Algebra. Chapter 4. 4a review of index laws

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1 Chpter Alger Chpter CotetS Review of ie lws B Str for sigifict figures C Trspositio Solvig lier equtios siulteous lier equtios e Applictios F Algeric frctios G Lier literl equtios igitl oc oc Quick Questios review of ie lws Bse uer Ie, power or epoet You lere i erlier yers tht es ultiply two y itself three ties, tht is, 8. The ie iictes the uer of ties the se is ultiplie y itself. I geerl, 8 Ie for Epe for Bsic uerl ties I sury, the ie lws re: Multiplictio: + Divisio: Risig to power: ( ) Risig to the power of zero: 0 Risig to the power of oe: Proucts quotiets: ( ) ; Negtive powers: ( ) Frctiol powers: Whe siplifyig epressios equtios with iices, we geerlly use coitio of these lws. Chpter Alger 87

2 Worke eple y yz Siplify yz 8. A the iices of the iices of y ultiply the costts together i the uertor. Sutrct the iices of, y z i the eoitor fro those i the uertor. Siplify to write the swer with positive iices use the rule z 0. y yz 6yz yz yz 6 6y 0 y z Whe uers with ie re the rise to other ie, the iices re ultiplie. Worke eple Siplify: y ( y ) ( ) 7 Reove the rckets y ultiplyig the iices. y ( y ) y 6 y A the iices of y. 8 y Siplify to write the swer with positive ie. 8 y Reove the rckets y ultiplyig the iices. 7 ( ) Chge the ivisio sig to ultiplictio sig 7 write the reciprocl of the seco ter. A the iices of i the uertors Sutrct the iices of i the eoitor fro those i the uertor siplify to positive iices. 8 8 Worke eple Siplify: Chge the 6 ito. 6 + ( ) + Reove the rckets y ultiplyig the iices. + A iices of uers with se iices of uers with se. + tutoril eles-80 Worke eple 88 Mths Quest Avce Geerl Mthetics

3 Chge ll uers to se of. ( ) ( ) Reove the rckets y ultiplyig the iices. 8 Siplify y ig iices of uers with se. 7 Worke eple Siplify ech of the followig, epressig the swer with positive ie y6 y 7 the epressio. 8 6 usig frctiol iices i ie for with se of. (7) 7 6 () Multiply the powers. Siplify write the swer. the epressio. y6 y 6 the epressio usig ie ottio. ( y6) ( y) Reove the rckets y ultiplyig the powers. y y Collect ters with the se se y sutrctig the powers. y Siplify the powers. 6 y 6 Rewrite with positive powers. 6 y eercise review of ie lws We Siplify the followig epressios. 8 e 6 c f 8 Chpter Alger 89

4 We Siplify the followig epressios. g j ( y) y ( y ) ( y) ( ) ( ) ( vw) ( vw ) 9w ( v ) e h ( y ) y ( ) ( v w) ( vw ) w ( vw) c f i ( y) ( y ) ( ) ( ) ( v w ) ( ) w ( vw ) C Epressig i ie for gives: B 8 e ( ) We Siplify ech of the followig. 8 C ( ) c e + f 7 6 We Siplify ech of the followig, epressig your swer with positive iices c ( y) ( y) e 8 ( ) ( ) 8 g h j ( + ) + k (6 6) ( ) f 7 9 i l ( y) y B Str for sigifict figures Str for Str for (or scietific ottio) ivolves prcticl use of iices. A very lrge or very sll uer c e epresse i str for s ore coveiet wy of writig it. This ottio ivolves epressig the uer s uer etwee oe te ultiplie y power of 0. 0, where < 0 A coputer y coplete sic opertio i pproitely secos. It is esier to write uer such s this i str for s secos. Likewise, light yer is esure of istce equl to kiloetres. I str for, this uer c e epresse s Str for is ot oly ore ecooicl es of epressig these uers, it lso kes clcultios ivolvig these uers esier through the use of ie lws. To write uer i str for: Move the ecil poit so tht the uer ppers to e etwee 0. Cout the uer of ecil plces the ecil poit hs ee ove (positive if ovig left or egtive if ovig right). Multiply y the power of 0 equl to this uer. The sig of the power will e positive if the gitue of the origil uer is greter th 0. The sig of the power will e egtive if the gitue of the origil uer is etwee Mths Quest Avce Geerl Mthetics

5 Worke eple Solve 70 epress s sic uerl. tutoril eles-8 Worke eple Epress the prole i str for Siplify the uertor usig ie lws where possile Divie usig ie lws where possile. 0 Epress s sic uerl. 00 Sigifict figures Ofte we will e itereste i ll the figures i prticulr uer. Sigifict figures re coute fro the first o-zero igit ( 9). For eple, hs two sigifict figures (9 ). Ay zeros t the e of the uer fter the ecil poit re cosiere to e sigifict. For eple, 0.0 hs three sigifict figures (, 0), wheres 0.0 hs two sigifict figures ( ). The trilig zeros t the e of uer re ot cosiere sigifict. For eple, 00 hs two sigifict figures ( ). All zeros etwee two o-zero igits re lwys sigifict. For eple, 0.87 hs 6 sigifict figures (, 0,,, 6 7). Worke eple 6 Stte the uer of sigifict figures i the followig uers c Sigifict figures re coute fro the first o-zero igit ( 9). There re two zeros fter the ecil poit tht re to the right of o-zero igit, so ll igits re sigifict. Sigifict figures re coute fro the first o-zero igit ( 9). All zeros etwee two o-zero igits re lwys sigifict. All igits i this cse re sigifict. c Sigifict figures re coute fro the first o-zero igit ( 9). The first o-zero igit i this cse is. Oly re sigifict. The trilig zeros t the e of uer re ot cosiere sigifict. Oly re sigifict hs 6 sigifict figures. 0.0 hs sigifict figures. c 0.00 hs sigifict figures. 000 hs sigifict figures. Chpter Alger 9

6 Clcultios ivolvig sigifict figures Whe perforig clcultios ssocite with sigifict figures, the followig rules pply. Whe ig or sutrctig uers, cout the uer of ecil plces to eterie the uer of sigifict figures. The swer cot coti ore plces fter the ecil poit th the lest uer of ecil plces i the uers eig e or sutrcte. Whe ultiplyig or iviig uers, cout the uer of sigifict figures. The swer cot coti ore sigifict figures th the uer eig ultiplie or ivie with the lest uer of sigifict figures. Worke eple 7 Evlute, epressig your swer to the pproprite uer of sigifict figures: This questio ivolves itio; therefore, fi the lest uer of ecil plces of the uers i the questio..87 hs ecil plces. 0. hs ecil plce. The swer will hve ecil plce. A the uers Rou the swer to ecil plce Iterpret this swer. The swer hs ecil plce sigifict figures. The questio ivolves ultiplictio; therefore, fi the lest uer of sigifict figures of the uers i the questio..00 hs sigifict figures. 0.0 hs sigifict figures. The swer will hve sigifict figures. Multiply the uers Epress the swer to sigifict figures. 0. eercise B Str for sigifict figures Epress the followig i str for c e f We Solve y epressig the uers i str for siplifyig usig ie lws. Epress your swer s sic uerl c e f g h i j k l p q o r Mths Quest Avce Geerl Mthetics

