)« « (ao)« + «(2«)»8 + 16(3a)<3»+ a0(2e)«3«+ 16<3a)*S4 + 6(2a)3« + 3« 8. (3 - 2my a 3» - 6 X 3*(2j») + 10 x 3»(2iii)" - 10 x 3"(2«)» f 6 X 3(2»)* - (2«)» 9. (3a - 3y)» ■ (3a)' - 6(3a)«(2y) + I0(3a)«(2y)«- 10(3a)\2y)" + 6(3a)(2y)*-(2y)» 10. (26 - 6c)« a (26)" - 3(26)«(5c) + 8(26)(5c)» - (8c)» 11. (3x-4y)* = (3a:)*- 4(3x)»(4y) + 6(3a:)"(4y)»- 4(3*)(4y>" + (4y)* 12. (ab + Say - (oft)" + 5(a6)*(3c) + 10(a6)»(3c)» + 10(o6)«(8«> r 6(o*)(3c)* + (3c)» 13. (2ac - «y«)» « (2ac)» - 3(2ac)«(«y*) + 3(2oc)(afy*)» - (xy«)» 14. {(a + 6)-c}»a (o + 6)>- 3(a+ t)»c + 3(o + &)€»-<:«■ «» + Za*b + 3a6« + 6» - 3c(o' + 2a6 + 6») + 3<:»(a + 6) - c» 15. {2o - (5 + c)}* a (2a)* - 4(2a)"(6 + c) + 6(2a)>(6 + e)* - 4(2o)(6 + c)» + (ft + c^* s 16a* - 32a"(6 + c) + 24a^6« -I- 26c 4- e^ - 8a(6» + 36»c + 36c» + c») + 6* + 46'c + 66 V + 46c* + c* 16. (2(0 + 6) - 3c}« a 2" X (a + 6)9 - 6 X 2*(a + 6)*(3c) + 10 X 2»(o + 6)»(3c)' - 10 X 2'(a + 6)»(3c)» + 6 x 2(a + 6)(3c)* - (3c)« « 32(a« 4- 6a*6 + 10a^6' + 10a'6> + 5a6* + b') - 240c(a* 4- 4a% 4- 6a%> + 4a6» 4- 6*) 4- t20c'(a^ 4- 3a'6 4- 3a6' 4- 6») * 1080c»(a» 4- 2a6 4- 6^ + 810c*(a 4- 6) - 243c* 17. {(1 4- X) - «»}* = (14- X)* - 4(1 4- x)»(x=0 4- 6(1 4- «)»(«»)» -4(1 4- x)(x»)»4-(x«)*= l4-4x4-6x»4-4x»4-x*-4x'(l4-3x4-3x»4-x>) + 6x*(l 4- 2x 4- x») - 4x«(l 4- x) 4- x» 18. {(o - 6) 4- 2c}» = (o - by + 6(a - 6)*(2c) 4- 10(o - 6)»(2c)» + 10(0 - by(2c)'+ B(a - 6)(2c)* 4- (2c)» = o» - 6o*6 4- 10a>6«- 10a?»6» 4- 5a6* - 6» 4- 10c(o* - 4a»6 4- 6a'6» - 4a6" 4- 6*) 4- 40c*(a^ - 3a»6 4- 3o6* - 60 4- 80c»(o» - 2a6 4- 6«) 4- 80c*(a - 6) 4- 32c» I \l 60 EST TO [fix. ZXXIZ. Binoin XZXIZ. (I) 4 -I- a« - 13«* i««-8«» a«-as' *9x* + x« (3) 4*»-12x»-2«* »«« + 8«« ia^-aa"4-a« + *«• 40*- 4a" + a« i + a*-**-**va«* «•-«•- «♦ + a** i««.«i + «• 40" - 4a'* + 8aM 0) 1 -f a6« - acx* aV-4aV W**-a6cx» + 4a'«* + c»x« (8) a' - aii6x - 2aex* 4- aodx* We" + 2&cx" - aMx* A* - ac(ix> + rf»8« 1 - 20 + 26»x" - ac"** + 2-48a'c*-l>32(iM- 8a>c' 86«*e*-48jx - ix» + jae* - ig» ■► y^x* x' «• (16) ix* V y X* + x\ «» «« + 2x -0— 2 + «» 1 •1 i I t ■i -xy-2 + ExiBOin XLII. (2) aS + 6a« - 40a' + 96a - 64(a* -f 2a - 4 a« ""' + «••. 4.. 6a» - 40a» 3a* + 6a» + 4a« ea» + 12a« + 8<^ 3o* + 12o3 + 12a« -12a« '-24a + 16 -12a«-48a>-l-96a-64 Sa« + 12a3 -24a -1-16 -lSa«-48a*-l-96a-64 64 TO (Bx. zui. a« - 6a> + 16a« - 30(^ •!■ 150^ - 6« -I- l(a* - 2a + 1 a« 3a« -6a» + 4o» 3o*-6a'+4a^ > 6a> + 15a« - 20i^ -6a< + 12a«- 8(^ 3a* - 12a» + 12o> 3a* - 6a .-arj M lFrT2?+ 16a* - 6a + 1 Sa*-12a»+15/i«-6a+l 3a« - 12d^ 4- ISo^ - 6a -I- 1 (6) |2««-3a« + 4 + 6> 3a« + 3a6 + 6« 8a*&-(-8a6« + &* 3(«+*)' + 3(a + A)c^.,, 8(a + 6)«c + 8(a + 6)c» + c» 3(o + 6)» + 3(a ■t'b)c + e* 8(a + 6)«c + 8(a + 6)tf« + c» 3(a+6 + c)«^3^^^^ + c)i^^. 8(a-l-6 + 0*<< + *c. 3(0 + 6 + c)« • H 3(a + 6 + C)d + (l« d(a + 6 -h cyd + Ac. [Ex. zux. (a*~2a+l c» - 8ax 4- 40^ 44a'x-t-64a0 - M + 64o« •*^144«''x + 64a« |6 + c + rs - «lrg»4n^ - 4«a - 4r* - Tif (10) (11) fl^ - 3a6* + 8a*6 - 6* (a* 4 x' ) - (ox) * *(a»fx5) + (ax)* c* + e^ a*- 6* c« + c« - Jft* + 3a6 - 30*6* + 6» •e>»i + ftc. a« - 40^6* + 6ab - 4a*6^ + b* ■ a' + 2tf^x* + x» - a*x' • a* + a*x' + X* 56 XBT 1!0 Obc < y e' i>i r^' + iaa'-ajr* 3««-4«y "+4««'-a«"y»+a«'y '« i«-JJ 4xy' + »-*« i 4««' + 2«"y ■«'-2x'*' + y-V-y -IJ ir'»4»'i/" J«-i.» *-.ii~i-.2x*2'-i/"*2* 8ar»-.4»'y 4'6»"y "t' + ly-V-y *-2x"«'-y (W) -8»"*y-3«"»\9«-*y-4«"'y->(-8x-» + 2x-*y -6«"' -4a:-3'y-i -6«"«-4x-'y-* a + a »6"*.aH-» -ft-»(fl*-. ) t*6~*+o*i"-*-6-* a+a *6'* + aH"* + a*6''*i-«*6"^+a^6'"^ ft f 'jr*» 7^ Obc fix. ZLltl.] ALQXB&A. 67 7m .-i + 1 + *' .-» (18) !-l + z "-l + ar'+x (.-«-.-»+i. .W -x"*-l+ «*+« .-J + «'+« + 1 + «* -l-x>-x» «* + «^ + * (18) (a^-a + o^ + l-a^^-a-i + a*"^) = a»- 2o* + 2a* + 2a^ - 2ff - 2a* + 2 a* - 2a* - 2a + 2a" + 2 - 2a*"" a + 2a*-2-2a-*+2a-» l-2a~*- 2a->+2a" I a-M-2a~^-2a"« a-«-2a"* a- 3 = 0^- 2a*+3a«-3a+ 2a* -KS - ea"" * -ha-* + 4a'" ^ -a-« -2a"* + a-» (IT) / a*+ 2a*-l-2a"* + a"'(a* + l«a'"* 2a * + l) 2a* -1 2a* + 1 2a* + 2-a'"*)-2-2a~* + a~' -2-2a~* + a"' 58 KBT TO [Ex. XLttt. (18) («* - ax* + 3 - 2*"* + x"^ 2x r2«*)-4x + 10«* J -4x+ 4x' 2x* - 4x* + 3) 6x* - 16x* + 19 6x'-12x*+ 9 2x'-4x* + 6-2x"*) -4x* + 10-16x"* + 10x"» .4x* 4x'+ 8-12x"'+ 4x -i -I 2x*-4x* + 6-4x"'*+x"') 2- 4x~* + 6x''*-4x-» + x"^ 2- 4x"'* + 6x"'*-4x-> + x"'^ (19) Sx'V' S+af^y^Xary-ax'^y + Sx^j^-'-xy-'Cx^^y-x^y'i fl?"V - 3x'''^ + 3x*y*- xy"* - 8x"''y + 3 x*y'*- xy"* <20) |x^ - 2x*y^ + 3y* •■-exi'y' + 21x»y*- 44xy*+63x'y'-54x'y*+27y (Ex. xLin. r*yi + 3y* 4x*y8+27y 4«*y^+27y 4a;*y«+27y Bz. XLxy.] ALQBBRA. 69 EzxBOiM XLIV. 1. 2' = (2»)* « 4* ; »* = (t^* = 343* } 2^ ■ (2«)*« 16*^ ; (|)' 2. a = J » (o«)* ; 3' = 9* ; (|)* ■ (V-)* J (2a)* ■ (4a«)* ; (3rt»6)^ = (9a*6»)* ; (4«y)^ = (16x*y«)* a=a'3»(a-)-*»(l)'*;(3-»)-*«(T^)-*j {(S)-}-^ {(4xy)-r*»(645^)"* o = a* = (a*)*; 3^ a (81)* ; (J)* « (^IH")* 'r (2a)^ ■ (16a*)* ; (So"*)* « (81a»6*)* ; (4«y)^- (256ar«y*«)* (1) -i. 3. a« a a* s (a-*)~ * = f^Y * } 3* = <3-»)~ {(2a%»)-«r * . ^^)" * ; {(a^ 'f * « (-^)"*; WTf^ = (-4V*)"*; 3-»»i«{(i)-»r*«8i-J; (5)-»«(|)««AV 4.(0^xa)*.(,,x,)*.(|)*;l(^V».i(^y. 2 /3a6\i a/6M a Ki)* « J(V)* - ! X iVii - W14 ; |(.V)* » Ki%)* « M^^', 3a/6\-* 3«/2\| 3a/4\i 3a/46\J 3a.— l(j) "a\j) 'T[r») -t(pJ =46^** ^ KEY TO (Ex. zuv. /200\i /»«V* 2 /8m\| 2 /9«»\i /82 9m«\i /IBm^i ft /am — pc/\ I am~pq)i 7. (136)* = (21 X 6)* a 3^5 ; Vi62 a '^Blxl « 9^2 i V80 = VlTxS = 2V6; 7^^324 = 1 ^27x12 = 7 x 3^12 « 21-^12 ; iV? = JVli*AV2r; ,.aAO*»'N* /704m»\i /m«\i 1 . ,i « f ab* 14 f6»xea(a + x)1i f 6« -i i 8-|6(rn0j =|-86(a + »)« I •t36(a + «)«'*««(«+*)}•* a / /c»«« 1 \ a em^ /T c»^ /n em , f aa - 2?)* 3V2 and 3V3, or island 81*, or island 81^; or (6832^^ and (6561)^ or(*|i)*} (44)* and (5103)*; or (»Wa)*,<8B184)* and (6103)*^, or (3528^)*, (85184)* and (6103)* Ex. xuv.] ALOBBIU. 61 10. 13V2 4- 12V2 - V2 - 8Va + 36^2 > 69^2 - 9v^ " B0V2; 8V(3) + 4a^ - VVn + >/Jf - 4V3 + iyflS - WIS + IVI5 = 4V2 + 4V^ ~ 2Vi^ = 4V» + 2VT5 11. 2V» + 8V7 + 3^3 - 4^3 - 8Vt - -1^3 ; 13. eViooTx 6V100 X 2 ■ 60V2 ; 36V60 = 3SV4 x 15 « ^0V16 ; ^h )«^« (3 X 6») X (4 X 60«) s (3 X 4) X (216» x 3600* - 12^64x12150 s 24^12160 S^xfti 14. 16« x8> B ^16 X 16 X 8 X 8 x 8 s ^4< X 32 s 4^32 ; 28a^ X a* a 28a^ a 28aVa ; 2 x 3^ x 12^* 2 x (2t x 72)* - 2^1944; 7^ x J^S « 14^3 x J^6 « 7^15 15. ax by c*d i i i j^ X jjj X — X {(ox)* x^byy X («)' } a xy{(cx)T°' X (6y)T^ x (c*)"'''} « xy(o64*c»xV*')^ (X - V*y + y)(V* + V») a «* + y* [See Alg. Art. 179.] 16. (2V3 + 5Vf )(3V2} - 4V3) = (2V3 + -ftVTSXlVIO - iy/B) s 3V30 + iVTsO - 8V9 - ■fr\^ = 3^30 + V6 - 24 - JV5 17. JVI « 5(1)^ « iV6 ; t(«)* = f (H)* « /fVn J 2(5 ^ 3)* = 2V^ = !Vio ; la + V)* = iVll = !Vm - ^130 = 4 X ^^^y = f V64827 3/V4\ 3/Vl6\ 3/16\i 3/16X125U , ., t(-V-5) = t(^)=t(i25;*-t(-5^) " ^*V2000 3 V Vl/ "3VV»y"3V|t7 "3 V243/ 3 V72»/ "OV**** 3" VVm/ ■ sAy^'Z " 3 UxV " 3 Va«x«; " 3«V« * 62 4^8 &V4 6V7 4^ KIT TO |^-4+-|--4 + 6-10; 5'V256 6'V49 6 (Ex. ZUT. « 2V3 - 4(Vy)^ + 3(1?)^= 2V3 - -5^-^6088848 + »V5«**57 J ( at*-*c» \^ /«i*:!ci\i /o6*-»c»*» 42 Multiplying by IVI %a we have (5V0'- (iV2)'= H-tt« 2,^ MuIUplyingbylV4 + iViwehaTe(!Vi)»-aVD'=A-u'f'»-r/*iT 2(V3~2V6) 2V3-i^/5 2V3-4v6 4V6-2y3 ^*- (V3 + 2VB)(V3 - 2y6) = (V3)»- (aVfi)' "^ (V2 » V3)(2V5 + 3V6) (2V6 - 8V6)(2V5 + 3V6) a-y^ + 2Vn + 6V3 + 9V2 ■ • -84 - (2y3 4- ynxye + sy?) i4y24 + vysg + leyn + 8yy? (V8 - 8V7)(7y8 + 8V7) " 392 - 448 28y6 + i4y25+ i6a^ + syr? 3-20 17 2yro + 2yi5 -t-^s^fn + syis 20 - 54 22. -66 8(y3+y«) Ac. 3V3 + syx (a'\/m'miJa)(a^mF-m'\/a') (y8-y«)(y3 + y«) * 3 - x • {a^m-^a)(a^m/\n'Jc^ (ai/m - mya)' a% - 2amVma •(- m'a a - 2Vflui 4- m j2jjy!)Gyi_+iyi) §Vi+ivn-yt + 2y» (jynTDoVi + ivi) '^ ix|-*x| 1V2 +tWT5 + W3 + lyio ^ B Tj ' :« 4kp Bi. XUV, XLV.J ALQEBBA 61 7-36> B-2ft -2= 1 48-90 3.42 ^t-H' ^a«b -^',s. ■T'/ira 4V5 -2V3 11 12 + 3VT8 n + 8V?7 3. (Vx»4-x+l - Vx'-x-l)' (V**+x+l)» - (Vx=«-x-l )• x' + x+ 1 -2(V»^4-x+ iXyx^-x- 1) +»»»«-l (x» + X + 1) - (x» - X - 1) 2x» - 2Vx* - x» - 2x - 1 x»-Vx*-x«-2x-l 2x4-2 «-l-l 24. V3 - V2 - V5 (V3 - V2 + V6)(V3 - V2 - V6) V3 - V3 - V5 V3 - Va - V5 {(V3 - V2) + V61{(V3 - a/2) - V6} ' (V3 - V2)' - 5 V3 - V2 - V5 V* - V2 - V6 (V3 - V2 - V^) X V6 3 - 2'/6 + 2 - 6 " - 2V6 Vn - -/n - V35 2V3 + V3<5 - 3V2 - 2V6 X V6 -12 12 (1 - 3V2)(1 + 3V2 + V3) V3-3V6-17 VS-SV^-lT (1 + 3V2-V3)(1+3V2+V3) ~ (l + 3V2)»-3 16 + 6V2 (V3 - 3V6 - iy)(16 - 6V2) 26V3 - 2TV6 + 6lV2 - 136 " (16 + 6V2)(16-6V2) ~ * 92 (2 + 3V3)(1 + 2V3 +^2) 20 + 1^/3 + 2V2 + 3^6 (1 + 2V3 - V2)(l + 2V3 + V2) 11-I-4V3 (20 + YV3 + 2V2 + 3V6)(11 - 4V3) 136 - 3V3 - 14^2 + 26V6 (n + 4V3)(ll-4V3) 78 EziBorai XLY. 6. Let V42 + 3V1 74| = V* + V Then V42 - 3VT74| = Vx - Vy Or Vl'^64 - 1568 = Vl96 = 14 « x - jf Also 42 + 3V1U5 = X + 2V«y + yor42 x + f .-. 2x a 66 ot X = 28 ; 2y = 28 or y = 14 .-. V42 + .SVmf = V28 + yl^; « 2^1 + vh »A 64 KIT TO PtbUV. 9. Ufc V« - aVJTTi • y« - Vy (I) then Va 4- aya - 1 ■ Vg •► Vy (n)| or V*^ - 4a + 4 ■ « - y ora-aax-y Squaring (i) w« get a x x + y /. aa > a ■ SJTf ^ 10. Let Vaa + av«" - 6" = V* + V» V aa - aVtt» . 6* « Vx - Vy V4a* - 40^ -I- 46' ■ V*^ ■ 36 « « - f And aa « X •(■ y 3x-«a6 + aa; orxa-a + 6 ayaaa-lft; orya»-6 11. Let Vs 4- V39 « V "i* Vy Then Vs - V39 = V* - Vk Or V64 - 39 = V^ft i" 6 " « -y Also 8 s X -f y 3x a 13 or X « Vi (^ii^d ay s 8 .'. y ■ I - V8T35 « V?" + VJ ■ I V55 + IV* la. Let ^ J + JftVtt" - 6» » Vx -1. VV t*?r vw Then ax « -r- - or X » -r - T AiUl7«x + y 6« b* ar-Ti ovy-T /.Ao/ Sfctx, Sim.] ALaiBRA. BnMin ZLVI. 1. -732 -V" « V8(a - V3) ••• VVSi - Va* ■ VV«(« - Vs) 2. 3V5 + ViJT* V8(3 + 2V2) ••• ^3V5 + /io « VV6(8 + V^) s V5Var+2Va « V6(V2 + 1) » Vb(V* + 1) - V^ + V5 3. 3V6 + aVn » V6(3 + 2V2) .'. Vs V« + 'Vl^ « VVt»(3 + 2V2) = VeVa+TTi » V6'+ (9 X- 3) = -29+6^6 ^2»-VT^ . V2~V^^ V2~VTT} ' (V2+V^)(V2-V^> "^ 2 - ( - 6) * » 7. cM»x-V"=Ts-a«V'=^} +1; V^^i -1 [SeeAlg«tet Art. 193 (m)] 8. (a - V"^' = a« • aoV*^ T ( « a) » <>>'^- 2a/^- a 60 KIT 90 [Ex. XtVlL 9. (V2-'/=T>*«(V2>"-8(V2)\aF1) + 3V2(aF4)*-(V"^)' 8 V8 - ,6V4/=1 + 3V2( - 4) - (V*/=^)' « 2V2 - 12V^=T- 12V2 + 8/=l « - 4/=! - 10V2 10. Let V - 2 - 2^"^^^ * V* - Vy I *t«n V - 2 + 2V'^^ = V* + Vy ••• V^ ~ 4( - iS) « Vs^ = 8 a « - y, and « + y s - 2 .'. 2* « 6 ; X a 3 ; 2y » '- 10 ; and y = - 6. Hence V* • VV 11. We are to find the square root of f V-1 Let Vo ± V~ « V* ± Vy then Vo T V^ " V* T Vy or vu'^-~r~^)'* V^ "* 1 '^ ' ' Vt *^ ' ^ y * ^ .*. X a }, and y a - 1 Henoe Vo ± V"^ 5 that is of i/^l » VJ iV-T * 1V2 + iV"^; o' iV2 -i-/^ 12. Let>/31+42V^»V« + Vyt thenV31-42V^sV'«-Vy . . V801 - 1^64( - 2) a V56rT3628 a Vii89 a 67 »x - y and 31 a X + y .'; 2x a 93, and x a 49 ; 2y a - 36, or y a - I8 Henee V* + Vy ■ V® + V"-^ a 7 + a/^^ 4 + V32 _ (4 + Ar^X2 •»• V^ _ 8 + 4V]r?H-2VT:i [-2 * a-v^~(2-V^)(2+V^) *-(-*2) 6 + 6V*^ 6 1 + V- 2 14. 7 - /rB\ 14 - Vl^ - 7V^ - 2/^/2 - 'v'^ 14 -2V^ ' 15. (a + b^f^Xa - AV"=1) « a' - (*V-^)^ » a« - (b» >f- 1> «a"-(-ft^aa^H.6> •Thus --/-6x-V-3a- Vsy - 1 X - V3V - 1 " V16(V - 1)* aVrSx-la--^ ISsLtLVH. ExMwma ZLYIII. 67 2. 3x - 6 3 2x ; or X ■ 6 — 3. X > 24 « X - 4>/x 4> 4 ; or 4^x s 28; or V« * ^i or x ■ 49 + ax"*aa + x, or- 2ya + «x " * = a j 4. X .2V«Y- V« v« JL VVvv^ f+6+«=-3;or^':;/^=ffr5+««»; 'x+123 + ^ or 7VVi+l2l+* + * ' ^ i ^^'^ VV» + 1W + * + B - »» 0' -^^x + 123 + *"*» or V«+1L23 + 4 « 16 ; orV« + lJ3i*M| «^ X + 123 « 144 ; or x « 21 A 6. vs^-V'^V*; o'V*(V<»"i)"V*; orv**' ^^, ^ or X (Va - 1)« 7. 2»+V**-**»«»+2»+l; orV**-*""*"*!; or«*-a* = «* + 2x«+l; or3x*«-l; or«*«-l; orxsiV"-! "i W"^ 8. X 4- 34V« + 168 s 152 + 42V« + xj or-6V»« 16 i or V« " 3 ; or X = 4 9. V*+V^ + 2 ^x + 2 0t^X'\/x + 2 + x + 2s4i 0rV*l/*+2 e 2 - X ; or x(x + 2)«4-4x + x*; orx'+2xB4-4Ji; +x* j oi 6xs4;orxs] 10. o + X + 2Va"-x* + o-x^«-«x ; or 2V<^ - x* « «x - 2a; o> 4aa.4x'"a>x*- 4a>x + 4a^} or flAB*-l> 4i>a 4aVj or^^f-t-ia 4i^ = 4a»:,or«(aF + 4)*4a)'j orarsgqp;! ^ • - ^ ^ ' " ■ MB ^^.i.ifc^- 68 KBY TC tlSSL, XLVtttt 11. a»+ 2«t + ««s <^+ xt/F^i Of 2>4a> or a;s 12. Vft' + ax + x^sa-J-x; ot6*+«p + «*» ^ -^ i-2*-2a + «*-2at + ^a *"+ «; otx" 2ax B2a<«c^'*l} or«« — .^«^ 15. X - 32 ■ 266 « 32 V "f* ' } or 32 V * 388 ; or V - 9 ; or * « ?1. 16. 4be \| ft ^/6c\J e / 4be \ or a+x a-x e ; or a(^ » ^x •»• ae-^ cz t 0; ot bx » c* » ab & ac; or a(6 + c) IT. x + V**- V**-*" 3>^* giTMl <\/* + V' equation multiplied x; orV«+ 1-V*-^"I (**i^WingbjV*) ; or 2V* by + 2 s 2V* - 1 + 3 ; or 2V* •^1«2V*-'1» or4x- 4Vx + 1 a 4x - 4 ; or 4 V* '' 6 i 0' 1^ « 25 ; or X ■ f |. 18. x + oac*-2c V*+"* + X + fr J or - 2c t^ x + b « « - 6 - c* | or 4c» (X + 6) ■ (a - b - cV} or 4c'x « (a - c" - 6)« - 4«»6 i Or X (^a-^i^- b)*^4 \ 2c ) Ex. zLVin, zuz.] AL6EBBA. 69 1 2 1 1 I 4 9 1 a |4 9 14449 149 X' ax or tr X' ' X a X ^ ' 4x = 8a] ot X = 2a tj/x + a + ^x ~ a _ »» Q_ 2^x+a ^ m+ 1 . ^x ■{• a ~ '^x - a 1 2tjx'-a **~1 ^x + a m+1 « + « m* -I- am 4-1 2z 2 (!»' + I ) X 2a iii=«+l o(««+l) ; or*s 2m 2fl» EziBOiM 2LIX« 1. x' = 9 ; or * = i 3 2. 18 - 18x+ 18 + 18x = 100 - 100x» ; or lOOx* s 64 J or.a!» = iVs » or X = f I / 3. 4x» = 3x» + 9 ; or x" = 9 ; or X = i 3 4. 4r» - 8 = 1 ; or 4x» = 9 ; or x" = I ; or X a 1 1 6. x' > 6x 4- 9 = 13 - 6x ; or x' s 4 ; or X s f 2 6. 3 (x'+ lOx + 26") - tx = 23x; or 3x':s - ^6 ; orx»=-26j or X = + 6 J^l ^ \ ^ , 216X'' + 36 216x« + 36 1. lOx' +11- 10x» + 8 " ^^^a,Q ; or 25 :^ ^^.2.8 ; or 275x» - 200 e 216x» + 36 ; or 69x» = 236 ; or.x» = 4 ; or x = ± » 8. V? + ^^ = 9 J or 9 + 2x' = 81 ; or 2x2 = 72 j or x» = 36 ; or x = ie 9. V(* - 3)(x + 3) a 3o; or x*- 9 = 9o'i or x" = 9a' + 9 i or » = i 3 Vft" + 1 / m ] 70 KIT TO [Ex. zux. 10. Vo" - *' « « «■-«»«' 2a a« a* «' 9a a* «* 2a a« ia^ a* 2a« /o* 4a« 4a»6» ^^ 4a* ^^ 4a5%» ^^ 4a' 4a* 2a 2a or-^«56»; or «» - gp- or « « + -y Vi = ± ggV^ 12. Sa** - r«* ■ i - 1 - 6 J or x»(3a - c) » c( • 1 * 6; or «" rf-1-6 ^//<'-l-*\ 13. ^/e^-g* ■ t^'^T^* - «V6 = (c - l)'V'; or *» . ^-^j-p ; or X « i ^-==' V2x . /2* — — — •jV3+l* + !«*i«-3; or-2Y-V3 + i« 2« */2»- « 2« ^ X* *-6- y; or y-jV^ •♦■ i« ■S + 't; or— (3+i*)=9+2*+- 8 Of-r»9+ '^\ orr||B0}or«*«Slx2i or««i9V2 1; Ex. ZLIK, L.] ALGEBRA. 71 16. o + X + 3(0 + x)*(a - x)* + 3(o + x)*( + 12x s 46 ; or x' + 12x + 36 = 81 ; or x •(• 6 « f 9 ; x = 3 ; or - 16 6. 3x»+2xs sn; orx» + |x8 Vi orx»+|x+Jaa^ 4.^^44; or X 4 J = i -"i^; s: 1 5 ; or - 6§ 6. 3x»-. 14x = - 16 ; or x»- V* = - 6j *' x"- V* + V = -6 + ^=j; orx-5=i: !; orxaS; orl| 1. 5x»-236x = -47; or x^-'jax «- V J orx»-4?fix + H^j«< = - V +- W* = HV*^ ; or X - 4i:v i 41; or X ^ ^•> or i^ ; tha is X = 4Y or i^ 8. 4x' - 8x = 320 ; orx'-2xs80; x'- 2x + I = 81 j x-1 = ±9; xs 10 or -8 9. x' - 2x K - o«; or x* * 2x + I a 1 - o' ; or 9; - 1 « V^-?! or X = 1 J Vl-fl^ T 1 EST TO [Ex. X.. ■ ^ii* 5 « + 1 ■ i V ; » « V or - t| U. ,7af« - 20* - 32 ; »« » «/»« -, 3^ ; «« ., >,fi, + i^ » »«gi *• W s V»* } a: - '/ » ± »^ ; ar : 4 or - I J 12. ar»-tr:sc -12; «2-.7x + <^«-V + V^=ii *-| = iJ; « - 4 or 8 13, 3«»- 11* « - 6; or «"~ V«^ « - 2 ; osr r»- Y* + W s-2 + 'i^'*|g; T-V«12; .« = 3or| 1. aax^ i bcx - adx a ftrf ; or *^ + be -ad bd ac X = ac hn-ad /be -adx* bd (be -ad)* „ OS - ad /be - ad\ '+-Tc--*+(-23r-) or«» + — — — + 2ae ]~ ac 4aV ' he -ad\* Aabcd r 6V - 2a6c(i + oW ae / be-ad y [2ac ) 4a»c» or « .= ± 15. 4oV 2ac 2ac 1 X-'^X /be + ady frc - (U< be + ad 2ac ■=i 2a The rejected facfor x + V* - 0^«3 ua x = or I 16. ••-XaaiO; X»-X+i 5*Jl + 4 a /tjl; X - J * ± -j' X a 15 or - 14. -.a X ^ C2. [Kx. X.. f^L .V ,V '.V ♦• '^ - 1 = i 1 ; s^* + W bc + ad a nation bj 2Vx + 1); - 2 or - 3 ' ^/x + 1 ; i V - 7) ; J :. i-a. Ex. L.] ALGBBRA. 78 11. Ax* + 36 = 3x* + 48 - llx ; or x" + llx = 12 ; x»-4x + 4 + x» + 4x + 4 a-2 x + 2 X — 3 x + 4 ^®' xT2"''x"-~2"xT3'*'x^» x»- Ta; + 12+x«+7x+ 12 2x» + 8 or x'' - X - 12 x' + 4 x» + 12 • x" - 4 x'^ - X '^'-^s^:t 2 x» 2x'-x x^-4 2x» •>• 24 ' x=» - X - 12 ' X 2x -> 1 ^5°"^ ¥ X + 24 ' °' 4 x + 24 ' or x* + 24 X = 8x - 4 ; or x" + 16x = - 4 ; or x^ + I6x + 64 = €0 ; or X + 8 = V60 = i 2 Vl^j or x = i 2 Vl6 - 8 The rejected factor x = gives us the other value. 19. 49x» + 42x + 9 = 10(2x2 + 4ar - « - 2x» + 3x + 9) = 10(Yx + 3) ; or 49x» + 42x + 9 = TOx + 30 ; or Yx' - 4x = 3 ; x»-^; x = ?x»-tx + -,V=li + A = 55r^-ni?; x=lor-? 20. ax' -/x' "bx-cx -"h -c, or/x' - ax' + 6x + ex = i + c (/-c)x' + (6 + c)x = 6 + c; x» + (^:^ J x = ^^ ; i^ t- X=rj. 21. /6 + c\ / ft + c y fr + c / 6 + c \» (^/•-aj^+(^27:r2H; =7^+ 1^2/- 2a; ^7:7^+ (^2/ -2a; -2/- c i. J_ JL . »»x — ax + am a- m-¥ x' a m ' x'a-m + x awix amx = 3omx - a'x + a^m - m^x - ani^ + mx' - ax' ; ox' - r/ix* + rt'x - 2amx + m'x = a^ni - am^ = a/n(a - m) ; (a - m)x' + (o' - 2a/.: + m')x «= am{a - ;») ; x' + (a - m)x = am ; x^ + (a - m)x + (a - m)' = affi + (a - m)' 4ani + (a - m)' o' + 2a»i + m' (a + m)' fl; = i 4 a + m ' a — ni •t; + —5— = i 4 a + m a - m - -n or - a 74 TO [Ex. z., u. — 2(a + b) (a-6y 22. a6«»- 2«(a + ft)Va6 « »; 2ax 4- 6 - x a i (6 4- c) ; X s' 12. 5J0X » 2c or - 3 J ; « * t ®' "" T a a Ex. u.] ALQEBBA. 75 9. 13«'+120»16«+135; 12*"- l«x«15; 3x»- 4x = a/ 36x>- 48x a 45; 36x*- 4lx 4- 16 » 4S •(• 16 » 61 ; 6« - 4 » i V^T 10. »x»-4x>V3 + (2-V3)*«2; (t - 4V8)x«+(3 - ^3)* » 3 x» + 2 - V3 T - 4V3 [since T-4V3 » (2-^3)'] 4x» + «x + 8 9 2 - >/3' ■*" 7 - 4V3 " Y - 4V3 "^ t - 4V3 " » - 4V3 2x4- = ± 3 2 -V3~^a -V3 2«»- or 2-V3'"^ 2-V3 X s 2 - V3 **'^ ' 2~ir73 11. x» + 6ax a 6«; x* -t- 6(» •!- 9a* s i' 4 8a*] x + 3a i V9a' + 6* ; X = ± V8»* + *■ -^ 3a 12. 45 -9x x-f3 - 3X a X - 63 + 36T 19(45 - 9x) 19 34-« -S40X-C1I; 855 - Itlx « 40x* + 5»x - 189 : lOx* + 5tx s 261 ; 400x« + 2280X + (57)* » 10440 :• 3249 « 13689 ; 20x + 57 s f 117 ; 20x s 60 or - 174 ; :; « 8 or - S^V 13. x2 - 5x «- ml; 4x* - 20x + 2r. * •. 4»* -1-36: 2x - 5 ^ + V25 - Am* ; 2x = 5iV26-4m*; x« |(6 £ V25-4»*) 14. mx« - fix* - 2mxVn s - mn ; (m - n)x* - (2m^n)x s ~ Mfi s«_ 2m^n m-n •X s - dx*- i» -n mn !Vn 4x*- SmV*) m-n 4m*n -X ae — 4mn m-n 4mn 4mn* 'Ji - (m-n)* (m-n)* m-n (m-n)* 2nyfm 4- 2MVn 2m-4a«-i-a*sa'-4, 2x-a = iV<"^-4) «-J(aiV«?-~*) 16. «* + 3*" + 6 a X* + 3x« + 13x2 + tx - 60 ; ISx* + t« ■ 66 ; 6T6«> + 36i« » 343S ; 676«'4- 364x + 49 s 3481 ; 26x + 7 s £ 59 ; 26x ■ 62 or « 66 ; « « 2 or - 2-|^ EXIHOIBI LII. 1. x«|'2»0/»ndJr + taO.-. (x + 2)(x+ t) = «' + 9x+14 = 2. (« - 4)(« + 2) (X - l)x = ; or x* - 3x» - 6x» + 8x = 3. (x - 2)<* + 2)(x - 3)(x + 3)x = ; or (x» -j4)(x»- 9)x = ; or x» - 18x» + 36x = 4. (X - 6)(» + 6\x - 2)(x + 2)(a; - 3 - V2)(« - 3 + V2) = ; or(x»- 25)(x»- 4){(x - :?/ - (V2)'} = («'- 26)(x»- 4)(xa- Cx + 7) ^ 0; or x« - ex" * f !x* + I74x» - lOSx" - 600x + TOO = 6. (X - l)(i 2)(x - 3)(x - 4>(x - 6 - V6)(x - 6 + V6) = ; or (x^ - 3x + "SXx^ - 'i c + 12)(x» - lOx + 19) = ; x» - 20x* + lC4x*- 590x»+ 1189x»- 1190x + 456 = 6. (X - 6)(x - 4)(x - l)x (jc - 2 - V'^)(« - 2 + V'^) - i . r >•» - 9x + 20)(x'' - x)(x' - 4x + T) = ; «« - 14x' + 76x* - 2064;» + 283x" - 140x = x*fx-x\ (l-x+««) ■ *» = ox; iVo" ^4; + T«« 66; H-Tsi 59; 9x+U = 8x = i»-9)x = 0; + V2) = ; •6x+'r)-0; + V6) = i /•-3) = 0; fix tit.l ALOBBfc-A. 77 7. (X - 5)(x + 3) » ; or x» - 3x - 10 a 0, And (x«- 6x» + Bx« + 12x - 60) + (x»- 3x - 10) girei ui x»- 3x + 6 » ; then x»-3x = -6j_x»-3x + |= I -V*-V; *-l = tW"^^; x = i(3lV-16) 8. (X - 1 - V -^)(* - 1 + V"-^) * i or *" - 2x + Y = ; and therefore (x* ^ 4x* + 8x« - 8x - 21) + (x"- 2x + 7) gire* ng a«- 2x - 3 = 0. Then x«-3xs3; x»-2x+l = 4; x-l=t3; X = 3 or - 1 P. (x» + 6x» - 3920) -f (X - 14) gives ns x» + 20x + 280 = ; x» + 20x = - 280 ; x' + 20x 4- 100 » > 180 ; « + 10 :^ . 6 V"-5 > x = -10±6V^ 10. X 3 is evidently another root, then (x*- 6x'+ 13x'-> lOx) 4 (x* - 2x), gives us x" - 4x + B = ; x' - 4x = - 5 ; x' - 4x + 4 = - 1 ; X - 2 = ± V^ ] x = 2± V^ 11. (x-3)(x + 4)x = 0} therefore (x»-2x*-25x»+26x»+120x) f (x* + X* - 12x) gives us x* - 3x - 10 = ; x' - 3x s lo ; 4x» - I2x + 9 = 49 ; 2x - 3 = f 7 ; 2x s 10 or - 4 ; x » S or - 2 12. X = is obviously another root. Then (x - V - 2)(x + V -^) r a;» + 2 = .-. (X*- x«- 2x - 4) + (x»+ 2) = x*- X - 2 « ; that is x»- x = 2, whence x ;s 2 or - 1 13. Alg. Art. 206, when the roots are equal 4*^ i .