Chapter 8 Page Decompose Where Has Non repeated Irreducible

Ch.8 SystemsofEquationsandInequalities

8.1 SystemsofLinearEquations:SubstitutionandEliminatio

n

1 SolveSystemsofEquationsbySubstitution

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Verifythatthevaluesofthevariableslistedaresolutionsofthesystemofequations.

1)

x+y=–2

x–y=6

x=2,y=–4

A) solution B) notasolution

2)

x+y=0

x–y=–4

x=2,y=2

A) solution B) notasolution

3)

4x+y=20

2x+4y=–4

x=6,y=–4

A) solution B) notasolution

4)

2x+y=1

3x+2y=3

x=–1,y=–3

A) solution B) notasolution

Solvethesystemofequationsbysubstitution.

5)

x+y=–7

x–y=12

A) x=2.5

,

y=–9.5;(2.5

,

–9.5) B) x=2.5

,

y=9.5;(2.5

,

9.5)

C) x=7

,

y=–9.5;(7

,

–9.5) D) x=7

,

y=2.5;(7

,

2.5)

6)

x+7y=-

2

3x+y=34

A) x=12,y=–2;(12,–2) B) x= –2,y=3;(–2,3)

C) x=3,y=7;(3,7) D) x=7,y=12;(7,12)

7)

x+8y=8

8x–3y=-

3

A) x=0,y=1;(0,1) B) x=1,y=0;(1,0) C) x=1,y=1;(1,1) D) x=0,y=0;(0,0)

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8)

5x–2y=–1

x+4y=35

A) x=3,y=8;(3,8) B) x=3,y=9;(3,9) C) x=2,y=8;(2,8) D) x=2,y=9;(2,9)

9)

5x+3y =-

4

5x =-

10

A) x=–2

,

y=2;(–2

,

2) B) x=2

,

y= –2;(2

,

–2)

C) x=–2

,

y=–2;(–2

,

–2) D) x= –2

,

y=0;(–2

,

0)

10)

5x+3y=80

2x+y=30

A) x=10,y=10;(10,10) B) x=0,y=10;(0,10)

C) x=10,y=0;(10,0) D) x=0,y=0;(0,10)

11)

3x+y=0

–3x+y=–6

A) x=1,y=–3;(1,–3) B) x= –1,y=3;(–1,3)

C) x=1,y=6;(1,6) D) x= –1,y= –3;(–1,–3)

12)

x+y=0

2x+3y=–7

A) x=7,y=–7;(7,–7) B) x= –7,y=7;(–7,7)

C) x=6,y=–6;(6,–6) D) x= –6,y=6;(–6,6)

13)

3x+y=13

2x–7y=24

A) x=5,y=–2;(5,–2) B) x= –5,y=2;(–5,2)

C) x=5,y=2;(5,2) D) x= –5,y= –2;(–5,–2)

14)

1

5x–y=–4

5

x+8y=–4

A) x=–4,y=0;(–4,0) B) x=4,y=0;(4,0)

C) x=0,y=–4;(0,–4) D) x=0,y=4;(0,4)

15)

1

2x+2

3y=32

1

4x–5

9y=40

A) x=100,y=–27;(100,–27) B) x=100,y=27;(100,27)

C) x=–100,y=–27;(–100,–27) D) x= –100,y=27;(–100,27)

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16)

2x+9y=–8

x+ 1

3y=13

3

A) x=5,y=–2;(5,–2) B) x= –5,y=2;(–5,2)

C) x=5,y=2;(5,2) D) x= –5,y= –2;(–5,–2)

2 SolveSystemsofEquationsbyElimination

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequationsbyelimination.

1)

6x+42y=42

5x–7y=-

7

A) x=0,y=1;(0,1) B) x=1,y=0;(1,0) C) x=1,y=1;(1,1) D) x=0,y=0;(0,0)

2)

5x–2y=–1

x+4y=35

A) x=3,y=8;(3,8) B) x=3,y=9;(3,9) C) x=2,y=8;(2,8) D) x=2,y=9;(2,9)

3)

x+y=-

2

x–y=13

A) x=5.5

,

y=–7.5;(5.5

,

–7.5) B) x=5.5

,

y=7.5;(5.5

,

7.5)

C) x=2

,

y=–7.5;(2

,

–7.5) D) x=2

,

y=5.5;(2

,

5.5)

4)

6x+8y=48

6x+2y=84

A) x=16

,

y=–6;(16

,

–6) B) x= –16

,

y=6;(–16

,

6)

C) x=–16

,

y=8;(–16

,

8) D) x= –6

,

y=16;(–6

,

16)

5) 6x+3y=51

2x–6y=38

A) x=10,y=–3;(10,–3) B) x= –3,y=10;(–3,10)

C) x=–10,y=3;(–10,3) D) x=3,y= –10;(3,–10)

6) 3x–5y=-

12

6x+8y=-

24

A) x=–4,y=0;(–4,0) B) x=4,y=0;(4,0)

C) x=0,y=–4;(0,–4) D) x=0,y=4;(0,4)

7)

2x+12y=-

60

8x+2y=36

A) x=6

,

y=–6;(6

,

–6) B) x= –6

,

y=6;(–6

,

6)

C) x=8

,

y=–8;(8

,

–8) D) x= –2

,

y=6;(–2

,

6)

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8)

5x+3y=80

2x+y=30

A) x=10,y=10;(10,10) B) x=0,y=10;(0,10)

C) x=10,y=0;(10,0) D) x=0,y=0;(0,0)

9)

7

12x–y=10

5

9x+2y=11

A) x=18,y=1

2;18,1

2B) x=16,y=1

2;16,1

2

C) x=–18,y=–1

2;–18,–1

2D) x=–16,y=–1

2;–16,–1

2

10)

2

5x+7

10y=31

10

4x+2y=46

A) x=13

,

y=–3;(13

,

–3) B) x= –13

,

y=4;(–13

,

4)

C) x=–13

,

y=7;(–13

,

7) D) x= –3

,

y=13;(–3

,

13)

Solvethesystemofequations.[Hint:Letu=1

xandv=1

y,andsolveforuandv.Thenletx=1

u,andy=1

v.]

11)

2

x

+4

y

=7

1

x

–2

y

=4

A) x=4

15,y=–8;4

15,–8 B) x=15

4,y=–1

8;15

4,–1

8

C) x=–8,y=4

15;–8,4

15 D) x=–1

8,y=15

4;–1

8,15

4

Solvetheproblem.

12) Aflatrectangularpieceofaluminumhasaperimeterof60 inches.Thelengthis12incheslongerthanthe

width.Findthewidth.

A) 9in. B) 21in. C) 33 in. D) 30in.

13) TheFamilyFineArtsCentercharges$20 peradultand$11 perseniorcitizenforitsperformances.Ona

recentweekendeveningwhen504peoplepaidadmission,thetotalreceiptswere$6840.Howmanywho

paidwereseniorcitizens?