7 C epresse i str for is: B e C We6 Specify the uer of sigifict figures i the followig c e 0.0 f g h 760 i j k l We 7 Clculte the followig to the correct uer of sigifict figures c e f g 0.. h C The solutio to with the correct uer of sigifict figures is: 0.7 B C e C The solutio to with the correct uer of sigifict figures is: 7. B 7 C e Coplete the followig clcultios, epressig your swer to the pproprite uer of sigifict figures. It is. kiloetres fro Je s house to school. Her verge step legth is 0.7 etres. How y steps oes it tke for Je to wlk to school? If cotier of sugr cues hs ss of 0 g whe full (ecluig the ss of the cotier), how y sugr cues woul e require to fill the cotier if they ech hve ss of. g? 9 The outer ski of hu cell, the cell ere, is pproitely etres thick. If the rius of the cell (icluig the cell ere) is etres, wht frctio of the rius oes the cell ere costitute? 0 A Olypic size swiig pool cotis litres of wter. The verge ily wter usge for fily of four is 6 litres. How log woul it tke for fily to use volue of wter equivlet to Olypic size pool? The Erth hs ss of pproitely 970 yottgrs (where yottgr, Yg, is 0 kg). The ss of the Moo is 7 00 zettgrs (where zettgr, Zg, is 0 8 kg). Wht percetge of the ss of the Erth is the ss of the Moo? C trspositio A forul is equtio or rule tht efies the reltioship etwee two or ore vriles. If forul escries reltioship etwee two vriles, oth of which re to the power of, oes ot coti ters tht iclue prouct or quotiet of those vriles, the such reltio is si to e lier. The grph tht represets lier reltio is stright lie, which is how the ter lier is erive. For eple: y 7 0 y + 6 re lier reltios, wheres + y y or + y 9 or 7 y re ot (s eplie previously). Lier reltios re ofte fou i prcticl situtios. For eple, the forul for the circuferece of circle, C π D, the forul for the coversio of teperture fro egrees Celsius to egrees Fhreheit, F 9 C +, oth escrie lier reltios. If we wte to fi y vlues of C give Chpter Alger 9

8 vrious vlues of F, it woul e ore coveiet to hve the correspoig forul the forul tht woul hve C o oe sie everythig else o the other sie of the equls sig. The vrile tht is y itself is clle the suject of the forul (tht is, forul escries its suject i ters of ll other vriles). I the forul I 0R, I is the suject. To ke R the suject, we ee to rerrge the forul. Such rerrgeet is clle trspositio. To trspose the equtio I 0R I 0R ivie oth sies of the equtio y I Siply. R 0 I the suject o the left-h sie. R 0 To rerrge or trspose forul, we ee to perfor the se iverse opertios to oth sies of the equtio util the esire result is chieve. Worke eple 8 Trspose the forul y to ke y the suject. the give forul. y A to oth sies of the equtio. + y + + y Divie ech ter o oth sies of the equtio y. Siplify oth sies of the equtio. y + + y y + To trspose the ove forul, we use the se ethos s those eploye for solvig lier equtios. The oly ifferece is tht i the e we o ot oti uique (or specific) uericl vlue for the require vrile, ut rther epressio i ters of other vriles. Most of the reltios tht escrie rel-life situtios re o-lier. Cosier, for eple, y forul for re or volue. A few eples re the re of circle, A π r (o-lier, sice it cotis r to the power of ); the re of trigle, A h (o-lier, sice it cotis the prouct of two vriles); the volue of cue, V s (o-lier, sice s is cue). No-lier foruls c e trspose y perforig ieticl iverse opertios to oth sies of the equtios. The iverse of is ±, the iverse of is so o. Worke eple 9 Trspose ech of the followig foruls to ke the prouerls iicte i rckets the suject. A π c r (r) P ( ) c pqrs ( s) tutoril eles-8 Worke eple 9 the equtio. A πr Multiply oth sies of the equtio y. A πr A πr 9 Mths Quest Avce Geerl Mthetics

9 Divie oth sies y π. A πr π π A r π Tke the squre root of oth sies. Note: Fro lgeric poit of view we shoul write ± i frot of the root. However, sice r represets physicl qutity (rius of sphere i this cse), it c tke oly positive vlues. A π r r A π the equtio. P c Multiply oth sies of the equtio y. P c Tke out coo fctor of fro the RHS. P ( c) Divie oth sies y ( c). P c the swer. P c Note: Cpitl P shoul e use i the swer. c the equtio. c pqrs The iverse of is, so squre oth sies. pq rs Sutrct pq fro oth sies. pq pq rs pq pq rs Divie oth sies y r. pq r rs r s pq r Multiply the uertor eoitor y (optiol). s pq r eercise C trspositio We8 Trspose ech of the followig foruls to ke the prouerl iicte i rckets the suject. (Where two prouerls re iicte, perfor seprte trspositio for ech.) y + 0 (, y) y + 0 (y) c () p k (k) e (, ) f 0 (, ) g 0.c (c) h ( ) () i ( ) 6(f + ) (, f ) j k ( + ) + (, ) l 7( ) ( ) 6y 6 0 (, ) (, y) Chpter Alger 9

10 We9 Trspose ech of the followig foruls to ke the prouerls iicte i rckets the suject. (Where two prouerls re iicte, perfor seprte trspositio for ech.) v u + s (, u) S πr (r) r c + ( R, R ) A A 0 + () r R R R 00 t e s u v t u ( + ) (, ) f T L π ( L, g) g g C + ( ) h s ut + t () v i P I v R (I, R) j R (, v) t Questios to 6 refer to the followig ifortio. A greer chrges $0 fie fee for ech visit plus $ per hour of work. C Which of the followig grphs represets the ove ifortio, where C represets the totl cost of visit t the tie the greer worke (i hours)? C B t (, 76) C C 0 0 t 0 0 C 0 0 t C (, 00) e t 0 0 t 0 0 C C Which of the followig represets the reltioship etwee t C? C + 0 t 0 B t + C 0 C t + 0 C 0 t C + 0 e 0 + t + C 0 C Whe the reltioship etwee t C is trspose to ke t the suject, it is the writte s: C + 0 C t B t + 0 C C t 0 C t 0 C 0 t 6 C If the totl ill ce to $79, for how log i the greer work? h B h i C h 0 i h i e h Questios 7 to 0 refer to the followig ifortio. The volue of squre-se pyri with the sie of the se s the height h is give y the forul V s h. 7 C The sie legth of the se of squre-se pyri with height h volue V is give y: v h h V V s B s C s s e s h V V h h 8 C The height of squre-se pyri with se sie legth c volue 7 c is: 8 c B 9 c C 0 c c e c 9 C If oth the sie of the se the height re oule, the volue is: oule B triple C icrese y ties icrese y 6 ties e icrese y 8 ties 96 Mths Quest Avce Geerl Mthetics