< 2 x e\ cr 16 a 8c; or c- 2 14. Alg. Art. 208 (Cor.), i8 + 7 = - — and /8y * — .-. -5-^ = - ~ c 6 11*,,. c 1 a - a — .'. -T- + — ='- — ] Also Since jSy « — , -^ = — a»/»y b a Hence — = the sum of the roots and --- ' their product and the b a equation isx' + - x + -— = that is ex* + 6x -f- « 3 78 tan to [tX. tit. Lift. 15. Alf. Ait, 208 fii'ym^p uid fiy q i9« + 7« - ^« + >• + a/8r - a/Jy « 0« + 2iBy + 7«) - 2/8y « (fl+7)« -2^t.p«-a«(l) O - Y)t . ^t . 2/ly + y« « 0« + y« ^ 2^) a J»« -. 29 - 2jfr0ln (I).-. -p«- 4^(11) ^»-'y*-(^ + T)(^-.7)«-p(iVf«-4«), aiace from (11) fi-y ± Vp* - *«) ; 11 ^+y p y 0y q BxiBom Lm. I. «-6V<-l-0»2S; V«-8«i6; V«« ior-a .•.xn64or4 "2. V*-*V» + *-l} V*-2«ilj V««8orl,.'.««81orl 3. «*-14x«»-40; x«-l4c> + 49»9; x'-7«i8; «•* 10 or 4, .'. X = ± 2 or ± VT5 4. *•+ 14Vi'« not; «>+ l4Vi^ + 49« 1166; »*+ 7-±34; «• = 27 or - 41 ; at" » 3 or ^-41; .••*«» or j^ImI 6. «-2Vx+6«2; (x + 6)-2Vx+638; (*+6)-2Vx+6 + l = 9; VxTe - 1 s i 3 ; V* + 8 « 4 or - a ; x + 6 » 16 or 4, .♦. x * 10 or - 2 6. x*-ia^=248; x* - Jx" + iV « 248 + -j*^ « 9f J* , x«-i«iV; x> a 16 or » ^i^'. X K £ 4 or i i'f^a 7. x«-8x»a6I3; x< -8x'+ 16« 629; x«-4«123; x*= 27 or - 19, .♦. X a 3 br ^ - 19 8. (X + 6) - V* + 6 * 6 ; (x + 6) - V«T6 + 4 ■ 6 + i « »^ ; Vx+l-J =ij; V«T6 3 3or-2; x-|-6«9 or4, .•.x9 4or-l tX. tit, Lltt. Iq - 2qtrota e firom (ii) h)<» - >) .« a 64 or 4 . « ■ 81 or 1 8 ; «« a 10 + 7-i34; r+6 + 1 = 9; , .'. X s 10 23; x»s27 ar9 4or-l fix. Lltt.l ALaitBilA. 79 9. V*(*" + * - 6) a 0, .'. V* ■ aad .'. X * Alio X>-i-X-6a0, .*. x'-fX«6i X" + X + J**^} X-t-iaff, .*. X « 2 or - 8 10. Clearing of fraotiont 2x + 2<\/x a le •• x ; 8x - 2Vx s 16 ; 36x - 24Vx -I- 4 s 192 + 4 « 196 ; 6Vx - 2 « i 14; 6V« - - 12 or 16, .*. V* ~ - 2 o' f» •'• » ■ 4 or tj \ 11. V* + 21 + Vx + 21 » 12 ; V -t- 31 + V* + 31 + i a V i Vx + 21 + I a f f; j^x + 21 a 3 or- 4; x + 21a81or2S6; T a 60 or 235 »■ 12. V*(* - 3 - V*) ■ ^t •••V* " <>i •*• * « 0. Also X - 1 . V - 0, .•.x-V« = 3; x-V* + i»f; V*-J " ±?; V* « 2 or* 1, .'. X a 4 or 1 13. c»+x* + 2 x* + x*-2 x»-x* x'-x* Then (by Art. 106 - vn) 2x» + 2 2x"-2 x»+l x"-l , 2i?T2 • 2^:72 i i^Ti = iJTlJ *\"' + x«- 1 . x' - x* + x'- 1, orx"-x»aa:»-x*j c*-x*-f«*-x««0; x*(x-l) + x»(x-l)aO; (x-l)(x*+x^aO,...x-l«Oorxai. Also X* + X« a 0, .-. X* + x« + i a i ; x« + i a ± I j" ajl s or * 1, .-. X a or i V - i 14. 9(6 -V«) 7x» - 3x + 4 23(x>2Vx) x + 2V« '^ (6 + V*)(« + 3V-.) ■•■ 6 + V*"" ' "°*'^''^^*°' by the denominator of the 2nd term we get ' 9(36 - X) a Yx» - 3x + 4 + 23(x« - 4x) or 324 - 9x a 7x' - 3x + 4 + 23x' - 92x ; 30x' - 86x a 320 ; or x» - 42x a V; x» - Hx -f (J8)» a Jj,a,-yi + i^iyi s x^^». «-J2--±^.Vj*'=5or-2f^ 15. x« - 3x» + 3x - 9 a 0; x"(± - 3) + 3(x - 3) a O; (x-3)(x»+ 3) a 0, .'. X - 3 a or X a 8; also x«+ 8 a 0, ,*. X* a - 3 ; or X a f ^111 80 KBt to ttx. ttn 16. (« - 3)(x - 4) = 2 - 2V2V(« - l)(a: - 2) + (x - l)(x - 2) or x' - 7x + 12 = 2-- 2V2Vx* - 3x + 2 + ar"'* - 3x + 2 ; or - 4x + 8 = - 2V2Vx* - 3x + 2 ; 2x - 4 = V2V-«^^ - 3x + 2 ; 4x»-16x + 16 = 2(x»-3x + 2) = 2x'-6x + 4; or2x''- 10x = - 12, .-. x« - 5x + 3^ a a^ - \* = I. Hence x - f = f }, .-. x = 3 or 2 17. x'-l-Sx + 2 + 1 = 0; orx«-l-3x + 3 = 0; or x» - 1 - 3(x - 1) = ; or (X - l)(x2 + X + 1) - 3(x - I) = : or(x-l)(x''+x-2) =0, .'.x-lsOorxs 1. Al80x'' + x-2 = 0, .-. x' + X = 2 ; or X* + X + i = f ; or x + J = ± |, .•. x = 1 or - 2 18. Since ("Vx* + ax + 6 + ^x'-ax + 6)(V-r^ + a« + 6 - V^^- '^^ ^ *) s 2ax, dividing these equals hj the given equation we have 2ax Vx'+ ax + b -Vx'- ax + 6 = -— , and adding the given equation 2CEX to this we get 2Vx'' + ax + b = —- + c ; or, by squaring, 4X' + 4ax + 46 = —5- + 4ax + r', .-. 4x'' - » = c* - 46 ; o^r x2(4c=*- 4a'«) = ca(c3 - 46), .-. x = i yV ^l g" 19. Reducing the terms of the first member to a common deno* x^ar-x-^/jt-x+x^x + x^a-x 6 minator and adding we get 2xVx 6 X - (a - x) Vx' 06 or iT^ ' W' °' ^''' ' ^^* * ''*' or x=« - 6x = - Y 6« 6« a6 62-2a6 x« - 6x + — = -: r- - whence x - — = + JV^^- 2a6, Orx=» J(6iV6*-2a6) 20. Clearing of fractions we get (Vx + 60 + ^Jjt^ + 9)* = 2Vx* + 60x^ + 9x + 640 + 89 ; that is X + 60 + 2Vx» + 60x''+ 9x + 540 + x»+ 9 = 2Vx'+ 60z^t 9x + 540 + 89; or f«+ X = 20 ; x"+ x + i = 20 + i = V } x + J = f | .-. x = 4 or - 6 -f llx + 20 a 0; multiplying by x we get X* - 8x" + llx' -i- 20x a ; or x* - 8x» + 16x» - 5x« + 20x a ; or (x» - 4x)» - 6(x» - 4x) a ; (x' - 4x)» * 6(a» - 4x) + »/ « 1/, ,'. x* - 4x - 1 a f I ; or x' - 4x a 5 or ; x(x - 4)* a 0, .*. X a 4. Also x* - 4x a 5, .'. X a 6 or - I 25. X + a - ft + 6 /2x + a + c + ft-6\* X +& ( ( 2x + 6 + -)■ or X + ft + a - 6 2x + ft -f e + a - ft\ ' 2x + ft + c )' or 1 + a- ft x+ ft (■• x + ft a-ft orl + a-ft = 1 + 2(a - ft) (a-ft)' 2x + ft + cy ' 1 )■ or x + ft~2x + ft + c(2x + ft + c)" x + ft 2x + ft + c -ft (2x + 6 + c) 2» or 2x + ft + c - 2x - 2ft ~(x + t)(2x + 6 +"0" "^ (2x + ft + c) 2> c-ft a-ft or x+ft''2x + ft + c' or 2cx - 2ftx - ft* + c* a ex - ftx ^• 06 - ft*; c'-aft or (a + ft - 2c)x = c^'-ab\ or x a rXT^Toc 82 KEY TO [Ex. uu« ■W II' 11 26. 3**- 14*'+ 2Lr - 10 « 0; multiplying by 3* yve have 9x*- 42x»+ 63*»- 30* « ; or 9**- 42x'+ 49a;'+ 14*'- 30x = ; or adding z* - 6x to each side (Ox*- 42x'+ 49x'0 + (15x'- 36x) B x"- 5x; or (3x»- 7x)»+ 6(3x»- Yx) + '^ * x« - 6x + »^, .-. Sx* - 7x + f = ± (X - 5) Tl»en 3r«-7x + §=x-f tliat 18 3x2 - 8x = - 6 whence x = IS or 1 Or Sx* - 7x + 5 = f - X that is 3x2 . 6x = whence x = 2 or 0» 27. Assame ^x + -^a s .^n, then cubing each tiide we have X + 3x'a' + 3x'a' + a = n\ orx + a+ Zl^lix{^x + ^a) = n ; or X + a H- S-^anx = n since {/x + ^a = ^n. But comparing this with the given equation we see that n - b, .■. ^n = ^b, 28. (4x2 _ 9j.) _ (4a.a - 9x + 11)' » - 5, or adding 11 to each side we have (4x2 _ 9a; + n) - (4x3 - 9x + 11)' = 6 ; or com- pleting the square (4x2 - 9x + 11) - (4x2 _ 9x + 1 1)^ + i = V i /. (4x2 ., 9x + 11)* - J = 1 4 ; or (4x2 ^ qx + 11)* = 3 or - 2, .-. 4x2 - 9x + 11 = 9 or 4, Then 4x2- qx*~2, whence x = 2 or J ; Also 4x2 - 9x = - 7, whence x « —(9 i V - 31) 29. Completing the square we have (x + 6)2 + 2x*(x + 6) + X ■ 138 + X + x', and taking the square root,x+6 + V^ = iV(138 + x'+x); or(x+V«) + 6=i'\/ir+x* +138; squaring, we have (x + ^/x)'^ + 12(x + ^x) + 36 = (x + x*) + 138 ; or (X + V*)' + n(r + V^) = ^^2 ; or (x + V*)* + 1 l(x + V^;) + ^{^ 9 102 + H^ ' '^5^ ,-. X + yx + V « ± as 1 1 X + ^x = 6, whence X = 4 or 9, or X + Va; « - 17, wben<^ /^x=J(-l±V- 67), and .'. X = J( - 33 T V^^t) •We throw away the root x = bocaose It a> .■ -m from the x by which we multiplied each side of the equation in the solutioui and i9 cpnee* qnently not a root of the gtvon equation. = 1 £x. LIU] ALGEfiiU 88 30. x^-ix^ + es^-ix + lse, OT extracting the square root x' - 2je 4- 1 s i V6 ; and again taking the aq. root x-ls± VTV^i whence « * 1 ± V ± V^ 31. Squaring we have 4j!^ - 4x^= a' - 2o'x* + a'x*, and divid« . . „ . 4 4x« 1 ^ 4 Ax* ingbya'x* we get, ^a" -^ = ^^I + 2 +x*; or^a--^-4 /, IV^/, 1\4 4 4- 4a* 4 2 Let - p(l T Vi - <"f'') ^e represented by 26'', then wo have x2--^ = 26'; or «*- 262x''»= 1; or x*- 26»x»+ 6* = 1 m 6*; or x'- 6' = i Vl + **i ••• *' ^' *' ± ; ' "+1* (A) But 26» = - ^(1 T vr^^^^), ,-, 6» - - ^(1 T vjn^^) •'• ^* = { - ^(1 ^ Vl^^j' ' l,il ^ 2Vrr^ + 1 - a*) = - i(2 T 2Vl - a* - a*) .-. 6* + 1 ^- 1 + --iH ? 2Vnri* - o*) or or 1 a* = 1 + ^(2 T 2V1 - a*) - 57 = ^(2 T 2V1 - a*) .-. 1 + 6* = -4(1 T Vl - <»*) Substituting these values for 6' and 1+6* in equation A, and then extracting the square root we have * =V - ^(1 T Vr^^ ± V^G ? Vi"^^^) or using only the upper signs V 84 KEY TO [Ex. Lin. 3 ,- or X = i — I - 1 + y) - a* + V'-i - 2Vl - a* } * ■ ± —{ «- I + VI - «* + V(i + a») - 2-^/1"^^"^ + (I - «')} * •. a? » i ~{ - 1 + VI - fl* + V(Vi + a^ - vn^^*} • ± "^i - 1 + Vi - <»* + VI + °^ - Vi - «1^ ■ ± jfvi - «* - va -"^ + vi + ^^ - 1}* « i -icvrrp - i)(vnr-^ + i)i* 32. ((« - 2)» - xp ~ {(X - 2)' - r} = 90 .-. {(* - 2)» - arf - |(* - 2)=» - a:} + J = 3^*; or ((X - 2)» - x| - I « i Y, •. (X - 2)2 - X = 10 or - 9 thai Ui x' - 4x + 4 - X « 10, whence x = G or - 1 ; or *" - 4* + 4 - X * - 9, wbeace x = J(5 ± 3V - 3) 5 a 33. Diriding tbrotigh bjr x* we have ox* + 6x + c + — + -, = ; or[ax'+-^j + (**+t) +vpr«W'«l in order, is etllod %recwrirtg equatUni. The above solution affbrds a g«.'ueral nwtlvod fi>r DOlrisg web recurring biquadratic egtiations. [Ex. LIII. L-a a + -5 = 0; l\ — l + c-0 ing these y = 2a-c, + 6" - 4flc a f, whenc? ienis follnw- rovPW^'l in ■ds a gi.'ueral Ex. Lm.] ALGEBRA. 85 34. Y fx^ - ^ j -^ = - V (<*' " i« )» "qiiafinfif ^ct^ lides ' have x^ - - + ~ - -_(^a:> - -j . «» - ^; 2xV o*\J /a* \ 2x ^ 1 X* - a* or z» (I* - a*y + jj— - ; or taking the square root we V(x* - 0*) get, z ■- = ; or transposing and squaring, x'= - ^ o* 6o« or «'af" = X* - B* ; or a:* - fl?x» ■= a* ; or r;* ~ a'x^ + -r ~ c' a:^ = -^ (1 i V5), whence x = ± aVi(l i V5) , , 2f(2x 4 4) -4(2 - «)l 35. V2x + 4 - 2V2 - X = -^^^ — - - ■-,' ' ; or factoring the second member, we have '^2x + 4 - 2V2 - x ~ 2{( V2x + 4 - 2V2 - x)(V2x + 4 + 2V2 - x)| Then dividing each aide by V2« + 4 - 2V2 - x we have 2(V2x + 4 + 2V2 - X) , . . 1 = -^ ... ; or 'JOx' + 16 - 2f \/2x + 1 + 2V^ - x\ Now squaring each side, we get Sx* + 16 = 48 - 8* ■»• i6V8 - 2x* .'. x3 + 8x = 32 - 8x» + 16V8~^x* ; or x' + 8x = 4(8 - 2x«) + 16V8-2x» orx'' + 8x + 16 = 4(8-2x'0+16V8-2x»+16.-.x + 4 = i(2V8-2x»+4) .-. X = 2V8 - 2r*, or x« = 4(8 - 2x*) : whence x = i jV2 or Also X ■= - 2V8 - 2x'' - 8, or x' + 16x + 64 = 4(8 - 2x'0 ; 9x* + lex = - 32, whence x Hty/' S(2 T V-H) And by equating the rejected factor *i/2x + 4 - 2V2 - x = t, . . . . 2 we obtain the remaimog root a? « —- -*# d6 KEY TO {£Z LTTI. 2*' + 1 + xJis^ +3 a 36. —- , - — = T, whence 2x* + 3 + a:V4«» +3 1 '•«, Article lOG 2a:'+3 + xV4a;^+3 1 or rV**' + 3 = in - 3 - 2x3 = = 1 - «r, .'. 2a;- : -♦•arV*** + 3 = j _^; 3a - 1 1 -a 2x^; eqimriDg /3a - 1\* /3a - 1\ each side we have 4** + 2x^ -^ K _ ^ j - 4i'( YT7 ) ^ *^ /3a - 1\ /3a - .n« / 12a - 4\ /9o - l\ (3a - 1)2 (3a - 1)" 1 - a 3a. 1 X = V(l - a)(9a - 1) 37. ((x-l)(«-4)}{(x-2)(x-3)} = 8; {x»-5x+4}{x»-5x + 6} = 8 .•.{(x«-5x) + 4}{(x»-6x) + 6},thati8(xa-6x)»+10(z«-5x) + 24 = 8 or (x« - bxf + 10(x» - 6x) + 26 = 9, .-. x» - 5x + 5 = ± 3 •,-. X* - 6x » - 2, whence x = J(5 i V^)- Also x* - 6x = - 8, whence x = i(6 ± V •" ') 38. {(X - l)(x - 8)}{(x - 2)(x - 7)}{(x - 3)^x - 6)}((x - 4)(x - 6)) s {(x»- 9x) + 8}{(x»- 9x) + 14}{(x2 - 9x) + 18}{(x=*- 9x) + 20} e (I* - 9x)(l £* - 163x + 230) + 401. For x" - 9x write y, then we have (y + 8)(y + 14)(y + 18)(y -i 20) -Uf-^ 230j/ + 401 ; that ifc y* + 60y» + 1308^" + 12l76y + 40320 = l7y' + 230y + 401, subtracting from each side 82^' + 17Cy + 320 we have V* + COi/« + 1300y' + 12000y + 40000 = 9^* + 54y + 81, or taking I i«j square root of each side y» + 30y + 200 = i (3y + 9) .*. y* + 27y = - 191, whence y = J( - 27 ± V - 35) Also y* + 33y = - 209, whence y = i( - 33 i V253) But y a x' - 9x, .'. x " -9x=i(-27±V- 35), whence ar « 1(9 i V27 ± V"^^ Also x»- 9x = J( - 33 i V263), whence » = 1(9 ± Vl5 i 2 V253) [Ex LTTI. tX. LIll.] ALGEBRA. IT rticle lOG _ 2 ; sqnanDg — ) + 4x* V 12a - 4 \ 1 -«; (3a - 1)'' 1 - a 5af + 6} = 8 5x) + 24 = 8 h 5 = ± 3 - 5* = - 8, -4)(x-6)} 9a;) + 20} >ite y, tbcn \0y + 401 ; ;30y + 401, we hare or taking (3y + 9) i), whence i 2 V253) 39. Multiplying aa indicated we havu x' - 6a;' -f llx - = *» + ea;* + llx + 6, whence 12** + 12 = 0, .-. x = ± V^ 40. Reducing aa indicated by the question we have X + 1 - 5Vx + 1 + 6 + 5Vl + X - 6Vx + 1 + Va; + 1 -1=0; of (x - 5^x + 1) + sVx - 6Vx + I = - 7 ; or completing the square (r - 5Vx + I) + 5(x - 5Vx + 1)* + V = - 5, whence (x - 6Vx + 1)* - 5 ± V^ =/ 2 » •"• * " ^V* + I = J(I1 ± 5V - 3) = a, suppose : Thenx-5Vx + l = a; orx-aa:6V« + l ; orx'-2ox+a'=25x + 25; or x» - (2a + 26)x = 26 -• a», .-. x = i(2a + 25 ± 5V4a + 29) But a = i(ll ± 5V - 3) by supposition .-. X = J{U ± Sy"" ^ + 25 i 5 V32 i 10V~^3 + 29J = 18 i 5(V - 3 + Vci ± lOV"-^) 41. Arranging the giren quantities, we have (4x*- 8x»- 4x»+ .'x - 1) - 2(2x2- 2x + l)V4x*- 8ar«- 4x»+ 3x - 1 + (4x*- 8x'+ Sx"- 4x + 1) = 0, and taking the square root V4x* - 8x' - 4X'' + 3x - 1 - (2x' - 2x + 1) = ; or transposing and squaring 4x* - 8x' - 4x' + 3x - 1 = 4x* - Sx* + Sx" - 4x + I , .-. 12x2 - Tx = - 2, whence x = ,'^(7 i V - 47) 42. Multiplying through by ax to cletr of fractions c'Ax-i + 2a"c-ix'- 2ab + 2a'c-*x»- 2ix = ac -»x(x»- a'^bcx + a*) multiplying now by c we have a^brx-'^ + 2o'x' - 2aftc + 2aV - 2icx = ax* - a-*6 Let V3 - VS = fl, then x* - 6ax' - o* - 2a', whence « = i V3o ± aV*" + 2a t 9 where a = ^3 - V^ Ex£ROiBB Liy. 1, (« + y)(aj - y) a 45, but ar - » = 5 ,', 5(« -J- y) = 48, or X + y = 9 and X - 3/ = 5 .-. 2x = 14, Ac, 2. C^ + y)(* -y) = 105, but a? + y = 21 ,•. 2l(« - y) = 106, or X - V = 5 .-. 2« s 26, &c. fix. uv.] AL0J2BHA. 89 a'x, 3. «' + 3x ; y" « 81, bat x« + y» « 41 .*. 2afy ■ 40, and . i' - 2xy + y' = 1, whence a: - y » ± 1, /. 2« = 10 or 8, Ac. 4. a= - 2xy + j/ = 225, but x» + y» a 113 .-. 2xy » - 112 .'. x*+2xy + y'= 113- 112 = 1, whence x + y = 1 1 and x -y = 16 ■. 2x s 16 or 14, &c. 6. x'' + y»=89andx= -- 3r 40 40* r + y*= 89 ; or y*- 89y»= - 1600 ; y*- 89y»+ C^)'=- 1«50( - la^iJ- = ^^^^ .: y'- V « ± ',«» whence 40 40 40 y = ± 8 or f 5, And x = — ^;or|-g-'i6orl8 72 24 24« 6. x»-y»=05,andx = — = — .•.-rr-y'=53; ory*+66y»=67e whence y' = 9 or - 64, and .-. y = i 3 or 1 8V - 1 24 24 24 And X = ^'°M7 = ±8orT3V-l y ± ^ ' + 8V - 1 1. a" + 3i/ = 14?, and y = 24 - 2« .-. x' + 3(24 - 2x>« = 148 ; or x^- 1728 - 288x + 12x2= 148. q^ I3x» - 288x = - 1680, whence x = 12^^ or 10 And y = 24 - 2x = (24 - 24i*;j> or (24 - 20) = - -j^ or 4 8. 3x'»-2y''= 115, or 19y' + 36y = 448, + 3y '2 + 3y And X = 2 + 3y Aua X = - 2 •". ai whence y « r4or -5H 2 + 12 • r 2 - 1' ma )'- 2y»= 115; * 7 or - 7U 9. 4x» + 3y' = 511, and x = b - 2y {»-!)' + 3y»= 511, or 43y' - 432y = 1683 ; whenoe y = 13^3 or - 3 And =,4»=(».?M) „,.,,, sj.jj or 11 10. a;« - y" n 26 ; also from 2nd cquat. x*- 3x'»y + Sxy"- y» = 8 by 2xu subtraction 3x'y - 3xy* = 18 ; or xy(x - y) = 6, but x - y = 2 or xy = 3. Then xy = 3 and x = 2 + y .-. y(2 + y) = 3 Oi- y^ + 2y -■ 3, whence j ' Oi- - 3. And x=2 + y = 3or-.l 90 KET TO [Ex. uv. IHI 11. « + y « 4 .•. (x + y)' = IG .-. »»+»»= 16, and from lat equat. t* + 3xh/ + SxyH f = 04, .-. by subtraction 3x^y + 3xy'^ =■ 48 ; orxy(x + y)al6,butr + y = 4.-.xy = 4.-.i/(4-2/) = 4ory'»-4y = -4, whence y = 2, and x = 4-y = 4-2 = 2 12. Squaring the Ist equat. i^x + 2 yxy + t/y^O, but"4 V^ = 8 .'. subtracting we have i^x-2 yxy + ^y=l', wlicnce "/x - \'y = f 1 and yx + Vy = 3j •'• by addition 2 Vx = 4 or 2, .-. yx = 2 or 1, whence x = 2GC or 1, kc. 13. v'+ 4x - 2y = 11, and X = 14 - 4y .-. ^3 + 4(14 _ 4y) - 2y s n, or y' - ley = - 45 ; .whence y = 15 or 3, and x = 14 - 4y = - 46 or 2 3y + 1 14. 2«' + xy - Gy2 = 20, and x = — r — - /3y + 1\» /3y + 1\ . .-. 2^— —j +y(— 2~) " ^y^ = 2^5 ^^ ^y' + Ty = 39, 3y + 1 whence y = 3 or - 6J and x = — - — =^ G or - ^i 16. 9x + 5y-4xy = 0,andx = 2 + y.-. 9(2+y) + 5y-4y(2+y) = 0, or 2y' - 3y = 9 ; whence y = 3 or - ?, and x = 2 + y=5orJ 10. xV + 4xy + 4=100 .•. xy + 2 -i 10; whencexy = 8 or- 12 From second equation x = G - y .'. 2/(G - y) = 8 or - 12 That is y'- 6y = - 8, whence y = 4 or 2 ; and y'*- 6y = 12, whence y = 3 iV^l .-. x = 6 - y = 2 or 4, or 3 fV2T 17. 9x* + 36xy - 85y^ = 0, and x = 2 + y .-. 9(2 + y)''' + 36y(2 + y) - 85y2= 0. That is lOy*- 27y - 9; whence y = 3 or - i'n> ftn<^ x = 2 + y=5or 1-,V 12 +y» 18. From Kecond equation x = — - — and substituting this /12 + y2\3 /12 + y»\ for X in the Ist equat. we get f — - — j + ( — r — jV =» 1 1 > 144 + 24y' + y* \ .. \ i or rpr + 12 + y2 = 77; or 2y* - 41y2 = - 144 .'. y' = 16 or Y> "W^hence y = i 4 or ± gV^ 12 T y» y ^i 4 Andx^ 28 i of 66 = ± 7or 33 11 V2 PV25 ;±^o^±-r [Ex. tiv. I from Ist 3-4y = -4, l"4 V^ = 8 :-V.V = il ar = 2 or 1, )-2y=ll, r = - 46 or 2 1y = 39, 2/(2+v) = 0, 5 or i a 8 or- 12 18 or - 12 6y = 12, 27y « 9 ; iting this \y = '^7; a - 144 Fjt. tiv.J AtGEBltA. dl ' lori 11 V2 19. Let X a V -f z and y = v — z Then x» + xy « (» + •)» + Co + «)(» - s) = 2t>'» + 2t>2 s 66 (i) Also x»- y«=r (» + »)a- (r - *)» = 4v« * 11 .•: 2vz » V (") From (I) we get 2v^ = = iji- .-. t>» a ^Ji or » = i V Prom (II) we get by - = ^ a j. n - ^ j. Then x = t> + «siJjiij=± t>-s- + llTJ«i5 20. From Ist equat. by clearing of fractions x' + v" » 18xy (i) and cubing the 2nd equat. we get x^ + 3x^ + 3xy' + y" = 1 728 (ii) and taking (i) from (ii) we have 3x^ + 3xy'= 1728 - 18xy; or xy(x + y) = 576 - 6xy ; or since x + y = 12, we hare 12xy = 576 - 6xy .-. 18xy s 676| and hence xy a 32. Then x s 12 - y .'. y(12 - y) = 32, or y' - 12y = - 32 ; whence y = 8 or 4. And xsl2-y = 4or8 21. Let X = o + 2 and y s v - £ Then x" + y» = (» + «)» +(v- zy a 2i>» + 20»«2»+ 10»3* a 33G8 ; or v'^ + lOv^z^ + 6»8* = 1684. But x + y = t> + 2 + »-s = 2t> = 8 .-. V = 4, and substituting this for v, 1024 + 6402*+ 202« = 1684 ; whence 2* + 322' = 33, .*. 2' « 1 or - 33 and s = ± 1 or ± V-33 Then x^r + 2s4±l; or4± ^-33 = 6 or 3 or 4 ± V-3^3 y = v-2-4Tl; or4T V-33 = 3 or 6 or 4 7 V-33 22. From 1st equat. x* + 3x'y + Sxy* + y' = 343, aud x* + y" = 133 .'. 3xhi + 3xy' a 210 ; or xy(x + y> = 70, but x + y a, 7 .•. xy a 10 And X = 7 - y .*. y(7 - y) = 10 ; whence y = 5 or 2, and x = 2 or 5 23. Let X = » + 2 and ysv~z Then x* + y* a (t> + 2)* + (t> - 2)* a 2t>* + 12»V + 2s* = 97 But X-yat> + 2-t> + 2a22al .'. Z = i Hence 2»* + 3»' a 97 - | a 96| ; whence v' = \^ or - a^ .-.r a i $ or i iV"^^ Thenxa|,+2ai| + J ; ori JV-3r+ i = 3 or - 2 or 1(1 ± V^TaT) Andya«-2=±f-}; oriiV-^-5 = 2or-3or }(-llV-31) N^ IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 11.25 liillS 125 ■tt l&i 122 WIKU 1.4 il.6 Photographic Sciences Corporation 23 WIST MAIN STMiT WISSTU.N.Y. USM (716) 172-4503 ^ o'^ n nf>v9^ {Bs. ttUt* 14.' MidtipijriBg tilt ted «QMt kf 8 and adding U lo tht^ft •fo^tioii m ^vn «• + 3»*» +««|* +y » 343 ; w)iene« x + » r 7 .% « a » - 1. Alw *V + ?>» mxy(» + y) » tzy » 84 .-. i^y * 13 andxs 7-y sSor 4 86. iflf f-i. c •!? y « 30, Mdijpg 3fy to Mcb tide of the ffiUKt. w» ftftTe («■ + 3»y +y> + (« + y) « 36 + tit^ or oompletiog the «V»«wOr+yjp+x»+^ <»**y»iV««l+*nf-||btti4(«4*^*8«y v;3*y ■ i 4^ +3^ - ^(traitpbfiiiguidiqitftriiigl^ I^ +-'i ^m*92iif .-l wy- mtfVtl9\ #1wim» »y .- fli ©rU Y ThfM A(x + j^ » lixy m 3t A»*y » 6, and W * 6 - > ••• Hfi - y) « «, <^y- 99 a - 8; f^nee y « 4 or 3, ind •. « « 3 or 4. Also y(6 - y) =«-*»•} wlieiic© y » ^ - 13 f yl^) uid.v«:s|(.i3f yaff) si dleiurbig tl» first eqiiwtioii of fr^ 8|(« + y)« + (» -y)^ » 3e(«^ - j^; <. |,18««+ Ifl^ a 38i»- 3l>y? «■ "' ■ ■ ''3»' '' ' '' Hinee 36!^ - 16««, or ey a i 4«, or y « jb -j- ; aobiUtoting this --■a:.. ■ '^' / '3*\'* " ' 4«« iBtho3adeq9|tion,weliaTes'4' ' 1835" (' tniiisposUl uir.} m «« » ao, Ut*«»jf; then ff^^ 3j«+ •,«« U, Of ^« -p-^iL or by yedaution •»+ « « |f • irbeaco « « f or - # : «> « -r-Sr~ ^^ • ^+r¥a M before y ^^ ±5. Tkenftire » « vy » i S x }; or i 5 x - f a ^ 3 orTS 30. Addinf the twoe^ofttiopa tof^^OT^we h(»Tf jf*+a*V>y« s 169, vheaee ** + »» . ± 18; bat «»+ J|«y + ,» « 85 .% bj f^ trtotion a*y a 1% or 98 .-. «V » 36 or 49, end «y « f 6 or i » Then » * -y or -j-, 1^ •afaftitatipg thie for «» w« h»v» / 6\» ■', .--^ .-'a* ■■■■ ■, ^^- '■■'* ■■''' •^i- ^^iyj ^.^*i»v«»*ti«Ji + Sf«-±13j orf«Tl8yi — ^, whene« v* « 9 or 4, and /. y s i 3 or ± 2. (Imfbttible^aef being rejected.) 31. Let X » «y ; th«it 8«V-f- Svy"- ^y* » 108 '**"Si»-f3f-4 Also »y - 3«y* - V 81 108 81 8»"+ 2«~4 ■ 8iT;?"+7 0' *>7 wbeaee y> redaction, 81 -. "■■ ■■■■■■ -'■- 'J-*- ' 18i>»- 6v B 40: whenee V s 2 or -• ^1 Theny*s tl 81 81 «»-^+»» " 6 - 4 » V ^ -: jf * f|g 4. t « ' * or 169 X at .'. y • jt t or « i SSV^I im* '< *m i . 94 TO (SX. UT. . ^^ And X a »f a 1 3 X 2 ; or ± 8 X - f J « i 6 or T 4)^ Alao « > vy s i 39^3 x 2 i or ± 39V3 x - f g » i 78V3 or 7 60^8 32. Faetoring the first equation, we have '^^'^ (*'- «D - (f + ») -» 12 .