A) 360seniorcitizens B) 144seniorcitizens C) 270 seniorcitizens D) 234 seniorcitizens

14) Aretiredcouplehas$180,000toinvesttoobtainannualincome.Theywantsomeofitinvestedinsafe

CertificatesofDeposityielding6%.TheresttheywanttoinvestinAAbondsyielding11%peryear.How

muchshouldtheyinvestineachtorealizeexactly$16,800peryear?

A) $120,000at11%and$60,000at6% B) $120,000 at6%and$60,000at11%

C) $110,000at6%and$70,000at11% D) $130,000 at11%and$50,000at6%

Page4

15) Atourgroupsplitintotwogroupswhenwaitinginlineforfoodatafastfoodcounter.Thefirstgroup

bought8slicesofpizzaand7softdrinksfor$38.83.Thesecondgroupbought6slicesofpizzaand6soft

drinksfor$30.54.Howmuchdoesonesliceofpizzacost?

A) $3.20persliceofpizza B) $1.89 persliceofpizza

C) $2.70persliceofpizza D) $2.39 persliceofpizza

16) Amovietheatercharges$8.00foradultsand$5.00forchildren.Iftherewere40peoplealtogetherandthe

theatercollected$272.00attheendoftheday,howmanyofthemwereadults?

A) 24adults B) 10adults C) 16adults D) 29adults

17) An8–cylinderCrownVictoriagives18milespergallonincitydrivingand21milespergalloninhighway

driving.A300–miletriprequired15.5gallonsofgasoline.Howmanywholemilesweredriveninthecity?

A) 153mi B) 147mi C) 132mi D) 168mi

18) ThePaperbackTraderisabookstorethattakesinusedpaperbacksfor20%oftheircoverpriceandsells

themfor50%oftheircoverprice.Patbringsinastackof15paperbackbookstotradeandgets$13.97

credit.Someofthebookshadacoverpriceof$5.99,therest$3.99.ShewantstogetsomeTomClancy

bookshavingacoverpriceof$5.99.Howmany$5.99booksdidshebringinandhowmanyClancybooks

canshegetwithoutpayinganyadditionalcash?

A) 5$5.99books,4Clancybooks B) 10 $5.99 books,2Clancybooks

C) 5$5.99books,5Clancybooks D) 5 $5.99 books,2 Clancybooks

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

19) Theperimeterofaparkinglotis500yards.Findthedimensionsofthelotifthelengthis50yardsmore

thanthreetimesthewidth.

20) Astorehasasaleonworkoutgear.MarkboughtthreepairsofshortsandthreeT–shirtsfor$70.35(before

tax).Later,hewentbackandboughttwomorepairsofshortsandfourmoreT–shirtsfor$63.90(before

tax).HowmuchdidtheshortsandT–shirtscost?

21) Ateashopownerismixingablendoftwoteas,oneofwhichcosts$6.50perpound,theothercosting$4.00

perpound.Theownerwantstohave20poundsofamixturethatwillsellfor$5.50perpound.Howmuch

ofeachtypeofteashouldbeused?

22)

J

ayahas$16,000toinvest.Sheinvestspartofitinanaccountpaying6%simpleinterestandtherestinan

accountpaying7%simpleinterest.Ifherannualincomeis$1080,howmuchdoesshehaveinvestedin

eachaccount?

23) Aboatonarivergoes77milesdownstreamin2hoursand45minutes.Thereturntriptakes3hoursand

30minutes.Iftheboatʹsspeedisthesameineachdirection,findthespeedoftheboatandthespeedofthe

current.

3 IdentifyInconsistentSystemsofEquationsContainingTwoVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequations.Ifthesystemhasnosolution,saythatitisinconsistent.

1)

x+y=8

x+y=–1

A) x=0,y=0;(0,0) B) x=8

,

y= –1;(8

,

–1)

C) x=0,y=7;(0,7) D) inconsisten

t

Page5

2)

x–4y=–10

2x–8y=–17

A) x=2,y=4;(2,4) B) x=2,y=3;(2,3) C) x=4,y=2;(4,2) D) inconsisten

t

3)

5x–7y=–8

5x–7y=2

A) x=5

,

y=7;(5

,

7) B) x= –8

,

y=2;(–8

,

2)

C) x=–8

7,y=2

5;–8

7,2

5D) inconsisten

t

4)

6x–6y=6

18x–18y=30

A) x=3

,

y=5;(3

,

5) B) x=3

2,y=–3

2;3

2,3

2

C) x=6

,

y=30;(6

,

30) D) inconsisten

t

5)

6x–7y=5

–18x+21y=–20

A) x=6

,

y=5;(6

,

5) B) x=4

5,y=–14

15;(4

5,–14

15)

C) x=3

,

y=4;(3

,

4) D) inconsisten

t

4 ExpresstheSolutionofaSystemofDependentEquationsContainingTwoVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequations.Ifthesystemhasnosolution,saythatitisinconsistent.

1)

x+4y=–5

2x+8y=–10

A) y=–x

4

–5

4,wherexisanyrealnumber

or{(x,y)|y=–x

4

–5

4,wherexisanyrealnumber}

B) x=0,y=0;(0,0)

C) x=–5

,

y=0;(–5

,

0)

D) inconsisten

t

Page6

2)

x

2

+y

3

=4

x

4

+y

6

=2

A) y=–3

2x+12,wherexisanyrealnumber

or{(x,y)|y=–3

2x+12,wherexisanyrealnumber}

B) x=–1,y=15

2;–1,15

2

C) x=0,y=12;(0,12)

D) inconsisten

t

3)

9x +y=7

–45x –5y =-

35

A) y=–9x+7

,

wherexisanyrealnumber

or{(x,y)|y=–9x+7,wherexisanyrealnumber}

B) x=–9y+7

,

whereyisanyrealnumber

or{(x,y)|x=–9y+7,whereyisanyrealnumber}

C) y=9x+7

,

wherexisanyrealnumber

or{(x,y)|y=9x+7,wherexisanyrealnumber}

D) inconsisten

t

5 SolveSystemsofThreeEquationsContainingThreeVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Verifythatthevaluesofthevariableslistedaresolutionsofthesystemofequations.

1)

x+y+z=–2

x–y+4z=2

2x+y+z=3

x=–2,y=–5,z=5

A) solution B) notasolution

2)

x–y+4z=14

4x+z=4

x+4y+z=12

x=0,y=2,z=4

A) solution B) notasolution

Page7

Solvethesystemofequations.