11 0 C If the sie of the se of pyri is oule ut its volue reis uchge, the height: ecoes twice s lrge C ecoes of the origil size B ecoes of the origil size ecoes ties s lrge e ecoes of the origil size The su of the iterior gles of regulr polygo is give y S ( ) 80, where is the uer of sies. Trspose the forul to ke the suject. Use the pproprite foruls to coplete the followig tle: Polygo Nuer of sies () Su of iterior gles (S) Trigle Hego 6 Doecgo Nogo 9 Heptgo The su of ters of rithetic sequece is give y the forul S [ ( ) ] +, where is the first uer of the sequece is the coo ifferece. Trspose the forul to ke the suject, hece fi the first ter i sequece tht hs 6, S 079. Trspose the forul to ke the suject, hece fi the coo ifferece of rithetic sequece with 0 ters, 8 S 0. igitl oc oc-996 WorkSHEET. Solvig lier equtios siulteous lier equtios A lier equtio is equtio tht cotis prouerl (ukow vlue) rise to the power of. Such equtio y lso e clle equtio of the fi rst egree. Eples of lier or first egree equtios iclue: 8, y 7 y +. Equtios of the type y, y, 8, + y y 8 re ot lier, s they coti prouerls tht re rise to powers other th i these cses,,,,, respectively. A lier equtio is equtio tht cotis prouerl rise to the power of. It y lso e clle equtio of the first egree. Solvig lier equtios Whe we re ske to solve equtio, we re to fi the vlue of the prouerl so tht whe it is sustitute ito the origil equtio, it will ke the equtio true stteet. Equtios re solve y perforig uer of iverse opertios to oth sies of the equtio util the vlue of the ukow is fou. Chpter Alger 97

12 Whe solvig equtios, the orer of opertios process, BODMAS (i.e. Brckets Of Divisio, Multiplictio, Aitio, Sutrctio), is reverse. We y therefore pply the SAMDOB process (BODMAS i reverse). This es tht the opertios of sutrctio itio re tke cre of first, followe y ultiplictio ivisio. Brckets re elt with lst. Worke eple 0 Solve the followig equtios. 0 the give equtio. (Optiol step.) Rule up tle with two colus to the sie of the equtio. I the first colu, ote ech of the opertios perfore o i the correct orer. I the seco colu, write the correspoig iverse opertio. The rrows iicte which opertio to egi with. Solve the equtio y kig the suject. A to oth sies of the equtio. Divie oth sies of the equtio y. Opertio Iverse Siplify. (or.) the give equtio. (Optiol step.) As i prt ove. Solve the equtio y kig the suject. Sutrct 0 fro oth sies of the equtio. 0 Opertio Iverse Multiply oth sies of the equtio y. Divie oth sies of the equtio y Siplify. 98 Mths Quest Avce Geerl Mthetics

13 Step i Worke eple 0 is optiol step tht y e use iitilly to help you ecoe filir with the process of solvig equtios. The swers y e checke y sustitutig the vlues otie ck ito the origil equtio or usig clcultor. If the prouerl ppers i the equtio ore th oce, we ust collect ters cotiig the ukow o oe sie of the equtio ll other ters o the other sie. Worke eple Solve for i the equtio + 6. the give equtio. + 6 Trspose to the LHS of the equtio y + 6 sutrctig it fro oth sies of the equtio. 6 A to oth sies of the equtio Divie oth sies of the equtio y. 0 Siplify. If the equtio cotis rckets, they shoul e epe first. I soe cses, however, oth sies of the equtio c e ivie y the coefficiet i frot of the rckets iste of epig. Worke eple Solve for i ( + ) ( 6). the give equtio. ( + ) ( 6) Ep ech of the rckets o oth sies of the equtio. Trspose 6 to the LHS of the equtio y sutrctig it fro oth sies of the equtio Sutrct 0 fro oth sies of the equtio Divie oth sies of the equtio y. 8 6 Siplify. 7 If equtio cotis frctio, we shoul first reove the eoitors y ultiplyig ech ter of the equtio y the lowest coo eoitor (LCD). Worke eple Fi the vlue of tht will ke the followig true stteet: +. + the give equtio. Deterie the LCD of. The LCD of is 6. Chpter Alger 99

14 Multiply ech ter of the equtio y the LCD. Siplify oth sies of the equtio ( + ) 0 6 ( + ) 0 Ep the rcket o the LHS of the equtio A to oth sies of the equtio Sutrct fro oth sies of the equtio Divie oth sies of the equtio y. 9 Siplify. (or.) 6 Soeties i equtios cotiig frctios, prouerl ppers i the eoitor. Such equtios re solve i the se er s those i the previous eples. However, cre ust e tke to ietify the vlue (or vlues) for which the prouerl will cuse the eoitor to e zero. If i the process of otiig the solutio the prouerl is fou to tke such vlue, it shoul e iscre. Eve though the process of ietifyig the vlue of the prouerl tht cuses the eoitor to e zero is t this stge erely precutio, this process shoul e prctise s it will prove useful i future chpters. Worke eple Solve the followig equtio for : +. Ietify the vlues of tht will cuse the eoitor to e zero. Note: Oce the equtio hs ee solve, vlues tht cuse the eoitor to e zero will e iscre. the give equtio. First frctio: 0 Seco frctio: 0 0 Thir frctio: 0 cot ssue the vlues of 0 or, ecuse they will cuse the frctio to e uefie. + tutoril eles-8 Worke eple Deterie the LCD of,. The LCD of, is ( ). Multiply ech ter of the equtio y the LCD. ( ) + ( ) ( ) Siplify oth sies of the equtio. ( ) 6 ( ) ( ) + ( ) + ( ) 6 Ep the rckets o the LHS of the equtio Collect like ters oto the LHS y sutrctig fro oth sies of the equtio A 7 to oth sies of the equtio Mths Quest Avce Geerl Mthetics

15 9 Divie oth sies of the equtio y. 0 Siplify. Note: The vlue of. is vli solutio. 7 (or.) Siulteous equtios It is ipossile to solve oe lier equtio with two ukows. There ust e two equtios with the se two ukows for solutio to e fou. Such equtios re clle siulteous equtios. Grphicl solutio of siulteous equtios If two stright lies itersect, the poit of their itersectio elogs to oth lies, hece the coorites of tht poit (, y) will represet the solutio of two siulteous equtios tht efie the lies. Whe we re solvig siulteous equtios grphiclly, the ccurcy of the solutio is highly epeet o the qulity of the grph. Therefore, ll grphs ust e rw o grph pper s ccurtely s possile. It is goo prctice to verify y swer otie fro grph y sustitutig it ito the origil equtios or usig CAS clcultor. Worke eple Solve the followig pirs of siulteous equtios grphiclly. + y y + 7 y y Rule up set of es. Lel the origi the - y-es. Fi the -itercept for the equtio + y y kig y 0. Fi the y-itercept for the equtio + y y kig 0. Divie oth sies of the equtio y. Plot the poits o grph pper joi the with stright lie. Lel the grph. Fi the -itercept for the equtio y y kig y 0. 6 Fi the y-itercept for the equtio y y kig 0. Multiply oth sies of the equtio y. /rw (See the grph t step 7 o pge 0.) -itercept: whe y 0, + y + 0 The -itercept is t (, 0). y-itercept: whe 0, + y 0 + y y y The y-itercept is t (0, ). (Refer to the grph t step 7.) -itercept: whe y 0, y 0 The -itercept is t (, 0). y-itercept: whe 0, y 0 y y y y The y-itercept is t (0, ). Chpter Alger 0