-. (y + «)(y - * - 1) = 12 ; or y + » • jTiTi 48 12 48 1 4 but » + * • ^^T^» ••• ^3iTi « (pif» i «' i^TT^ or (y - *)" » 4(y - x) - 4 .•. (y - x)' - 4(y - x) + 4 » ; whence y-x = 2, andy+x=^^— ^s\4= 12, .■.ys:14or2ys7,andx 6y Taking x 4- y' » 4y = x + 8, (by 2nd given equation) we have y* « 8 .*. y « 4, and x » 4y - 8 » 8 Tr^-'^g X -i- y* s > 5y, and subtracting this from the 2nd glren \ ..,wehaTe8-y'»«»y/.8-8yay+y«; or8(J-y>y(l+y') and dividing each side bj 1 4- y', we have 8(1 - y^ « y ; that is y + 8y^ « 8, whAioe y' > . 4 Jt 2V6, and .*. y^iO^ 19^9, end c s 4y - 8 s 152 7 64V6 , also 1 + y' s .-. y = 1 34. x« + y»=35(i),x« + y«al3(ii). ; From (I) (X + y) (x» - xy + y») = 35 ; but x« + y» ■ 18 35 .*. (« + y)(13 - xy) = 36 .-. X + y = ^3 , j.y , squaring we have 1226 x» + 2xy + y" = i^9.26xy + x»y' ' "ubstituting (11) in this,\ wt 1225 1226-219t + 3S8xy-13xV ^^^'^'m^2exy + xy^^' i69.aoxy-i-xy 338xy - 13xV - 872 .-. 2xy » i6»^26xy4-a^y» ' ^^^^^^« <>' lifaotloMi we have 8«y- 39xy +972 » i •ttdHWtpfiiit (xy - 6)(3«y - n»y - 168) BX. UT. ax. UT.) ALOEBKA. 95 F60V8 i&d«-6 ') Vy -30 w« bftTe ■ .*. «y - 6 « (m) ; and abo 9xV - 2lxif - 162 « i 5, or i 7, or i V^ ; limilarly tnbtraoting theie from (ii), and tbea extraoting the square root « - y " i l^ or i V- J3, or i 2. Hence by addition and subtraction we-hare a? ■ i 8 ; or ± 1(7 + V^^) ; pr ± i(2 + V23) yl2; oriJ(7--/::r23); or±|(V23-2) Otherwiiei thus : Let zmv + Xf and y s « - s Then from (i) 2«*4- 6o«' » 36 (ni), and from (n) 2o> -I- 2«^ « 13 (it) Multiplying (}r) by 30, and subtracting from (lu,) we hate 40* - 390 ■ - 35. Multiplying by v, we have 4v* - 399* a - 35« Dividing by 4, o« -^ ^^M « - V^ ; add v' to each side, and 0«.*/vt. ,;«. 8^i>...»4_pas a;(t,»_i,) .♦. (©a -©)(©» + »-«/) a .•.«■- »« (r), and v' + v^^^ (n) From(T)|«aOorl; andfrem(7i), «sf or~{. But2v'4-2«'-«13 .*. z m 1^; or i iV5^; or ± J ; or i i^'^^H Then «■« + «=!£ JV2T; or j t i j or - J i V"=^ y ■ V - « B values as obtained above. ■*, ^ Kora.— The values « s O, uid s = iV26 are derived firom the v, by whieh wftmultlpUed eqaetion (it). _^ 36. By Algebra Article 106, we have Vy* ->* ^ whence « ■ 9y' ; substituting this in the 2nd equation, we got 9 W + 2y + 1) = Sey* + 64 ; or 9^* - 18y» + Oy» = 64 ; or 9y»(y«-|y + 1) » 64 .-. 3y(y - 1) « i 8; or y»- y = ± & whence y ■ i(8 ± VT55), or ^3 ± aT^^^) And « ■ »>• « 5(19 i -/T^S); or J( - 13 ± -Z^^) . 30. Multiplying the 2nd equation by x, w« have «* + xy* « «, buta;*+y*«af.'.«*f|*=«*+x/.-.y«-jt/aO, thati8y'(y-x)»,0 wboneey'mO.'. y>iO,andhtaee«» 1. Also v-'-O.*. ys>«^ whence a«* • 1, and y ■ « ■ 1^4 96 Kir TO IBs. Uf. 87. Dividing (he Ist eqofttioa hy x*, we iMre/** '^pjlf'f'^^ .'. «■ 4- ^ ■ y 4- -^ (i). AgAln diiriding Om Sod «qa»tloD % y*, we or Now Adding equations (i) and (n) togetlier, we hare -y^-j^ + ay + y = (» + 7) « 3(* + 7) .or fitraoting the cube root of eaoh| y + -r- ■ (« * t)^'* ^"^ 0) f "♦■ 7 ■** + "ir«*»** + "5r* (« + "J")^* »»d foetoring ^« + l^^««-l + ~|)«^» + -i-)V3.-.* + 7"0iOr«»«-l, ot«*iV"^. AliO»"-l+j;*V8» or*« + 3 + --|«S + ^8(lli) bj adding 3 to eacli side, .-. x + -^ >■ yz + i(/9, sinilarly by talcing 1 from each side of (m) and then tailing Vi « - ^ ■ i VV " * Then by addition, we hare * « i i{V3 + V^ + V^^JTVTJ And y + y - ^« + ^^^3 « i ^3 x VpTl . V^ + ^9 .«. y" + 3 + -J » 3 + ay», or Uking 4 from each y*- 3 ••■ .rj y y = 3i^9~^l .'. y - — = i yaya - 1. Hence by addition y « i i{V3 4- 3^9 + V3V* - l|« Alio since « + 7 ■ Oj is. unr. tCz. htr.) ALOBHEA. frr t + tT Kirftoilnf (Olf + 7 fiMtorlng 1 ftddliion 38. Bj trangpoaition ji + 5?. + -r + ~ = Vi <>' »ddio» * to each side, l;j?+* + Jil + {'r + ~)'*V; completiof the iqnare + i « 9 .*. 3 ^^-2 7 + i + l i 8 6«« *lxy .'. 2xy + 4 s -5- or - -y ; whence jcy = 8 or - A- Again by squaring the 2nd equation, we get x* + y' ■ 2xy •(• 4 Then since a' - 2xy -f y* 3 4, and 4«y « 32 or •» \\] we hare by addttio.: »a + 24fy + y* = 36 or |^ .*. x + y = ± 6 or i ^?\^/S^, and « - y s 2 .'. xs 4 or - 2; or 1 f -iW^; add similarly y « 2, or > 4, or - 1 i -jifVSa '^ 39. Prom the istequat., we get(Va"+Vy)+V*(V*+Vy) =10, completing the square (V« + Vtf) + V5(/* + Vy) + i = V .'. (V* + Vy)* + 1 V5 = ± iV5 i whence (V* + V*) - V5 or - 2V6 .'. V + Vy * 6 0' 30 (u). Taking the former of these values, and raising to the 5th power, we have x^ + 6x^* + 10x*y+10xyH5zV+y^ = 3125 '^' But x^ . ; + y'a 275 . .*. Sxly* + 10x*y+ lOxy* + 5xV = 2860 x*y" (x« + 2xy' + 2x*y + y' ) = 570 (iii; But cubing equation (i), and multiplying it by* x'y', we bare x'y'(«' + 3»»' + 3x*y + y*) = li.26xM (!▼) Subtracting (in) from (it), we hare «lyi(aryi + x^y) ; that is xy(x* + y*) = ia5x*y* - 670 But X* + y* « 6 .S 5xy « 125x*y* - 570 Hence xy - 26x'y' » - 114 .*. «"y" a 19 or 6 ' Then x -f 2x*y' + y = 25 ; and 4x'y" a 24 or 76 a 98 Kit TO (Ks. ttr. V .'. * - 3**ir* + y ■ lor -51; or**->" »i 1 or ±V- Wj and at* +y* « 6 .', »* a 8, or 2, or |(6 i ^/-6l) ; whtnoe « « »i or 4, or J(^ 18 i V"^^^ J ilmlUrly y « 4 or 9, or |< - 18 T ^"^^Tfl) B7 Qflng throaghoot the rftlae V« ->■ V^ 20, other Yalaei of * and y mvf be timilarly found. 40. Prom the let equation (x + yX*" -«y + y*)s«-y X — y .'. x' - xy + y* = jT-r» and from the 2nd eqoat. x" - oxy + y* « x-y ./x"-2xy + y* .'. by SttbtracUon, we have (a- l)xy = j^ = y g^-f axy-t-y* ^'VV^ ^/oxy-axy ^/ (o-2)xy ^/ S"^ Y%_^A . «xy + 2xy ~ I' (a + a)xy 6 b* suppose. Then y - •- , and y" * -j; 8ub8tUo.iing these valaes X' b* in the 2nd equation, we have x' + ^ « oft ; or x* - oix" « - *■ ; Whence x» « - 2 '2 a f IV*{2(« i V*?"^^)}* " i JV*(V«+a i V«^ by Algebra 6 Art. 189. And y = ~ - ± 2Vft(V« + 2 T V« - 2). See AIg«b»a Art. 181 41. From the Ist equation xy + ox -• ay - a' ; that is (x - a) (y + a) = .*. X - o = ; or x a a. Also y + a = .♦. y » - if From the 2nd eqnat., substituting x » a, we hare a-^y' + a^Bf, .'. y a i a'V - (a* + l). Again substituting - a for y in the 2nd equation, we have x + a» + a' s ; whenoe x «. - • 2(9 - a") ** « <^(9 -I- <^ «^-9i3V9-llM^+2a« _, whence «' « |y ' ' " i ana .•.* = £ tM17(«? - 9 ± 3V9-16fl^ + 2a«)l* Andy« j— ■ - VfCfla" - 3 i V^ - lBa»+ au«) 44. Ratoing the Ist equation to the 4th power, we hate X*- 4x»y + 6«y- 4xj"+ y^ » «* + y*- 2*y- 4ary(«»- 2a?y + y^ = o* Bat ** - 2«y + y" = a*, and «* + y* = 6* .-. 6*- 2**y*- 4d^y » «• ; . 20* f VSo* 4- 26* that is 2«y + ^'''^V -**-«*! whence a^ = j— Then x" - 2xy + y^ y* s - S, whence y* « IB, or 1(1 ± -/^rre), and a * ■ y* - l « 14, or 1( i V'^HS - 1) * whence y ■ i ^IB, or i V|(l 1 V-19) = *«• ' 4t. Prom the lit glren eqnat. y* - a«*y* + «* 4- 4(y* - ««) 4- 6 • a V4(f • - *•)" + S(y* - **)« ; that ie (y* - a««y« 4- x*) - a(y* - »*)V4(y* - **) 4- S 4- {4(y« - x*) 4- 8} « .•. extracting the iquare root, we have y* - «* - V4(y* - «*) + 8 « 0; or y* - «* ■ V4(»" - «*) 4- 8 .-. (y* - a?*)* - 4(y* - x«) a 6 whence y* - «* ■ S or - 1, taking y* - x* s s, we hare from thu and fiilren eqoatlon y* - 8y« + 1 « Sx« - 8x 4- 8*V»' - a» 4- 6 4- 4 «Sx*-.8x4-4xV4«'-8x4-a04-4»0x*-8x4-4xV3y*4-x*-8x4-84-4 ifnce 16 ■ 3y* - 8x*. Hence by transposition, we hare y* « 8y* 4-x* - 8x 4- S 4- 4xV3y« 4- x» - 8x 4- 8 4- 4**, and taking the square root y* ■ ± (V3y* 4- x* - 8x 4- 8 4- ax); nslng the positive sijgn, y* - ax ■ •*• •xtrMtlnf tht iqu»r0 root, we bare aty - S^f* - »• » ± 4yy" - «* '. »y ^ TViF^ or- VF^^i »»d .'. » V - «(f ■-«") or (y« -*•) From the lit glren jqnfttion «* - y* ->-4s* -f ijf* « 4»* - 13 .-. ** - 8*« + le « >♦- 4y« + 4j whence «• - 4 - i (y« - a) that ii y' « «• - a, or 6 - «• .-. y*- *• « - a ; x«y« » 49(y* -»') s>r (y> - «•); that ii «■(«' - 3) • 49( - a) or - a ; that b »* - a»« - - »8 or - a ; whence «• - I i V"=T7 or I 1 V^ and y'«x'-a«-li ^p^^t or - 1 i V*"^* -^Im >inee y* » 6 - X*, we hare by inbititntion x*(6 - x*) « 49(6 - ax') or 6 - a«< ; that ii «« - 6x* » 98«' - 394, or - ax* - 6 that is x« - 104x« « - 394, whence x* « 83 i VaUo ; or x«- 8x« » - 6, whence x' « 4 i '^. And y« « 6 - x* » - 46 7 -^19 or a T /TS i whence x s 4c. EznoiM LY. -1 1. Let * s one part, then 19 - x a other, and x(19 » x) a 84 361 — 336 *« - I9x a - 84, x« - 19x + (V)« = -^ « V .-. « - V^« i I X s 13 or Y ; whence the numben are 13 and 7 2. Let X 3 greater, then 17 - x » less ; x - (17 - x) « ax - 17 = diflferencei then r(ax -^ 17) » 80, 3x' - 17x » 80, whence X X 10, and the numbers are 10 anil 7 3. Let X s length, then x - 13 = breadth, and x(x > 12) » 2080 that is X* > 12x s 3080, whence x s 62 and sides aro 63 and 40 rods. 4. Let X s greater, then x > 9 s the less, and x' + (x - 9)' « 353 that is ax* - 18x •(• 81 « 353; x' - 9x s 136, whence x b 17 or-8 and the nnmbera are 17 and 8, or - 8 and - 17 loi ur 10 itx. tT. 5. Ltt « ■ oat purt| tbtn le— a ■ otter, thtn 9(16— a) ^ «* 4- (16 - «)* ■ a08r that is «* - ie« « - 46,. wteB«e« « 19 or 4, Mid the Bimiben mo 13 and 4 a 6. Let » ■ gain per cent, s bajlng price of wlieot ; then -^ m gain per dollar on bajing priee, and x x Yqq * Jqq ■ g^tn on a dollar!, i. e. gain on whole trantaetion ; bat 1 1 1 - x a whole gain, x' whence jg^ " 1^1 - 'i or «' + 100« « ITIOO, whence a * $90, bajing priee of wheat. FBoov. 681 + t90=6in:alaoifhegaln681oii600bhefaii 100 i whence x -f x >■ 10 (it). Redacing (in), we hare 99x - 99x a 694 , or x - « s 6 (r). Adding (it) and (t) together, we haTC 2x « 16; whence x s 8. Alio x + x « 10 .'. 2 a 2, and y' « 2xy + 4 s 32 4- 4 « 36 i whence y a 6. Hence the required namber is 862 240 9. Let X s number of sheep boaght, then -^ a price per sheep. 240 + 1 600 4- X 0*. selling price; then j(x-16) » 108; X - 15 a number sold, and /240 2\ (X - 16) {jr^T}" 2**' *^- " V ^ or X* 4- 4Sx s 9000 ; whence x s t5, and *^ « $«;*20. Hence number bonght was 76, and price per sheep $3*20 10. Let a " one number, then 10 - x s other, and x*4' (10 - x)", that is X* 4- 1000 - 300X 4- SOx' - x" » 280, that is 30x« - 300x Vtx.tr. • 13 or 4, 100 ItAn on X liole ff«in, « * $90, Bx. tv.j ^Wl»|u. 108 •«»<»^-*-*), that I. aJ!:.' ' •^••'' -- •(34 . .) or .• . 94, . - a40. wb.o L/. Vo"oT ^ ''^ " '^^ ' »<>• '^y »*'-y, Whence iilmllV ''"^^''p^. »^2*-y^y...,../.,'';^'''>*y*y.Ki^y);or ''^••l io 7«^,| then H? ^. 120 ' "^ «""«** of f«, ^- ^^-. lao ,.^. ;"'"5"-''^-'«"-«--eb,..e, 120 120 rf« * F-M 20 20 7 "M(Objdirldln|fb/e an r ^ « -— - fl -x 20 20 Then * y n T • T -n( 120 120 >i or 9 n - •«« of Uirtr «»ip,o«l. . J. . >» • W-' 104 KEY TO (Ex. LT I 436«« - 841* « - 848 ; or «« - f Ji * - H* ; *" - tti* + (IH)* - ?«M* - mm « miJ^, ▼hwce * mh i »« - w- or m » jorf, a0dff-» = ?»-il^ = ^'V « I; or f| - f ^ « H « |. Heace ftactions are i and | ^> 46800 15. Let X = nnmber of children, then — - — = share of each ; 24 then 46600 46800 + 1950, that is 24 ■¥ 1; whence X - 2 X ■ ' *- 2 x' - 2x = 48 .*. a; = 8 a number of children. 16. Let X » number of hours the clock is too fast, then, since the shadow on the dial moves from 1 to 6, the clock will strike the hours from 2 + x to 5 + x inclusive ; i. e. will suike 2 + x + 3 + x + 44-x + 6 + x=14-f 4x strokes, and last stroke will be 5 + X. Then (5 + x)* - 41 = number of minutes the clock is too fast above the x hours ; i. e. 25 + ||0x + x* - 41 ; i. e. r* + lOx - 16. Bat hours too fast + liii inutes too fast s whole number of strokes ; that is x + x« + lOx - 16 = 14 + 4x, whence x« + Tx = 30 .•. x = 3 or 10, and x» + lOx - 16 » 9 + 30 •> 16 s 23 .-. the clock is 3h. 23 m. too fast. The second answer 10 is excluded by the limitation that the clock does not strike 12 ^uring the time. 17. Let X = hours travelled by each - miles per hour travelled by slower, then x + 3 =; miles per hour travelled by faster; X' + x(x + 3) = 2x' + 3x a 324, whence x = 12. Hence slower travelled 12 x 12 » 144 miles, and the faster 12 x 15 » 180 miles. 144 . , . 144 , 144 18. Let X = number, then — = share of each .*. ^^j "•" ^ * "x~ whence x' + 2x = 288 .*. x = 16 ^ number at first. 19. Let X 3 left hand, and y = right band digit, then 10x+ y « 2 (i), and lOx + y + 2t = lOy + x (ii). Prom (i) lOx + y = 2xy (ui) Prom (ii) 9x - 9y = - 27, whence x-ya-3, orx = y-3; substituting this in (iii), we have 10(y - 3) + y s 2y(y - 3) .'. 2y' - I7y « - 30, whence y = 6, and x = y - 3 = 3, Hence the required number is 30 m; Ex. I.T.] ALQBBBA. 105 20. Let X = price of eoilbe, and y - price of sogar per lb. In 800 cts. ; then 60« + 80y s 2600 (i). Alio — — ^ Ibe. of f ag»r for $8, 1000 . ^ .800 1000 and = lbs. of coflfee for $10 : then -— - « —-- + 34 (n) X ' y X ^ 12S-4y Prom (ii) by redaction lOOx- 126y = 3xy (in). Prom (i) x » — ^ — , 100(126 - 4y) substitnting this for x in (lu), we hare 5 126jp (126 - 4«) - 3y X 5 , whence by redaction Cy* - 5^6y a ^ 6250 3 .'. y = 12) cents, and x s 126 ~ 4y 3 s 26 cents. 21. Let X and y = number of days reqnired by B and C respcctiTcly to finish the work; then in 1 day jt does -^th; 1 1 11 36 x + 18 B,-th; and C,yth of the field] -Ar+J : - :: $36 ■-*-j^ 648 = " • : iQ - what B would have jreceired, had C not been called X + 18 ' 10 360 in; but B worked 10 days .*. he did receire — x 36 = — - Then / 648 \^x + 1! 36d\ jg - -j^ J dollars = $1'60 « $J ; whence by reduc- tion x« - ll4x = - 4320 .'. x« - lt4x + (87)« 3 ^669 - 4820 » 8249 .-. X s 30 K days B would require. And +J + i8 + — « 1 9 4 .-. -7 = 1 - (H + iH) - 1 - if ■ ft^ •*• 2y « *2, and y = 36 = days y C would require tocradle the field. pRoor.r-lf C had not been oallod in, thej would have taken lU days to finish the work, and A'» share would have been 88 x 11| = 82260. Hence B'» share would have been 818*60, but since, when C is ealled in, B only works 10 days, be receives only ^3 = } of 836 = 812 = 81*60 less than be would have otherwise received. 22. Let X and y = the number of feet in the side of the base ; then 6xy - 4xy » xy s 80 4- x -K y (i) ; also ^26 + x* + y* ° ■M''* + y*)(«). Pf om (11), we get 3(x« + y'> « 40^26 + «* + y« 106. or TO [Ex. LT. That if (»« + y» + 26) - VV«* + y* + 26 » 26 («» + y« + 26) - i^(»* +y*+ 26)* + T -AJtt + ag& = Ap ... V«* + y* + 26 - V « ± y .-. V«" + V* + 26 = 16 or - f ; Bqaaring, we hare »> + y' + 26 » 226 or ^^ .*. taking the fbrmer valae, «* + y* « 200 (in), but bj (i) 2«y = 160 + 2(x + y) (it): Adding (iv) and (m), we have a?» + 2xy + y* = 360 + 2(a! + y). Hence (a: + y)* - 2(x + y) « 360 .-. (x + y)« - 2(a? + y) + 1 « 361 ; (x 4- y) - 1 B i 19, « + y s 20 or - 18 .-. again taking the former ralae, x. + y » 20, and hence xy s 80 -l* 20 « 100 x« + 2xy + y' = 400 ^ X s y .'. 2x s 20, and X a 10 s y the base li a square whose side is 10 ft 4xy » 400 / " « - 2xy + y» » f .*. X - y » J " 23. Let X s distMce B has travelled when he meets jS, then X +16 s distance «4 has travelled; Also since ji has yet to travel x miles, and accomplishes it in 2 hours, his rate of X X f 16 2x4-30 travelling is y "»»1«* P«' J»onr ; also JTs rate is -'-rj — s — - — X + 15 2x 4- 30 Then time A travels before they meet s hx time B toavels before they meet - X 9x 2x 4- 30 2x 4- 30' 2x4-30 9x 9 2x4-30' that is X* - 24x = 180 ; whence x s 30 = rkte Hence distance = x4-x4-16s76 miles. B'b rate = 2x4-30 90 9 9 « 10 miles per hour. A^b rate « -^ s 15 miles per hour. 24. Let X and xy be the two numbers, the latter being the greater ; then x'y = x«y* - x«, whence y' - y = 1, and y *= j(l ± V*)« -A^lso x«y« 4- X* a xy - x" .'. y» 4- 1 « xy" - X ; «w„,.,.?!±i.*(3iV5)+l J(6±V5) }V5(V5fl) ^ whencex-y3_, J « (TTvePT ' ~iTV6~ ' liV6 ^'iiVS Ex. LV.] ALQBBftA. 107 25. Let X and y & hours required bj Baochni «nd Silenuf respectively ; tlien Baeohni would drink -~- of it in 1 hour, y lience in y hours he would drink ---ths of it, and in }y hours 2tf 3« Bacchus would drink :^th8 of the cask full .*. 1 - ^z ' PArt 3«' Zx drunk by Silenus, and since he drinks — - of the cask in 1 hour, / 2y\ 1 the time he required to drink part remaining, was ( ^ ' §: ] "^ 'IT 2y« c y - -=— . Had both drunk together, Bacchus woutd only hare / 2y\ 9 consumed Ml'~o~)~i~3^> <^>'^^ Silenus would have taken y \ ^ oZ] hence when drinking together, time taken by Bacchus / y\ \ X y ^*' (^""Sx/* — "'T'"'a'» ****^ *^* ^™® taken by Silenus 2y / 2y«\ Also ~o "^ [y " "Sx I ~ ^^^^ taken when drinking separately X v = 2" - "T + 2 (u). From (i) 3x* - 5yx = 2y*. From (a) By 25y' 25v* 2v' 12xy - 4y« = 3a;« + 12* (m) «* - J* + -gf- = "g- + "f- 49y^ 5y 7y 12y ~ "sis' ' * " T ~ * "6"' ^®'**'* * ~ "e" ~ ^y » substituting this value of X in (ni), we have 24y* - 4y» « 3(2y)« + 12(2y), that is 24y« - 4y« = 12y» + 24y .-. 8y» - 24y = ; y* = 3y ; y - 3, whence a; = 6 108 KBT V> (Ex. hn. ExtROiBi LYI. ab 1 (g-5)(g-t-3) (X + 2)(J - 1) (X + iXx + 6) an- 7 ^- (x-6)(x + 2) ** is greater than 2a6 (Algebra Art. 134, Note 2) •*• a« + 6« -^ a + 6 + y» > (« + y)' r^ < X* - x»y + x«y* - xy» + y*» »««0»din» »k x6 - x''y + 2x^« - 2xy + 2x»y* - xy» + y« ^ x« + 4x''y + 6«*y» - 5xV-4xy» -y» ; or as txV+3xy« + 2y< ^6x»y+2xV+8iB*y»; or as Tx'y" + 3xy* + 2y» ^ 5x» + 2x»y« + 3x*y ; or a8 y»(7x« + 3xy + 2y«) ^ x'(6x« + 3xy + 2y«) i Now since x^5 > y^T, cubing we hare 6x' > V »*•*•> ^ .-. y'(3xy + 2y*) < x»(3xy + 2y') ; also 7x V < 5x' •/ 7y* < 6** /. y»(Yx« + 3xy + 2y«) < x«(6x« + 3xy + 2y'' ) x' +y' x« -y* < -i (x + y)< X* - x"y + x'y* - xy" + y* 6. Let X = the quantity to be subtracted from eaeb term; then a-x c 6-x ~ rf (h{.(2xs6c*cx: cx-dxs:bc-ad,:xi Ex. tn, tTii.] ALOKBKA. 109 7. Let X « the qnantitj to be added to etch term; tben m-¥ae TTg" l|Wheiiee« + ««n + »j x-xsm-n', x(l-l) = iii-ii; Wl •■ Jl III «• fl X B 1 - 1 = oc (g + ft) /tt« tt«\ y (a + b) /a« \ »» (a + i) o« - i X — rs — X 6« (o + 6)» 'a+b ' b : a + b — ( H c)* a* + 2ac + «* y. Sinoe a : « :: c : 6, c « Vo* ; then jjp;:^ « PTWTP * i« + 26V35 + fl* * 6(* + 2Va6 + «)'*"**' a* w^' <^ a — 6 + «&• - fr" J or 01 o*ft - «i6« ^ oft* - o«ft J or M 2o*6 ^ 2o6* ; or Ai ^ ft Ensoul LVII. 1. Letxatheqaantity tobeadded; theno4-x:ft-fx::c+x^:C» 100 equals by equals ^^ - j^i > ^I^'b Algebra Art. ma* - 2nb* pe* - 2qd* . . . ' ma* -f 2nb* " j>c« + 2grf« •'• "»«' " *'»**'^' " %qd*'.:mi,^ + 2n6« :|»c« + 2qi* 4. Let X s one number, then » other And ^'(jf ' ("Tf'-'-^^'-^ Hence x» - ^"y = 19|a:» - 72* + ^ - ^H^)*! . /24V - 82882 /24\* xs- (^-j = »9*'- 18e8x + -j~ - J9(^-j /24\' . 32882 18f ~j - 18«" = ' ^ - 1368X /24V _« "24 ^^ /24\» ■ ^ /24 \ 24 /34\« Dividing each side by •— - x, we have f — • J + 24 + *• « 76, 676 ^ ' thati9^ + x« = 5a; x* - 62x« = - 676 ; x*-82» +(26)«- 100; X* = 26 i 10 = 36 or 16 .*. x s f 6 or i 4, and the nombers are i6andj:4 6. Let X = one part, then 20 - x s other part ; then X : 20 - X :r 9 : 1 .'. X = 180 - 9x; or * ■ 18, and 20 - ar a 2 Let y be the mean proportional between these ; then 18 : y :: y : 2 } or y' = 36 .•. y « 6 a' c + x X e -¥ X X a* a ^c + X 6. — = Fa i *i80 -^ = ..—^=^ ,'. y rd: : xy : p « x+y , rf «-y , Mid — = xy ' j> xy •. K* + y) ■ f*y» f^t^dpi* - y) = ^^y •*• ^ ■ « 1 1 ^ • + << 2 . — a — + — J By Addition « -^j whence y = P d y Ilk 2 ~ = y - — jBy 8ubtWM5tion ~ 2P • -d . 2|» -^-— , whenee x « ^ — 3 9. Let X s ipecd in yards of ^ter train per second, and y = speed of slower ; then in 2" the former passes orer 2x, and the latter 2y yards, flbnseqaently 2x + 2y » length of the faster train ; also 36x- 30y » length of faster train, .•. 30x - 30y = 2x + 2y, or 28x » 32y, or tx « 8y, .*. * : y :: 8 : t 10. Let X s J's money, and y = Fs ; then x + 150 : y - 50 : : 3: 2, whence 2x + 300 = 3y - 160 ; or 2x - 3y = - 450 (i). Also X - 50 : y + 100 :: 5 : 9,^ whenee 9x - 450 « 5y + 600 j or 9x - 5y a 950 (n). Maltiplying (i) by 9, and (u) by 2, we Lare \9x - 27y 3 - 4050 (in)l ' ISabtraeting (ui) from (it) I7y » 5950, 18ar-10ya 1900 (i?)J "^ ' v / jr 1 whence y « $350 = JPa stock ; 2x ~ 3y s 2x - 1060 s . 450, whence 2x s GOO, and x * $300 3 JTs stock. 11. b = ^/ae .-. 6* = flc; 26*= 2ac; 6'= 2(ic - 6» .♦. adding a^ + c» to each, «^ + 6» -h c» = a» + 2ac + c* - 6'; or o» + 6» + c* (fl + c - 6)(a + c + 6) = (a + c)«-i»=(a + c-6)(a + c + 6).