3)

x+5y+5z=1

3y+2z=-

1

z=4

A) x=–4

,

y=–3

,

z=4;(–4

,

–3

,

4) B) x= –4

,

y=4

,

z= –3;(–4

,

4

,

–3)

C) x=4

,

y=–3

,

z=–4;(4

,

–3

,

–4) D) inconsisten

t

4)

x–y+2z=3

2x+z=2

x+5y+z=7

A) x=0

,

y=1

,

z=2;(0

,

1

,

2) B) x=2

,

y=0

,

z=1;(2

,

0

,

1)

C) x=2

,

y=1

,

z=0;(2

,

1

,

0) D) inconsisten

t

5)

x–y+3z=5

5x+z=0

x+4y+z=-

20

A) x=0

,

y=–5

,

z=0;(0

,

–5

,

0) B) x=0

,

y=0

,

z= –5;(0

,

0

,

–5)

C) x=0

,

y=–5

,

z=5;(0

,

–5

,

5) D) inconsisten

t

6)

3x+4y+z=–19

4x–5y–z=31

2x+y+4z=–11

A) x=1

,

y=–5

,

z=–2;(1

,

–5

,

–2) B) x=1

,

y= –2

,

z= –5;(1

,

–2

,

–5)

C) x=–2

,

y=–5

,

z=1;(–2

,

–5

,

1) D) inconsisten

t

7)

7x+7y+z=1

x+8y+8z=8

9x+y+9z=9

A) x=0,y=0,z=1;(0,0,1) B) x=1,y= –1,z=1;(1,–1,1)

C) x=0,y=1,z=0;(0,1,0) D) x= –1,y=1,z=1;(–1,1,1)

8)

x+y+z=6

x–y+2z=13

3x+y+z=10

A) x=2

,

y=–1

,

z=5;(2

,

–1

,

5) B) x=5

,

y=2

,

z= –1;(5

,

2

,

–1)

C) x=5

,

y=–1

,

z=2;(5

,

–1

,

2) D) inconsisten

t

9)

x–y+z=7

x+y+z=3

x+y–z=–5

A) x=1

,

y=–2

,

z=4;(1

,

–2

,

4) B) x=1

,

y=4

,

z= –2;(1

,

4

,

–2)

C) x=–2

,

y=4

,

z=1;(–2

,

4

,

1) D) inconsisten

t

Page8

10)

x+y+z=4

x–y+2z=0

2x+y+z=9

A) x=5

,

y=1

,

z=–2;(5

,

1

,

–2) B) x= –2

,

y=5

,

z=1;(–2

,

5

,

1)

C) x=–2

,

y=1

,

z=5;(–2

,

1

,

5) D) inconsisten

t

11)

2x+2y+z=17

2x–2y–z=-

1

5x+y+2z=23

A) x=4

,

y=5

,

z=–1;(4

,

5

,

–1) B) x=4

,

y= –1

,

z=5;(4

,

–1

,

5)

C) x=–1

,

y=5

,

z=4;(–1

,

5

,

4) D) inconsisten

t

12)

x+y+z=7

x–y+2z=7

5x+y+z=11

A) x=1,y=2,z=4;(1,2,4) B) x=1,y=4,z=2;(1,4,2)

C) x=4,y=2,z=1;(4,2,1) D) x=4,y=1,z=2;(4,1,2)

13)

x–y+z=8

x+y+z=6

x+y–z=–12

A) x=–2,y=–1,z=9;(–2,–1,9) B) x=2,y= –1,z=9;(2,–1,9)

C) x=–2,y=–1,z=–9;(–2,–1,–9) D) x=2,y= –1,z= –9;(2,–1,–9)

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Solvetheproblem.

14) Findrealnumbersa,b,andcsuchthatthegraphofthefunctiony=ax2+bx+ccontainsthepoints(1,1),

(2,4),and(–3,29).

15) Findrealnumbersa,b,andcsuchthatthegraphofthefunctiony=ax2+bx+ccontainsthepoints(1,2),

(2,11),and(–3,–14).

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

16) TheFamilyArtsCentercharges$25foradults,$14 forseniorcitizens,and$9 forchildrenunder12for

theirliveperformancesonSundayafternoon.ThispastSunday,thepaidrevenuewas$12,020for764

ticketssold.Therewere40morechildrenthanadults.Howmanychildrenattended?

A) 294children B) 254children C) 216 children D) 284 children

17) Ronattendsacocktailparty(withhisgraphingcalculatorinhispocket).Hewantstolimithisfoodintake

to123gprotein,110gfat,and168gcarbohydrate.Accordingtothehealthconscioushostess,the

marinatedmushroomcapshave3gprotein,5gfat,and9gcarbohydrate;thespicymeatballshave14g

protein,7gfat,and15gcarbohydrate;andthedeviledeggshave13gprotein,15gfat,and6g

carbohydrate.Howmanyofeachsnackcanheeattoobtainhisgoal?

A) 9mushrooms,5meatballs,2eggs B) 5 mushrooms,2 meatballs,9eggs

C) 2mushrooms,9meatballs,5eggs D) 10 mushrooms,6meatballs,3eggs

Page9

18) Aceramicsworkshopmakeswreaths,trees,andsleighsforsaleatChristmas.Awreathtakes3hoursto

prepare,2hourstopaint,and10hourstofire.Atreetakes17hourstoprepare,3hourstopaint,and4

hourstofire.Asleightakes4hourstoprepare,16hourstopaint,and7hourstofire.Iftheworkshophas

128hoursforpreparationtime,97hoursforpainting,and138hoursforfiring,howmanyofeachcanbe

made?

A) 9wreaths,5trees,4sleighs B) 5 wreaths,4 trees,9sleighs

C) 4wreaths,9trees,5sleighs D) 10 wreaths,6 trees,5sleighs

19) ThreeshrimpboatssupplytheshrimpwholesalersonHiltonHeadwithfreshcatch.TheAnnabelletakes

50%ofitscatchtoHudsonʹs,20%toCaptainJʹs,and30%toMainstreet.TheCurlyQtakes40%ofitscatch

toHudsonʹs,40%toCaptainJʹs,and20%toMainstreet.TheSloJoetakes30%ofitscatchtoHudsonʹs,40%

toCaptainJʹs,and30%toMainstreet.OneweekHudsonʹsreceived261.9poundsofshrimp,CaptainJʹs

received227pounds,andMainstreetreceived169.1pounds.Howmanypoundsofshrimpdideachboat

catch?

A) Annabelle181lbs,CurlyQ283lbs,SloJoe194 lbs

B) Annabelle283lbs,CurlyQ194lbs,SloJoe181 lbs

C) Annabelle194lbs,CurlyQ283lbs,SloJoe181 lbs

D) Annabelle194lbs,CurlyQ181lbs,SloJoe283 lbs

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

20) Toraisemoney,agroupofthreefriendsiseachtryingtosell100candles.Eachfriendhasapackage

containingthesamenumberoftapers,votives,andtealights.Thetaperssellfor$6.00each,thevotivessell

for$1.50each,andthetealightssellfor$0.75each.Mayasoldallofthecandlestoraise$431.25,Lorrinsold

allofthetapersandtealightstoraise$378.75,whileSarasoldallthevotives,allthetealights,andhalfof

thetaperstoraise$251.25.Howmanyofeachtypeofcandledideachpackagecontain?

21) Meishahas$25,000thatshewantstoinvest.Sheinvestsitinaccountspaying12%,7%,and6%simple

interest.Theaccountpaying12%isahigher–riskaccount,soshewantstheamountinthataccounttobe

halfoftheamountshehasintheaccountpaying6%simpleinterest.Ifherannualinterestis$1945,how

muchisinvestedateachrate?