16 7 Plot the poits o grph pper joi the with stright lie. Lel the grph. y 0 y + y (, ) 8 Fro the grph, re the coorites of the poit of itersectio. 9 Verify the swer y sustitutig the poit of itersectio ito the origil equtios. Fi the il itercepts for y + 7. Let 0, the y 0. The poit of itersectio etwee the two grphs is (, ). Sustitute y ito + y. LHS + RHS + LHS RHS Sustitute y ito y. LHS RHS LHS RHS I oth cses LHS RHS; therefore, the solutio set (, ) is correct. 0, y y 7 y 0, Fi the il itercepts for y. Let 0, the y 0. 0, 0 y y y 0, 0 Plot the itercepts o grph pper. Joi the il itercepts for ech equtio. Lel the grph. 7 y (, ) y 7 y + 7 Fro the grph, re the coorites of the poit of itersectio. (, ) Verify your swer. Sustitute y ito y + 7. LHS + () RHS LHS RHS Sustitute y ito y. LHS () () RHS 0 6 LHS RHS I oth cses LHS RHS; therefore, the solutio set (, ) is correct. 6 the swer. The poit of itersectio etwee the two grphs is (, ). 0 Mths Quest Avce Geerl Mthetics

17 prllel lies If two equtios hve the se griet, they represet prllel lies. Such lies will ever eet so ever hve poit of itersectio (tht is, there is o solutio). The pir of equtios y + y + efie two prllel lies; hece, there is o solutio. The grph t right eostrtes tht the stright lies ever itersect. Coicietl lies If two lies coicie, the there re ifiite uer of solutios. For eple, cosier the two stright lies give y the equtios y + y. y + y + Rerrgig the seco equtio, y we oti the se lie. y y y 0 y + The two equtios whe grphe represet the se lie they coicie. Therefore, every poit o the lie will represet the solutio, s there is ot oe uique poit tht stisfies oth equtios. lgeric solutio of siulteous equtios Whe usig lger to solve siulteous equtios, the i is to oti oe equtio with oe ukow fro two equtios with two ukows y vrious lgeric ipultios. This c e oe i two wys sustitutio eliitio s outlie elow. Sustitutio etho The etho of sustitutio is esy to use whe t lest oe of the equtios represets oe ukow i ters of the other. To solve siulteous equtios usig the etho of sustitutio:. Check tht oe of the equtios is trspose so tht oe of the ukows is epresse i ters of the other.. Sustitute the trspose equtio ito the seco equtio.. Solve for the ukow vrile. y y 0 y + y Worke eple 6 Use the etho of sustitutio to solve the siulteous equtios y + y. the equtios oe uer the other, uer the. Sustitute the epressio for y ( + ) fro equtio [] ito equtio []. Note: By sustitutig oe equtio ito the other, we re left with oe equtio oe ukow. Solve for. () Ep the rckets o the LHS of the equtio. () Siplify the LHS of the equtio y collectig like ters. (c) A to oth sies of the equtio. () Divie oth sies of the equtio y. Sustitute i plce of ito [] to fi the vlue of y. y + [] y [] Sustitutig ( + ) ito []: ( + ) Sustitutig ito []: y + Evlute. 8 + Chpter Alger 0

18 6 Aswer the questio. Solutio:, y, or solutio set (, ). 7 Verify the swer y sustitutig the poit of itersectio ito the origil equtios or usig CAS clcultor. The swer ws checke usig CAS clcultor fou to e correct. If either of the equtios give oe ukow i ters of the other, we c still use etho of sustitutio y first trsposig oe of the equtios. eliitio etho As the e suggests, the ie of the eliitio etho is to eliite oe of the vriles. This is oe i the followig wy.. Choose the vrile you wt to eliite.. Mke the coefficiets of tht vrile equl i oth equtios.. Eliite the vrile y either ig or sutrctig the two equtios. Oce this is oe, the resultig equtio will coti oly oe ukow, which the c e esily fou. Worke eple 7 Use the eliitio etho to solve the followig siulteous equtios. + y y the equtios oe uer the other, uer the. A equtios [] [] to eliite y. Note: y is eliite ecuse the coefficiets of y i oth equtios re equl i gitue opposite i sig. Divie oth sies of the equtio y. Sustitute the vlue of ito equtio []. Note: y e sustitute i either equtio. Solve for y. () Sutrct fro oth sies of the equtio. () Divie oth sies of the equtio y. + y [] y [] [] + []: + y + ( y ) 6 6 Sustitutig ito []: y y y y 0 y 0 y 0 6 Aswer the questio. Solutio:, y 0, or solutio set (, 0). 7 Verify the swer y sustitutig the poit of itersectio ito the origil equtios or usig CAS clcultor. The swer ws checke usig CAS clcultor fou to e correct. If there is o pir of equl coefficiets, we c ke the the se y ultiplyig or iviig oe or oth equtios y pproprite uer. 0 Mths Quest Avce Geerl Mthetics

19 Worke eple 8 Solve the followig siulteous equtios. + y + y 0 Nuer ech equtio. + y [] + y 0 [] To eliite the vrile y, ultiply equtio [] y equtio [] y. + 6y 8 [] 9 6y 0 [] A the ew equtios. Solve for. To evlute y, sustitute solve for y. ito equtio [] + y [] ( y ) + + y 0 y y y 8 6 the solutio. y 8 or 7 Verify the solutio y sustitutig the vlues for y ito equtio []. y + y 0 [] ( ) ( ) Sice the LHS RHS, the solutio is verifie. 8 Aswer the questio. Solutio:, y 8 or, ( ) 8. eercise Solvig lier equtios siulteous lier equtios We 0 Solve the followig equtios c e f g h i 7 j + 7 k 6 l 7 0 Chpter Alger 0

20 We Solve for c e f g 7 + h + 0 i 8 j 6 k 9 + We Solve for. ( 0) 6 ( + 6) c ( 7) 8( + ) (7 ) e ( + ) (7 ) + f ( ) + 7( ) 0 We For ech of the followig, fi the vlue of tht will ke the stteet true c ( ) 7 + e + f ( 6) (6 ) ( + 7) g h 9 6 We For ech of the followig: stte which vlue (or vlues) of will cuse the equtio to e uefie solve for. i + ii + ( 6) 7 iii + iv ( )( + ) + (7 ) v vi 8 6 We Solve the followig pirs of siulteous equtios grphiclly. Verify your swers with CAS clcultor. + y 6, y + y, + y c y +, y 0 + y 6, y 7 C The pir of siulteous equtios y y hs: solutio B solutios C o solutios ifiite uer of solutios e oe of these 8 C The pir of siulteous equtios y y + 0 hs: solutio B solutios C o solutios ifiite uer of solutios e oe of these 9 Coplete the followig stteets. If two lies with ifferet griets go through the origi, the the solutio to the pir of siulteous equtios efiig those lies is (give coorites). If two lies hve the se griets ut ifferet y-itercepts, the the pir of siulteous equtios efiig such lies will hve solutio(s). c If two lies re efie y the equtio y + c hve the se vlue of c ut ifferet vlues of, the the solutio to the pir of siulteous equtios will e (give coorites). 0 We 6 Solve the followig pirs of siulteous equtios y the etho of sustitutio. y y y 6 y 0 c y 6 y 6 + y 6y 6 e y y f y y 06 Mths Quest Avce Geerl Mthetics