-. 1 = ^^ at+f^ + ^ 1 a-i-e-b a + e + b a-b + e °' a + e + b ' <^ + 6« + c*' ®' (a + c + 6)» * a? + 6» + c> .-. a + 6 + c : (a + 6 + c)» :: a - 6 + c : a« + 6* + c* 112 TO Vtx. vm. B t ill ! , 13. a: c i: a: c, .•. malUpJIjiog etch term of the Utter ratio by a - e, we hft?e a : c :: «i{a-e) : e(a - c) .'.arc : : a('\/a - i/eXi/a + VO : «(V<» - V«)(V« + V*) /.arc:: V + m«j> + |s2 III 3 9 iL 4. X* « my"; 4 « 64fli .*. »» « At ** = iV^* •'• * * Wv 5. X a m 4- nxy .*. 2 « nt •(- 6n (i), and 3 « m •^ 9i» (ii) Subtracting (i) from (ii), we have is. isn .•. n « ~ -^ m--)'a = 2 .*. »= 2+-^* V" Then x - nxy = to ; x(l -ny)sm TO -^ V 36 X = 1-ny l-(-T^y) 15+ y 15 15 + y 6. y = w + nx + px'7 then s m -f 3n + 9p (i) - 12 = TO + 5n + 25p (n), and - 32 = to + 7n + 49p (m) Subtracting (i) from (u), yre get - 12 « 2n + IGp (it) " (i)from(m), " -32 8 4n + 40p (r) 114 XBT TO [£Z. LTUI DiTidiog (ir) bj 3, Md (▼) hj 4, we bave - 6 ■ n -i- 8p (ti) * . 8 ■ M + lfl|p (Til) Subtnctlng (ri) from (rii), we b»Te - 2 ■ 2ji .*. p > - 1 ; - 6 • n - 8 .'. n ■ 2 ; <■ IN 4- 6 - 9 .'. »i <■ 3 /. y « 3 4- 2« - «' n n n 7. y " mx» + — ; then 1 « 26iii + y (0; B ■ 81w + -^ (ii) ♦ 2SA n Diridiof (ii) by 6, and (i> by 9, we hare i ■ "J" + 45 ("')» 8lM n 2 604m and I ■ -T- + ?r (ir). Sabtraoting (in) from (it) -r- ■ -^g- n S«* 9045 .M»-7iii.'.n.86-126*i-32Hi; **»«ny*«' + -i-"802'*'802i a* 1 •* 8. y ■ «6* ♦ m*« .•. -r- ■ m&* + «ia« - mb* ■ «io« .♦. y « m b* «• »• 9. * - » - y ■ >«, and (« + y + £)(* - y - «) « ny« ; that is \*^(y-¥ «)}{« - (y + «)} or «• - (y + «)" ■ ny«. Adding 4y« to each tide, «• - (y - «)• ■ (n + 4)yaf ; that is (x - y -t- z)(x + y - a) » (» + 4)y« ; but «-«-y«wi.*. » + y-«s-m.*. - m(x - y + «) n4-4 ■ (» + 4)y« /. « - y + s « — ^tf* ••. X - y + « oc y* 10. Let x' ■ number of cars attached, then decrease of .speed oc X*, and is .*. ■ mx' ; then 24 - mx s speed of train, .*. 20 ■ 24 - 2m .*. 2m « 4, or m c 2 ; then 24 > 2x* ■ speed when X* waggons are attached. Now if. speed is reduced to 0, we hare 24 ~ 2x* ■ .•. x » 12, and .*. x' « 144 « nnmbei'^.of cars required to completely stop the train, .*. greatest number it cm more « 143 [Ex. vmi j» - - I J 2«-«' » + -g (") + ^ (t«). 2 eOAm •9 ■ "46" 6x* 9945 ■ 802* 802* t ...... m nyz', tbat is Lddlng 4yx to re»teof*peed >eed of traiO) ipeedwben aced to 0, ve nmbei^ of cars number it cab JoamtLk. Ixnoiu LIZ. 115 1. S,,«{ia6 + (81-l)3)V 126 -I- 60 X 81 > S888 5.= {l26+(il-l)a}-r-»(ia6 + 2»-2)"r- n 124 4- in xn»n(/ii^n) 2. 5,, • { - 400 + (22 - 1) X 12}V « ( -400 + 2S2)11 • - 1028 n (-400+12n~12) X n « fi(6ii - 200) 2m + p 5, = l-400 + (i.-l)xl2}j 3. 5iT « {4 •!■ (17 - ])|)V » (4 -I- 24)1/ - 238 5,.o - {4 + (3* + ji - 1)1}— j-^ ■ » + (a« + ^)l)^ 4. «8,.*(|+(ii-i)x-|}V"a-V)V— VxV— ^I'—aoi 5. nui « 2 -I- (17 - 1)3 - 2 + 16 X 8 » a -I- 48 - 60 28»> « 2 •!■ (28 - 1)3 a 2 ■(• 27 X 3 « 2 -I- 81 - 83 nu> a 2 + (n - 1)3 « 2 + 3n - 3 « 3a - 1 6. 17» » 3 + (17 - 1) X - 6 « 3 + 16 X - 5 « 3 - 80 « - 77 28'» s 3 -t- (28 - 1) X - 5 a 3 + 27 X - 6 « 3 - 135 - - 132 nti>s3-l-(n-l)x>5a3>6ii-l-5s8-5n 7. 17* « 21 + (17 - 1)4 ■ t + 16 X ? « f + y ■ -W « IS/'^ 28*» « 2i -I- (28 - 1)? " i •!• 27 X f « f 4- ^?A » 2IH ntfcs21 + (»-l)Hf + 5»-4"t-4 + 4»»U+^« = A(5 + J»») 33 — 3 B. dm -g-jTX a '^ e 7J ; hciico series = 3 + lOJ + 18 + 25i + 33 hence series 9. d' --66-9 6-1 -Va-16] - 51 - 66 10. d = 100 - ( - 1) 100 + 1 9-1 8 s 9 - 6 - 21 - 36 = x^L s i2{ ; henco series s - 1 + Hi 4- 24i + 36| + 401 f 62i -I- 74| + 871 + 100 11. JT,, « {2 -I- (73 - 1)1JV «= (2 + I2y^ * 37 X 73 = 2701 12. n«*tcrm=l + (n-l)2sH-an-2 = 2»-l \ 116 ti^. u* \m •' II 13. i\ » {a + («- i)ajj - (2 4- an - a)y ■ a» x j ■ »• 14. 8t m [2a •(■(/- l)aa) J - {2a -f ao/ - 3a)y » aal x ~ » fU^ 10. aOtk term <■ (1 4- (30 - 03 ' « a + 19 x 3a ■ a -f 38a « 80a l>k ttMi - tt 4- (< ' l)3a 0/ 3a«3a<-a«a(ai-l) 16. Let « - 3y, « ~ 7, :: v y, x t- 8y copreient the numberi Then (* - ry)» r ;x v)« - a«» + X8y« - 300 (»- /' + («+ y)«« 3«*4- ay*«iaf6 .-. iey'« 64 Henee y' - 4 or y « i 3 .•. 3«' * 136 - 3y' ■ 136 - 8 « 1|8 ; or «- « 64 ; or x « i 8 .*. the lerlet Ii i 14 i 10 j; 6 i 3 ^ IT. Let X - 8y, X - y, X -f yi X •!■ By represent the nambera ; then (X - 3i^)(x - y)(x + y)(x + 3y) - (x« - 9y«)(*« - y«) = (x« - 36)(x« - 4)* ; X* - 40x« + 144 « 1680, or x* - 40x« ■ 1636 ; X* - 40x" + 400 « 1936 .-. x* - 30 « i 44 .-. x" « + 64, or - 34 Rejecting the latter ralue, we hare x' ■ 64| oi^ a^ ■ i 8; hence the lerlei is i 14 ± 10 f 6 i 3 18. Let « - 3y, x - y, x, x + y, x 4- 2y represent the nnmbers, then x-3y + x-y + x + x + y + x + 3y«6x«36 .*. x«6 (X - 3y)(x - y)(x + y)(« + 3y)i « (x« - 4y«)(x» - y«)a: » 5(35 - 4y<)(25 - y«) » 6(4y« - 135y« + 635) « 945, or 4y« - 128y» « - 436 ; y< - H*»"+ (^l*)» - *S1* •*• V* - ^1* * ± V whence, y* « 4 and y « i 3. Hence the series is I. 3, Q, Y, 9 or S,?,r,3,l t. ' s (a + :, . » (60 + 1)^« B 61 X 30 = 1830, i. e. since the principal on interest is $60 the first daj, and only $1 the 60th dny, the whole interest is equiralent to that of $ 1830 for 1 dajT; Interest of $60 for 360 days = $3*60, or of $1 for 360 dAjs » $0*06, or of $1 for 1 day = jj^ s ^^ of a cent j hence tae Inter'^st of $1 for • Thseomaon difbrenoe is giren ss4 .*. y =9. i. 0. since the ^T Addition ) + 8ii| + . "'=3(1' + 2.+ . „**•'-'"(-» -"«-" i« question 30, ti.»t : r,' Ta" '' " '' ^ "^" Tberefofei,«.3^.3nr« + i) 3 + « 118 KEY TO [Ex. LIZ. m 1 1 aod iuTerting the series 35, 32, 29, 26, Ac , w« have 2, 5, 8, 11, 14, 11, 20, 23, 26, 29, 32, 35 24. / = 5, and I- 2d = 7 .'. the sixth term or last term but Y+ 6 ■ one = l-d = —z— = 6, and the series is found by reversing the series 5, 6, 1, 8, 9, 10, 11 26. S = lm+en^=(b-¥ m)n = (26 + 2«»)y a (26 + 2 1)<2}. v^^lso the m^ term = a + (m - !)i. Therefore, &c. 27. (t)+o)*»'term = a+Cp+o-l)(iam'1 . m-n vu V. ^.•.23 - l)d, when^ a = J^/, and d = - i. Then sum of n terms a S s [2a + (n - iMIy .n n n /27 n\ n . |13 + (« - 1) X - }}- = (18 - Y + J)y- (y - yJY i! n i Ex. IXX, LX.] ALGBBRA. 119 29. Let X > y, x and X +2/ = the numbers; thcn(x-yy+x(x+y) = x»- 2xy + 5'+x'+ xy = 2x'- xy +y»a 16, and x*+ (x - y)(x + y) = x' + x* > y> = 2x' - y' = 14. Subtracting tbe second from the 2y*-2 first, 2^* - xy = 2, or X = y Substituting this for x in the 676 i _ v» equation 2x' - y* » 14, we have ^ or 7y* - SOy* = - 8, 196y* - 840j/« + 900 = - 224 + 9U0 Uy^- 30 = ± 26 ; 14y* = 66 or 4, y* = 4 or f ; rejecting this latter 8—2 6 value we have y * i 2. Hence x = — — - = — = ± 3, and the three numbers are 1,3 and 5, or - 5, - H and - 1 30. Let X - 3y, X - y, X + y and x + 3y represent the numbers ; then x-3y + x-y + x + y + x + 3y = 4x = 20.'. x=5; 1 1 1 1 4x» - 20x»« a: - 3y ■*■ X - y **" X + y ^ X + 3y * x* - 10x*|^ + 9y* ~ ** .-. 25(625 - 250y« + Sy*) = 24(600 - 100y») ; or 9y* - 154y» =» _ 145 ; 324y* - 6644y» + 23716 = - 5220 + 23716 = 18496 .\ 18y«-154 = + 136; 18y» = 290 or 18 .'.f=l or if*, and y = i 1 OT±yTis Rejecting the latter value, we have 6 ^F 3, 5 ? 1, 5 f 1 and 6^3; that is 2, 4, 6 and 8 or 8, 6, 4 and 2 for the suries. EXIBCIBI LX. '^'6 = 1. 6"»^term = 3 x 3" = 3 x 243 = 729; 5(3<» - 1) 3 X (729-1) ~ 2 1092 3-1 2. 9»>» term « 1 x 2« « 1 x 256 = 256 ; 1(2» - 1) 612 - 1 ^9- \.i '" -^- = 511 3. 7* term « | x 2« « ^ x 64 « ^f* s 18f j *7 = ^^fj^ = ?x(128-l)»36f 120 KMTt TO [£x. LZ. 4. 12"» term = 3 x ( - 2)«V= 3 x - 2048 = - 6144 S 18 3{(-2)»-l} 3(4096 - 1) -2-1 -3 409S 6. 6*tenn = 4x(-J)»= 4 x - ?Jf 4 = - VcV =" 12,?<» 1A«C8 — 40416 1084 -i-1 -« -it -It -8.= 6. 8'»» term = 30 x ( - J)» = SO x - ji^ s - -1% s - 30{l-(O'} «i4 I 1 + J = L27A 3 ». 5r„ = -u i-(-§) -♦! -l-IJ. 8. S„ = 1-1 " I = f=U i S ! ¥m i ,.;; i 9. S^ a 10. s« 11. s„ = 12. 5^ « i-(-j)~ 1+i ~ a 64 64 = — = lA = 41 64 l-(-J)" 1 + 1 " i AJ* = 421 Jii. '.Ar Toinr ■AV m= 1- Ti^inr 1000 - I ~ JSA9^ ^ ili A 1 - l^J " Vtf 13. 5«. s .,ftr + oc l-TilT Tcooinr lUOU IV ~ TT + ¥jf5' - 5SC 14. s^^m*^^"m*^-mu,^%- uyuo 15. 5. = 1(3* -1) 3* - 1 3-1 " 2 --1(3*-1) 16. S, 2{(-|)"-l> 2(1 - ( - D"| -1-1 = ¥{i-(-D"l 17. Sio« 2[(V2)»» - 1| 2(32 - 1) 62 62V2 + 62 i62(l+V2) V2-1 V2 - 1 V2 - 1 2-1 18. S. = aVj(a«)'» - 1 } qP(a*« - 1) gW^p^gP a«-l a«-l o«-l [£x. LZ. = - 6144; = -i2,V«; H«««-4rt96 1»<4 -» ^ r^-iS; £x. LX.] ALOBB&A. 121 -m 62V2 + 62 19. r = ^yj = (If)* = i .-. series = 1 + ! + J + /f + i? 20. r = (Aai38)'^ = (65Cl)* = 3 .-. series = 2 + 6 + 18 + 64 + 162 + 480 + 1458 + 4374 + 13122 21. r = f-~\ = (^S)^ = J ••• "eries = 9 + 3 + l + i+J 22. Let X, ary, xy' and arj^ represent the four numbers ; then X + x^a = x(l + y') = 148, and xy + xy» = *y(l + y») = 888 ,148 . 888 148 888 .-. 1 + «2 _ and 1 + y» = — .-. — = .'. 148y = 888 * xy X xy 148 .-. y = 6 ; then 1 + y" = 1 + 36 = 37 = .-. 37x = 148 .*. x = 4, and the series is 4, 24, 144 and 864 23. Let X, xy, xy' and xy" represent the numbers ; then x + xy = 15 (i), and xy» + xy« = y'(x + xy) = 60 (n). Diriding (ii) by (i) we have y's 4 .'. y « ± 2, and since x(l + y) = 15, we have X = I/, or :^( = 6 or - 15 ; hence the numbers are 5, 10, 20 and 40, or - 15, 30, - 60 and 120 24. Let xy*, ay and x represent the number of dollars they severally had ; then xy'= x + 135 (i), and xy" + xy + x = 315 (n) 135 .-. xy + 2x a 180 (III). From (i) x = , ^ . , and from (lu) 180 135 180 3 4 , X = .*. — 5 r «= — ^- ,*. — 5 T = .*. 4ir - 3v = 10, y+2y«-l y+2y'-l y+2^ ^ ' . , 180 ,.„ 180 whence y = 2 or - 5 i hence x » ^77-5 » ^f " « 45, or x = „-- r 180 y+ 2 2-5 -r- = 240 ; hence the shares W(>re $180, $90 and $45. Taking the negative value u above, gives us x = 9240, and the shai^ would bo 9876, - 9800 and 9246, whloh implies that the second leeeives S300 less than nothing for his sluire, pr in other words, instead of receiv- lug anything he gives 9300 to be divided in addition to the IW5 among the other two. 122 KBT TO (Kx. tx. m it mi 25. Let X = the first number, and y a the common rutio of the Ist three numbers ; then the numbers are x, xy, xy\ xy'-f a^, and xy" + 2xy And xy + xy' + (xy* + xy) + (xy* + 2xy) = 3xy* + 4xy » 40 (i) ; also xy(xy' + 2xy) = xV + 2xV = 64 (ii). Multiplying (i) by xy and (n) by 3, and subtracting, we have 2xV = 192 - 40xy .-. xV + 20xy = 96 j whence xy = 4 or - 24. Prom (i) xy(3y + 4) 40 = 40 .-. 3y + 4 = — = V = 10 •*• 3y * 6, and y = 2 .*. x s 2 ; hence the numbers are -2, 4^ 8, 12 and 16 26. S = o + (a + h)r + (a + 26)r* + [a ■¥ \n- l)6}r*' * Sr = or + (tt + ft)r'+ {a + (n - 2)6}r" ' > + (a + (n - l)6}r* S>- Sr= a + hr + br* + br'+ 6r"-i - Jo H- (n - l)6jr» S(l-r) = o + 6r(l rn-I - (a + (n - l)6}r'' 5 = 1-r 6,»( 1 _ y" - X) {« + (n - l)5}r» 1-r (l-r)» a-{a + (n-l)ft}r» 1-r ftr(l-r"-») (l-r)V 2Y. (i)a« + 6» + c*-(a-6 + c)«= o«+ 6«+ c*- (o»+ 6«f e«) + 2a6 + 26c -^ 2ac = 2a6 + 26c - 2ac = 2a6 + 26c - 26', (since ac = 6') = 26(a + c - 6). Now (a + c)' - 6' » a' + 2ac + c» - 6« = a' + 26' + c' - 6' = a' + 6' + c' a positive quantity .'. (a + c)' > 6', and .'.a + c > 6, and .*. a + c - 6 is a positive quantity, and .-. 26(a + c - 6) is positive, .-. a« + 6» + c'- (a - 6 + c)' is poiltive, .-. a' + 6' + €2 > (a - 6 + c)» ^ (n) (a + 6 + c + rf)»=(a + 6)'+(c + rf)' + 2(a + 6)(c + d)j but o 6 c a +6 6-f c c+d (o + 6)(c + d) (6 + cY bd .-. (a + 6)(c + rf) = ^(6 + c)*- (6 + c)« •.• 6d « c* .-. (a + 6 + c + d)' = (a + 6)' + (c + d )' + 2(6 + c)« Ex. tx, tzi.] ALQBBBA. 128 28. (I) (p + 9)«« term = (»•»♦«-» = i» (P - tf)* term = ar*'^-^ = n .'. oV""' = m»», .'. ar'"* « V"*" "P*^ t*™" (^)'^."^^=(v)' .♦. the ^«» term « a»^-» = — j^^— = jy = »» x jj: = «»( ~ 1 29. Let X, x^ and xy* represent the numbers ; then x + xy + xy' 3 8ft, and xy : xy» - x :: 2 : 3 .-. y : y* - 1 :: 2 : 3, or 3y « 2y' - 2 .-. 2y* - 3y « 2, whence y = 2 or - J .*. X + xy + xy" a X + 2x + 4x a 7x = 35 .-. X - 6 ; or x + xy + xy' K X - |x •(■ |x s )x s 35 .'. X s i|a s 463, hence the numbers are 6, 10 and 20 ; or 46!, - 23} and 11} 30. Let X, xy and xy* represent the digits ; then lOOx + lOxy + xy' a the number, and j; + xy + xy' « sum of its digits ; then lOOx + lOxy + xy" : X + xy + xy" :: 124 : 7, that is 100 + lOy + y" 100 + lOy + y" : 1 + y + y' :: 124 : 7, whence 124 . 99+9y llf 1 + y + y* ll+y 13 1 + y + y* ~ 7" .'. 77 + 7y a 13 + 13y + 13y', or 13y» + 6y = 64 ; whence y = 2. Also lOOx + lOxy + xy'+ 594 = 100xy'+ lOxy + x ; or 99x - 99xy' ■ - 694 } or X - xy* = - 6 ; or x - 4x = - 6 ; or - 3x = - 6 .*. x = 2, hence lOOx + lOxy + xy* = 248, the number required. SXBROISB LXI. 1. (i) J.S. > 7, 5, 3 ; hence d = -2 .-. 13, 11, 9, 7, 5, 3, 1, -1,-3 inverted, give H.S. -jV, iVi i, h h 1, 1, - 1| - i (II) A.8. ■ 18, 14, 10 ; hence << « - 4 .-. 30, 26, 22, 18, 14, 10, 6, 2, - 2 inverted, give H.S. ^, ^, if, tV, tV, tV, i, i, - 1 124 Kunr 10 [Ex. LSI, ii ' Ip <'i!!i ! (Ill) ^. and .-. H.S. = 2, U, ¥,!,? = 2, 2^, 2f, 2!, 3 , (11) Insert 3 A. means, between i and j. Here <{ ■ r^-r * ^—^ = - T^j ; tence J.S. ■ |J, |Bi H, H »ttd jg, and Invert- ting these we have H.S. = 6, S^jf, 5f, O^^f and 7 *-tV (in) Insert 3 J. means, between -j^ and }. Here J <■ , ;j ' ; ■ 'A = X ~ ^' ^®'*°*' *^''^* ~ ^» ^*^' "^'» ^*'^' ^^* *'*'* iaverting these we have H.S. = 11, Oj, 4?, 3|, 3 (iv) Insert 3 ^5. means, between % and ^^ . Here (/ ■ - g^ . = "T^ = - ^W; hence ^.5. = ?§J, \%\, m, \\\, ^\ and inverting these, we have H.S. = 2^ + 2 t^^ + 2 -ftV + 2 |?f + 3f (v) Insert 3 ^. means, between \ and - \. Here rf ■ . ^ . -- —^ = - 1 ; hence J.S. =* i, - ?, - J, -Vi - ^t% ^'^^ Inverting these, we have H.S. = 6, - 2, - f, - i*f, ~ I 3. Corresponding ^.5. « ], 1, \. Hence<{>*2i 6'^tormot ^.S. = f + (5 - l)g = ? + "i* a J^ ; hence 6* term of H>B. ■ -f^ ; Ex. LZI] ALQBBiU. 125 a* term of -4.5. « I + (U - 1)J = | + V » V i l»«nce 11* term ^ - 3n 3n-l ofH.S. '^an^ term otJ.S. « | +(n- 1)| a | +y ^ » a — ^ ; 5 hence n"* term of H.S. = g^ _ . 4. Of covretponding J. S. -^j -fg^ -^ the 6* term * ^j + (6 - 1)t'» » i'o + A « A; 10* term -■,»,+ (10 - 1)^1, * A + A =* iJ. •nd nti» term =* A + (» - l)iV * t?* + 13 ~ iS ' "fa" •'• required S**, 13 10* and «* terms of fr.5. » 1| ; l-^ and ^-^ 6. Of the corresponding ji.S. 10, 12, 14, the A^^ term e 10 + (4 - 1)2 a 16, and the 8* term ^ 10 + (8 - 1)2 « 24 .'. the 4f^ and 8*^ term of the ir.<9. * ^g and A 1-i J 6. Insert 2 ^. means, between i and 1. Here d = j^ ~ I" ~ i hence ^.8. » \ + I + i + ^, and inverting these, we have H.S. s44'2 + U4-]; hence unknown terms are 2 and 1} 1. Of the corresponding J.S. — , -r-, the 8* term I /I 1\ 1 7 7 "? £ 7a -66 = ~ + (8-l)(^y- ^j= — + y- Y ' 6 - a = ab ' ab I /I 1\ hence Stbterm of H.S. a y^^gv i n* term -■T + ('*-1)(t'"'"j) 1 /_l_J_VJI_2.i.i.iLiL- ^"^ a l6 a J b a ~ a " b 6""o~a 6(2 - n) + fl(n - 1) a6 nth term of H.S. = n 2-n n- I a ab 6(2-n) + a(n-l) 2 8. //.Jtf. = 2ub a+ b m^ - »»'•* 1 1 + nH- 1» m — n 1 9. wJ.Af. = i(a + 6) = i(4 + 9) = V = 6J ; G.JIf. = ^Jab - V4 x 9 — 2ah 2x4x9 V36 . 6 i H.M. = j^^ = -jp^ = U = 5^f 126 OY TO [Ex. hxi. 10. J.M. s i(6 + 4l)»i of 10} » 6t^; (?.iir. - ^^"731 « V35 2x6x4^ 80 11. a:e::u-&:&~ff.\a6-ac»ac-&c.'. 2atf s a6 + 6c & ft(a + c) 2ac /2ae\* /2ac\' .•.6 = 7— .-. 26» » 2(~--) .•.u«4-c»-26»»a» + c»-2(~^) u+c \*''*'V ^ \ ■'"v / 2ac \" ^, f 4oc 1 = (o - 0' + 2ac^ . .A = a positive quantitj if a and c have like signs .-. o' + c' > 26" | 12. 6 s i(a + c)| and mb s ^^; substituting tlie taIuo of 6, m .— ni we hare -r-Ca + c) = V<»c .'• "T'C* + 0' " <*« •'• «»'(« + 0' * 4*^j 4flc and dividing each by a + c we get m\a + c) = ~rTi but a + c = 26 4ac 2ae .'. 26m" « -T^-Tj or6Mi'a -— r— ; hence Art. 261, 6w' is the H.M. between a and c .*. a, 6m" and c are iu H. Prog. 13. Let a, 6 and c be any three quantities in H. Prog., and let X be the quantity which, when subtracted from each, leaves remainders in G.P. ; then (a - x)(c - x) » (6 - x)', that is ac - ex - ax + x* = 6' - 26x + x' .•. 26x - ex - ax a ft" - ac 6''' - €u: .'. X ■= 2b - c - a ' ''^"*' ^'^^^^ **» * *"^ '^ "® *** ^.-P-j a:c::a-6:6-c.'. a5-ac = oc-6c; a6 = 2ac - be ab .-, c = 5 — ^^. Substitute this for c in the above value of x, a»6 2a63 - 6! - a26 62- and we have x b. w^i 26 - a - 2a -> 6 2a -6 4a6 - 2a* - 26» + a6 - a/» 2a - 6 2a -6 X = 2a6^-. 6» - o'6 6(2a6 - b'^ - o») 6 4ai 6 - 2b* - 2a» " 2(206 - 6» - a'') " T = * of middle term. Ex. txi, uii.] ALOBBRA. 127 a and c have 14. J.M. B i(a + 6), and G.M. » ^/ab .-. |(a + b)+^ab*\B (i) and i(a + b) - Va6 » 4 (n). From (i) a -f 2V«i6 fl •. 28 + 146 + 146 + 76» a 1286 .-. W - IOO6 s - 28, or 2600-196 6a_J4a6.-4; 6«-->-^«6 + 4^1{a, ^^ s ajgA ... ft _ A/i = ±V.-.6-Uorf ^^ 16. o + 6 » 30, and tth 49 306 « 131 .'. -30- « 13i 2a6 « 400 ; a'^+ 2a6 + 6"= 900, and 4a6 « 800 .'. a'- 2a6-)-6's 100, or a-6 s i 10 ; a -I- 6 s 30, and a - 6 » i 10 .*. 2a a 40, or 20 ; a » 20 or 10 ; 26 = 20 or 40 .*. 6 s 10 or 20 .'. the numbers are 20 and 10 17. a-b- 16i, and 'Jab « 9, since the O.M. between the J. and H.M. of a and 6 s O.M. between a and 6, (see Art. 261) Then a'-2ab + b^= ^)|&, and 06 = 81 .*. 4a6 > 324 ; a'+ 2a6 + 6^ = 4t|5 + 324= ^ti}^ .-. a + 6s*^. Hence 0-6 a V, and + 6 = af ... 2a = V .-.asV-SOJi 26a a^ =8 .-. 6 3 4 EXBftCIBB LXII. 1. Fg a l-2*3'4'5-6 a 720 2. (1) r* = 8-7-6-6 = 1680; (11) Fg = 8-7-6'6-4'3 = 20160; Fg = l-2-3-4-5'6-7*8 = 40320 3. We are to find the permutations ot 13 letters of which 5 are as, 4 are 6's, and 3 are c's \n l-2-3-4-5-6-7-8-9'10111213 Th^« ^= i^ft ' 12-3-4.6xi2»3-4xl2>3 = 3^0360 128 KBT TO [Ex. uai. 4. Vit* 1'3'3'4'6*6'Y'8'9'10'1M2 « whole munber of changes l'2'3'4'5'6'Y'8'9'10'ni3 iq.gQ.10 « 49896 « number of daji required ■ 136 yeari 322 daji. 5. F, - n(n - 1)(» - 2)(n - 3)(n - 4), and F, « n(n - l)(a - 2) Then n(n - l)(n - 2)(n - 3)(n - 4) = 6 x n(n - l)(n - 2) .'. (n - 3)(n - 4) « 6 ; that is n' » 7n s - 6, whence n s e 6. K,^ • I'2'3'4'5'6't-8'9'10 ■ whole number of days .'. l'2*S'4'r<'6'8'9'10 w 618400 « number of weeks be had to board tbcm, and since board is worth $5 per week for one per-\ son, it is worth $50 per week for 10. Hence total ralne of board <■ $50 x 618400 « $26920000 ; and $25920000 - $6000 a $26916000 « loss when the $5000 ia not paid till the expira- tion of the term of the board. And amount of $5000 at 6 per cent. 3628800 for ^g. . >■- /ears, i. e. for 9935*112 years = 6000(1 + ri) » 5000(1 -f 690'112) s 6000 x 597-112 - $2985533-60. Hence bis loss when the $6000 is paid at once, and put out at interest until the expiration of the term s $25920000 $2986533'60 » $22934466-40 1. F; ■ 15-14-13 (15 - n + 2)(16 - n + 1) and F„.ja 151413 {15- (n - 1) + 1}; then 1514-13""(16-n + 2)(15-? f-l) = 151413«"-(l,'»-n+2)x 10 .'. cancelling same factors of both sides, we Lave 15 - n 4- 1 = 10 .-. n = 6 8. (i) Permutations of 14 letters whereof 2 arc o's, ?. are n'g, I? 1-2-3-4-5-6-7-8-9 10-1112-1314 and two are Vs = \p \a \r 1-2 X 1-2-3 X 1-2 = 3032428800 (ii) PermutationB of 1 2 letters whereof 5 are i's '- 123'4-5-0-7-8-9-1 0-ll-12 ' ~\f ' 1-2-3-4-6 = 3991680 [Ex. LXII. of changes ri required »-l)(ii-a) . l)(n - 2) t s 6 ir of days 8 he had to for one per-\ tal ralne of )00 - $5000 i the expira- at6 per cent. SOOO(l + ri) $2985533-60. and pnt out $26920000 £z. LXII, txtn.j ALOIB&l. 129 1680 (ui) P«rmotoUoni of 8 letters whereof 4 art o^i ' ~\jf ' l'a-8'4 (it) Permntatipni of 18 letters whereof 8 are 0*1 and 8 artia'i (n ia'8-4-6-6f-8'9101112I8 = YY ia-3 X 1-23 " 1M9T2800 9. (1) Permntationi of 7 letters of which 2 are a's If* 1-2-8-4-5-6-T = Ip " 1-2 (II) Permntations of 13 letters whereof 2 are o's, 9 are n's, I? l-2-3 4'6'6^7-8-9-101M2-13 ^_ _ •°^ * ■" ''• " iFFlr 1-2 X 1-2 X 12 * "8877600 (III) Permntations of 7 letters whereof 2 are /'s and 3 are o's I? 1-2-3-4-6-6-7 2520 (p Ij 1-2 X 1-2-3 420 6n/5n \ /6a \ 2ii/2» A /2» \ t\ /6» - 4\ 290n /2n - 3\ /2« - ON ■)[-^)-—{-i-){-r-)- /6n - 2 IK- f (6a - 2)(6n - 4) = V^(2fi - 3)(2n - 6) ; or 136(2&»' - SOn •(- 8) - 2320(2n' - 9n + 9) ; or 2&3n' - 3366n » - 3960, whence n s 12 |->-n+2)xl0 n + l = 10 >'8, 3 are n's, • 1M213H Ix 1-2 ExiBOisi LXIII. 