22) Lexiewantstohaveanincomeof$9000peryearfrominvestments.Tothatendsheisgoingtoinvest

$90,000inthreedifferentaccounts.Theseaccountspay7%,10%,and14%simpleinterest.Ifshewantsto

have$10,000moreintheaccountpaying7%simpleinterestthanshehasintheaccountpaying14%simple

interest,howmuchshouldgointoeachaccount?

23) Craig,Jenni,andJadegotoastorehavingasaleonworkoutgear.Craigbuysthreepairsofsocks,two

T–shirts,andonepairofshortsfor$35.90.Jennibuysfourpairsofsocks,threeT–shirts,andthreepairsof

shortsfor$71.35.Jadebuysonepairofsocks,fourT–shirts,and2pairsofshortsfor$59.30.Whatisthe

priceofeachitem?

24) Ahealthshopownermadetrailmixcontainingdriedfruit,nuts,andcarobchips.Thedriedfruitsellsfor

$5.50perpound,thenutssellfor$7.50perpound,andthecarobchipssellfor$8.50perpound.Theshop

ownermixed50poundsofthetrailmixandsellsitfor$6.70perpound.Iftheamountofnutsisfive

poundsmorethantheamountofcarobships,howmuchofeachitemwasusedforthetrailmix?

Page10

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

25) AnapplicationofKirchoffʹsRulestothecircuitshownresultsinthefollowingsystemofequations:

I2+I3=I1

20+40–10I1–10I2=0

40–10I1–5I3=0

FindthecurrentsI1,I2,andI3.

20V40V

8Ω

5Ω 3Ω

2Ω 7Ω

A) I1=3.5,I2=2.5,andI3=1B)I

1=5.5,I2=4.5,andI3=1

C) I1=4.5,I2=3.5,andI3=1D)I

1=2.5,I2=1.5,andI3=1

6 IdentifyInconsistentSystemsofEquationsContainingThreeVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequations.Ifthesystemhasnosolution,saythatitisinconsistent.

1)

x+y+z=-

7

x–y+3z=-

21

5x+5y+5z=-

23

A) x=–4

,

y=2

,

z=–5;(–4

,

2

,

–5) B) x= –5

,

y=2

,

z= –4;(–5

,

2

,

–4)

C) x=–5

,

y=–4

,

z=2;(–5

,

–4

,

2) D) inconsisten

t

2)

x–y+2z=2

4x+z=0

–x+y–2z=-

4

A) x=0

,

y=–2

,

z=0;(0

,

–2

,

0) B) x=2

,

y= –2

,

z=0;(2

,

–2

,

0)

C) x=0

,

y=0

,

z=–2;(0

,

0

,

–2) D) inconsisten

t

Page11

7 ExpresstheSolutionsofaSystemofDependentEquationsContainingThreeVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequations.

1)

x+4y–z=3

x+5y–2z=5

3x+12y–3z=9

A) x=–3z–5,andy=z+2,wherezisanyrealnumber

or{(x,y,z)|x=–3z–5,andy=z+2,wherezisanyrealnumber}

B) x=z–2,andy=–3z–5,wherezisanyrealnumber

or{(x,y,z)|x=z–2,andy=–3z–5,wherezisanyrealnumber}

C) x=3z+5,andy=z–2,wherezisanyrealnumber

or{(x,y,z)|x=3z+5,andy=z–2,wherezisanyrealnumber}

D) inconsisten

t

2)

–x+y+2z=0

x+2y+z=6

–2x–y+z=-

6

A) x=z+2,andy=2–z,wherezisanyrealnumber

or{(x,y,z)|x=z+2,andy=2–z,wherezisanyrealnumber}

B) x=2–z,andy=z+2,wherezisanyrealnumber

or{(x,y,z)|x=2–z,andy=z+2,wherezisanyrealnumber}

C) x=z+2,andy=z–2,wherezisanyrealnumber

or{(x,y,z)|x=z+2,andy=z–2,wherezisanyrealnumber}

D) inconsisten

t

3)

2x–y+5z=-

7

x+y–2z=-

2

 x–y+4z=-

4

A) x=–3–z,andy=3z+1,wherezisanyrealnumber

or{(x,y,z)|x=–3–z,andy=3z+1,wherezisanyrealnumber}

B) x=3z+1,andy=z–3,wherezisanyrealnumber

or{(x,y,z)|x=3z+1,andy=z–3,wherezisanyrealnumber}

C) x=z+3,andy=3z+1,wherezisanyrealnumber

or{(x,y,z)|x=z+3,andy=3z+1,wherezisanyrealnumber}

D) inconsisten

t

Page12

4)

2x+4y–2z =0

3x+5y =1

A) x=2–5z,andy=3z–1,wherezisanyrealnumber

or{(x,y,z)|x=2–5z,andy=3z–1,wherezisanyrealnumber}

B) x=2+7z,andy=–3z–1,wherezisanyrealnumber

or{(x,y,z)|x=2+7z,andy=–3z–1,wherezisanyrealnumber}

C) x=–2–7z,andy=–3z–1,wherezisanyrealnumber

or{(x,y,z)|x=–2–7z,andy=–3z–1,wherezisanyrealnumber}

D) x=1–5z,andy=3z–1

2,wherezisanyrealnumber

or(x,y,z)|x=1–5z,andy=3z–1

2,wherezisanyrealnumber

8.2 SystemsofLinearEquations:Matrices

1 WritetheAugmentedMatrixofaSystemofLinearEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writetheaugmentedmatrixforthesystem.

1) 3x+5y=46

6x+7y=77

A) 3546

6777 B) 46 5 3

77 6 7 C) 3646

5777 D) 3577

7646

2) 8x–2y=50

4y=28

A) 8–250

0428 B) 8–250

428 0 C) 4028

8–2–2D) 50 –28

28 0 4

3)

4x–y=0

–4x+y–8=0

A)

4–10

–418

B)

4–10

–41–8

C)

4–40

–118

D)

4–48

–110

4)

2.4x+0.3y=-

11.1

0.6x–0.6y=-

4.8

A)

2.4 0.3 –11.1

0.6 –0.6 –4.8

B)

2.4 –11.1 0.3

0.6 –4.8 –0.6

C)

2.4 0.6 –11.1

–0.6 0.3 –4.8

D)

–11.1 0.3 2.4

–4.8 –0.6 0.6

Page13

5)

5

2x+4

11y=–3

8

2

11x–1

2y=1

2

A)

5

2

 4

11

–3

8

2

11

–1

2

 1

2

B)

5

2

4

11

3

8

2

11

1

2

 1

2

C)

5

2

4

11

–3

8

2

11

1

2

–1

2

D)

5

2

4

11

–3

80

2

11

–1

2

1

20

6)

–2x–2y+7z=9

6x+5y+5z=47

6x+4y+9z=55

A)

–2–27 9

6 5547

6 4955

B)

–2–27

655

649

C)

–266 9

–25447

75955

D)

97–2–2

47556

55946

7)

4x+7z=41

3y+4z=9

6x+9y+3z=30

A)

40741

034 9

69330

B)

40641

039 9

74330

C)

47041

340 9

69330

D)

407

034

693

8)

6x+3y+3z=18

5x+7y+9z=69

4x+7y+4z=25

3x+5y+7z=4

A)

63318

57969

47425

3574

B)

6543

3775

3945

18 69 25 4

C)

6334

57925

47469

35718

D)

6545

3774

3945

18 69 25 3

Page14

9)

3x+2y–6z+w=–1

9y+z=4

x–y–10z=11

12x–12y+7z=–7

A)

32

–61

–1

09104

1–1–10 0 11

12 –12 7 0 –7

B)

32

–6–1

0 916

1–1–10 11

12 –12 7 –7

C)

3 019

29

–1–12

–61

–10 7

1 000

–1411

–7

D)

3261

–1

09104

1 1 10 0 11

12 12 7 0 –7

2 WritetheSystemofEquationsfromtheAugmentedMatrix

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethesystemofequationsassociatedwiththeaugmentedmatrix.Donotsolve.