21 We 7 Use the etho of eliitio to solve ech of the followig. + y y 9 y 7 y + c y 6y 9 + y 7 y 9 e + y 6 + y 8 f y 0 y 9 C Nth is solvig pir of siulteous equtios, y [] + y 0 [], usig the eliitio etho. To eliite oe of the vriles, he coul ultiply equtio [] equtio [] y: respectively B respectively C respectively 0 respectively e respectively We 8 Solve ech of the followig pirs of equtios usig the eliitio etho. y 8 y c + y + y + y 9 y y 0 e y f + y + y 6 + y y + 0 C Without solvig the equtio, we kow tht will ot e equl to: + B C 0 0 or e or or 0 ( ) C To solve, ech ter of the equtio coul e ultiplie y: B C e 6 6 C To solve the equtio, the opertios tht ust e perfore re: oth sies y, the y B oth sies y C oth sies y oth sies y, the y e oth sies y, the y 7 C The equtio of lie [] i the igr t right is: y [] [] y B y C y y + e y + 0 The equtio of lie [] i the igr is: y + B y + C y y + 6 e y + 6 c The poit of itersectio of the two lies hs the coorites: (, ) B, C (, ), e, 8 Fi the vlue of z such tht the solutio to the followig equtio is. z 8 + ( )( + ) 9 Solve the followig equtio Fi the vlues of, y z i the followig three siulteous equtios with three ukows. + y z 7 + y + z y + z igitl oc oc-988 Ivestigtio Coprig prouctio costs Chpter Alger 07

22 e pplictios Lier equtios c ofte e use to help us i prole solvig. This is usully oe i the followig wy.. Ietify the ukow choose y coveiet prouerl (usully ) to represet it.. Use the ifortio give i the prole to copose equtio i ters of the prouerl.. Solve the equtio to fi the vlue of the prouerl.. Iterpret your result y reltig the swer ck to the prole. Worke eple 9 If the su of twice certi uer is ultiplie y the ivie y 7, the result is 9. Fi the uer. Assig the prouerl to the ukow vlue. Let the ukow uer. Buil the equtio ccorig to the ifortio give. () Twice the uer this es, so write this. () The su of twice the uer this es +, so this o. (c) The su is ultiplie y this es ( + ). A this o. Note: We iclue rckets to iicte the orer of opertios. ( + ) () The result is ivie y 7 this es. 7 A this o. (e) The result is 9 which es tht ll of the previous coputtios will equl 9. this. + ( + ) ( + ) 7 ( + ) 9 7 Solve for. () Multiply oth sies of the equtio y 7. ( + ) ( + ) 6 () Divie oth sies of the equtio y ecuse they re oth ivisile y. ( + ) 6 + (c) Sutrct fro oth sies of the equtio. + 6 () Divie oth sies of the equtio y. (e) Siplify. 8 6 Aswer the questio. The ukow uer is 8. Soeties the prole cotis ore th oe ukow. I such cses oe of the ukows is clle the other ukows re the epresse i ters of. 08 Mths Quest Avce Geerl Mthetics

23 Worke eple 0 Srh is uyig tulip uls. Re tulip uls cost $.0 ech, yellow tulip uls cost $.70 ech. If uls cost Srh $07.0, how y of ech type i she uy? Defie the vriles. Note: Sice there re uls ltogether, the uer of yellow tulip uls is the uer of re tulip uls; tht is,. Let the uer of re tulip uls. Let the uer of yellow tulip uls. epressio for the cost of the re tulips. Note: Oe re tulip costs $.0; therefore re tulips cost.0. epressio for the cost of the yellow tulips. Note: Oe yellow tulip costs $.70; therefore tulips cost.70 ( ). Forulte equtio reltig the totl cost of the re yellow tulips the epressios otie i steps. Solve the equtio. () Ep the rckets o the LHS of the equtio. () Collect the like ters o the LHS of the equtio. (c) Sutrct 0. fro oth sies of the equtio. () Divie oth sies of the equtio y 0.. Totl cost of re tulip uls.0. Totl cost of yellow tulip uls.70 ( ).7( ) The totl cost of the re yellow tulip uls is $07.0. Also, the totl cost of the re yellow tulip uls is. +.7( ). Therefore,. +.7( ) (e) Siplify. 8 6 Iterpret the swer otie. There re 8 re (tht is, 8) yellow tulip uls. 7 Aswer the questio. Srh purchse 8 re yellow tulip uls. Chpter Alger 09

24 Worke eple A tri (eote s tri ) leves sttio A oves i the irectio of sttio B with verge spee of 60 k/h. Hlf hour lter other tri (eote s tri ) leves sttio A oves i the irectio of the first tri with verge spee of 70 k/h. Fi: the tie eee for the seco tri to ctch up with the first tri the istce of oth tris fro sttio A t tht tie. Defie the vriles. Note: Sice the first tri left hlf hour erlier, the tie tke for it to rech the eetig poit will e the spee of ech tri. Tri : v 60 Tri : v 70 the istce trvelle y ech of the tris fro sttio A to the poit of the eetig. (Distce spee tie.) Equte the two epressios for istce. Note: Whe the seco tri ctches up with the first tri, they re the se istce fro sttio A tht is,. Let the tie tke for tri to rech tri. Therefore, the trvellig tie, t, for ech tri is: Tri : t + 0. Tri : t Tri : 60( + 0.) Tri : 70 the equtio. 60( + 0.) 70 6 Solve for Sustitute i plce of ito either of the two epressios for istce, sy ito. Whe the seco tri ctches up with the first tri,. Sustitute ito Evlute. 0 9 Aswer the questios. The seco tri will ctch up with the first tri hours fter levig sttio A. Both tris will e 0 k fro sttio A. Siulteous equtios re use to solve vriety of proles cotiig ore th oe ukow. Here is siple lgorith tht c e pplie to y of the:. Ietify the vriles.. Set up siulteous equtios y trsforig writte ifortio ito lgeric seteces.. Solve the equtios y usig the sustitutio, eliitio or grphicl ethos.. Iterpret your swer y referrig ck to the origil prole. Worke eple Fi two cosecutive uers tht up to 99. Defie the two vriles. Let the first uer. Let y the seco uer. Forulte two equtios fro the ifortio give uer the. Note: Cosecutive uers follow oe other iffer y. Hece, if is the first uer, the et uer will e + tht is, y +. + y 99 [] y + [] 0 Mths Quest Avce Geerl Mthetics