10-9-8 , ^ 10-9-8-7-6 1- 0) ^3 = -fi^ = 120 ; (n) C, = l-2-8'4-5 2M I (ni) C, 10-9 8 - Cf« = 1-2 = 45 1514-13-12-11 1514-13-12-11-10-9 (")^^^ 1-2-3-4-6-6-7 ''g^^e; ^ 16-14'13 (m) C„ := C3 = -pjTg- s 466 X ,y* m 8. C, ia-lM0*9-8 KIT YO 192 [KZ. UEUI. 1-2-3-4-5 4. Whole namber of combiofttioni of in thlogf, 1, 9, 8, 4, kt.f . . . 2n together ■ 2** - 1 ; timilarlj the whole number of comblnfttloni of n thingi, 1, 2, 3, 4, kc, . . n together ■ 2* - 1. 2"»- 1 2"»- 1 Then 2*» - I « (2* - 1) x 618 .-. -jjrrr * *^^ » ®^ ■*"*• yt-1 ■ 2* -f 1, we hare 2* + 1 ■ 518 .*. 2* « S12, and .*. by- inipection n« 8 6. (I) C, r a6'35-34*33-32 439824-« No. of different lelections l-2-8-4'6 (n) Taking away one man from the 86 there remain 35, and 35-34-82'81 theee combined together, 4 and 4 giro ^.^.g.^ « 62860 com- binations to each of which the reierred man must be attached. 6. Number of combinations of 21 consonants, 4 together 21'20-19-1 8 ■ 1-2-3-4 3 together a = 6;885 ; also number of combinations of 6 rowels, 5'4'3 1'23 8 10. Hence there can be formed 5986 x 10 m 59860 different sets of seven letters, each set containing four consonants and three vowels. But each of these 69860 sets can be permutated, l'2-3'4'5'6'7 » 6040 ways, each forming a different word .'. the required number of words » 69860 x 6040 a 301644000. t. The different arrangements of 9 of the persons while the tenth remains 6zcd = 9'8'7'6'5*4*3-2'l « 362880 s whole number of different arrangements of the ten persons, so that no one has the same neighbours in any two cases. But one half of these arrangements will be similar to the other half if the position of neighbours on the right and left hand sides be not regarded as making a difference. So that it Jis said to 4Htve the iamt neighbours in the arrangement BjiC that he has in the arrange- ment CjSB, then the correct answer will be i o/ 362880 =181440 Ex. Lsm.] amuaa. 181 8. n<* - l)<» - a) : *^'' " ^y,]^.^^* " :: e : 1 .-. cmmI- n-S Hog, —J— ■ 1, wheDoe « ■ t 9. 11(11- l)(«i-2)....(ii-p+l)-10n(«-l)(«-2)....(ii-p+a); or diTidinf Meh by n(n - l)(n - 2) ... . (n - p •!■ 2), wo get n - ji + 1 ■ 10 .*. n - ji ■ 9 (i). Again fi(n-I)(ii-2).... (n-p-f 1) fi i'2'Z'A'5 ° ^^' * 10'9-8-t number of signals with foar flags out of 10 = t.o.^.j " 310; 10-9'8 10*9 number with 3 flags = "t.o.^ " 130 ; number with 2 flags » -zj^ ~ '^^ and number with one flag s 10. Therefore whole number of signals ^ 10 f 46 + 120 -f 210 + 262 > 637 12. There are in all nine coins and they may be combined, any number together, to make a sum ; then the combinations of 9 things 1, 2, 3, 9 together « 2*- 1 * 2« - 1 » 612 - 1 « 611 i 182 XST TO BziBOira LXIT. [Kit. uav. 3 8-4 3-45 «• + 1. (1 +«)-»:! 1-Y« + Y:^a:«- a 1 - 3a: + 6x" - lOx" + 16ar* - Ac. 3-4S-6 1-2-8-4 «♦- 4c. 2-3'4 l'2-3^ 2>3-4-5 l'2-3-4 X* - Ac. - 1 - 2x + 3«» - 4a:" + 6** - Ac. 3. (1 -2x)-» = l+^(2a:) + il?(2«)»+ ^:(2«)»+ 1-2' i; 1 + 2r + 4x" + 8x' + 16x* + &c. 6-6 1-2-3 l-2-3^ l'2-8'4 1-2-3-4' (2x)* + Ac. 6'6-7 4. (1 - Ix) - » - 1 + -(jx) + -(Jx)» + r:r:^Ox)» + 1-2" l-2-3^ 5»et'8 1-2-3-4' d')* + &c. = 1 +ix + Y*' + V*' + V** + *c. 2-3 2*3-4 6. (1 + 3x)-« - 1 - Y(3x) + --(3x)»- T:5r,(3»)» + 1-2' l-2-3^ 2-3-4-5 1-2-3-4' (3x/ - ke. = 1 - 6x + 27x» - 108x« + 405x* - Ac. 5 5-6 5«C'7 6. (1 - 2x) - » = 1 + r(2x) + -;(2x)« + ^:^,(2x)« + 1-2' l-2'3^ 6-6-T8 1-2-3-4 (2x)« + &c. a 1 + lOx + 60x' + 280x» + 1 120x* + kc. 7. (I - X) 1 + rrx + 4-S x» + 4-6e . 4'6-6-7 1-2-3' x» + 12-3-4 X* + Ac. a I + 4x + lOx" + 20x' + 36x* + Ac. 1-2-4 8. (l-4x)* = l--r-(4x) + <4x) ,»- l(~l)(-3) 12-38 (4x)» l(-l)(-3)(-6) 1-2-3-4-16 9. (1+x) ' = l-^x + (4x)* - Ac. = 1 - 2x - 2x»- 4x» - lOx* - Ac. 2-5 rra- 2'5'8 x» + 2-6-8-11 1-2-9 l-2'3-27* ^ 1-23-4-81 ar*-&c. 1 - 5x + 5x« - Mx» + HSa:* - Ac. 4 4 4(~1) 10. (l-ix)«-l-^(«x)-^ (|x)« - 4(-l)(~6)(>ll) 1-2-3-4-62& 4(-lK"6) 1-2-3-125 (Jx)» (|x)* - Ac. 1 - S* - ih'* - iS Jo*" - t/oVju** - *c. Ex. txir.j ALGBRiu. 183 12. (l-x)-^si+i.^^ 4-9 4.9... 134 KBY TO [Ex.ixiv. s a Wfi _ ~/'a-«i:-a (« 4*5 4-5-6 ) + ^(«-'*-'/ - i:5r,(«-'*-») «« - a'k* 45-6-t 1-2-34' 1-2 1-2-3' (a-'x-y-Ac.) = a-i«(l-.4a-»a?-'+ 10a-«x-*- 20a-9x-« + 35a-"i:- » - &c,> = a-"-4a-»'x-»+10a-"x-*-20a-«x-«+ SSa""**-' - Ac 18 y-x-*)"* = {«*(l-«'VO}l=a-^V{i.«-t,i)-J = o"Tf^{l + -o-(flX) 'J + (ax)"* + &c.} 14-7 10 1-2-3-4-81 1-4 1.2-9 (ox) \ 1-4-7 1-2-3-2 7( to 'x + |^ '^«-VJ TO + iia« 'i^m'^x' + Ac. 20. (o + x-»)^ = {a(l + o-»x-»)}^ « a*(l + a^^x'^y = J = a' (1 + l/_l_\ 2(-3) /J_Y ,\^ax»y ^ l-2'26 [ax* J H - 3)( - 8)/ 1 l'2-3-125 2(-3)(-8)(-13)/ 1 l-2-3'4-626 Uv (ax») + &C.} [Ex. txir. V, ix-' - Ac »x'' + Ac. Ex. UUY, LZV.] ALQEBBA. 185 \ ir^ 'Z'2l\flhnJ 125 \^a«V = «"ii + ^(siif) + ^(s^) + 155(5^) - ^(55^5) +*« J 21. (a - 6x) " * » {a 6^ term « (1 + r)tb term .*. r s 6 . 4-6-6-7-8 . Hence 6tt» term = ( - 1)' x ^.2.3.4.5 ^ = - 66»' 3. (i) General term of (1 -x)"' P(P + 9)(f + 2tf) fp + (r - 1)7) 2-6-8--"(3r-l) , (11) Since general term = (r 4- 1)^1 term s %ih term .*. r s 5 2-6-81114 . 308 , Hence 6th term « ( - 1)» x ^.g.g.^.g ^ 3^ *' - - ^x" 186 KEY TO [Ex. LXV. m 4. (i) General term of (1 - x)^ p(p - g)(p - 2q) " »'(p -. (r- l)g| , 41(- 2)' •••(•: -3r) '- . 4i(-2)(-6)(-«) . (II) As before r = 5 .-. 6th term = ( - 1)» x — 1.2.3.4-6.248 ' S. (i) General term of (1 + x) -1 (-ly «(-!/ P(P + 9)(P + 2q)-" '{p + (r - 1)^1 J i |rxy 7'912 (5 + 2r) II x2' t'9lM816 (11) As before r « 6 .*. 6th term 3 ( - 1)' x fTjrsTs^* = « »AO^a:» 6. (i) General term of (1 +a;) ! (-1)' = (-l/x 8lll4«'«-(6 + 3r) jrxS* 8-lM4'l7'20 (II) As before J* = 5 .-. 6th term = ( - 1)« x ^.^.^.^.^ i ^g J 7. (a-x) » = {a(l - a-**)}-* = a-»(l-o-»x) (i) Gen. term of (« - x)-* = «-* x ii(»+l) (n + r-1) ifl'^xy O"* X 1-2-3 -a-'x'* = a-» + a-'x'" = a" f*iV » (II) .-. 6th term = a" ^x" 8. (a + Jx)^ s {o(l + ia-ix)}» = a' (0 .'. Gen. term of (a + }x)l ' ^ ^ \rxq^ U«' '(-^) Ks. unr.] AI«aBB&A. 187 I gi(-*)(-o)'-"("-gy) /«^\ ^ fi 6-l-4""(5r-ll) « 61-4*9'U (n) .'. 6th term = ( - 1)» x o» x 1.2.3.4.5. lo i'*' ^'^ 9. (I) Oen. term of (I - 2x)-» = ^ '-^ j^ — ^ ^(2*)' 23.4.... (r+i) 2-3'4«"«r(r+l) " }^- • 2'»'" = 123. ..-r ^2;*' = (r + 1)2V (II) Since general term = (r + l)ti> term « 5U> term /. r« i Hence 6«» term = (4 + l)2*a;* = 5 x 16x* = 80x* 10. Oenerftl term of ( 1 + §«»)" * ^ 6 7*9<"(3 + 2r) 2''x** 6-7-9""(3+2r) , 6-7-911 (11) At before r = 4 .-. 5th term = ( - l)* x . „ - ..-. x' e + 1 X |Ji|x» « |f^x« 11. (a-« + «') = {a-!'(l+a»r"3)) saHl + a^x"*) -I -i .'. (1) General term of (a-* + x ') ^ . ,s, 2-7l2-'-'(5r-3) .& . a* X ( - !)• X ^-^r a-x • . ,vr 2''ri2'"(5r-3) ^^^ -fe "(-irx jfTgi: a»-*?x • 2'ri2l7 4 ,4 119 1* .8 (n) .•. 5th term = ( - 1)* x 1.3.3.^.^25 '»' * (* ' " ^S*^* ^ 12. (a" * - x" *)" « |a" *(1 - o*x" *)f" = a(l - a*x " *)"' • .'. (i) General term of (a ' - x ') KE7 TO [Ex. tM,y, uti. w(n4-l)(n + a)""(n + r-l) ^ i .1 r max — ■ .y (a'g ■) •3'4*'*'(r+l) 1 ^1 r l'2'8'4""rx(r+l)^ 1 .1 ? j~ («*« *) -ax it2 .3.4 - ...r («** ') sax * (r + l)a" a; • 13. r»» « 1024 14. r^ » 128 16. O** = 16. r" ■ 4096 17. r is the least integer eqaal to or next greater than IH » ' 2 (n + l)^f^i or (4 + l)f:jr2' <>' '^ >* ! o"^ ^*^ i *>>»* tJ»« *"* Integer > V^ is 4 .'. the greatest term of the expansion is the 4th term «32 18. r is the least integers or next >(n-l)r~; or (6 - 1)/^; or 4 X 1 ; or 4 .'. r « 4*1^ term « 5*^ term ■ 4| 19. r is the least integer » or next > (n f l)rT~' : or 3 (20 + 1)2^ ; or 21 X ? ; or V which Is 13 .'. the greatest term is the 13U> term - 125970 x 2* x 3» ' s 20. r is the least integer = or next > ^n - 1) 73^1 or (t - l)r71l or 6 X } ; or 9 .-. the 9^ term « ^f^l^UP " the lOtb term. ExiRCisi LXVI. 1. 7x<36.-.«<5 2. 16«-84>108, or 16* > 192 .•.«>!» 3. 4«<12.*.x<3 4. 4x+10>x-20; 3»>-30.'.«>->10 6. ox + 6bx ~ 606 > a' ; ox - a" -f 56x - 6ai > ; a(x -> a) + 66(x - a) > ; (x - a)(a + 66) > .*. x - a > ••. x > «. Also *x - 7ax + 7a6 < J», bx - 6» - 7ax + 7a* < 0, 6(x - 6) <- 1a(x - 6) < 0, (6 - 7a)(x - 6) < .-. x-6<.0.-. x<6 6. a»+l 5 «'+«! according a?*+l5a(a+l); oral d^-af l^a; or.asa"+l52l then Art. 134, a" + I > 2a .% a?-l-a, if a'4-l>2a; bat Art. 134 for all Tftloef of a, «xcept a a i, a> + i > 2a ,*. d^ 4- 1 > a* -i- o, when a is a negratire improper fraction. a b 8. -^ + — > 2, if a" + 6» > 2aft ; but a» + 6» > 2a6 by Art. 134 9. Httltiplying each by 12| and reducing, we hare 7x + 6 < 8« -I- 12, and 7x 4- 6 > 6x + 10 .'. X < 6, and x > 4 .*. x s 5 10. a» + 6«>2a6,Art. 134; also o» + c» > 2ac, and 6»+c»> 36c. Then by addition a" + 6« + a" + c" + ft" + c» > 2a6 + 2ac + 26(?; that is 2a«+ 2**+ 2c'>2o6 + 2oe + 26c .-. a»+6»+c»>a6 + ac + 6c 11. a''> tf*- (6 - c)', since (6 - c)' is necessarily positive .'. a» > (a - 6 + c)(a + 6 - c), these being the fiiotors of ^- (b - c)" similarly 6' > (a + 6 - c)(6 + c - a), and (^>(a + c- 6)(6 + c-a). Multiplying nnequals by onequals, a'b't^ >(a~b + c)'(a + 6 - c)' (6 + c - a)' ; extracting sq. root oftc > (o - 6 + cXa + b- c)(6 + c~d) 13. Let b" a + m, and c = a + n,a being the least of the three quantities ; then a6(a -k 6) « a(a -f tn)(2a + m) s 20^ + 3a"fli + am' ac(a + c) = o(a + n)(2a + n) a 2«' + 3a'» + an' 6c(6 + c) a (a + m)(a + n)(2a + m + n) a 2a* + 3a\m + n) + a(m + «)•+ mn(m + n) .'. by additioin (I) ab(a + b) + ac(a + c) + 6c(6 + c) = 6a» + 6a*(i» + ») + 2a(m' + n') + 2amn + tnn(m + n) (II) Also 6a6c a 6a(a + m)(a 4- n) = Ga" + ea\m + n) + 6amn ; subtracting (n) from (1) we hare (i) - (a) a 2o(m' - 2mn + »") + mn(m + n) a 2a(m - n)' 4- mn(m + n) ; bat since by sapposition a< b and < c, it follows that m and n are positive quantities .', 2a(m - nf 4- »»n(m 4- n) is positive .♦. a6(a 4 6) 4- ac(a + c) + 6c(6 4- c) - 6a6c is a positive quantity .•. a6(a 4- 6) 4- ac(a + c) + 6c(6 4'c)>6a6c (III) Also 2(a» 4- 6» + c») a 2o» 4- 2(a 4- m)» 4- 2(a 4- n)" = ea" 4- 6o"(»» 4- n) 4- 6a(m«4- n') 4- 2(m»4- n») ; subtracting or < 1 .*. their product is positive. Hence 3(1 + a' + 0*) - (I + o + a')* = a positive quantity .-. (1 + o + a*)' < 3(1 + o' + a*) unless a » 1 14. xV - (ac + 6rf)' = (a" + 6«)(c« + tP) - (ac + 6rf)* = oV - 2o6(;rf + ftV = (ai - 6c)' which Is necessarily positive, unless ad -be; but xh^ - (oc + 6«0' = {'V + («c + ft*^'i^« quantity .\xy>ae + bd 16. V^'^rp + V2a6 -6i» > a, if V3a6 - i* > a - V«?"rii; orif2a6-6»>a«-2aVa*^r5'+a»-6»; orif 2a6>2«J»»-2aVo»^rp*; orif6>«-V«^^^; orifV^^>«-6; orifa«-6«>a»-2a6+6>, orif2o6>26»; orifo>6 16. Making the same supposition as in Ex. 13 (I) (a + 6 + c)" = (3a + m + n)« » 21(^ + 2'7a'(m + n) + 9a(m + h)* + (m + n)* (11) 2Ta6c » 27ora + m)(a + n) = 27a» + 2Ta«(m + n) + 2ta«» (ui) 9(0"+ 6'+ c») = 9{a»+ (a + m)»+ (a 4- n)>| = 2ltfi+ 2la*(in +n) + ?.7a(m» + n«) + 9(m» + n') .«. (1) - (n) = 9a(m + n)' - 2tamn + (m + n)» = 9tt(m + n)' - 36amn + 9amn + (m + n)* s 9a(m - ny + 9amn + (m + n)' s a positive quantitv Ex. LZIT.] ▲LOIBBA. 141 That is (a -t- 6 f c)' - 27a6c a « positire quaatlty .'. (a + 6 + c)'>27oic Again (m) - (i) = 2 7o(m» + «») - 9a(jii + n)« + 9(jji" + n») - (m + «)• = 9o{3(m" + n») - (m + n)»} + (m + n)|9(»" - «n + n») - (m + n)«| = 9a(2fli' + an' - 2mn) + (m + n)(8»ii' - 7iiin + 8n*) = 18a(m»- mn + n*) - ISamn + ISomn + (fli + n){(8«'- 16»in + 8r.») + 9ninJ = I8a(m - n)' + 18amn + (mi + »){8(ni - n)* + 9mn) = a positive quantity . That is 9(0^ + 6» + c") - (a + 6 + c)" » a positive quantity .-. (a + ft + c)» < 9(a» + 6» + c») 17. (a + 6)(6 + c)(c + a) ^ 8aftc, according as o»6 + flft» + o»c + ac" + 6=^ + ftc* ^ 6aftc or as (oft* - 2a6c + oc») + 6(c» - 2ac + a') + c(a» - 2a6 + ft«) ^ or as 0(6 - c)'' + ft(c - a)» + c(o - ft)« ^ But a(ft - c)" + h(c - a)» + c{a - 6)» > unless a = 6 = c .-. (a + 6)(6 + c)(c + a) > 8aftc ar»+34x-7l 18. Let a 4. 2x - 7 ° "* ' thenx* + 34«- 71 = a»x' + 2ma;-7Tn; that is (m - l)x' + 2(»i - l7)x « 7/» - 71, whence X = . (17 -m± ^/8(m - 6)(»i - 9)}, where if x is to be real, m - 5 and m -> 9 must both have the same sign : i. e. m must be > 9 or < 5 .*. the given expression can have no value between d and 5 n» - n + 1 1 19- First -TT-r-T-, > T* if 3n» - 3n + 3 > tt' + n + 1: or if 2/»'' - 4n + 2 > ; or if n' - 2n + 1 > ; or if n» + 1 > 2n ; but n" + 1 is > 2n .•. Ac. Secondly ^a "| ^ ^ < 3, if n» - n + 1 < 3n« + 3tt + 3 ; orifO<2n« + 4tt + 2; or if + fr'(x - a) ' (X - a)(x + 6*) " x + &> ' a + b* a(x - c )(x - e) a 5(x - c)(x - c) " "ft" x(a - x) -r a cc (a - x)(tt' - o'x -^ ax' + x*) ~ a* - a* - a' + «• x(x'-tt') + 2a(x'-o') (x + 2a)(x-tt)(x+a) (x + 2a)(x+a ) (x-a)(x»+flX-12o»)" (x-a)(x«+ox- 12a^ * x« + ox - l2o» Sax 2a 60^ 3 s » I)" .-. *c. l+l+l+l a» 3a" _ 3a — ""ao " 2 a + 6« a (g + 2a)(g+g ) fix. Lzyrn.] ALQinU. 148 Bxnoui LXVIII. 1. DiTidiog by 3, w« bare x + y + y-S+l.*. -j- it integral ^- t saj. Then x « 3< + 2 ; inbstitnting tbis for x in tbe giren eqaation we bare 3y « 11 - A(3t + 2) .*. y ■ 1 - 4f ; letting f ■ we bare x ■ 2 and y ■ 1. * 3y 1 -I- 3y 2. DiTide by 8, and we hare x - 2y - ■=- ■ 2 + J .♦. — r — Is 2 + 6y 2-fy integral .•. lo also — r — integral .*. —g— « *, wbenee y « 5*- 2. Sdbstitntiug tbii for y in tbe g^ren equation, we bare 6x ■ 11 + 13(6^ - 2) .*. x» ISt" 3. Henee taking in ineceMion f s l, 2, 3, kc, we bare x a 10, 23, 36, 49, Ac, and y « 3, 8, 13, 18, 4e. y y*"! 3. Diride by 2, and we bare x + 3y + y « 29 + j .♦. — r- » f ; whence y«2t+l. Sobstituting tbis for y in the giren equation, we bare 2x s 69- 7(2t -t- 1) .*. x s 26 > 7^, and taking in luc- eession < « 0, 1, 2, Ac, we bare x s 26, 19, 12 or 5, and y « 1, 3, 6, or Y y y~i 4. Dividing by 6, and we bave x + 2y ■»• -r- s 6 + i .•. — r- « t ; whence y « 9f 4- 1. Substituting tbis in tbe giren equation for y, we bare 6x ■ 26 - U(5t + 1) ; wbenee x ■ 3 - llf, and hence wben t > 0, we bare x = 3 and y » 1 ~ 8y 2 2 -I- 8y 5. Diride by 9, and we get x - y - y = — , .•. — g — is integral, 1 + 4y V -f- 28y so also is — g — integral, .•. so also is x — integral, y+7 y+7 .-. 80 also is —T" integral. Let —r— » t, then y = 9< - 7; substituting tliis for y in tbe giren equation, we bare 9x « 2 + 17(9^ - 7> .'. X ■ 17^ - 13. Now writing in snceessiou ^ = 1, 2, 3, ftc, we bare x s 4, 21, 38, 65, ftc, and y^ 2, 11, 20, 29, 4c. 144 KIT TO (fix. tZTUl. 8y 6. Difide b/ 18, and w« g«t « 4- y + rx ■ 6 4- t? .*. 11 8y-ll 18 13 18 is 40y - 65 » - 8 . . Iot«gr«I, .•. — Tg — is integral, .-. Sy - 4 + -jg- if integral, y-3 .'. \q ° ^ wbenee y > 13/ -f 3 ; lubstitating thii for y in the given equation, we have 13« s 89 - 21(13/ + 8), whence V s 2 - 21/. Now writing / > o, we bare x « 2 and y 3 3 6y 12 - 1-12 7. Divide by 12, and we get x - 3y - 6y.5 y — 1 y " 1 .'. ii integral, .•. so alio is integral. Let ■ ■ ■ = /,, then y = 12/ + 1 ; sabstitnting this in the^iven equation for y, we have x 3 41(12/ + 1) - 17, whence x s 41/ + 2. Now writing in succession, 0, I, 2, ko., for /, we have x » 2, 43, 84, 126, Ac, and y « 1, 13, 26, 37, ko. • ' . By 24 ey-24 8. Divide by 37, and we get x f y + r= « 9 + r= .*. — ^ — is y- 4 integral, .*. so also is -^r- w'aich say s t] then y sit + A Then 37x 3 357 > 43(37/ + 4), whence x « 6 - 431 ; wherefore taking / = 0, we have x > 6 and y > 4 " ' * 21y 6 21y+6 . 9. DiTide by 22, and we get x - y - -^g- = 32 •"' — 22~~ " 21y+6. . 22y-21y-6 integral, .-. so also^ is y - — 22 — Integra ; that is 53 , y~ 6 or — — - is integral > /, say then y = 22/ + 6. Hence 22x c 6 4- 43(22/ + 6) .'. x = 43/ + 12. Now writing in successiou, 1, 2, Ac, for / we get x s 12, 55, 98, Ac, and y « 6, 28, 60, &c. 4y 2 4y - 2 10. Divide by 7, and we hare x + By + y = 25 + y .*. — s — 8y - 4 y - 4 is integral, .-. so also is — » — integral, .•. — r— = /| whence y ■ 7/ + 4. Then 7x = 177 - 26(7/ + 4), whence x « 11 - 25/. Hence tailing / = 0, we have x & 11 and y » A E.X. LXVIU.] ALOSBBA. 145 11. Dmde by 99, and we gtt * -y " ■99" " ^ '•' 99 •*• — 99 — 80By + 190 . . li integral, .•. go aUo is Zg integral, .*. 10 alio U 8»+ 91 , . I04y+I183 . 3i/ + I + — gg— integral, .•. lo also is ^ integral, 6y + 94 . . . lOOy + 1880 .-. so aldo IS y + 11 + — rr — integral, .*. ao alio is == y + 98 integral, .'. so also is y + 18 + — g^ integral, .•. y ■> 99/ - 98 ; Bubdlituting this in the giren equation for y, we hare 99« = 335 •«- 160(99< - 98), whence x ■ 160< - 1S5. Now sabstitnting in succession 1, 2, 3, &c., for /, we hare x ■ 5, 165, 326, 485, &c., and y = 1, 100, 199, 298, Ac. » 3 «-2 12. Diride by 4, and we hare 4* - y + ~ •« 5 + -r .*. — r— « /, whence x • 4< + 2 ; then 4y « 17(4/ + 2) - 22 /. y ■ 17< + 3. Taking < = 0, 1, 2, 3, Ac, we hare x ■ 2, 6, 10, 14, Ac., and y -3, 20, 37, 54, Ac. 13. Multiplying the first equation by 3, and the lower by 4, and adding the results, we have 18x -f 29y ■ 123. Deride by lly 15 lly- 15 . 18, and we hare x + y + -^ " ^ "*" li *'' — 18 — *' integral. B5y-75 y-3 .'. so also is — , g — integral, .•. so also is 3y - 4 4- -j|- integ., .-. y = 18/ + ?^ Hence 18x = 123 - 29(18/ + 3) .-. x » 2 - 29/. Now taking / = 0, we have x « 2, and y-3, and consequently *-4 \ 14. Multiplying the upper equation by 11, the lower by 6, and adding the results, we get 56x - 49y = 469, or 8x - 7y » 67. x-4 Diriding this by 7, we have x-yf-ysO + y whence x = 7/ + 4 ; then 7y = 8(7/ + 4) - 67, whence y = 8/ - B. Now taking / = 1, 2, 3, Sec, wo have x s n, 19, 25, Ac, and y = 3, 11, 19, kc. ] but since z must also be positive and integral. 146 KEY TO [Ex. LXVIU. we find upon trial that the o^ly admissible values are x & 11, and y = 3, and consequentlj « = 2 ^^ • 15. Let X = bhe number of $3 notes, and y = the number of 2y 1 2m -1 $5 notes ; then 3x + 5y = 69*7, or x + y + y = 232 + y .-. —g— 4t/-2 y-2 is Integral, .•. — 5— is integral, and .-. also - .— - t, that is y s 3f 4- 2 ; then 3x « 69*7 - 5(3/ 4- 2), whence x = 229 ~ 6/. Hence 5/ < 229, or f < ^^ ; i. e. < 45| .*. the given sum can be made up of $3 and $5 notes only in 45 different ways. 16. Let X 3 the number of 25 cent pieces, and y " the number of 10 cent pieces ; then 25x + lOy 3 2t30, or 6x + 2y » 546, X .'. 2x + y + y = 273 .-. X = 2t. Also 2y = 546 - 10* .-. y = 2Y3 - 61. Hence 5t < 273, or < < 54 J .*. the given sum may be made up as directed in 64 diffisrent ways. 17. Let X s the number of guineas paid, and y = the number of half-crowns received in change; then 21x > y s inoj, or 2x 1 2x •- 1 42x - 5y a 301, .*. 8x - y + y = 60 + -r- .*. — g — is integral, 16x- 8 X- 3 • .*. so also is — g — .'. — g— = /, or X a 5< + 3. Also 6y s 42(5/ 4- 3) - 301 = 210f - 175 .-. y = 42/ > 35 ; and taking / « 1, we have x r-- 8, and y = 7 18. Let x^ and y^ be the two square numbers required, and assume x' + y* =(nx- yY = nV - 2nxy + y' ; then «' = n'x" - 2nxy ; 2ny or X = n'x - 2ny .'. (v? - l)x = 2ny, or x s — — - where n and y »"— 1 may be assumed at pleasure, and it will be found that x^ + y' is a complete square. But if only integral values are required assume in the expres- 2ny sion X = "a" Tj ^'*** y = n' - 1, then x «s 2n, where n majr be ■I I £Z. LXTin.] ALQEBBA. MT^ taken « sny integnl nuttber, and it will be fovtid ilutt «* 4 y* is a complete iqaare. 19. Let x* and y" be the two squares reqaired, and assome £> - y> s (X <■ nyy u 3^- 2nxy + nV. Then y» = anxy - iiV ; n' + 1 or y « 2nx - n"y ; or 2im; ■ («* + l)y .•. x » ^ x y, where n and y may be assumed at pleasorei and it will be found that x!* - y> is a complete square. Bat if only integral values are requwed, assume in the abore expression y s 2a; then x x n' + 1, where it will be found that when n is taken s any integral number, x" - y' will be a com- plete square. 20. Assume that the basket contains x parcels of 4 with 2 over, or y parcels of 6 with 2 over. Then 4x 4- 2 s 6y •!■ 2 ; or y y 4xw6y30; or2x-3y = 0; orx-y - ^ = .•.—=*} ory = 2<. Also 2x » 3y .*. x 3 3^ Hence taking t = 1, 2, 3, Ac, we have X = 3, 6| 9, 12, ke., and y a 2, 4, 6, 8, Ac. But X and y must be taken such that both 6y 4- 2 and 4x + 2 are > 90 and < 100 .