1) 7–8–2

432

A)

7x–8y=–2

4x+3y=2 B)

7x–8y=0

4x+3y=0 C)

–8x+7y= –2

4x+3y=2 D)

7x–8y= –2

3x+4y=2

2) 15 17 12

17 10 14

A)

15x+17y=12

17x+10y=14 B)

15x+17y= –12

17x+10y=–14 C)

17x+15y=12

17x+10y=14 D)

15x+17y=12

10x+17y=14

3)

6860

6296

A)

6x+8y=60

6x+2y=96

B)

6x+6y=60

8x+2y=96

C)

60x+8y=–6

96x+2y=–6

D)

6x+96y=–8

8x+60y=–2

4)

100–7

010 4

001–8

A)

x=–7

y=4

z=–8

B)

x=7

y=–4

z=8

C)

x=0

y=–3

z=–15

D)

x=1

y=12

z=0

Page15

5)

–3–2–9–2

–3–503

–408–9

A)

–3x–2y–9z=–2

–3x–5y=3

–4x+8z=–9

B)

–3x–2y–9z=–2

–3x–5y+3z=0

–4x+8y–9z=0

C)

–3x–2y–9z=–2

–3x+–5z=3

–4x+8y=–9

D)

–3x–2y–9z=–2

–3x–5y=3

–4x+8y=–9

6)

397–2

5044

630 2

A)

3x +9y +7z =–2

5x +4z =4

6x +3y =2

B)

3x –9y +7z = –2

5x +4z =–4

6x +3y =–2

C)

3x +9y +7z = –2

5x +4z =4

6x +3z =2

7)

10003

0100–1

00101

00010

A)

x1=3

x2=–1

x3=1

x4=0

B)

x1=–3

x2=1

x3=–1

x4=0

C)

x1=0

x2=2

x3=4

x4=–3

D)

x1=2

x2=–2

x3=0

x4=–2

Determinewhetherthesystemcorrespondingtothegivenaugmentedmatrixisconsistentorinconsistent.Ifitis

consistent,givethesolution.

8)

100–6

010 7

000–1

A) consistent;x=–6

,

y=7;(–6

,

7) B) consistent;x=6

,

y=–7;(6

,

–7)

C) consistent;x=–6

,

y=7

,

z=–1;(–6

,

7

,

–1) D) inconsisten

t

9)

1000

010 0

000–5

A) consistent;x=0,y=0;(0,0) B) consistent;x=0,y=–5;(0,–5)

C) consistent;z=–5;(–9) D) inconsisten

t

Page16

10)

1 33–4

054–1

001 1

A) consistent;x=–4

,

y=–1

,

z=1;(–4

,

–1

,

1) B) consistent;x= –4

,

y=1

,

z=–1;(–4

,

1

,

–1)

C) consistent;x=1

,

y=–1

,

z=–4;(1

,

–1

,

–4) D) inconsisten

t

11)

10–98

018 –2

000 0

A) consistent;x=8+9z,y=–2–8z,zanyrealnumber

or{(x,y,z)|x=8+9z,y=–2–8z,zanyrealnumber}

B) consistent;x=8–9z,y=–2+8z,zanyrealnumber

or{(x,y,z)|x=8–9z,y=–2+8z,zanyrealnumber}

C) consistent;x=8

,

y=–2

,

z=–9;(8

,

–2

,

–9)

D) inconsisten

t

12)

1001 2

0105 –2

001–30

0000 0

A) consistent;x1=2–x4

,

x2=–2–5x4

,

x3=3x4

,

x4anyrealnumber

or{(x,y,z)|x1=2–x4,x2=–2–5x4,x3=3x4,x4anyrealnumber}

B) consistent;x1=2+x4

,

x2=–2+5x4

,

x3= –3x4

,

x4anyrealnumber

or{(x,y,z)|x1=2+x4,x2=–2+5x4,x3=–3x4,x4anyrealnumber}

C) consistent;x1=5

,

x2=–3

,

x3=–2

,

x4= –3;(5

,

–3

,

–2

,

–3)

D) inconsisten

t

3 PerformRowOperationsonaMatrix

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Performtherowoperation(s)onthegivenaugmentedmatrix.

1) R1= 1

3r1

312 3

52

–6

A) 141

52–6B)

14 1

5

3

2

3

–2C) 14 1

66–5D) 14 3

52–6

2) R2=4r1+r2

1710

–43–5

A) 1710

031 35 B) 1 7 10

0–25 –45 C) 4 28 40

031 35 D) 4 28 40

–43–5

Page17

3) R3=4r1+r3

–7–5–1–10

6–295

28 –6618

A)

–7–5–1–10

6–295

0–26 2–22

B)

–7–5–1–10

6–295

016 2–22

C)

–7–5–1–10

6–295

0–26 10 –22

D)

–7–5–1–10

6–295

016 10 –22

4) R2=–2r2+r1

246–8

1236

4671

A) 

246–8

000–20

4671

B) 

24 6 –8

0000

4671

C)

2 4 6 –8

–4–8–12 6

4671

D)

2 4 6 –8

–4–8–12 –20

4671

5) R2=–2r1+r2

4–415

–50 5–3

–11

–2–1

A)

4–41 5

–13 8 3 –13

–11

–2–1

B)

4–415

3–877

–11

–2–1

C)

14 –4–911

–50 5

–3

–11

–2–1

D)

–13 8 3 –13

–505

–3

–11

–2–1

6) (a)R2=–3r1+r2

(b)R3=–2r1+r3

(c)R3=5r2+r3

1–3–5–2

3–5–45

2546

A)

1–3–5–2

0 4 11 11

0316965

B)

1–3–5–2

0–8–93

0–29 –31 –5

C)

1–3–5–2

0 4 11 11

0152521

D)

1–3–5–2

014194

0 81 109 30

7) (a)R2=3r1+r2

(b)R3=–3r1+r3

(c)R3=6r2+r3

1–3–52

–3–525

3–546

A)

1–3–52

0–14 –13 11

0–80 –59 66

B)

1–3–52

0–14 –13 11

0 28 121 66

C)

1–3–52

0–14 –13 11

0–98 –89 66

D)

1–3–52

0–4–13 11

00 611

Page18

8) (a)R2=3r1+r2

(b)R3=–2r1+r3

(c)R3=4r2+r3

12–34

–3–58–10

20–14

A)

12–34

01–12

0014

B)

12–3–4

01–12

001–4

C)

12–34

011–2

0014

D)

12–3–4

01–12

0014

9) (a)R3=–5r1+r3

(b)R4=3r1+r4

11

–11 5

0–51

–40

50

–4–24

–330 1

–4

A)

11

–11 5

0–51

–40

0–51

–7–21

06

–3411

B)

11

–115

0–51

–40

0–51

–7–21

–33 0 1

–4

C)

11

–115

0–51

–40

10 5 –9329

06

–3411

D)

11

–11 5

0–51

–40

0–51

–7–1

06

–344

4 SolveaSystemofLinearEquationsUsingMatrices

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequationsusingmatrices(rowoperations).Ifthesystemhasnosolution,saythatitis

inconsistent.