25 Sustitute the epressio ( + ) fro equtio [] for y ito equtio []. Sustitutig ( + ) ito []: Solve for. () Siplify the LHS of the equtio y collectig like ters () Sutrct fro oth sies of the equtio (c) Divie oth sies of the equtio y. Sustitute 9 i plce of ito equtio [] to fi the vlue of y Sustitutig 9 ito equtio []: y + 6 Evlute. y Verify the swer y checkig tht the two vlues re cosecutive tht they su re cosecutive uers The otie vlues stisfy the prole. 8 Aswer the questio. The two cosecutive uers tht up to 99 re 9 0. Worke eple Two hurgers pcket of chips cost $8.0, oe hurger two pckets of chips cost $.90. Fi the cost of pcket of chips hurger. Defie the two vriles. Let the cost of oe hurger. Let y the cost of pcket of chips. Forulte equtio fro the first setece cll it []. Note: Oe hurger costs $, two hurgers cost $. Thus, the totl cost of cost of two hurgers oe pcket of chips is + y it is equl to $8.0. Forulte equtio fro the seco setece cll it []. Note: Oe pcket of chips costs $y, two pckets cost $y. Thus, the totl cost of two pckets of chips oe hurger is + y it is equl to $.90. To eliite the vrile, ultiply equtio [] y. + y 8.0 [] + y.90 [] y.80 [] A the ew equtios. + y 8.0 [] y.80 [] y.60 6 Solve for y. y.0 Chpter Alger

26 7 To evlute, sustitute y.0 ito equtio [] solve for. 8 Aswer the questio iclue pproprite uits. + y 8.0 [] A hurger costs $.0 pcket of chips costs $.0. It is etreely iportt to e cosistet with the use of uits while settig up equtios. For eple, if the cost of ech ite is epresse i cets, the the totl cost ust lso e epresse i cets. eercise e pplictios We 9 The verge of three cosecutive o uers is. Fi the lrgest uer. Hlf of certi uer is sutrcte fro 6 the result is the triple, the swer is 8. Fi the uer. The su of oe-thir of uer is 7. Fi the uer. We0 Fio is uyig tulip uls. Re tulip uls cost $6.0 ech yellow tulip uls cost $.0 ech. If 8 uls cost Fio $67.0, how y of ech type i she uy? A rectgle is. ties s log s it is wie. Fi the iesios of the rectgle if its perieter is 6 c. 6 I isosceles trigle, two sies of equl legth re together 8 c loger th the thir sie. If the perieter of the trigle is c, wht is the legth of ech sie? 7 I sclee trigle, the first gle is ties s lrge s the seco, the thir gle is 0 sller th the seco. Fi the size of ech gle; hece, e the trigle ccorig to its gles sizes. 8 All ites t clothig store hve ee reuce y %. If Stephie purchse shirt t the reuce price of $8., wht ws its origil price? 9 C If 7 ties uer sutrcte fro gives, the the uer is: 7 B 7 C 8 6 e The su of oe-qurter of uer 0 is. The vlue of the uer is: 00 B 0 C 0 0 e 0 0 I ties s ol s y cousi Crl, who is ties s ol s y ughter Ni. If our totl ges re yers, how ol is y cousi? Aother cousi, Zr, is Crl s oler sister. Zr is s y ties s ol s y ughter Ni s the uer of yers tht she is oler th Crl. How ol is y other cousi? Sio is oly 6 yers ol, ut he hs lrey live i four ifferet coutries ecuse of his fther s jo. He ws or spet few yers of his erly chilhoo i the USA, the the fily ove to Gery, where he stye oe yer loger th he h i the USA. After tht, he live i Loo for twice s log s he h i Gery. Filly, they ce to live i Meloure. So fr, he hs ee i Austrli for yers less th he live i Aeric. At wht ge i Sio leve his coutry of irth? For how log i Sio live i ech coutry? Mths Quest Avce Geerl Mthetics

27 I the storeroo of fruit shop there were two oes of pples, oe of Gole Delicious the other of Joths, which were to e sol t $.80/kg $.0/kg respectively. The pples, however, were ccietlly ie together, iste of sortig the out, the ower ecie to sell the s they were. So s ot to ke loss, he sol the ie pples t $.0/kg. How y kilogrs of ech type of pple were there if together they weighe kg? We Ale Nt re goig for ike rie. Nt c rie t 0 k/h, Ale c evelop iu spee of k/h if he ees to. Nt leves hoe t 0, while Ale stys ehi for iutes the sets out to ctch up with Nt. Whe will Ale rech Nt, ssuig tht oth of the re riig t their iu spee? Suel is plig with costt spee towrs certi plce he hs rke o his p. With the i of curret (which hs spee of k/h) it tkes hi oly h 0 i to rech his estitio. However, o the wy ck he hs to ple gist the curret, it the tkes hi h to rech his strtig poit. Fi Suel s spee o the still wter. Oe iistrtive ssistt c type. ties s fst s other. If they oth work together, they c fiish certi jo i 6 hours. However, the slower oe workig loe will ee hours to fiish the se jo. How y hours will the quicker ssistt loe ee to coplete the jo? 6 My ees to reovte her house. She hs eough oey to py pluer for 8 ys or crpeter for ys. For how y ys c she py the reovtors if they oth work t the se tie? If My s et py cheque will coe i weeks, c she ffor to hire oth specilists util the? 7 I prticulr school, uer of VCE stuets otie tertiry etrce score higher th 99., % ore stuets otie score higher th 99.0 ut lower th 99.. If there were stuets whose tertiry etrce scores were ove 99.0, how y of those otie score ove 99.? 8 We Fi two cosecutive uers tht up to Whe three ties the first uer is e to twice the seco uer, the result is. Four ties the ifferece of those uers is. Fi the uers. 0 Hlf of the su of two uers is 6 less th the first uer. Oe-thir of their ifferece is oe less th the seco uer. Fi the uers. Five ties the first uer is twice s lrge s four ties the seco uer. Whe the ifferece of the two uers is ultiplie y 0, the result is. Fi the uers. A rectgle s legth is c ore th its with. If the perieter of rectgle is c, fi its iesios, hece, its re. ( ) c I the rectgle t right, fi the vlues of y. Hece, eterie the perieter. ( + ) c 0 c ( + y) c The sies of equilterl trigle hve the followig legths: ( + y) c, ( ) c (y ) c. Fi the perieter of the trigle. The perieter of rhous ABCD is 0 c loger th the perieter of isosceles trigle ABC. Fi the legth of AC, the igol of rhous, if it is c sller th its sie. 6 A tle cosists of colus rows. Ech of its cells is rectgle with legth c with y c. The perieter of the tle is 70 c the totl legth of iterior lies is 6 c. Drw igr to represet the ove ifortio. Fi the iesios of ech cell coet o its shpe. Chpter Alger