*. y > 16, and x s 24, and the number of apples » 16y + 2 = 98 21. Let the number s6x-fl = 8y + 5sl0s4-9 y 2 Then 6x-8y«4; or3x-4y = 2; o^'-y-Y^'J .'. y a 3< - 2, and x « 4f - 2 Also 6x + 1 = IO2 + 9 ; or 6x - 10« s 8 ; or 3x - 6z = 4, but 2/ x = 4t-2 .-. 3(4«-2)-6« = 4; orl2^-5s«10 .•.2i-s+- = 2, whence ^ = St' and 2 > 12f' - 2. Then x = 4f - 2 = 20/' > 2 ; y = 3/ - 2 « isr - 2, and z = Ut' - 2, whence taking r » 1, we have X ■ 18, y « 13, and x ■ 10, and .*. the least number divisible as requhred » 6x + 1 « (18 x 6) 4- 1 « 108 + 1 « 109 148 KBY CO [Ex. LXVllI. X y X y. 38 22. Let j^ and -^ be the two firactions, then tq "** 16 * 60 ' X 1 or cleariug of fractious 3* + 3y » 19 .-. a + y + y = ^ + Y> *nd conseqneatly x*2t + lf whence y = 8 - 3^ Now taking t ~ 0, 1 and 2, we haye x s 1, 3 or 5, and y = 8, 5 and 2 /. the required fractions are -^ and -ft- i I'a and ^ ; and ^.^ and yg Note.— We cannot take t = 8, since then y = 8-3< = 8-9 = -l=:a negative quantitj 23. Let X, y and z '= barrels respectively ; then x + y -{-'s s 60 (i), and 2x + 6y + 42 » 260 (u). Haltiplying (i) by 2, aud subtracting the result from (ii), we bavf >y ' 2z = 150, whence y = 2/ and z='J5-3t^ Also X = 50 - y - 2 = 50 - 2/ - (76 - 30 = < - 26 Then in order that z may be positive 76 - 3t must be posiiive, and .'. 3^ < 75, or t < 25, and in order that x may be positive, < - 26 must be positive, that is / > 26 ; therefore t is both less than and greater than 25, which is impossible. 24. Let X, y and z * the number of pieces respectively ; then X + y + 2 « 100 (i), and lOOx + 20y + 6z s 2000 (ii). Dividing (ii) by 5, and from the result subtracting (i), we have 19x + 3y = 300, whence x = 3t and y = 100 - 19/ .-. s = 100 - X - y = 100 - 3/ - (100 - 190 = 16/. Now taking / = 1, 2, 3, &c., we have x = 1, 6, 9, 12 or 15 ; y = 81, 62, 43, 24 or 6 ; and z > 16, 32, 48, 64 or 80 « 36. 2x 4- 3y = 26, whence x = 11 - 3/, and y = 2/ + 1. Now taking / = 0, 1, 2 or 3, we have x = 11, 8, 5 or 2, and y = 1, 3, 5 or 7, and hence the parts are 2x and 3y = 22 and 3 ; 16 and 9 ; 10 and 15, or 4 and 21. 26. Let X, y and z be the three parts ; then x + y + 2 » 24 (i), and 36x -f 24y 4-82 = 616 (n). Dividing (n) by 4, and multiplying (i) by 2, and taking the diflference of the results, :. Lxviii. Ex. Lzvin.] ALQBBRA. 149 y. 38 [6* 60' ig t = 0, by 2, and 0, whence >e posiiive, e positive, both less ipectively ; 2000 (ii). ig (i), we 100 - \9t ow taking , 62, 43, 24 + 1. Now y= 1,3,5 16 and 9; we hare U + 4v = 81, whence * = 4/ - 1, and y « 32 - 7« .-. a = 24 - (4< - 1) - (22 -. It) -3 + 3t. Now taking * = 1, 2 or 3, we have x = 3, 7 or 11 ; y « 16, 18 or 1 ; and « * 6, 9 or 12 27. Assnme y"x to be a perfect namber; then its diviaora are 1, y, V', •••• y") «i a:y, xy», .... «y*-» .•. y*x « 1 + y + y» + .... y" + x + xy + xy' + .... xy"-*. Now 1 + y + y*+ .... y" j,n + i . I y-1 y"x - , and X + xy + xy" + .... xy' i»-i = y*" I X X y**l - 1 + (y* -. l)x y-l y-i or clearing of fractions yn+ijj _ y»jp = yn+1 _ n. yWjp _ ar ; or y***x - 2y*x + x « y***- 1 .•• X = -TfTi — 5r» — ?• Now in order that x may be a whole y — 4y - 1 number, let y*** - 2y'* = 0, or y s 2 ; then x = 2*** - 1. Also let n be so assumed that 2" *^ - I may be a prime number ; then it will be found that y"x = 2*x (2**1- 1) will be a perfect number. Thus if ns 2, we have 2''x (2»- 1) « 4 x (8 - 1) 34x7=: 28 » 14 + 7 + 4 + 1 + 1= sum of all the' divisors ot 28. 28. Let the number = lOx + 7 s 12y + 9 = 14z + 11 ; then lOx - 12y = 2, or 5x - By = 1, whence x > 6^ - 1, and y a Sf - 1. Also lOx - 14s = 4, or 30t- 7» =< 7, whence t s 7^', and z « SOf - 1. Then x = 6/ - I = 42^ ^ 1 ; y = 6t - 1 = 36<' - 1, and««30«'- 1. New assuming f' « 1, we have x = 41 ; y - ?4; and x s 29. Hence the least odd integer =/10x + 7 = 410 + 7 » 417. 29. Let X, y and z represent the numbers respectively ; then X + y + a a 100 (i), and 50x + 30y + 2« = BOO (n>. Dividing (ii) by 2, and from the result subtracting (i), we have 24x + 14y = 160, Tihenco x = 7/ + 1, and y = 9 - 12^ .-. « = 100 - (It + 1) - (9 - 12/) - 90 + 12/. Now / must be < 1 *.* y = 9 - 12/ must be positive ; also / must be > - 1 because x s 7/ + 1 must be positive, and since x, y and z must be integral, / can only » 0. Hence, when / = 0, we have x ■ 1, y s 9, toki. s ■ 90. 150 nnr to !M».ls.M. IfisoiLLAniovB EnRonjis. . 1 a a 2 IT-2U 2. {(xa - «-») + 1}» - {(a:* - x-«) - lj« B (x»- «-«)' + 2(x»- x-») + l-|(««- «-«)'- 2(«*- Jf-*)*!} = 4(x*-a:-») ^ 3. The O.C.M. of the first three qtiantities is evidently a + ft, and 'as it is also a measure of the remaining quantity, it is their G.C.M. b'^ b* fr'-^H-ofr ab 4. Since x = r — -, x - 6 = t — - - 6 = b-a d-a' and X - o = 6« ft» - tfft + o^ 6- a "x - 6 ' X - a b t« - oft + 0^ 6«-.y+a»~tt« gft-a« 6 - a (b~a)b 'a(b -Ti) o " 6(0 - o) ■ F 5. X + y + « a 16 (i), X - y + « a 5 (n), -x<-y + «B8 (m) Adding (i) to (m), we have 2z s is .: z - 9 Adding (i) to (n), we have 2x + 2z = 20 .*. 2x a 2, and x s l Hence x + y + 2= l + y + 9 = 16.'. y = 6 6. 6-^x1 - 3^'8T8 + 2^121} X 5 - iVeTxl « lfi{^5 - 6(^6 + 10^6 - 16^5 = (15 - 6 + 10 - 16)^6 = 3^5 1, x« f I = 0. Divide each aide bv x' : then x' •f -s ^ »■ .*. x« + 2 + -J = 2 .*. X + —- s ± ^2 ; clearing of fractboa x«?xV2c-.l.x«TxV2+fY) ''*-^'-*«*-**vi" V^ Hence X a — .^ ' • Hit. Bx. 8-11.] ALaXBRA. 151 a+ b b + e ^, b+c c + d b+e e be ^^"^l r ••• cTd M-- »"*T ■ T ••• bTi * FTd Sencc a + b : b + c :: b + e : c-t-d g-t- ft 6 c •*• ft + c ' c a+ft ft+e ently a •!• ft, mtity, It If oft ab ■ ft- a' a" oft - fl« « « 8 (in) md X 8 1 9. a : c :: 2a - ft : 2ft - a e 2a 20 a : 2a -r ft 2a 2ft - e 2c 2a - ft * 2ft - c ®» 2a - ft ° 26 - c -6-"3Fr2A tbe H. a + mean between a and -r .*. a, -r- and -?- are in H.P. 2 3 10. Sam to n terms when r is a proper fractioa 1-r Sam to 9C pa It (-t)' '{■ - (■ - t)'} • P times the snm to n terms. \l, X* - X » /xi - « • \ x" + 1 + * -• x« - «• M Sa x« - x' 1 ' »« - X* » X* • - x"T x'T - x"T 152 mST TO [Xm. ts. lUtl. X* + «*x^ + J)x^ + a'*' + o^(«* - a*x* + a' X* + o'x + a'x* - a*x + a^ - o'x - a'x' - flx' a'x* + ox* + o* I o'x* + ox* + A' 12, 1« trial dir. = UYx* !•» comp. dir. = 147x* - 63x^ + 9xV 2««« trial dir. = 14tx* - 126x»y + 27xV 2»« comp. div. = 14'7x* - 126xV + lllxV - 36x^ + l«y« 13, g3m-n*sn-p*ap-in _ ^mtn + p = abc X x" = oAc x 1 = oic abc X x'"«*«"*'**'"i» 14. {(2x»+ Jx- V) + y}{(2x'« + Jx- V) - y} « (2x«+ |x-"y«)'-y» V *'=4x* + 2y»+ix-V-y'=4x* + y*+4x-V {(x»+ 6«) + flx}{(x»+ 6=0 - ax] = (xa+ 1«)*-«'*'= ** + 6*+ 26»x»- a^x" (x» + y*)(** + y") = x"*** + x^y* + x*y' + y' ♦« " " 15. (3V6 - 2V3)« (2V5 - 8V3)(3V5 + 2V8) 33 33 ,ij(46 - 12V15 + 12 + 30 - e-ZIB + 4VTF - 18) « ^(fi9 - 17/13) 1 12x«+l * "rornr " 24xT6i 16. 2x + 12x»+l 2X -1- TT 4x 11. 2x + 1 - (2x - 1) 4{4x»-l) 2x + 1 2(2x - l)(4x» + i) ' 4x* 4- 1 + 4x» 4- 4X 4 - 1 4x» + 2x + 1 2(4x«-l)(4x*+l) * 16x* - I 12x» + I 12x» + 1 2x+l 1 2(2x - l)(4x''' +1) 2(4x* - 1) (4x» 4- 1) + (2x + l)(2x + 1) 2(4*" - 1)(4«» + 1) SS. 11-17. M». fix. 18-ao.] ALGEBRA. 158 y»+19y* 18. (I) x(a - c) o-c (« + o)(x + c) ■" af + o - c • • (X + a)(x + c) " » +o-c '. x* + (u: - ex = x' + «* + fx + ac ; 2cx » - ac .*. x = - -r- (II) V(* - l)(x - 2) - 2 » V(* - 3)(x - 4); squaring {X - l)(x - 2) + 4 - 4V(« - l)(* -~2) » (X - 3)(x - 4); /. 2V«'-3x+2 = 2x - 3 .-. 4x«- 12x + 8 = 4x='- 12x + 9 .% 8 = 9 which Ib absurd .'. the equation has no possible roots. 1 I 1 ^'"^ (X + 3)(x - 6) ■•■ (X - 5)(x + 7) ~ (X + 3)(x - 16) ^ ^ .: (X - 7)(x - 16) + (X + 3)(x - 16) - (x - 6)(x + 7) = x» - 9x - 112 f r« - 13x - 48 - x» - 2x + 36 = ; x» - 24x = 126, whence x = 12 i V269 6+cl b-e 19. Since n = r — -, — ^wlll = r-— ■; then H. mean between nand '• b' JL - 2 ^^ 6»- c" ynr^' Buto:6::6: c« • • a« + 6« 6« + c» .*. also is the ff. mean \ 9-17/nJ) 12x« + 1 24x* + 6a 1 8<4x'' - 1) )(2g 4- 1) 1) between « and —• 20. Let 10 B work and x, y, s » times in which J, JS and C can separately perform it ; 10 Then ~ » JPb daily work + B's daily work (i) 10 6 --^B " + Ca i( (") 10 c + Cb (1 (ra) 19 W - Ci II (nr) 10 10 10 Then adding (xii) and (iv), we hare r = 2Fi daily 154 work = 2- 10 KBT TO „ 2 1 1 1 Heuce — *T""T+"7* mm. Ex. be - ac i- u b • 2abe 2abe iabc "^2/-A,.^ + ^; similarly x = ^^^^_^ ,andz»;;g:;^^;^^ a» fe« c» signs. Hence /. c. m. = (a - &)(a - c)(c - 6) 6c2 3=1 o''(c - fe) + 6''(g - c) ~ <:'(a ~ b) t?e'd*b^aJI»*^cli^-ac*-^bc (tt - t)(o- c)(c - b) " a»c-o»6 + o4»-c6"-ac=' + ftt- 22.«'-(--46— ; = (-+ — 46— )(« 46—) a« + 4a6 + 46" - 9c» 9c* - fl? + 4a6 - 46" X 46 46 (a + 26)3 _ (3c)a (3^)* » (a - 26)« 46 46 (g + 26 + 3c)(g + 26 - 3c) (3c + a - 26)(3f - a -f 26) 46 '^ 46 (tt + 26 + 3c)(tt + 26 - 3c)(a - 26 + 3c)(a|6 - a f 3c> ^ 166* 23. o* + 2flW + 6* - 2o»6« = (a» + 6»)' - («&V2)" s (a» + 06V2 + 6»)(a« - 06V2 + 6") Similarly a* + 2o»6> + 6* - 3a^ = jf •.3d=|f-4s-Jf; hencerf = -if .•.a=4-2e/='4 + Vi' = 6H /. A series = 6Ji + 45f + 4 + 3^ + &c. 166 KBY TO [Hit. Ex. 2SK}1 (III) 3«^ term = flr«a 4, and 6'' term s or" s |f .*. ar* ♦ or"* J^ t 4 4 .'. r* a W ; whence r = J. And o s — ■ 4 4- J - 9 .*. G. series ■ 9 + 6 + 4 + 2} + Ac. 80. Let X m length, then x - 60 s breadth in yards, nad «(x - 60) :: 5500 ; that is x* - 60x 3 5500, whence x « 1 10, and 81. (I) (x»- y»)" + (X - y)> = (^^y) = (x« 4- xy> y»)' (n) 1 + - a + 1 7 + + + + 6 - 4 + + 8+9 + + + + + + \\ - 14 + + 28*- 14 - 66 + 64 + T + 0-14 + 7 + 33 - S2 t + - 14+7 + 33-33 - 69 + 100 - 23 (in) 7«' - 14x» + 1x^ + 33x - 32 - x^-x"** S9x> - lOOx + 23 X-X' x" + 2x - 1 -.- B x"*"* + X™"' + x"*"* + X*" ' + x"*'* + &c. We observe here that each term is derived from that preceding It bj dividing by x>. Let us now assume that this is true to r - 1 terms, and we have then left as remainder x""'''" ■*-' - x ". Dividing thiii by x - x ~^, and we get as first term of the quotient xw-ac+i) which will be the r»'» term of the quotient of x"» - x "« ♦ X - x.-». But x"*-*** = x"*-*'"- »^*» ^ x» = (r - I)*" term ^ x' .'. if the law is true for r - 1 terms, it is true for r terms. Kow it evidently holds for 5 terms .*. for 6 and .*. for 7 terms and so on, and .*. it is generally true, and since the first term is a^' S and each term is derived IVom the preceding by + by x' .'. the r^term is x'""*"'''''-^ » x"**"*!^ jf j^ be an even number the quotient will contain an even number of terms, and will be x'''**+ *"*"•+ x**"' + Ac. + X**" ("*"*) + x^'C***) + x**" ("*■*■') + Ao. + x**»-*» a. x'»(x-* + X-' 4 x-» + Ac. to X* ") + x-*+ X"* + X-" + &c. to x*-"* .*. first part of quotient « second part x x*» [Mlt. Ex. 2!K8i ir»for'« J^t4 a 9 .'. G. series in yards, nud 36 X- 110, and :«* + *y> f)* 3+9 W 64 33 - 32 100 - 23 OOx -f 23 2x- 1 that preceding ia true to r - 1 • a/r-x; _ jg -M_ of the quotient intofx*-x " - 1)* term ^ ^ i« for r terms. .*. for 7 terms he first term is ing by t by x' eyen numbei 18, and will be •*) + x^-C™*') ftc. to x* ") t of quotient Mis. £x. 0-07.] ALGEBRA. 167 32. (I) Let V37 + 20V3 ■ V* + Vy i then V37-20V3- V'-Vy .'. V1369 - 1200 ■ Vl69 ■ 13 ■ » - y. Alfo 3T + 20^3 = X + y + 2Vxy .'. x + y ■ 37 ; hence x ■ 25, and y « 12. Then V* + Vv ■ V^ + >A^ * * + W^ (II) Let V4x + 2V4x«-l * V** + Vy 5 *^en V** - 2V4x» - I = V'' - Vy ••• Vl^Jf'-iex'-t'i « V* " 2 = x' -y.^ Also 4x + 2V4x»- 1 = X' + y + 2\/x'y .*. x' + y « 4x .-. 2«' » 4x + 2, or x' » 2x + 1, and 3y = 4x-2 .'. y = 2x-l. Then V*' + Vv = Vax + 1 + V2x - I 33. (o* - X*)"' » {o*(l - o-*x-*)]'' « a-"(l - a-*x-*)'" 3-4 3-4*5 3-45-6 *l-2-3 ^l-2-3-4 '3'4-6-6 +&C.) Hence S*" term =o-Wx,-r-s-;«'"x-", a- »xl5o-»«x-»« = tt-"(l+70-*x-* + --o 1 1-2 c 15-"x-" 1-2-3-4 34. C, 28 X 27 X 26 X 25 X 24 X 23 X 22 1184040 1-2-3-4-6-6-7 35. (X* - 4x" + 10 « 12x-» + 9x-*)* « ((x* ~ 4x' + 4) + 6x-»(x»- 2) + 9x-*)* = {(x»- 2)»+ 2 x 3x-»(i'- 2) + (3x-»)^* = x'-2 + 3x-» 36. Letx = ^l; then x'«l and x*-ls0.'.(x- l)(x'+x+l)a0 .-. x-l = Oorx= 1. Al80x'+x = -l, whence x = i(-l ±V-3) .-. ^I » 1, or J( - 1 i V"^- Also 1» « 1, and {K - 1 T •/^l" ■ l + l(-l± V-3), i. e. sum of the cube roots of unity = sum of their squares. 37. (i)bx + ay = ab = ax-a?+b^-b'';ax-bt-ajf + by = o'+ 6', o»+6» ft(a« + 6») orx(o-6)-y(a-i) = a'+6*i x = y+ ^_^ - .•. ——^ — + by+ay = ab iu2 + 6» 6o«+6» i»(o + 6) .-. -^-j^ + y(a+ 6) = a6; y(a + 6) = o6 - -^^g- = - ^_^ iTT— ••• y = - iTTt = 6^' and X = y + -— 6» a* + 63 -j2 + a» + 6» ft-a"^ 0-6 a-6 a-A 158 KEY TO [Mm. Ex. 37-12. (u) If ia these equations wo write x for y, we shall obtain ralues of x and y, whioh will limaltaneonslj satisfy the giyen equations. Thus x' ■ 6as + 4x ; or x" - 10* ■ .-. x* - 10x« is a factor of the reduced equation in x. Now from first equation, x' — 6x y a _ — . substitute this for y in the second equation. Then i^y a Ax + 3(x» - ex) •. x« - 13x" + nx' + 80x s 0. But we have shown that x' « lOx is a factor of the left hand member of this .*. (x* - 10x)(x' •> 2x - 8) = 0. Hence x' • lOx = ; whence x « or 10. And x' - 2x « 8 ; whence x s 4 or 4 2. Then x = y » 0, or 10, or - 2, or 4 88. Let X = yds B sold for $1 ; then x + i » jds J sold for $1 ; 1 -- a what B received for I yard, and Then 90 40 x + J what ji received. ^ + — r-T = 42 : whence 2lx'- 6Sx = 15 .•. x » 3, and '"•*'" 2* ; Tor I" a 7-2 5 39. Insert 5 Jl. means between 2 and 7 ; a « - — r a = — r = — . Hence Jl. series is 2 -i- 2g 4- 3| + 4) + 5} •(- 6^ + 7 ; that is 3 + y + Y + S + ¥ + V + ''• Therefore the H. series is 40. The /. c. m. of denominators = (a -b)(x - o)(x - 6) ; then clearing of fractions (a + c)(x - 6) . (d 4- c)(x - a) = (x + c)(a - b) ; or ax + ex - a6 - fee - 6x - ex + aft + ac B ax + - tf)(x - 6)(a - 6). Hence i(a - 6) - o(x - 6) + ft(x ■ (X - a)(x - 6) ; that is ox - 6x*+ ab - ax + bx ' ab 'K 3^ ax - bx + cb\ that if i»-(o + 6)x = -aA •. a'-(a+6)x + f-j-j ■ (-J- ) - «* «3 - 2ai + 6» a' - 2a5 + 6* a + * — ab * ; .'. x - a- b 3 '± 2 -, whence 4 x = aotb (III) Multiplying by 168 168 + 63x - 48x + 8 s 186 .'. 16x « 10 .•. x > } 46. Haltiplying by (x - y), (y - z) and (z > x) refpeetirely, we hare i» - y» ^ 3t(x - y) \ / - «■ ^ b(y - «) { .-. by addition 18x - 9y - 9z = :' - a:» a 19(z - x) ) .'. 2.r - y a z; sobstitating this in third given equation, - (21 - y)> + x(2x - y) + x" » 19 .♦. 7x« - 6xy + y» = 19 ; subtract I from this the first equation, and we hare 6x' - 6xy s . ig x' + S ■. y ; substitute this in the first given equation, and we I have x'^ + /x»+3\ /x«+3\' « x»+ x"+ 3 + i^y- 31 I clearing of fractions ; 2x* + 9 + 9x' + x* » 3tx* .*. 3x* - J8x* = - 9, x* + 3 9 + 3 whence x*s 9 or J .*. x = i3ori JV3; y» i3 ±*, 16C i +3 10 or ¥ KEY ro ilOV3 i W3 ■ ij/3 " ± V3 3 [MlB. Ex. 4»-4?. 3 ; «= 2x-y«±6T4 = i2, or = ±!V3?W3 = T|V3 4T. The qaestion amounts to finding the least real value of x, which will satisfy the given equation b^x' " 2ahx = m - a'6^ - 2a'6 - 2a', where m represents the least , 2a m-a'6»-2o»6-2a=' value which makes X rational ; x'- -r-x = -^ 2a a« a» + i» - o»6» - 2a% - 2a' m - a't» - 2o'6 - a^ ^'-T + i' 6» 6» a ±Vot_--^'(6»+26 + 1) X ^ y :s X = a i^/m-u\b+iy b • * - b Therefore the least value of m that will render x rational, is m s a\b + 1)', and .'. the least possible value of the given a expression is found when x = -r-, and is therefore a\b + I)' 48. {(x6i» + 6 + 9x-6») - 4x»(x"' + 3x-»0 + 4x2P|^ = {(x" + 3x-V>» - 2 X 2x»(x»' + 3x-V) + iZxPyf s x"' + 3x-"» - 2xP 49. (1) S = {2a + (n - l)d}|- = {6f + (8 - 1)^}| s (6f +"20)4 s 4 X 26f = lOYf a(l - r") 81x"{l -.(-}a-ay)»} l-(-ix-V) (11) fir Ux 1-r r ' /256x-'V\1 ,./ 256x-»V\ "Y - (■243ir2f)} 243xM^i . -^^jj^) 243x1* 1 + Ix-'y 2 56X-V 27 3x* -I- 2y 3x* + 2y 656 Ix^- 256x-'y8 Six' + 64y 243xi*[l-(-}x-'^)''} 243(1- (-n *i (III) S^ = as above ^^^r^ -'^ '-jy-^ 243 X I 243 s -^ « 60J Uu. Ex. 6O-60.J ALGBB&A. 50, a(r'- 1) Y~ = sam of first three terms, ana af*(r9 - 1) r-l = sum of next six. Then 72 a(r»-l) I r-l 1= r-l I or dividing each eal value of x, | ^y -^p, we get 72 = r»(r» + 1), whence r» » 8 or - 9 .-. r » 1 .'. any series having r s 2 will answer. 51. x""* ""*''•■'*'"''"* *"**'"*' = x" = 1 («* + 2x'' •♦• 1) + x(x» + 1) (x* ^. i)» -1. 4r(JC« -t- 1) ^^' (X* + 2x» + 1) - x(x» ;• 1) ~ (x» + 1)» - x(r' + 1> (x'-UXx' + x + l) x» + X + 1 ' (x»+l)(«»-x + l) " x^-x + l 63. x' - 2(a + b)x = 3a? - lOab + Bb' ; complete the square i» - 2(a + b)x + (a + 6)' s 3a* - lOaft + 3i» + o? + 2a6 + 6» = 4a» - 8afr + 46^ = 4(tt - ft)!* .-. x - (o + 6) » i 2(a - 6) ; or x = o + 4±2(a-6) = 3a-6 or 36-0 • 54. Because VF-^ • V^o"-^ ;: 2 : 2 .*. Vy~-^ = V'*^^ - ^ .'. y - X a 20 - X ; hence y - 20. Then from first given equation, \ly - V20-^ = V20-X .". Vy = 2V20 - x, and hence y = 80 - 4x, but y s 20 .'. 20 = 80 - 4x, or 4x = 60, and x a 15, and y = 20 55. {(X* + 1) - 2x}{(x» + 1) + 2x} + 2(x» + 2x + 1) = (x»+l)»-4x«+2(x'+2x+ l) = x*f 2xH I - 4x» + 2x'+4x + 2 = x*+4x + 3; (X* + 2xhf^ + y3)(x* - 2x*y^ + y*) - 2y»(x* ^- 2x'y^ + y») = (»' + y^)'(«'' - y*)* + 2y'(x* - 2xy^ + y») = (a:* - y*)* + 2a;V - 4a;y^ + 2y« = x' - 2xV + y' + 2xV - 4xy* + 2ye = x« - 4xy* + 3y^ 86. Multiply S"* given equation by 3, then 3x2y + 3xy» = 36». Add thiS to 1*^ given equation, and we get x» 4- 3x=|y + 3xy» + y» = o» + 36» .-. x + y = ^T"3J*. 6» But xy(x + y) a 6" .•. xy(yo* + 36") » 6' .•. xy j^o» + 36» 162 KBT TO Then «» + 2«y + j» = (aP + Sft^'i and 4«y » ! [lizs. Ex. 66-60. 4i" .*. x» - 2xy + »" = (<^ + 3ft") I. 46» (a» + 36») J (a» + aft")* Then a: - y = ± V (a» + 3ft»)' (l - 55^) x-.y = ±(«» + 36«^*V5rf35i « + y = (a» + 34»)* .•.x«K«^+36^*(ll\/$T3p) 35 57. Let X = number at first, then — ^ whAt each had to pay, bnt two left, therefore the number remainining = x - 2, and con- 35 35 36 consequently - — 5 = what each paid. Hence - — ; = -7 + 2 .*. x' - 2x = 35 ; or x' - 2x + I = 36 .*. X - 1 » ± 6, and x s 7 68. (a* + 6*)* * «"(l +a"M)* . «Kl , la-M . i-^(«-M)* . J|:|(aM)' + *c. Hence 4t«» term = «» x 77^75(0' *6* ) » «P x 4a ^6^ a 4a*6^ 69. ( - 2a»x*)* X 4o*x» - 2o»x* x - 3ax» x 2 » I2a»x" - iox' x Jax> X 2 s - lo'x* - x« x Jax* x 2 » - ox" a« X - Sax' X 2 = - 6a»x« (12a^-o)x" * -V«^' .*. coef. of «" a (12a? - a) (4a« - ^tf)x* .'. coef. qf x« = (4a* - ^a*) tiB. fix. 66-69. (tt» + 36»)* ihad to pay, . 2, and con- and « = T 4- Ac. X 2 = I2a»a:" 2 a - \ x» + x+g + x-^ [y* - ij \^X» - 1 + X~') ° X* - x" + 1 - «» + 1 - x-» x'+ 2x + x-* X*- 2x' + 2 -x~*» x + 6 62. x" + 2x' + X x' -2x*t2x''- 5 X - 4 x(x' + 1)» J-^ + x6 - 2x* + 2x» - 1 x + 2 (X + 7)(«! - 6) ^ (X + 1){x -<• 3) ~ (X - 5)(x + 3) .-. I. c. m. of denum. = (x + Y)(x - 5)(» + 3) ; then reducing to (X + 6)(x •}• 3) + (X - 4)(x - 6) - (X + 2)(x r 7) common denom. = O^TlXx - 6)(x + 3) x» + 9x + 18 + x' " 9x + 20 - x" ^ & .t - 1 4 x*~9x + 2 4 x»+6x»-29x- 105 ' x»+6x«- 29x^06 (n) {(X* - 2x» + x») + J (x» - X) + -|i^}* = {(x=' - x)» + 2 X }(x* - x) + (i)«}* = x» - X + J 64. (x"* — 2y'')(«'* — y*) = x** — 2x'''y* — x'*y* 4- 2y''* = x^- 3x*y* + 2y* ; jx"' + (»• - 6)}{x*' - (ax« - 6)J = (x"*)' - («m:« - by = x'«''-o»x»»+2a6x«-6» 164 EBT TO [llit.Es.66-n. 6b. (i) 12(a* - 2*) ^ 3(x - 2) = 4(«» + 2*« + 4* + 8) = 4a:" + 8x» + 16* + 32 (It) 4-24- 1)20 • 22 + 11 - 3(5 - 3 /. qaotltnt • 6aV - tab* 20-10+6 -12+ G-3 *12+ 6-3 66. Let a ^ a a b c a b e a a a* r b '.: b : c '.: e : d\ then a : i a a c .'. o : 2 .'. valaei are 8, 4, 5, 6 Md *l \ or thus(-x2+10x-16)>0j -(x»-10«+16)>0; -(«-2)(«-8)>0, .'. X - 8 must be negative, .*. x < 8 and x - 2 muit b6 poiitirc, .*. X > 2 68. By Art, 106, (vu) 26 20 1 V*-5 4Vx - 5 .*. 6 = V^;- 6 ; or X - 6 = 25 .•. X - 30 a & 69. y + — > 2 J if a'' + 6» > ^a6, but o' + 6» if greater than a b 2ah by Art. 134, Note 2, .-. y + ~ > 2 70. Let n - 1, n and n + 1 be Hbe three numbers; thtn (n - 1)* ^ V? -^^ (n •{■ 1)' « the sum of th«ir cubei « n» - 3n» + 3» - 1 + n» + n» + 3»' + 3n + 1 = 3n» + 6n « 8«0, It bd poiltive, 1 r a v*^ • greater than iree uttmbers; of tb«iir cubei 6»«8»- + .(I)-') • •.. (x* + o'). Now the product of the first tvrc ♦erms /i\n-3 /n*-* a x^"' - a^*^ ; and the product of this by the third factor s x^" - o^" , and so on. Hence the product of the first n- 1 factors will be (x' - o*), and the product of this by the n** factor (x' + o v will be x > a which .