1) 2x+6y=16

5x+3y=-

8

A) x=–4

,

y=4;(–4

,

4) B) x=4

,

y= –4;(4

,

–4)

C) x=–4

,

y=–4;(–4

,

–4) D) inconsisten

t

2) –4x–7y=1

–8x–14y=-

1

A) inconsisten

t

B) x=-4

,

y=-1;(–4

,

–1)

C) x=1

,

y=-1;(1

,

–1) D) x=-4

,

y=-4;(–4

,

–4)

Page19

3)

x–y=8

x+y=1

A) x=9

2,y=–7

2;9

2,–7

2B) x=7

2,y=–5

2;7

2,–5

2

C) x=9

,

y=–7;9

,

–7 D) inconsisten

t

4)

5x–4y=20

x+y=1

2

A) x=22

9,y=–35

18;22

9,–35

18 B) x=2,y=–3

2;2,–3

2

C) x=22,y=–43

2;22,–43

2D) inconsisten

t

5) 4x+6y =4

2x =-

10

A) x=–5

,

y=4;(–5

,

4) B) x=4

,

y= –5;(4

,

–5)

C) x=–5

,

y=–4;(–5

,

–4) D) inconsisten

t

6) 2x+y=4

8x+3y=18

A) x=3

,

y=–2;(3

,

–2) B) x= –2

,

y=3;(–2

,

3)

C) x=–2

,

y=–3;(–2

,

–3) D) inconsisten

t

7)

3x+2y–z=11

x+8y+7z=99

5x+y+z=28

A) x=3

,

y=5

,

z=8;(3

,

5

,

8) B) x=3

,

y=8

,

z=5;(3

,

8

,

5)

C) x=–3

,

y=5

,

z=6;(–3

,

5

,

6) D) inconsisten

t

8)

9x–y–2z=31

–8x–7z=-

104

2y+z=22

A) x=6

,

y=7

,

z=8;(6

,

7

,

8) B) x=6

,

y=8

,

z=7;(6

,

8

,

7)

C) x=–6

,

y=7

,

z=12;(–6

,

7

,

12) D) inconsisten

t

9)

–4x+4y–z=-32

x+7y+8z=86

9x+y+z=82

A) x=8

,

y=2

,

z=8;(8

,

2

,

8) B) x=8

,

y=8

,

z=2;(8

,

8

,

2)

C) x=–8

,

y=2

,

z=16;(–8

,

2

,

16) D) inconsisten

t

Page20

10)

–3x–8y+2z=-

72

12x+32y–8z=288

–9x–24y+6z=-

216

A) x=–8

3y+2

3z+24,yisanyrealnumber,zisanyrealnumber

or(x,y,z)|x=–8

3y+2

3z+24,yisanyrealnumber,zisanyrealnumber

B) x=–9

,

y=24

,

z=6;(–9

,

24

,

6)

C) x=8

,

y=8

,

z=8;(8

,

8

,

8)

D) inconsisten

t

11)

x–y+5z=9

5x+z=2

x+3y+z=5

A) x=0

,

y=1

,

z=2;(0

,

1

,

2) B) x=0

,

y=2

,

z=1;(0

,

2

,

1)

C) x=2

,

y=1

,

z=0;(2

,

1

,

0) D) x=2

,

y=0

,

z=1;(2

,

0

,

1)

12)

x+y+z=–2

x–y+3z=–4

5x+y+z=10

A) x=3

,

y=–2

,

z=–3;(3

,

–2

,

–3) B) x=3

,

y= –3

,

z=–2;(3

,

–3

,

–2)

C) x=–3

,

y=–2

,

z=3;(–3

,

–2

,

3) D) x= –3

,

y=3

,

z=–2;(–3

,

3

,

–2)

13)

3x–2y+z=–7

x+y–2z=12

3x+y–z=10

A) x=1,y=3,z=–4;(1,3,–4) B) x=1,y= –3,z= –16;(1,–3,–16)

C) x=5,y=13

2,z=–3;5,13

2,–3 D) x=2,y=3,z= –7;(2,3,–7)

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

14)

–5x+2z=13

x+3y–2z=–15

4x+5y–z=–18

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

15)

x–y+2z+w=1

y+z=5

z–w=1

A) x=3–4w,y=4–w,z=1+w,wherewisanyrealnumber

B) inconsisten

t

C) x=3

,

y=4

,

z=1

,

w=0;(3

,

4

,

1

,

0)

D) x=–1

,

y=3

,

z=2

,

w=1;(–1

,

3

,

2

,

1)

Page21

16)

x+3y–2z–w=–2

4x+y+z+2w=28

–3x–y–3z–2w=–35

x–y–3z–2w=–23

A) x=3

,

y=3

,

z=5

,

w=4;(3

,

3

,

5

,

4)

B) x=–2

,

y=28

,

z=–35

,

w=–23;(–2

,

28

,

–35

,

–23)

C) x=3+w,y=3–2w,z=5+2w,wherewisanyrealnumber

D) x=2

,

y=5

,

z=1

,

w=6;(2

,

5

,

1

,

6)

17)

5x +y=6

–2x –y+z–w=7

z+w=8

A) x=5

3

+2

3w,y=–7

3

–10

3w,z=8–w,wherewisanyrealnumber

B) x=–7

3

–10

3w,y=5

3

+2

3w,z=8–w,wherewisanyrealnumber

C) x=5

3

–2

3w,y=–7

3

+10

3w,z=8–w,wherewisanyrealnumber

D) inconsisten

t

18)

3x+5y–2w=-13

2x+7z–w=-

1

4y+3z+3w=1

–x+2y+4z=-

5

A) x=1,y=–2,z=0,w=3;(1,–2,0,3) B) x=4

3,y=–13

20,z=0,w=5

2;4

3,–13

20,0,5

2

C) x=3

4,y=–2,z=0,w= 3

4;3

4,–2,0,3

4D) x=–1,y=–20

13,z=0,w=2

5;–1,–20

13,0,2

5

19)

x+y+z–w=6

2x–y+3z+4w =-

4

4x+2y–z–w=-13

–x–2y+4z+3w=12

A) x=–4,y=3,z=5,w=–2;(–4,3,5,–2) B) x=4,y= –3,z=–5,w=2;(4,–3,–5,2)