28 igitl oc oc-9966 WorkSHEET. 7 Phuog coucts survey. She sks 7 people whether or ot they use the iteret t hoe. There were three ties s y people who swere Yes s those who swere No. Fi the uer of people i ech ctegory hece help Phuog to coplete the followig stteet: Accorig to the survey, (isert frctio) of the popultio uses the iteret t hoe. 8 We At the e of the y, two shop ssistts copre their sles. Oe sol tosters swich-kers for totl of $9.6, while the other sol of ech for totl vlue of $.70. Fi the price of ech ite. 9 I qutic cetre, pool sp etry is $.0, pool, sp, su ste roo etry is $.0. At the e of the y, cshier fis tht she sol 9 tickets ltogether, her tkigs re 0c short of $800. How y of ech type of ticket were sol? 0 Spiro epties his piggy k. He hs cois, soe of which re c cois soe of which re 0c cois, to the totl vlue of $.0. How y c cois how y 0c cois oes he hve? My Rose re uyig et for picic. My s fily likes l ore th pork, so she uys kg of l oly hlf s uch pork. Rose s fily hve ifferet tstes, so she uys. kg of pork oe-thir s uch l. If My spes $.0, which is $8. less th Rose spes, wht is the cost of kg of ech type of et? Bell Boris re celertig their th weig iversry. Toy, their coie ge is ectly 00. If Boris is yers oler th Bell, how ol ws his rie o the y of their weig? Ssh is kig i sis sprig rolls for his guests. He is goig to prepre everythig first the cook. O verge it tkes 0. hours to prepre oe portio of i sis 0. hours to prepre oe portio of sprig rolls. He ees 0.0 hours 0. hours to cook ech portio of i sis sprig rolls respectively. If he spes hours o preprtio iutes o cookig, how y portios of i sis sprig rolls oes Ssh ke? F lgeric frctios Algeric frctios re frctios tht coti prouerl. Perforig opertios ivolvig these frctios ivolves the se rules s those for uericl frctios. itio sutrctio Frctios c oly e e or sutrcte whe they hve coo eoitor. Worke eple Siplify: + +. Fi the lowest coo eoitor. 0 Lowest coo eoitor 0 Multiply the first ter y the seco ter y A the uertors. 7 0 Fi the lowest coo eoitor. Lowest coo eoitor Multiply the first ter y the seco ter y. 6 + A the uertors. + 6 Mths Quest Avce Geerl Mthetics

29 Worke eple Siplify: + +. tutoril eles-8 Worke eple Fi the lowest coo eoitor. Lowest coo eoitor ( ) Multiply the first ter y the seco ter y. ( ) ( ) ( ) A the uertors. ( ) ( ) Ep siplify the uertor. ( ) Fi the lowest coo eoitor. Lowest coo eoitor ( + )( ), or 9 Multiply the first ter y ( ) ( + ) + the seco ter y A the uertors. ( ) + ( + ) 9 Ep the uertor ultiplictio ivisio Whe ultiplyig, ccellig verticlly or igolly helps to siplify epressios efore ultiplyig top otto. This y ivolve fctorisig epressios to ietify coo fctors. Worke eple 6 Siplify: y 6 y + 0. Ccel coo fctors etwee uertors eoitors, the siplify. Multiply uertors together eoitors together. Coo fctors:,, 0 9 Fctorise the eoitor. + y 6 y ( + ) Ccel coo fctors siplify. y Multiply uertors together eoitors together. y Chpter Alger

30 eercise F lgeric frctios We Siplify the followig epressios. + + c 7 e h + f h k + g 7 Siplify the followig epressios. + + p ( p+ ) q ( q+ ) c 7 e ( s ) ( s+ ) (t+ ) ( t ) f w ( ) ( + ) g h + ( w ) We Siplify the followig epressios. 7 + ( y+ ) ( y ) ( z+ ) ( z ) c + e + ( y) ( y+ ) ( + ) ( + ) f h g g + ( r+ ) ( r ) v + (v ) 8 + ( ) ( ) 7 ( ) ( ) C Whe siplifyig the epressio, the lowest coo eoitor is: ( ) ( ) B ( ) C ( ) ( ) e ( ) + C Siplifyig ( ) gives: B C () () ( ) e () (+ ) 9 e ( e + ) 6 C Siplifyig gives: e e e e e( e) + e + e B C e e e 7 We6 Siplify the followig epressios. ( ) + ( + ) ( + ) 8 c e e g( g) g 7 hh ( + ) h e f e ( + e) 6 g 7h + h 8 Siplify the followig epressios. ( j )( j+ ) ( j )( j+ 7) ( k+ ) ( k) ( k+ )( k) ( j + 7) ( j + ) ( k ) ( k ) + + ( + ) 9 c 6( ) ( ) 6 p+ ( p+ ) ( q )( q+ ) ( q ) e f 7 p q + q + r r ( s ) s g h (r + r+ ) (r + ) 8s 7 ( s 9) 6 Mths Quest Avce Geerl Mthetics

31 G lier literl equtios Literl equtios re those tht re writte i ters of prouerls such tht o uique uericl solutio will e possile, ut rther epressio cotiig these prouerls. A equtio such s p coul e escrie s lier literl equtio i, s it is lier cotis prouerls rther th uers. (Note, tht i this cse is efie s the vrile.) A solutio to literl equtio c e eterie lgericlly y the use of iverse opertios just s for uericl equtio. The ifferece is tht the solutio will e geerl oe tht is, i ters of the prouerls. I the eple ove, the solutio to this equtio c e otie y isoltig s the suject s follows: p p + p + ( to oth sies of the equtio) (ivie y o oth sies of the equtio) Note tht literl equtios lwys coti t lest oe prouerl (prt fro the vrile), ut they y lso coti uerls. Worke eple 7 Solve for. c ( ) A c to oth sies. c + c Multiply oth sies y. ( + c) ( + c) Divie oth sies y. Multiply oth sies y. ( ) ( ) Multiply oth sies y. ( ) Ep. Collect ters. Fctorise. ( ) Divie oth sies y. Solvig siulteous literl equtios requires the se etho s uericl lier equtios, ely, sustitutio or eliitio ethos. The solutios will e i ters of the prouerls. Chpter Alger 7

32 Worke eple 8 Solve for y. y y 6 tutoril eles-8 Worke eple 8 Assig uer to ech equtio. y [] y 6 [] Multiply equtio [] y. Assig uer to the ew equtio. Sutrct equtio [] fro equtio [] to eliite the ter. y 8 [] y Solve for y. y Sustitute this vlue of y ito equtio [] + solve for Stte the solutio. 8, y eercise G lier literl equtios + y + We7 Solve for. ( ) c y c p w + r k t e f g f l s + l g + c h c i ( ) j k r ( s ) l p( q) ( + p) e + + c f( + ) c+ e o + p f e C The solutio to y c i ters of is: c y c ( ) c( c y) B C c ( + y) c( c+ y) e We8 Solve the followig siulteous equtios. + y + y y c y y y y Mths Quest Avce Geerl Mthetics

33 e + y + y f ( ) + + y + y ( ) + 0 g y + h y y + + y The su of ters of rithetic sequece is give y the forul S [ ( ) ] +, where is the first uer of the sequece is the coo ifferece. Trspose the forul to ke the suject, hece fi the first ter i sequece tht hs 6, S 079. Trspose the forul to ke the suject, hece fi the coo ifferece of rithetic sequece with 0 ters, 8 S 0. Chpter Alger 9