•. = the pro- duct of the given factors. Note.— All this will be perbfeps more ~ ident to the student, if ho tukei a nomerlcal example, and exatnines how the indioes are affbcted by ni'iltiplyi&x them as In the above question. Thus suppose n = 5. .Tben{x'"') {x' "^i 73. 13(2x + 3) - 7(2x - 3) X- 4 12X + 60 X - 4 12(4x8 _ 9) 4x' + 9 " 12(4x'»-.9) 4x^+9 x + 6 4x»-9 ~ 4x* + 9 30x'4- 18x 4- 9 16x* - 81 16/*- 81 -4 4X» + 204i^ «x + 46 - 4x»+ IGx'' + 9x - 16 wm (2x* + 5[y*)(2x* - 3y*){(4«* + 9yO + e«V) {(40-* + 9y*) - 63- V! - {ix^ - 9y*)(^3'' + "^Zi-^y* + Sly - 36x*yO X (4x* - Oy ) {\Gx- ■! 36a:*y* + Sly) = e-kc^ + Uixy^ + 324x*y - 144»y* - 324**y - t26y* = 64x* - V29y^ 75. Multipljiag first equation by V^t a>^<1 second by V^) ^^ h'ive 2x^6 - 6y « 6V2 (i); 3xV6 - 6y ■ 6^15 ■ 16V2 (ll), 9V2 9-/n Suibtracting (i) from (11) xV6 = 9V2 /. x ■ -j^g ■ -^ •- 8V8 ; then 2xV3 - 3yV3 = 6V3 x V3 - 3yV2 « 18 - 3yV2 « 6 12 .-. 3|V2 = 12 and •'• y = 3J2 = V2 76. S„ of 1 + 3 + 5, &c., = (2 + (» - l)2}y = (2 + 2n - 2)j ■ n" i» ft S to J« terms = {2 + (4» - l)2|y « (2 + n - 2)-^ « i»»; n» then sum of last half of the series = n' - -7 « }n' » 3 timet in* 77. The ./I. iu bsiitute ibis for b 2ac 0+7 Then 0+ c s « a + c 2ac a-ftf 2ac a+ c - a 2ae a+ tf a - c (c - «) «(« «- r) by V3> we 16Va (II). li n 2 2)-r ■ n" '■J " i»'j timet in'' H. mean ft A %ac ■ HTc' c(tf - r) ^ut* £x. 78^1.| ALaBBRA. lt»< * 1 I •c-i^»a<-»c* 2tu — a' - c* Now (c-a)(a-c) reversing the steps of this operation, we Bhall have proved the point reqnired, 19. » a r + « + *, and $ » m — , and t = nxy' .•.» = r + — - +nxy* = ri-m+n \ 8 s r + m + 2ln \ From these equations r s 1, m s - ^|, and n s -^ 1-r+O+O ) •••» = »■ + Y + »»«y' = 1 - By + iVy" 80. {(a + b)x + (a - ft)H(a + 6)x - (a - 6)} = (a+6)V- ( - 4a6 „ *(* i V^'+ 4a'- 4a6 X - 2o(a-ft) 2a(a - ft) X = 2a(a - ft) 81. Product of first two factors ^= d^- ft'; hence product of first three factors = a* - ft*, aLd product of first four factors = a* - ft". Hence it is evident that the exponent of a or of ft in first term is 2", in product of first two terms 2\ in prouuct of first three terms 2", of four factors 2', of five 2*, and so on ; hence the exponent in the product of first n factors will be 2**'^, and of the series to n + 1 factors, the exponent will be 2^. Hence the required continued product Is a*" -- ft"* 168 KBY TO {Mis. Ex. 82-87 83. + p -?1 1 - (a + i + j») + (aj» + 6p - c + 5) - (a^ + 6g - '•p) - qi 9 ~ep + (aq + 69) + 9C + i 84. l-(a + 6) - c + Hence quotient > x" - (a + i)x - c 83. {(o»x« + 2a6x* + 6V) + (2acx^ + 26c) + c** " »} = {(ox" + 6«)' + 2 X ctf-»(a«« + 6x) + (ex -*)^* = ax* + 6x + cx'^ (x-oX x 4-6) + (x + u)(x-ft) (x - a)(x - 6) + (x 4- q)(x 4- ft) (T»-o»)(Jt*-6''') * (xa-o'K*'-*") x'-ox-t-ix -ct+x'+ox-ftx-fl ft (x' - »)(x" - px + ;>»), and (x' + px + p'Xx* + pa: - p») is evidently x' + px + p'. Otherwise by ordinary rale, thus : x*+p»x''+p*)x* + 2px»+p»x»-p*(l x'+px+p')x»-p»(x-p x*+p*x»+j>* _x»+ px» + //^a: 2px*-p* = 2p(x» -p*) -px* -p^x-^ 3* - p«)a:* +p'ar» + p*(x -px»-p»x- /)" x*-p'x ' pi»x''+p"x + p* =i»"(x'+j)X +p») /. G.C.JIf . = x»+ px +/ 86. \(x + 5j(x - 4) ; y»(x - e)(x + 5) ; and V (* - 6)(x - 4). Hence l.e.m. « V(* + 6)(« - 4)(x - 6) s a^(x« - 6x» - 26x + 120) 87. x» - 2(a6 + l) 1 - 6» ■ 2(06+1) a»-l ' * " a«-l + /o6_+_l\ /a6 +2y » 1 - 6» + o»6» + 2 16 ■•- 1 + a' - o«6=« - I + 62 a»-l (a»-l) ffJi + 2a6+_6* (a"-l)' (fi ! + (a+l) 06 + 1 a + 6 ' . J-l) + (a+l) 06 + 1 i (a J (6 +:}> + !) or = '•'-I ~ (afl)(tt-l) (a-1 (flf+l) o^ ., 6 - tt + 1 (fr - l)(a - I) »-^ ~ a» - I ~ (o + l)(o - 1) ° o + 1 6+Jl a-i' i If IS. Ex. 82-87 Mm. Ex. 88-00.] ALQEBBa. 169 i 88. Multiply both nnmerfttor and denominator of the first factor by x ; then x * + x-'+2(x^+l) /x»-iy x-\x* + \) + 2(x'+\) /x»-iy ,*_x"-2(r'-l)' \*'+V ""x »(x«-l)-a(«"-l) \^*»+lJ /x • »(x *-x»+l) + 2 \ / x"- 1 \ / x'-l+x-'-f 2 \ / x»- 1 \ (^x-2(x* + x» + 1) - 2y \^*'+ 1/ * \x» + 1 - x-=» - ay \f^) x« - 1 x« + 1 /X » 4- I + X - » \ / x* - 1 \ / x* + x»+ 1 \ / x» - 1 \ (^i'-i+x-y ^^j?T^y '^ ^^x* - x» + ij (^x» + v ° »-l Then 89. Let n represent any square number ; then ^^-~ will be half the next lower number, and —r- will be half the next higher. /n-iy n»-2n+l 4n + n»-2n+l n» + 2n+l '»+(-rj =«+--i 4 — = —4— ^>0. Let X, y and z represent *^je number of hours takou ^y Jt, S and C respectively to fill or ompty the cistein ; consequently, it. 1 hour J will fill -th of it, B, —th of it, anr' C will tmn y X y 1 — th of t. z • Then3(i--|)+^=1 5f— -- — j+^--l>i.e, once the contents of the cistern A i. li »x'^2y ~V 7 3 27 6 6 1* Hence--~=l;4^--=I,and3j^ + 5p=l. Multiplying the first of these by |, and the third by | 35 £ 6 1 35 7_ ^i '' iy"^ z 'T" 3 ' 6 6x'''4y'" 2 J = 1 Also^-J if = ij «= V ; whence ,»- - i, or y 3 ITO KST TO [Xu. Ex. 00-03. 21 6 21 6 21 15 Then t-- — ■rr-T" 1 •*• -~ = "rj-l = T;;;Of"r" 4y 12 la 12 x-i 91. 8 as 4 8 1 2x or 2x x = 3 ax» + 2at* - 6x« + 4x" - 9 8 3** + 3x* - 10r» - * + 3\ 6»» + 6x* - 16x» + Ux* - 2l(2x / ea?" +6a*-20x»~ 2x' + ex 6«»+14*»- 6X-27 8x* + 3x» - 10x» - X + 3(3x - 27 (te» + II**- e«- 2T\ 16x* + 16x» - 60x» - 6x + 16 y 15x« + 42x»-18x»-81x _ 27x« - sax" + 78x + 16 - 136a ' - 160x» + 880x + 75 - 13Px» - 378x» 4- 162x + 729 218x> + 218x - 654 i» + X - 8\ 6x» + 14x2 -ex - 27(Bx + 9 ') = 218(x* + x-3) 6x» + 6x» - 16x 9x»+9x-a7 9x» + 9x-.27 92. /. c. m. of (ax + 6)(jw + q), and (ox <¥ 6)(^ -> jt) « (oi + b)(px f qXqx -p) m apqjfi + (oj" + bpq - aji^)x^ - (apq + dp" - bq*)x - ipg I. cm. of x(«ir- y); (« + V )(* - » ); and y(* + »)» that Is of x(x* - y*)(x* + xV + v') > (^f* + y*)(x* - j^*), and y(x' + y*)(x' - z'y* + y*) = u;(x - y)(x + y)y B xy(x' - >*) ■ x"y - xy" 93. (i) Maltiplying by 273, we have 91x- 182a a 78x + 234a - 21x - 43«; or 34x - Zlia, whence XB Ua 9. Ex. 0(M>3. (2x \x (x-2'7 + 3(3* - 27 6* + 16 81g 'ie«+ IB f 380x + 75 ^ 162g 4- 729 218x - 654 x-8) Mk. Vjs.. oa-&7.] 6)(3X - y) h(x + y)y ALOKARA. 171 (ii) Reducing flrti m«mbtr, and «Iso the Moond member, jc-6 x-6 11 ~6~ ^ x»rT » °'' dirldlng by x - 5, we have j = irTl* whence x* - 1 « 6, or x' = 7, or x = ± ^7 (III) Squaring eaeh side, x 4- 4 4- 2V2x' + 14x + 24 4- 2x + 6 = 3x4-34 ••• 2V2x»+ 14x 4- 24 = 24; orV2x*4- 14» 4- 34 = 12 .-. 2x« 4- 14x 4- 24 = 144, or x« 4- 7x = 60 ; x» 4- 7x 4- (J)' = 60 4- ^ = »ja .'. X + J - ± V^ .-. X t. 5 or - 12 (iv) xhf - X* + Zxhj - 3y = <^x* + By, but x^ e 6 .-. 5 - x» 4- 16 - 3y -Jxi'-v 3y, or 20 - (x» 4- 3y) = ^/xTly .: (x» 4- 3y) 4- V*' 4- 3y = 20 .-. (x' 4- 3y) 4- (x" 4- 3y)* 4- i = V .-. (x'4- 3y)' -£1-1=4 or -6; squaring these, x»4- 3y » 16 or 2S. But since x^y = 6, x* = — 4- 3y = 16 or 25. Hence 3y" - 16y = - 5 ; or 3y''' - 26y = - 5. From first of these eqnat. y = 6 or }. Hence x = 1 or ± ^1^ 94. (X - 2)(x - 3)(x 4- 2 - V - 3)(* 4-2 4- V - 3) = (x» - 5x 4- 6)(x' 4- 4r 4- 7) = X* - x» - 7x» - 9x 4- 42 = 95. a-a-\-m + a-m + m-af*m 96. a» 4- 69 = a" 4- 2a*6* 4- 6» - 2a*6* = (a* 4- ft*)* - (o^V^)' = (a* 4- J* 4- a»6V2)(a* 4- i* - a'6V2) ^, '^ 97. Since Art. 260, .i = i(o 4- 6) ; fl = jr-j, and G = V^^i we have by substituting the^t values for J, H and Q 2ah 0+6 a + = 1 4- Vo4- 6 /\a4- 6 J or or 4a6 (o 4- by' (a 4- 6)» = 1 4- (2o6 - o^ - a6)(2a6 - i^ - a6) a6(a 4- 6)» ab(a + 6)' 4- (a6 - o»)(o6 - 6») ai(a 4- by ab(a 4- 6)2 4- a(b - aXa - 6)6 a6 + 3a6 + 6> + a6 - fr> - a'f 4i6 or iab = 4a6 98. Let X = minute divisions the hoar hand pasies over ; then 12« == diviaions pftssed orer by minute band. Also 60 + x » minute dirisiona passed over bjr minute hand between two successive transits .-. 12x = 60 + x; or llx = 60 .*. x = 6,\- = minute divisions passed over by hour band, hence time in minutes = 6-^f x 12 = 1 h. 6i*> m. 99. Let X and y = sides of rectangle ; tben Xjf s afciv (x + o)(y - 6) a xy + ay - 6x - 6a ■ xy (x + cXy ~d) = xy-¥cy-dx-cd»xy-e ay - bx ^ab "j ady - bdx = abd cy-dx-cd-e] bey - bdx » bed - 6c /. (6c - ad)y = bed - 6c - abd b(cd - e - ad) whence y - — j^^_^ Also aey - bcx = a6c acy - adx s ac(2 - a« whence x = (6c - ad)x s acd-at" abe a(cd - e - 6c) be -ad If ai = 6c, and be i- f ed aled - (6c 4- e)| a X ' Then x = - — ^^2 ad ° ~0~ * T * *"^ ® whfttevef b(cd - e - ad) b(ed - « ^ 6c) Also y = 6c -ad 6c -ad 6{cd - (c + 6c)} 6x0 be -ad ° since ad = 6c \ a- 6 / o-6\« 3(a-6) '*x-6 o-6 i^^'^-^i-^ =^-^rr26+^"^-Try;forjr6writey ^'*"x-6 3tf 3v Then (I - y)»= 1 - ^; orl - 3y + 3y'- j/» = j - ^^^ Atn. Ex ino-i02.j ALOBBRA. 1T8 - 3 h 3y - y -«'= - aloe whatever 1+y .-. - 8 + 3y -y" - 3y + 3y» - »• a - 8 J a-b or 2y'- j*= .•. 2y"a y"; or 2 ay .♦. — — r ■ 2 .*. 2x-26«a-6; or 2x s a + 6, whence x 't i(a + b) 101. .1 + 2 - 3 5+ 0+ 0- 3+ 0+ 1 + 10+20+10-46 -16-30-16 + 69 + 0+ 0+ 0+ - 122 - 104 + 1S8 + 628 + 183 + 166-*237-942 6 + 10 + 5 -28:- 61 + 70 1-61-62 + 79 + 314 + 391 61X-70 6x»+10x»+6T-23-^irr2rr3 or 6x«+ 10x'+ 6x - 23x'' - 61x-»- 62x-> + 79x » + 314x-* + 39lx-''+Ac. 102. G.C.M. of '(y - 3)*' + (y* - 9)x - y(2y« - 3y - 9) and (y + l)x» + 2(y + l)»x - y(3y' + 6y + 2) = G.C.M. of (y - 3)x»+ (y - 3)(y + 8)x - y(y - 3)(2y +, 3) and (y + l)x» + 2(y + l)'x - y(y + l)(3y + 2) = G.C.Af. ofx»+(y + 3)x-y(2y + 3)andx» + 2(y + l)x-y(3y + 2) = G.C.M. of (X - y)(x + 2y + 3) and (x - y)(x + 3y + 2) = x-y. See Algebra Art. 73 If the student does not clearly understand this method of factoring, he may obtain the G.C.M. bj rule. Thosx' + (y + 3)x - y(2y + 3))x''+ 2(y + l)x - y(3y ! 2)(l x» + (y + 3)x - y(2y + 3) (y - l)x - y( y - I) = (y-l)(x-y) Then X -^^x' +.. (See Ex. LIX, Exaiople 13). 5, of (2»i + 1) + (2fli + 3) + (2iii + 5) + &c. a {2(2111 + 1) + (n - l)a}-j ft n = (4m + 2 + 2n - 2)^ = (4to + 2n)y s (2i» + n)n = 2TOn + n% and it id manifest that the latter sam ecceeds the former bj 2mn, i. e. bj twice the product of m and n fin 110. Let j3 and y be the roots, then /8 : 7 :: fli ! n .•. — = •- ' ' Y fi & HndiB= — xy = — ii x — ;— =- — x — — ' a m+n' "^ ti ' n a m+n a m + n / b n \ / b m \ b* mn e a^c mn b* (m + ny* e b* mn ■ ' "7 ~ a* " (m + ny ' "' iiP " (m-ny " 'ac "^ mn 111. The denominator = a\b - c) + i^c -» ic^- 6'a + ac" = a\b - c) + 6c(6 - c) - '«°d^y=(T=^::6)---^y+^=(Ti^r6) (a(> + l)(j'+ 1) a; -HI I (^ " ") J l~a '^^^"(":nr+T)(a^+l)"y-H= [ 2(1+06) i.TT'^T" t(l-a)(l-6)j(«^+^> TTi 2(a6 + 1)(1 + o') (l-o)" 1 - b (1 >- c)(l - b) 1 - b '^ 2(1 •>■ a^^(a' + 1) " I - a" (1 - a)* "" 1 - a (1 - a)(l 1-6 1-6 *) 1 -a 1-a = 115. 1 -1 1 > 1 -1 T + 21 + 35 + 35 + 21 + 7 - 7-14-21-14-7 7 + 14 + 21 + 14 + 7 = 7x* + 14x»y + 21xV+ 14xy»+ 1y^ 7 + 14 + 21 + 14+7 -7-7-7 - ,7- 7-7 7+7+ 7 = 7a;"+7ary+7y* iliB. iSat. U6-ia0 ] ALGEBRA. 177 no. 8 + 4-6-1 + 1)72 + 36-68-16 + 16(0 72 + 36-54- 9+ -14-7+7 -7) 2+ 1- 1 2 + 1-1)8 + 4-6-1 + 1(4 + 0-1 g+4- 4 +0-2-1+1 ■*2-l + l .'. G.C.M. » 2x2 + a - 1 117. (ix' + x- l)(9a^ - 4) » (2x — 1)(« + l)(3x + 2)(3x - 2) (2x» + « - l)(4x» - 1) = (2x - 1)(« + l)(2x + l)(2x - 1) 118. /.cm. of denomiioators * 8(1 + a;')(l + a?)(l - x)' .'. the given ezpressiou » 6(l+x)(l+xO+3(l-x)»'•', and ?; = tlie n^^ terms of a G. scrien; also let a ^ l»» term, and r • common ratio, then m - ar^'^ Rud — . ™ . ... - jW -i-n + i-.~.m-n . y_ m ~ n Hi?> a a If 00 12s 126 and S, «m » ~ an ' 1 ^ 3-<2 127. i', + «• » 1 + ; Also iS .'.(«- -iStc. t Whei .*?! + A', (^, - .N » {1 r 2 Ex. 19^124. Ml- ^x.m-m.] ALOBBSA. 179 ire In A. V. by fl + 6 "f t : «i(rftdius)^ ig c'.rcle » I- [ of l»' 3 tcnoa of l»'l)tertu3 I. nd s - 23 a G. series ; ' [;n) • = m* '. first term s 1-1 and common ratio , where in - the m^ and n the n^ terms of the series. If one of the terms be taken as the first, the above becomes 125. a = 3 ; ar* = Jf .-. r* « H .*•»•* ± !• Art. 254, r - 2^-3 >n 2«V - a^? " I? < -a 2 4 8 16 •J .-. the series is3-2 + — -- — + — 126. 5„ = {2a + (» - l) + Ac. to (n - I) termsf '- [1 r 2(n - 1)Y = (2n - 1)^ .-. S, + S^ = (5, - S,y 180 KEY TO IMM.^Z. 128-138 128. ( - 1) ^('lyx mof (1 + »-»)'* \rxf 2S-8 {2 + (r-l)3] 2-6-8' \rxdr '(3r-l) (ar-»/ When (r + 1)«> term = Y"« term, r s 6 2'6-8-IM4*lt '•• 'f*^ term = ( - l)« X ^.2.3.4.5.6 x 8»' -u 2618 6501 « - ili «-!•=- 5S9130 •in When (r - ly* term = 10"» term, r as 9 2'S8'U'14-1T'20'23'26 .•.10*»«term = (-l) x i.2.3.4.6-6.f8'9 x 8«» -^ 1594328" 129. (z - l)(x -H l)(a: - 2)(« + 2)(x - 3 - V^)(* - 8 + V"^) B (x* - l)(x» - 4)(t2 - 6x + 11) = *8 - 3«» + 6x* + 30«» - 51x^ - 24x 4- 44 = 130. (x + l)(x - l)(a: -l)«x»-«»-j?+lsO; then x« + 2a;* - 3x8 - 3x2 + 2x + 1 ■{• x* - «* - « + 1 « «' + 3* + 1 .-. x» + 3x + 1 = ; or x2 + 3x = - I .•. «» + 3x + (J)» « | - 1 = ^ .-. X + J = ± iV5, and X = i( - 3 i V5) ilf JL X ^C 151. Let X = the quantity, then .^^ s -j .-, a - 25)(y - 49)' or 5y - 12- + 3y - 2Y = ; or 8y = 152 (X + 1)» = 19, X = 1 ^ ± Vl9 and X B 1 ± VT9 or y - 9 ~ y - 26 . y a 19. Then _. 1 y-4 1,1 I/- 16 1 2 y-86 , 2 9 y-25 9 13 y-49 ' ia~685"6"9'^18 or i v- ^-y-^^ . i y-ig-»+ ^ 6 y~» "^ 9 • y-25 2 y-8C-y + 49 13 y-.49 -0 1 or — , — 6 y + 2 18 9 ' 9 y-25 13y -49 = 0, Ac. i 182 EET TO [llTit. Ex. 187-189. 137. Let X = V + z, and y = i> - z; then 2xy * 4"' + x' Also x' - y' = ©* + 2»« + 2* -V + 2»« - a' » 4»* a 36 .*. r« « 9 .-, 10v5 - tr^ - 6«» = 90 - »2 - 5»» = 4 /. »' + B«» « 88 .-. tr i 2i>zV5 + 5«' = 86 ± 18V5. Extracting square root (right band member bj inspection, and left hand member bj Art. 189) we have v ± Z'^5 = 9 ± V5 .*. Art. 186, » * 9, and ± «V6 =■ ± V^ .-. s « 1 ••. X = 10, y = 8 '^^- ^ = S^ = ^'* = <^'^" ^>"'- 'Expanding by binomial theorem, we have (10 - 1)"='= 10"'+ 2 x 10*' + a-lO"* + &c. + M0 1 + 810-'' +9-10-" + 1010-" + Ac. + 1710-1^ + 18-10-" + i9-10-»o + 20-10-a+ 4c. to infinity. Now 810- 9 + 9-10-" + 10-10-" + 1110-w + 12-10-"+ &c. = 810-9 + 9.10-10 + 1-10-" +(1-10-" + l-lO-i*) + (1-10-12 4- 2-10-") + (1-10-13+ Ac.) « 8-10-9 + 10-10-1" + 10 -11 + 2-10 -" + 3- 10 -" + &C. = 810-9 + 110-9 + 10-11 + 2-10-" + 3-10-i» + &c. = 910- 9 + 010-" + 1-10-11 + 2-10-"+ 3-10-"+ Ac. Similarly for IS-lQ-i^ + 1910-'o + 20-10 "» + Ac. and generally for(10»-2)10-i»«+i, Ac. .•.(10-1)-=':-- 10-2 + 210-8 + 310-* + Ac. +7.10-* + 9-10-9 + 10-11 + 210-12 + Ac. + Y-io-iT + 9.10-" + lO-'o + 2-10-21 + Ac. to infinity = -012345679012345679, Ac. to infinity. Naric.— the point in this operation is the sign of maltipUcation, and not tho decimal point. 139. ax* - 6x = o*x - ab .-. ax* - a*x - 6x + 06 = j or (fifx - b)(x - a) = 0. Now if we assume tfx - 6 » ; or x - a = ; the equation will be satisfied .- ax - J = : or ax ■ 6 .-. x = — a b Also X - a = .-. X ■ a. Therefore the roots are — -, and c, which are rational if a and 6 are rational. Ex. ia7~189. i /. «« s 9 6«" s 86 root (right J Art. 189) sV6 = ± V5 Y binomial L0-* + ke. to infinity. 10-" + &c. + (110-12 1 i generally » + 910-9 ' + 2-10-=« ty. ; ioation, and b - 0; or . b b — , and CL Vjs. Ex. 140-144.] ALGEBRA. Ida 140. x» + (a + fr)a: = (» - l)a6 ; 4.x» + 4(a + 6)* + ( .*. by addition xz + yz + z* = c^ J x' + 2xj/ + 2X2 + y* + 2y2 + a' » a* + 6* + c*, and extracting the square root x + y + 2 = ± V*" + *^ + c* 6« y z * X = i y - ± 2 = ± a> Va'' + b' + c^ Va* + i» T' c" ^d^ + b* + c^ 144. Leta,10* + ajl0"-i + a3-10''-2 + Ac, +ffn.ilO + ffn be any number = (10**- 1)0^ + (10"-i - IK + (lO''-^- l)a, + &c , f (10- iK-i + fli +a« + a3'** +<'n-i-t-<'n Now 9 = 10 - 1, and each of the coef. (10" - 1), (10" ~i - 1), (lO"-2 - I)- -(10 - 1) is divisible by (10 - 1), i.e. by 9 .-. the nunber = 9ni + a^ + Og + fls + • • • • On-x + ''n) where >v is the qiotlent by dividing (10" - l)aj + (10" -1 - l)tfj t.(10"-2_ 1)0^ + •-.. (10- lK_xby 9 Similarly the number reversed = 9m^ + ^n + <^n -i + "n-a I- rtj +Cj + ai. N^umber x 4 = 36m + 4a, + 4flQ + 4aa + • • • *n -i 4% :'l,-','i I? Mi i it. :!/ I! IM KEY 10 tMxi. Ex. 4649. Number roversed x 6 = 45w» + So^ + 5(t„ .i + 5a,|.a + • • • • +603 •f Sag + 6a| ••. lum a 36m + 46m* + 9a, + 9aa + 9a.3 + • • •• + 9(in x + 9rtn ■ 9{4m + 5m* + tti + o, + aj +••••+ fl,.i + o„| - Thif statement may be generalized as follows :— OiNiRAL Thiomm. — Let r be the radix of any tyitem of numben, then if any number in that ay$tem be multiplied by any nunUnr n and the same number reveraedf a$ to its orders, be multiplied by t - (n+ 1); then the sum of the two products thus obtained it divisible by (r - 1). i 146. (a + b)(b ^ c) - (a ■{■ l)(c + 1) - (a + c)(6 - 1) (147) (i»g'+2afta;y+ftV+<»x»4-2cdgy-H'^'<» « (»i - l)'^^+2(n-l)hs + (n- l)i + 4» *^ -' K«-i>^ ?^'^' ■.2(1.-1)^ -yrr A («. I)i(P. l)a*+ (n + 1)»(P - I) , - 2(« . 1)»(P + 1),. «>i'(«-l)»(l-iV-2(n - l)«(i ,+ /.), . - (, + i)«(i . pj. 4(».1)*(1.JP)%«- 8(n - l)4(i» -/»)* + 4(« ^ 1)«(1 + P)t • -K* - l)*(l ♦ f)* - 4(« - i)i(n + i)>(i - P). Dividing bj <(» - 1)", <« - 1)*(1 - 1«^««* S(« - 1)^(1 _ pi), + („ . ,S,>| ^ p., A(«-l)(i-/>-(«-l)(l+P)=iV(«-l>V+/')'-(iH.l)»(l-#)« ' (*- W"i^* « (n- l)(H.f)f V 4>V^ 4i«,i - 4IH. 4# A (« .^1)0 -/>)* . („ . 00 + P) ± V (A-lXn^i») '' ' " nr? i ^rri)( -nrp)V^^ > IXn - f > ,• where in order !^' \T' "t "•' V(>n - l)(n - IT) „„, ^ „,,, ^, J, (F« -IXn-P) tanit be poiitire, and tbAt n *»»v.. t^ itXi TO piis.Ex.i«i-ie6. Mis. 161. 4 s i(^ term « a -K (n - 1)<( .*. 37t» term of the series V + V + V + *c.xV t-l)(-i)«3/-y = *», « (2 X 6 + (31 - l)v ^;jV- « (12 + 30 X- J)V« Tx V« 108 J »*, » {2x6 + (42 - IX - i)|V » (la + *1 X- t)2l » (12 - 6j)21 » Bi x 21 » 1081 .-. 5,1 •!• ^4, H I08i + 1081 « 21? . « • 1 1 ef a(V-»*) l-(-.-4)«» l-(-'4)» 6 ^'4{'-(4)*} ^•* l-r -l-(--4) 166, Of l«»geries, fi, = {2 +(n- 1)1}^« (♦«+ l)^i •«»«» S, « (p+ 1)| Of 2»* series S,»{4 + (n-l)3}ya(3« + l)-5> and 5y«:(3j»+i)|- where BX.U1-1W. » flft - J» H-H+b '' H-b - (g - ft) g - b) In A. prog, latifl, JSTis the aeries :ia- 68)21 p (p+iy (3>+l)|- Mtt. Ex. 166-170.] ALQBBBA. 189 Of S"* wriM 5» ■ {6 + (n - 1)6} J « (6« 4- 1) p told fi^ « (dllp + l)j Of 4*k seriei 5« « {8 -t- (n - 1)T)~- « (7n 4- l)j, and «, » <7p + 1)|- n fi n . .-. of the Beries (n + 1)^ + (8ii + 1)^ + (8« + l)-^- + Ac, n where the first term is (n + 1>^ and the oommon difRsrence is 2nx-~afi«, the Sy s {2(n+l)y+(|r-l)»i?|ja(n^+i84'|m»-«^j «(n+|m«)-|-«(l+jm)y Also of the series Qt + 1) j + (3p + 1) j + (6p + 1) j + Ac. « p where the 1** term is (jp-f l)r and the eommon diibrenoe is Spx ^ s j)S 5n = {2(p+ l)|- + (n- l)p»||- » (p»+l»+np»- ji>)y « (p + np^ J pit a (1 +p»)y •'• '^p of the former series s 8^ of the latter series. 1<*T. {(« + ») - V*5M(* + y) + V*5} * ^*' + 2«y + y*) - *3f {(ls» + y«) + *»}{(«» +V)-a:y} = (*» + y»)*-xy « a« + «y + ^ 168. 8(B» -3x8)*+ 6(6'+ 3x8)* = 8(26 - 24)* + 6(26 + 24)* B8Vl + 6Vi9>»8 + 36«43 169. Vl6a^- 9Ba^b + 216a%>- 216a6* + 816* - 4a?- 12a6 + 96*, and y4o» - 12a6 + 96" = 2a - 36 11 3c 26 Also -rT-;rr^--:;r:--:nLr ITO. — + a 4d ^{h^y 26 3e 6de 66c 3c 26 bat since T" = "Ti i' follows that 6c « ad .*. -. -^ - -rr- 3c 26 dad" 60(2 _f_ * /^JL i.^ /A JL\ '~2«l~3aa''''\^2 **ady~\^3 ^ ad) — T XT - =- + 26 3c a(l\^4 " 3 " 1 I/'. 4i " ad\^^ :t-)-t orf 04 + (lj 190 XBT TO [Mil. E)^. 171-176. 111. Multiply by 4(x + 1), and 8« + 12 « ,4* + 5 + "^^|^ - Redocing «nd then clearing of flractioni, we baTft 12«'-i- 2Sx + 1 »12«»+24e+12.-. »»5 2x + 6 2a + b 172. oleariog of fractionf^ .-. 2o'x + 2abx + 0^6 + c6» ■ 2a*» + 2± (2e?+2ab + ft») s 4(^ + 2aft, or s - 2ab - 26' a 2a(2a + 6), or s . 26(a + b) \ 'X s 2s(2a + h) a a, or X a -. £(a-l-&) 2(2a + 6)""'"* 2a + 6 .„ afl ab-\ra a-¥l ah^a m. Smce.T = 55^,andva5j+Y; * + y = 5TT+56TT 2a + o6 + 1 <•*+! 2a + aft + 1 X + y - 1 aft + l X + y + 1 * ^IflTf^TTT 2a oft + l a - 1 + I 2a -i- aft 4- 1 ' aft - 1 oft+l 2a'i-aft4-14-a6+l oft •hi 2a + 2a6 + 2 ~ a + aft + 1 174. 2x2 _ 2xs - 2xy + 2yz + 2^^ - 2xy - 2y* + 2x2 + 3i' - 2x« + 2xy - 2yz s 2x* + 2^" + 2a' - 2xy - 2x« - 2y« = (x=» - 2xy + y») + (x» - 2x« + ««) + (j/» - 2y« + a') *(x-y)« + (x-»)2 + (y-z)» 175. (a + ft)»-.f'\(tt=» - ft")' + 4afte»- c^fCa - ft)^ + c« y(a''-ft'/-( a-.ft)VV (Ca + ft)V-c* (a + ft)V-c* £i. 171-176. li(x+iy 3X+1 :»+ 26» + 1 2»(a + 6) r + o^-fa ab + l - ab - 1 I ¥ ab + I !i»- 2xs Mis. £z. 170, 177.] ikLOIBEA. 