C) x=–1

4,y=1

3,z= 1

5,w=–1

2;–1

4,1

3,1

5,–1

2D) x=1

4,y=–1

3,z=–1

5,w=1

2;1

4,–1

3,–1

5,1

2

20)

3x+5y+2w=–12

2x+6z–w=–5

–2y+3z–3w=- 3

–x+2y+4z+w=–2

A) x=–1,y=–3,z=0,w=3;(–1,–3,0,3) B) x=1,y= –3,z=0,w=3;(1,–3,0,3)

C) x=–1,y=3,z=0,w=–3;(–1,3,0,–3) D) x=1,y=3,z=0,w=–3;(1,3,0,–3)

Page22

21)

x+y–z+w=–5

3x–y+3z–2w=7

–2x+2y+z–w=16

–x–2y–3z+3w=–22

A) x=–2,y=3,z=4,w=–2;(–2,3,4,–2)

B) x=–2,y=–3,z=5,w=1

2;–2,–3,5,1

2

C) x=2,y=–3,z=–4,w=–2;(2,–3,–4,–2)

D) x=1

2,y=–1

3,z=–1

4,w=–1

2;1

2,–1

3,–1

4,–1

2

Solvetheproblemusingmatrices.

22) Findthefunctionf(x)=ax3+bx2+cx+dforwhichf(0)=–2,f(1)=5,f(–1)=3,

f(2)=4.

A) f(x)=–10

3x3+6x2+13

3x–2 B) f(x)=10x3–18x2–13x+6

C) f(x)=8

3x3+4x2+5

3x–2 D) f(x)=–8x3+12x2+5x–6

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

23) Findrealnumbersa,b,andcsuchthatthegraphofthefunctiony=ax2+bx+ccontainsthepoints

(–2,–4),(1,–1),and(3,–19).

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

24) AnapplicationofKirchoffʹsRulestothecircuitshownresultsinthefollowingsystemofequations:

I2+I3=I1

12+24–6I1–6I2=0

24–6I1–3I3=0

FindthecurrentsI1,I2,andI3.

12V24V

5Ω

3Ω 2Ω

1Ω 4Ω

A) I1=3.5,I2=2.5,andI3=1B)I

1=5.5,I2=4.5,andI3=1

C) I1=4.5,I2=3.5,andI3=1D)I

1=2.5,I2=1.5,andI3=1

Page23

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

25) Melodyhas$45,000toinvestandwishestoreceiveanannualincomeof$4290fromthismoney.Shehas

choseninvestmentsthatpay5%,8%,and12%simpleinterest.Melodywantstohavetheamountinvested

at12%tobedoubletheamountinvestedat8%.Howmuchshouldsheinvestateachrate?

26) Acompanymanufacturesthreetypesofwoodenchairs:theKitui,theGoa,andtheSantaFe.Tomakea

Kituichairrequires1hourofcuttingtime,1.5hoursofassemblytime,and1houroffinishingtime.AGoa

chairrequires1.5hoursofcuttingtime,2.5hoursofassemblytimeand2hoursoffinishingtime.ASanta

Fechairrequires1.5hoursofcuttingtime,3hoursofassemblytime,and3hoursoffinishingtime.If41

hoursofcuttingtime,70hoursofassemblytime,and58hoursoffinishingtimewereusedoneweek,how

manyofeachtypeofchairwereproduced?

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

27) Thetablebelowshowsthenumberofbirdsforthreeselectedyearsafteranendangeredspeciesprotectio

n

programwasstarted.

x(Numberofyearsafter1980) 1  23

y(Numberofbirds) 41 63 95

Usethequadraticfunctiony=ax2+bx+ctomodelthedata.Solvethesystemoflinearequations

involvinga,b,andcusingmatrices.Findtheequationthatmodelsthedata.

A) y=5x2+7x+29 B) y=6x2+14x+24

C) y=7x2–7x+32 D) y=10x2–21x+25

28) Aceramicsworkshopmakeswreaths,trees,andsleighsforsaleatChristmas.Awreathtakes3hoursto

prepare,2hourstopaint,and10hourstofire.Atreetakes16hourstoprepare,3hourstopaint,and4

hourstofire.Asleightakes4hourstoprepare,17hourstopaint,and7hourstofire.Iftheworkshophas

114hoursforpreptime,95hoursforpainting,and108hoursforfiring,howmanyofeachcanbemade?

A) 6wreaths;5trees;4sleighs B) 5 wreaths;4trees;6sleighs

C) 4wreaths;6trees;5sleighs D) 7 wreaths;6 trees;5sleighs

29) Therewereapproximately100vehiclessoldataparticulardealershiplastmonth.Thedealertrackssale

s

byagegroupformarketingpurposes.Thenumberof36–to59–year–oldbuyersandthenumberofbuyers

60andoldercombinedexceedsthenumberofbuyers35andyoungerby30.Ifthenumberofbuyersinthe

oldestgroupisdoubled,itis23lessthanthenumberofusersinthemiddlegroup.Findthenumberof

buyersineachofthethreeagegroups.

A) 35

,

35andyounger;51

,

36–59yearolds;14

,

60andolder

B) 37

,

35andyounger;48

,

36–59yearolds;15

,

60andolder

C) 14

,

35andyounger;51

,

36–59yearolds;35

,

60andolder

D) 29

,

35andyounger;53

,

36–59yearolds;18

,

60andolder

30) Ronattendsacocktailparty(withhisgraphingcalculatorinhispocket).Hewantstolimithisfoodintake

to144gprotein,122gfat,and171gcarbohydrate.Accordingtothehealthconscioushostess,the

marinatedmushroomcapshave3gprotein,5gfat,and9gcarbohydrate;thespicymeatballshave14g

protein,7gfat,and15gcarbohydrate;andthedeviledeggshave13gprotein,15gfat,and6g

carbohydrate.Howmanyofeachsnackcanheeattoobtainhisgoal?

A) 7mushrooms,6meatballs,3eggs B) 6 mushrooms,3 meatballs,7eggs

C) 3mushrooms,7meatballs,6eggs D) 8 mushrooms,7 meatballs,4eggs

Page24

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

31) Theperimeterofapictureframeis11feet.Ifthreetimestheheightisequaltoeighttimesthewidth,wha

t

arethedimensionsoftheframe?

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

32)

J

ennyreceives$1270peryearfromthreedifferentinvestmentstotaling$20,000.Oneoftheinvestments

pays6%,thesecondonepays8%,andthethirdonepays5%.Ifthemoneyinvestedat8%is$1500less

thantheamountinvestedat5%,howmuchmoneyhasJennyinvestedintheinvestmentthatpays6%?

A) $1500 B) $8500 C) $4500 D) $10,000

8.3 SystemsofLinearEquations:Determinants

1 Evaluate2by2Determinants

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthevalueofthedeterminant.

1) 65

96

A) –9B)81C)9 D)

–24

2) 6–8

77

A) 98 B) –14 C) –98 D) –97

3) –12

23

A) –7 B) 1 C) 7 D) –8

4) 12 –7

–43

A) 8 B) 64 C) –8D)4

Solveforx.