34 Sury review of ie lws 0 + ( ) ( ) ( ) Str for sigifict figures trspositio Solvig lier equtios siulteous lier equtios pplictios usig lier equtios The uer of sigifict figures i uer c e eterie y cosierig ech of the followig rules:. Sigifict figures re coute fro the first o-zero igit ( 9).. Ay zeros t the e of the uer fter the ecil poit re cosiere to e sigifict.. The trilig zeros t the e of uer re ot cosiere sigifict.. All zeros etwee two o-zero igits re lwys sigifict. Whe perforig clcultios ssocite with sigifict figures, the followig rules pply:. Whe ig or sutrctig uers, cout the uer of ecil plces to eterie the uer of sigifict figures. The swer cot coti ore plces fter the ecil poit th the lest uer of ecil plces i the uers eig e or sutrcte.. Whe ultiplyig or iviig uers, cout the uer of sigifict figures. The swer cot coti ore sigifict figures th the uer eig ultiplie or ivie with the lest uer of sigifict figures. Trspositio is the rerrgeet of ters withi forul. The suject of the forul is the vrile tht is y itself o oe sie of the equtio, while ll other vriles re o the other sie. Lier equtios c e solve y usig iverse opertios. Whe solvig lier equtios, the orer of opertios process, BODMAS, is reverse. Siulteous lier equtios c e solve either grphiclly or lgericlly.. Grphicl etho: Drw the stright lies represetig the equtios fi the coorites of the poit of itersectio.. Algeric ethos: () Sustitutio: Trspose oe of the equtios so tht oe of the ukows is epresse i ters of the other sustitute ito the seco equtio. () Eliitio: Equte the coefficiets of oe ukow eliite it y either ig or sutrctig the two equtios. To solve wore proles usig lier equtios, follow these steps:. Ietify the vriles.. Set up equtio y trsforig the writte ifortio ito lgeric stteet or stteets.. Solve the equtio.. Iterpret the result y reltig the swer ck to the origil prole. 0 Mths Quest Avce Geerl Mthetics

35 pplictios usig siulteous equtios lgeric frctios lier literl equtios. Ietify efie the vriles.. Trsfor writte ifortio ito lgeric stteets.. Solve the pir of equtios grphiclly or lgericlly usig the ethos of sustitutio or eliitio.. Iterpret the result y reltig the swer ck to the prole.. Alwys ke sure the uers i the equtios re i the se uits. To or sutrct lgeric frctios, first fi coo eoitor solve y ig or sutrctig the uertor. If the prouerl is i the eoitor, you c geerlly fi the coo eoitor y ultiplyig the eoitors together. Whe ultiplyig or iviig, ccel ow efore ultiplyig through uertors eoitors. Lier literl equtios re equtios tht hve vrile, such s, with costts coefficiets tht re either uers or prouerls. To solve lier literl equtios, use the iverse opertios to oti equtio with the vrile s the suject i ters of the other prouerls. The solutio will e i ters of the prouerls. Siulteous lier literl equtios c e solve usig eliitio or sustitutio ethos. The solutios will e i ters of the prouerls. Chpter Alger

36 Chpter review Short SWer Siplify y ( y ). the followig i str for siplify c Clculte the followig to the correct uer of sigifict figures Trspose ech of the followig foruls to ke the prouerl iicte i rckets the suject. (If two prouerls re iicte, ke seprte trspositio for ech.) 6 y + 0 () 7( ) 8( e+ 7) (, e) Solve for Fi the equtios of the two lies show o the igr. y 0 Fi the coorites of the poit of itersectio (the igr is ot rw to scle). 7 Solve the followig siulteous equtios. 6 + y 8y y 7 + y 8 c y 0 + y 8 Before opeig the store, cshier kes sure tht his register cotis t lest $ i chge. He couts uer of 0c cois, twice s y c cois ties s y 0c cois, to the totl vlue of ectly $. How y cois of ech type oes he cout? 9 A uilig copy chrges $00 set fee plus $00 y while it is workig o project withi the tie liits tht re specifie y cotrct. If the project is coplete erlier th the set tie, the copy will still chrge $00 for ech of the reiig ys. However, if the project is ot coplete y the ue te, the copy will py $ pelty for ech etr y util the work is oe. Fro the give ifortio, costruct set of foruls for the totl cost of work, T; the uer of ys it tkes to coplete the jo ccorig to the cotrct, ; the uer of etr ys, e. 0 Jessic is yers oler th Reecc. I yers she will e ties s ol s Reecc ws yers go. Fi the girls preset ges. Siplify the followig. + 6 c + ( + )( + ) ( )( + ) ( g )( g+ ) ( g )( g+ ) Siplify. ( g ) ( g ) w w 9 Siplify w 8 ( w 6). Solve for. p w + pk g k Solve the followig siulteous equtios for y. y + y Mths Quest Avce Geerl Mthetics

37 is equl to: B 7 e C The uer of sigifict figures i 0.0 is: B C e 6 The solutio to with the correct uer of sigifict figures is: 0. B 0.0 C 00 0 e 000 The solutio to with the correct uer of sigifict figures is: 6. B 6. C e 6.78 B + If A is trspose to ke B the suject, the: B A A + B B C B A Ultiple ChoiCe B A e B A + Questios 6 7 refer to the shpe t right. 6 Usig π, the perieter of certi shpe is give y 7. Whe trspose to ke the suject, is: 7 B P P 7P C 7P 7P e 7( P ) 7 If the perieter of the ove shpe is 8 c, the is equl to: c 7 B c C c c e.7 c 8 The solutio to the equtio is: B C e 6 9 To solve 6, the followig opertios coul e perfore to oth sies of the equtio: A, ultiply y, ivie y B Multiply y, ivie y, sutrct C Multiply y, ivie y, sutrct Sutrct, ultiply y, ivie y e Multiply y, sutrct, ivie y 0 A equtio tht is the se s ( ) + is: 6 + B C + e Chpter Alger

38 The vlue for tht stisfies the equtio + is: B C e 6 8 The vlue for tht stisfies the equtio + is: B 6 C e The su of solutios of the pir of siulteous equtios y + y 6 is: 6 B C 0 e 8 If y y +, the the vlues of y, respectively, re: 9 B 9 C 9 9 e 9 The poit of itersectio of the two lies i the grph t right is: y (, ) B (, ) C (, ) ( 9, 9 ) e (, ) Which stteet elow is ot true for the pir of siulteous equtios y + y 6? The su of the uers is. B Three ties the first uer is 6 lrger th the seco uer. C Three ties oe uer is 6 sller th the other uer. The ifferece etwee ties oe uer the other is 6. e Whe oe uer is sutrcte fro, the other uer is otie. 7 If ties uer sutrcte fro 6 gives 9, the the uer is: B C e 8 The perieter of regulr hego is.6 c ore th the perieter of squre with the se sie legth. The legth of the sie of hego is:. c B. c C.6 c.6 c e 6. c 9 Whe hlf uer is sutrcte fro 8, the result is the se s ig oule tht uer to. The equtio tht tches this ifortio is: 8 + B 8 + C e The su of two uers is their ifferece is. The sller of the uers is: B 7 C 8 9 e Be is yer short of eig twice s ol s Esther. If their ges totl 0 yers, Be is: B C e ( ) ( + ) Whe is epresse s + + c, the, c, respectively, re: 6 0, 9 B 0, 9 C 0, 9 0, 9 e 0, 9 9 e ( e + ) Siplifyig gives: e e e e B e C e( e) + e e + e e Mths Quest Avce Geerl Mthetics