191 Ite. «»-l)«»<»+»»+»« + a«» + 2«*+a«»+ae*+x+l(*«+jf + i *»•-*■ ?^+I? !:•-; «* + aaB»+2* ' <» - 1 a«' •I' 2x« -f 2«*-l- 2s*'l- 2« -f 2 •"^ + »* + «•+«■+« + l)««-l(« «•+»» + «* + *» + *»+* ar- •k-a:* + -c»+a» + x+ l)«' +«* + «• + »» + »+ 1(««-« ^^ ■¥x* + x'+i*'¥3f + :^ **+«» + «* + 2x+l jf* + a:' + a* + 2« + l)«» +** + *• + »■+« + l(« j:^-. 1 W + x> + «« + a» + l(«a + » + 2 CI.C.lf,sjf+'l)»»-l(x-l jc'+2«* 2«* + 3x ax»-2 ie» + * -«-l 3x + 3 3(x + 1) TIT (to + 3)(» - 1) (ag - r)(g 4- 1) « -f a ^^* (X + 6)(x« - 1) " (X + 6)(x» - 1) " WT~l 2x' + x-3-(x»-6x-7) X't-2 x*-Tx4-i x 4-2 (x + 6)(x»-l) x>+ 1 ■ (X + 6)(x>- 1) " x»+ 1 (x» + Tx -h 4)(x« 4- 1) - (X 4- 2)(x« - l>(x + 6) (« + 5)(x*-l) (x« + 7x + 4)x» + x« + 7x 4- 4 -» (x* + Vx + IO)x» + 3fl+ tx -t- jO 2x» + 14x + 14 - 6x» (X + 6)(x* - 1) 14x-4x*+14 (X + 6Xx* - I) " (X + 6)(x*- 1) 102 KIT TO (]fl«.JSz.l78.17&. 1?9. Ltt « « By b« th« fint of any four potitire qvantltiM in J.P.f Md lit 3|f Im thilr oommon diflbrenee. Tbra tlM torn qaftatlUM ave c - ^, « - y, » ■»• y and » •!• 8y. And ilit fom of th« «xtr8nief « « <- 3y 4- « + 8y ■ ax. Alio fbt f am of th« mMns "V-y-t-x + ytajr. And 8x f 2x /, tlM ram of «bt extremes » the ram of the m«uis, X Again let -f he the finiof foor positire quantities in G»P*f and let y 1^ their eommon ratio. St SR Then the four quantities are 3, •~-, x and xy. y y ^ . ' X x+xy" . X «+ay Bum of extremes ■ jjy + «y ■ ^ ; snm of means 3 --.+a; a — —- X + «y« Then — ^p- X 4- xy y , according as x + xy* ^ xy + xy*; orasl-t-y"^ y + y"} or as (I + y)(l - y + yOS^y (1 + y); or as I - y -f y* ^ y ; or as 1 + y' ^ ay. But 1 + y» > 2y by Art. 184 Note a, .«. — 3— > — -— , that is the sum of the extremes is greater than the sum of the means. Lastlji if as before x » By, x - y, x 4- y and x + 3y are in A.P., 1 1 1 IV their reciprocals i^-^^-g-, ^^, -^^ and ^^-^-^ »«» in H.P. 1 1 2x Then And X - ^ 1 m - y + X + 3y 1 x« - Sy* ax X + y ~ »■ - y* 3 sum of extremps. « sum of means. to. Ex. 17»-1M.] ALOKBRA. 198 (Tow whether y be poiUire or negatire, Sf" it neeeaiarilj potitire, jud therefore at* - j* > *• - Oy», and .•. ^ ^ ^^ > ^, , ^i i ikhat is the sum of the extremes is greater than the sum of the ineans. . , 2n - 1 180. 8f^,i « {2a + (2n - 2)rf} — ^ — , and when d = o *i»-i * f** - (2* " 2)<»}^Y- ■ {2a + 2an - 2a}— r— « na(2« - 1) Also (2« - 1)*^ tern s a + (2» - 2)d « a + (2n - 2)a « a -f 2an - 2a « a(2n - 1) .*. snm of 2n - 1 terms » the (2» - 1)*^ term x n when the series is asoending, i. e. when the first term Is the least and the last term is the greatest. 181. ab + V**-** « *" •• Va" - ** « x« - aft ; oV - 6V = X* - 2oft** + oV .*. «* - 2a&e» + ftV « ; or «"(»" -2ab-¥i^»0 .'. X = Q or « « ± V*(2a - 6) 182. 3x*+«^«3104; 36*^ + 12x^+1 = 37249; 6x§ + l = ±193; 6x» = 192 or - 194 ; x* = 32 or - 32i ; x« » 2, hence x s 64; or x» = ( - 32J)«, whence x - VP^l? « 32* ^/SaT » ^^^^ V^ x'+ 2ax + x«- x"+ 2ax - a' 6»+ 2ix + x*- 6»+ 26x-x' 183. 4ax 4&X 6»-x» 6 grria; a6«-ax»»6x»-te», • • x« - a» 6' - x» • • x» - o» or 6x* + ffx* a od* + fttt' .-. (6 + o)x' = aft(i +a) .•. x' = aft, whence X a iV^ 184. Vx» + V** + 96 = 11 -« .-. x» + Vr' + 96= 121-22x + x" .-. V«'+ 96 ■ 121 - 22x. Again sqaaring x> + 96 s 14641 . 6324X + 484x3 . 483xX. 5324x = - 14545; , 6324 14545 *»'^*-483*''-"48r J 6324 /2662Y ^0862 44 14545 7086244 - Y025235 •'•* " 483*"''\^483y * 233289 " 483 * 233289 N m KBY TO [Mil. XX. 184-180. « • X m 9963 ■iV 61009 388289 ■ * 24T 483 483 3862 f 247 2909 . 2415 , , . "ief ~ 483 » "* 483 185. Let X s the left hand digit, and y " th^Tight hand digit ; then the namher is lOx + y \OX'¥y •'• •jrjr"^^^^*'*®* 10x + y«21x-21y,or22y« lla-jOrxaZy 10« '22^^ + 10y> + 4f .*. 72 « 86y, whence y « 2 And « ■ 2y a 4 /. the required number is 42 186. Let X a minutes per mile talcen bj B, then x -f 1 60 ■ minutes per mile talcen hjJ] —• s miles per hour of S, and 60 —77 ■ miles per hour talcen hy Ji, 60 The second time round the rate per hour of B *-. --•^. 2 60 - 3* . . ^ 60 62 + 2x « — - — , and rate per hour of J a ■ . ■■ , ■ + 2 ■ ^ . , X ' X + l X+l And since the course is 2 miles long, the time in hours taken bj £ to go round - -gj-^ = -^^—f- " sTT^ time in x + l x + l 60x -f 60 minutes required to go round « g^ .*. minutes per mile 30x 4-30 talten by .4 « ». . ; similarly minutes per mile itt2»* round 30x required by £ a _ ^ , and since Ji does the two miles in two minutes less than B, his time per mile will be one minute less tbun B T*. 30x + 30 31 4- X + 1 = 30x 30 ■;—, whence by reduction ;.lM-m iddigH; orx32y wo ht^a 56 y ■ 2 n X + 1 }f Bf and 62 + 2x X + 1 I taken by time in per mile i^ ronnd BS in two mte less leductioQ Mn.Ex.m-m.] I ,30j! •(> 80 •»• 81 •!• « 8l«4'6l ▲LQlBftA 80« 195 ■ 553^ .'. »8©» •!> 1880 - 8lJ« - 6li i 3l4-« 81+« N980« + 80c>; or-61«*-6U ■ - 1880, whenet «*•!■ jr « 80; x> 4- s 4- 1 " ^1^ .*. » + i « i Vi *nd * " B .■. J's rate !■* round « 6 -f 1 ■ mtn. per mile ■ 10 milei per hour F§ rate !■* round ■ 5 minntei per milei or 12 milei per boor jf§ rate 2*< time round ■ 10 4* 2 ■ 12 »Um per honr £f§ rate 2'>* time round « 12 - 2 ■ 10 mlei per honr Whole time of B for boUi rounds ■ 10 -I* 12 ■ 22 minntei Whole time of J for both ronndi ■ 18 4- 10 ■ 22 minntei .'. neither hone wins. 18Y. Let », s 4- 1, « 4- 2, « 4- 8 and « 4* 4 be anj fire oonse- cntire integers; then x(a + 2)(« 4- 4) 4- (x 4- 1)* 4- (x 4- 3)" ■ (X 4- 2)(x« 4- 4x) 4- x» 4- 3x» 4- 8x 4- 1 4- x» 4- 9x» 4- 2tx 4- 27 B(x4-2)(x*4-4x)4-(x*4-4x'4-Sx4'2)4-(4x4-8)4-(x*4-8x*4-21x4-18) » (X 4- 2)(x*4- 4x) 4- (X 4- 2)(x»4- 2x 4- 1) 4- (x 4- 2)4 4- (x 4- 2)(x»4- 6x4-9) = (X 4- 2)((x« 4- 4x) 4- (x> 4- 2x 4- 1) 4- 4 4- (X* 4- 6x 4- 9)) = (X + 2)((x> 4- 2x 4- 1) 4- (x» 4- 4x 4- 4) 4- (X* 4- 6x 4- 9)} - (* + 2){(x 4- 1)' 4- (X 4- 2)» 4- (X 4- 3)»} « prodact of middle number by the sum of the squares of tlie middle three. i 188. X* 4- y* r .T* 4- 4x»y 4- 6xy 4- 4xy" 4- »* = 2x* 4- 4x"y 4- 6xy 4- 4xy" 4- 2y* = 2(x* 4- 2x'y 4- 3xy+ 2xy»4' y*) = 2(x» 4- xy 4- y^* / 189. (x» 4-V 4- x»y + xy^( x» - / - x«y 4- xy^ «={(*■ + «y^ + (!/• + '^M'^ + *yO - (»• + «*y)} s- (x» 4- xff - (y» 4- xV)' = x« 4- 2xV 4- x»y* - (y« 4- 2x»y* 4- x***) = x« 4- x*y' - xV - y* 190. X* a (Vo+T ± Va"=T)» = a 4- 6 1 2Va^'P + « - 6 = 2a ± 270" - *» ; ax*- ix* s x"(o - ix») » {2a i 2V^^|(a - (Ja f lVtf'-6»)l = {2a ± 2V5rrFj{jaT J)/arr-5»| « {a i Va* - 6»)(a ? VS'TT^ = a» - (a» - 6») =• 6» 196 KIT TO pfu.Kx.in-l». , ♦ « + (y + s) a f (ay 4- «s -I- Scy) •*• (fty* 4- acy* 4 3^«) •<• (fry* •»- VO - («y -t- «) " (agy* ■*• agy«) » (fty* » V«) 1 .(y+«) •(•aey Tli?' .% qaoUtnt ■ ««* 4* aeyj» 4- ^ 19a. (**)• - (I*)' ♦ (*• - 1*) ■ «• 4- 1* « «* 4- 1 193. l-l4-l-«4-a«-34-Sx4-a4-4-S««4 4>x 194. 0(6" 4- Zbe 4- c*) 4- b(jc* + 2ca 4- o") 4- c(<^ 4- 2a6 4- 6") . |(a> ~ afr •. oc 4- *c)(6 4- c) 4- (6» - fte - a* 4- ac)(a 4- c\ 4- (c*- oc - 6* 4- fl6)(a 4- 6)1 = o5"4- a<;"4- 6c'4- 6a" 4- co«4- e6" 4- dabe - 0% - a^ - 6'a - 6^ - c^ - c'6 4- 6a6c a iaa6e 195. ((6 4- c - a) 4- (c 4- a- 6) 4- (a 4- 6 - c)]x 4* {(e 4- a - 6) 4- (o 4- 6-c) 4- (64-c-a)Jy 4- ((a4-6-c) 4- (6 4-c-a) 4- (C4-0- 6)}« ■ (a 4- 6 4- e)x 4- (a + 6 4- c)y 4- (a 4- 6 4- c)s = (.a 4- 6 4- c)(x 4- y 4- «) 196. (X 4- 2y)'x (X - 2y)" » (x»- 4y^' = x« - 12x*y»4- 48x«y* - 64y« (a 4- 6V"^)(a - 6V"^) » a' - 6"( - 1) = o» 4- 6» 197. {(a 4- 6 4- t')(« 4- 6 - c)}((c - o 4- 6)(c 4- a - 6)| = {(tt 4- 6)" - c»|{c» - (a - 6)»} n c»{(a 4- 6)» 4- (a - 6)»J - (o» - 6V - c* ■ 2aV4-26V-o*4-2oW-6*-c*« 2a«6»4-2ttV4-26V-o*-6*-c* (X 4- 1 4- x-»)(x - 1 4- x-») = {(X + x-») 4- l}|(x 4- x-») - IJ. = (X 4- X-*)' - 1 = x" 4- 2 4- x-» - 1 - x» 4- 1 4- x-» 198. (2x* - Sx'y 4- 4iy - 6xy» 4- 6y*) + 6xV = lx«y"'-lxy-»4-|-§x-»y4-x-y (x*4' 4x 4- 3) + (xH 2x 4- 1) « (x'4-2x4-l)(x»-2x4-3) «" 4" 2x 4- 1 s x»- 2x 4- 3 199, « 8 =»-^ a 8(1'' + iX^S-Hin » 8*' + 4X*» + if Umm. Ux. IQMM.] AMSBBA. 107 f (x« + ax + «•) - a/KT « x" + (I - |>)ax + tfl aOO. (I) (»" -Sjc -4) ■ (* + IX* - 4)»'»n* <»" - 2* - •) ■ (r + IX* - 4) ;(*«+«- 20) - (» - 4X* + 6) /. O.C. Jf. ■ x - 4 (II) 3^+ 4«"- 3* - 4 « jr'CSx + 4) - (3» + 4) ■ (*»- 1X3«-I- 4) 2x* - 7*" + 6 ■ (3*' - 6)x" - (f*« - 6) « (2*" - 5)(»» - 1) .'. G.C.M. ' X* - I (ni) Let J) be the O.C.M. of m «nd a; then the Q.CM. of («■• + a"*), and (x* + o") » x' + «*, and Of (x* - aP), an'' x* - o* ■ x» - «» .-. required Gf.C.jV; » (x» + «»)(x» - «») ■ x** - <^ 201. (i) /.cm. of (X - 2a)(x + o), x^x + a), and a(x + a)(x - a) « ax'(x - 3«)(x + a)(x - a) = ax» - 2«Af* - oV + 2«*x* (n) X* - x*y - tflx + a^y => (x' - a')(x - y) ; x" + «x" - xy" - ay" » (X + a)(x» - y") .-. /.cm. = (x» - a«)(x« - y«) a X* - xy - rfx" + ay (g 4- 6 - c •» d)(a + b-e-d) (fe + c -a + " 6(g + ft) "■ g« - ft> ' ft(ga-ft") a» + o»6.a» + 2g'»~gft»-2g6* 3g»ft-3gft» 8g6(a-ft) ft(a?'-.l») ' ft(rf'-ft") 'ft(a-ftXa + 6) 3g g + ft 198 205. KBT TO /a' ~ ax + ax a!*+ ax - aat\ I a- X a + X J (His. Ex. 20&-210. (o •»• a;)" 4- (g - g)' /a* **' V 2a' + 2x» g* o«-a' o* (^gTi ^ afxjj ,a*:'X* ^ a» -x«J; .2Xa'+a:^ " 2(g" + x^) Aab g + 6 + 2a 4gA a + 6 + "43r + 26 %ab + 2a« 6a6 + 2 6* 2a6 - 2o« ■*" 2a6 - 26« ' — TT - 2a — — r - 26 g + g + o dividing numerator and denominator of l**by 2a, and of 2<><) by 26, 36 4- a 3a + 6 36 + a 3g -I- 6 26 - 2a ^® get ~fc — :; + ~:, — a - ~t — z " o -> a a — o — a b - a k'O^ = 2 207. (I) VjJ* - 4z» + 4ar« - 41" + 8x + 4 e ^(x* - 4x* + 4x=') - 4(x=« - 2x) + 4 = x» - 2x - 2 > f x" Tx^X^I / x"\ x^" x*«|4 - 4y + (^yj I = x»(^2 - yj . 2x» - -3- /a* 2a6 2af 6' 26c c*\' a 6 Cs^ ("0 \J*-~bF''~W^'^'^lic"^a*) '^ T'c^T 208. The sqnare of which g'x'^ 6x are the first and second 6« terms, is aV + 6x + -^ .•. in order that a'x' + 6x + 6c + 6? may 6« c 1 be a perfect square, we must have 6c + 6" = j-^ .*. -r- + 1 = t-jj . 1 <^ and.-.4^-y=l 209. (i) mnx + amn = n'x + am^ .*. mnx - n'x s om' - «mn, . . .^ » owi(jii — n) am that IS (mn - fr)x - anr - amn .•. x » — 7 r a (11) 2x' - I3x = - 6, whence x s 6 or J . , ,^ _ 7x+l 400 /x- A 400X-200 210- C^ 13^76^ = -r{x^i) ' 3x^2 whence 2421x» - 6411x = - 25&8, or 807x» - 2137x * - 866; ' , 2137 /2137y 4566769 - 2795448 1771321 * " 807* ■*■ \l6i4j ' (1614)=' " (1614)" s. 206-210. (a - xy X* o* 5(a » + x=') + 26« - 26« ' 2'">by26, -2a - i J -an V V I second 6? may 1 4i«' ' - amn, - 866; / ni321 ; 1614)" i Mm. Ex. 210-214.] ALOEBRA. ' 2137 V1771321 ± 1330-9 •'• '^"1614''* or - 0-49 1614 1614 .'.X9 3137 i 1330*9 1614 199 >-f2'14 "-"•f (n) x« - 2(u + 6)x + (a + 6)» = 4(a» - 2o* + 6») « 4(fl - &)» . X - (a + 6) = i 2(a - 6), whence x = 3a -> 6, or 36 - a 211. ex - acy = abx + 6y .•. x(c * ab) ■ y(ac + b) x(c - a6) ^ , ttr(-1 1-1 =3V2+2V3 •a»'rt. •tf„ = a, a„ = a.i^-f i-ia forioed from 0^-1 l>y multiplying it by a," .•. a, + a^, •!• a, + ttci, is a Geom. series having a, for first term, and Oi' for common (a,2)« - 1 ratio. Then 8^= « 1 • -—a — j- = a 216. Squaring each side and transposing, we get z* - 20a' + 94x2 ^ gQa; + 9 = j extracting the square root of each side, we have x'- lOx »- 3 = .•. x' - lOx + 26 « 28: X - 5 = i 2Vt .-. X = 6 1 2Vt 217. Multiplying, we have - 30x* + 46x" + tx* - 23x + 4 » 4 .-. x(30x8 - 46x2 - Tx + 23) = .-. x{30x» - SOx-* - lex* + 16x - 23x + 23]*= ' .-. x{30x2(x - 1) - 16x(x - 1) - 23(x - 1)} = .-. x(x - l)(30x2.- iQj. _ 23) = .-. X = 0. Also X- 1 = .% « » I Also 30x' - 16x = 23, whence x' - -^x + g'g", « 3JJ .-. X = Va(4 i WJ54) 218. The given series is double, i. e. is equal to the J series 1+2 + 3 + 4 + 5 + &c., + tht G series 1 - 2 + 4 - 8 + 16 - *C# Then sum of ^ serieu as follows : — 54, = {2+(4»-l)|y = (4n+l)2« S^n^x = {2 + (4« +1^- 1)}^^^ = (21 + l)(4fl +.J) , 4n + 2 « -3 = tU-(-2)*«*T /. of gJren scries 5,n = 2n(4n + 1) + §(l - 16*) '^4n*i = (2a + l)(4»i + 1) + J{1 - ( - 2)^»*»| ^\»*2 = (4n + 3)(2n + 1) + J(l - 4=***) '»4«*» = 2(n + l)(4n + 3) + J( 1 - ( - 2)*"*"} 219, Let X « number ia width, and y = number in the length ; tbefl xy « whole number in the bunch. Also, since y> 10 but < 20, V « « number of two digits .-. when x is written to the left of y H must occupy the third or hundreds place .-. lOOx 4- y * the number in scale of 10. * Also since * < 10, it consists of but one digit, therefore when written to the left of y, the number will be represented by lOff 'f X which .'. « number in scale of 10 Again in similar rectangles the perimeters are as the cor- responding sides, and whole perimeter of first bunch = 2(x + y), 3c*y and ot second bunch xy .-. 2(x + y) : xy :: x • ofx + v) ~ ^'^*** xu" of 2nd bnnch, and 2(x + y) : xy :: y : ^. ^ v = length of 2nd bnneh /. whole number of matches in the second bnnch x* y xy* ay * Hx + V) ^ 2(x + y) '^ 4(x + y)« Tlien from Anf- condition lOOx + y : xy :: a : 2 (i) ** second " lOy + X : xy :: a - 10 : 4 (it) xy u third 11 202 KET TO (Mis. Ex. 219, 220 Krona (u) 20y + 2x : xy :: a - 10-: 2 .-. 20y + 2* + 6y : «y :* a ' " .'. 20y + 2x + 6xy : xy :: lOOx + y : xy .'. 20y + 2x +.6xy = 100* + y .•.' 6xy = 98x - i9y (iv) Also from (in) xy = 16(x + y)» .-. xy = 4(x + y) .'. 6xy = 20(x + y) Substituting this in (iv), we have 20x + 2y = 98x - 19y, wlience 2x = y. Again substituting this in (in), we have x^ = 16(x + yy, that is 4x* = 16(x + 2x)» .-. 2x» = 4 x (3x) .-. 2x» = 12x, or x = 6 .-. y s 12 ; and xy = 6 x 12 = Y2 = number of matches in the bunch. ^ 220. Since the conditions giving the equations (i) and (n) remain the i^ame, these equations and .'. also (iv) which is derived, from them independently of (in), remain the same. '. we have but to solve in positive* integers the equation 5xy = 98x - 19y, remembering that x < 10, and y > 10 but < 20 5xy a 98x - 19y .'. 6xy + 19y fe 98x Stx * • ^ * 6x + 19 400x 1862 •'•5y=6r+l9=^8-B?Tl9 1862 K&w since y is an integer, - . ,q is also an integer. And since x is integral, 6x + 19 must equal an integral divisor 4>f 1862, and further since x is finite, positive and less than 10, 6x + 19 will be > 19 but < 69 and will end in 9 or 4 according as X is even or odd. Now the only divisor of of 1862 fulfilling these conditions .*. 6x + 19 = 49 .% y « 6 98x y%x + 19 = ^2 , .*. XV = 12 is 49 But ^ Thoj must be positive froth the nature of the problem. X. 219, 220 ty : • a • " I9y (IV) s20(x+y) j/, whence 6(x + y)', r, or X = 6' ihes in the » [) and (n) ti is derived > e equation 10 but < 20 Mis. Ex. 221.] ALOEBRA. 208 Br. pral divisor 88 than 10, according tions is 49 221. Let X = rate per hour of the express down ; y = rate per hour of accommodation down, and d s distance from Stratford d d to Toronto. Tlien— 'hours s time- dowh**)^ express, and -— * y d = time down bjr accommodation. Also -7 » cents per mile in d d* express fare, and — ■ x rf = -— = whole fare by express. X - -— = rate of expressing going up .•. 4? goin d X - — X - hours on road But if the fares had varied as the velocities ; then fare at X : fare at y :: z : y .*. fare at x- fare aty : fare at x :: x-y : x But in this case, fare at x - fare at y = d cents, and since fare d* by express to Toronto remains the same, d : — :: x - y : x (i) / d\ d Also fare at x : fare at ( x ) :: x : x .-. fare at \ ^/ * X - fare at ( x - — j : fare at x : : — : x / d\ d But fare at x - fare atfx j =*-~ cents d d^ d , ^ *^ XX X ^ ' Using the formulas now found in expressing the remaining d 1 /dY statements in the problem, we obtain — = — f — J (ni) ; Then from (111) dy = 2x' (iv) from (1) x^ = d(x -y)~dx-' 2x», by (iv) .-. rf = 3x (v) from (II) x\x^ -d) = d? .-. by (v) x\x^ - 3x) = 2Tx» •. x' - 3x = 2Yx .-. X - 3 = 27 .-. x = 30 .-. + 20} R. 6(Vi?+2S + 5)(^/j?+T5 + 4) (n) But (Vx' + 26 + 5)(Vx^T5 - 6) = z» And (Vr'+25 + 4)(Var'+25 (VrH^'S + 5)(Vx^T2B - 5)(Vx^2"6)(Var' + 25 + 4)(Vx'+l5 - 4) (Vi^l!&- 1) = 5(VFTr5 + 6)(VPT25 + 4) .-. (Vx'f+ini - 6)(Va?+25)(Vi^^2"5 - 4)(Vx^2"6 - 1) =^ 6 (ill) .*. (X'' + 25 - 5Va:"+ 26)(x'' H- 25 - 6Vx'' + 26 + 4) = 6 .-. (x' + 25 - SV^'+TS)' + 4 (x» + 25 - 6V«' + 26) = 5 " /. (x» + 25 - R^+B)' + 4(x» + 26 - VF+2]}) + 4 s 9 .-. x» + 25 - 5Vx='+25 = - 6 or 1 .-. (x2 + 25) - 5Vx^+l5 + V = f or V iZT-^TT gJVS 6iV29 /. V^ + 26 = — 2 — O"^ 2 /. x» + 25 = i(30 i 10V5) or i(54 ± 10^29) Whence x = i (V ± 10V5 - 70) or i(V ± 10V29 - 46' Also V-c''+25 + 5 = 0, whence V^^+'-sB = - 6, or x* + 25 = 26 .*. a? - ) VxTlTS + 4 = 0, whence Vj^+25 = - 4, or x»+ 26 = 16 .*. x'* = - 9, or X = ± 3V- 1 223. Let X = number of yards dug at $1*26 ; then 100 - x a number of yards dug at $0- 75 .-. l'26x = 50 = '76(100 - x). Therefore, we hare two independent equations containing only one unknown quantity, and any solution obtained from one equation is inconsistent with the other ; consequently the problem is impossible. 224. Let X = length of one side of rectangle and y = other ; then xy = area, and 2(x + y) " perimeter of the rectangle ; and X2^ = area and 4\/iy *! perimeter of the square. Mi8. Ex. m, 326] ALGEBRA. 205 (i) .'. xy a 4miJ^ (I), and xy ■ 2n(x + y) (n). From (i) i/xy F 4m .*. xy » 16111* (ni), substitiito this in (n), and ve get 16»' F 2n(x + y) .'. 8i»" « n(x + y) (it). Squaring (nr), we hare to4m* = n«(x + y)« ; multiplying (in) by 4tt", we hare 64»W » 4n^ ;•. by subtraction 64m'(i»" - n") « n\x - y)" .*. ± 8mVm' - «■ = n(x - y) (t) 4m Adding (iv) and (v) and reducing, we get x a -r-(*» ± V"** - ^ 4m Taking (r) from (iv) and reducing, we got y a — (m :f ym" - n*) (ii) When the perimeters are equal ; thon talcingx and y as before, (x + «)* we hare 2(x + y) = perimeter of the Square, and — j — = its area ; (X + yy = 2m(x + y) (i) ; xy = 2n(x + y) (n). Prom (i) x -h y s 8m (III), substitute this in (ii), and xij s 16mn (it) Square (in), subtract 4 times (iv) and then take the square root, and we have x - y = ± 8Vm' - mn (v) Adding (iii) and (v) and reducing, x = 4(m ± ^/m? - mn) Subtracting (v) from (iii) and reducing, y = 4(m T ^/ri?~-~mn) 225. Let X = age of younger at first trial, and y = age of elder. Let r = ratio of throw to age at first trial, and r^ = ratio of gain of one to age of the other at second throw .*. first throw of younger = rx, and first throw of elder = ry ; gain of younger = r (y + 1) ; gain of elder = r^(x + 1) ; second throw of younger = rx + r(y + 1); second throw of elder = ry -i- r,(x + 1). 2(x+l)(y+I) Also H. mean of their ages at latter trial = — , . .. , « — ; " X + y + 4, ' r(x + y) J. mean of first throws = — ^ — > ^^^ *^- ™caa of 2nd throws r(x + y) + r,(x + y + 2) r,(x -l- y -f 2) = -^ 2 •*: di|ference of -«. means = s Longest throw = second throw of the elder = ry^-r^(^x + I); value of ratios compounded of ratio of throw to age and gain 206 KBT TO^ lM».£jc.226 to Agt of Other ■ rr^ ; valae of ratio formed bj maltiplylng AOteeedent of this eomponnd ratio by I product of ages at leoond trial » irr <^x -^r \){y ^ \) ) ralao of tbe ratio of which this if the dcplioate = Wrr^(x + l)(y +1) ; value of the ratio compounded of the ratio of throw to age of one with gain of one r to age of other ■ — Then using the values thus expressed in stating the problem, we bare the four equations : — ry - rx a 24 ; or r(y - t) = 24 (i) \ {ry + r^(x + 1)} - {rx + r,(y + 1)} = 26 ; or (r - r,)(y - ar) = i2 (u) ry -f r,(3i + D 2(g + l)(y + 1) ^ ,. , s 2 IVrr,(x + l)(y + 1) = — ; or (x + l)(y + l)r» =» 4r (n) Then (i) - (u) gives r^(y - x) = 2 (v) ; substituting (m) in (u) r^(y - x) a 22 (vi) ; dividing (iv) by (v), we have x = 11 ■ age of younger at first throw (vii). Substituting (vii) in (ui) r ■ 12r^, and in (v) r^(y - 11) s 2 ; also substituting (in) in (iv) and reducing, rj'^y + 1) = 4 (viii). But r^(y - 11) = 2 •'• *','(y *■ 11) = 2r^; subtracting this from (viii), we have 12r,»a4-2r, .-. 6r,='+r, = 2 .-. r,2 + ir, + TiT= 4 + tH = "A^ .'. f, " i -{g - h = J- But »■ = 12rj •'• »• = 6, and since r^(y - 11) = 2 ; i(y - U) = 2 .-. y - 11 = 4, or y = 15 s age of elder /. throws at first trial = 11 x 6 = 66, and 15 x 6 = 90 ; and throws at second trial = 66 + i(16 + 1) = U, i^nd 90 + i