5) 8x

25

=32

A) 4 B) 3 C) –4D)

–3

6)

59

–2x

=8

A) –2B)

–4C)2 D)

–5

2 UseCramerʹsRuletoSolveaSystemofTwoEquationsContainingTwoVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

SolvethesystemofequationsusingCramerʹsRuleifitisapplicable.IfCramerʹsRuleisnotapplicable,sayso.

1) 2x+4y=16

5x+y=-

14

A) x=–4

,

y=6;(–4

,

6) B) x=6

,

y= –4;(6

,

–4)

C) x=–6

,

y=–4;(–6

,

–4) D) x=4

,

y= –6;(4

,

–6)

Page25

2) 3x+4y=23

3x+2y=13

A) x=1

,

y=5;(1

,

5) B) x=5

,

y=1;(5

,

1)

C) x=–5

,

y=1;(–5

,

1) D) x= –1

,

y= –5;(–1

,

–5)

3) 3x+3y=15

2x–3y=5

A) x=4

,

y=1;(4

,

1) B) x=1

,

y=4;(1

,

4)

C) x=–1

,

y=4;(–1

,

4) D) x= –4

,

y= –1;(–4

,

–1)

4) 4x–7y=5

2x+5y=-

3

A) x=2

17,y=–11

17;2

17,–11

17 B) x=23

3,y=–11

3;23

3,–11

3

C) x=–2

17,y=11

17;–2

17,11

17 D) x=2

3,y=1

3;2

3,1

3

5)

3x–8y=2

6x–16y=10

A) notapplicable B) x=2

,

y=5;(2

,

5)

C) x=4

3,y=–1

2;4

3,–1

2D) x=2

,

y=10;(2

,

10)

6)

4x+2y=8

3

3x–3y=3

A) x=7

9,y=–2

9;7

9,–2

9B) x=7

9,y=16

9;7

9,16

9

C) x=–7

9,y=–2

9;–7

9,–2

9D) notapplicable

7)

x+4y=28

1

2x–y=–6

A) x=4

3,y=20

3;4

3,20

3B) x= – 4

,

y=4;–4

,

4

C) x=52

3,y=44

3;52

3,44

3D) notapplicable

3 Evaluate3by3Determinants

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthevalueofthedeterminant.

1)

322

–321

444

A) 4 B) –12 C) –4D)12

Page26

2)

641

623

643

A) –24 B) 288 C) 24 D) –168

3)

–254

3–21

16–3

A) 130 B) 80 C) –90 D) –12

4)

600

554

472

A) –108 B) 228 C) 108 D) –103

5)

125

224

125

A) 0 B) 76 C) 1 D) –16

Solveforx.

6)

x–4–1

–220

–1–28

=10

A) 5 B) 2 C) 1 D) –2

7)

5–31

–2–2x

82–1

=28

A) 0 B) 1 C) –1D)2

8)

x19

1x–5

019

=6x

A) 0or1

9B) 0or–1

9C) 1

9D) –1

9

Solvetheproblem.

9) Theequationofthelinepassingthroughthedistinctpoints(x1,y1)and(x2,y2)isgivenby

xy1

x1y11

x2y21

=

0.Findtheequationofthelinepassingthroughthepoints(3,5)and(–1,4).

A) x–4y+17=0B)x+7y+17=0C)

–x+4y–17=0D)x+4y+17=0

Page27

10) Theequationofalinepassingthroughtwodistinctpoints(x1

,

y1)and(x2

,

y2)isgivenby

xy1

x2y21

x3y31

=0.Usethedeterminanttowriteanequationforthelinepassingthrough(5,4)and(9,9).

Expressthelineʹsequationinstandardform.

A) –5x+4y+9=0B)4x+9y+45 =0C)5x–4y+9=0D)9x+5y+36 =0

11) Determinantsareusedtoshowthatthreepointslieonthesameline(arecollinear).I

f

x1y11

x2y21

x3y31

=0,

thenthepoints(x1,y1),(x2,y2),and(x3,y3)arecollinear.Ifthedeterminantdoesnotequal0,thenthe

pointsarenotcollinear.Arethepoints(3,–6),(0,10),and(6,–22)collinear?

A) Yes B) No

12) Determinantsareusedtoshowthatthreepointslieonthesameline(arecollinear).I

f

x1y11

x2y21

x3y31

=0,

thenthepoints(x1,y1),(x2,y2),and(x3,y3)arecollinear.Ifthedeterminantdoesnotequal0,thenthe

pointsarenotcollinear.Arethepoints(9,–10),(0,–8),and(27,–11)collinear?

A) No B) Yes

13) Theareaofatrianglewithvertices(x1

,

y1),(x2

,

y2),and(x3

,

y3)is

Area=±1

2

x1y11

x2y21

x3y31

,

wherethesymbol±indicatesthattheappropriatesignshouldbechosentoyieldapositivearea.Usethis

formulatofindtheareaofatrianglewhoseverticesare(9,2),(6,–8),and(–4,–6).

A) 53 B) 106 C) 9 D) 18

4 UseCramerʹsRuletoSolveaSystemofThreeEquationsContainingThreeVariables

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

SolvethesystemofequationsusingCramerʹsRuleifitisapplicable.IfCramerʹsRuleisnotapplicable,sayso.

1)

3x+6y–z=32

x–7y–2z=-

18

–5x+y+z=-

21

A) x=5

,

y=3

,

z=1;(5

,

3

,

1) B) x=3

,

y=1

,

z=3;(3

,

1

,

3)

C) x=5

,

y=–3

,

z=–1;(5

,

–3

,

–1) D) x=6

,

y=1

,

z=1;(6

,

1

,

1)

2)

6x–3y+6z=42

–7x+3y+5z=10

–3x–5y+6z=11

A) x=3

,

y=2

,

z=5;(3

,

2

,

5) B) x=2

,

y=5

,

z=2;(2

,

5

,

2)

C) x=3

,

y=–2

,

z=–5;(3

,

–2

,

–5) D) x=4

,

y=0

,

z=5;(4

,

0

,

5)

Page28

3)

4x+6z=52

–6x+5y+6z=22

9x–2y=47

A) x=7

,

y=8

,

z=4;(7

,

8

,

4) B) x=8

,

y=4

,

z=8;(8

,

4

,

8)

C) x=7

,

y=–8

,

z=–4;(7

,

–8

,

–4) D) x=8

,

y=6

,

z=4;(8

,

6

,

4)

4)

x–y+4z=5

3x+z=0

–x+y–4z=-20

A) notapplicable B) x=0

,

y= –5

,

z=0;(0

,

–5

,

0)

C) x=4

,

y=–5

,

z=0;(4

,

–5

,

0) D) x=0

,

y=0

,

z= –5;(0

,

0

,

–5)

5 KnowPropertiesofDeterminants

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethepropertiesofdeterminantstofindthevalueoftheseconddeterminant,giventhevalueofthefirst.

1)

xyz

uvw

1–32

=9

1–32

uvw

xyz

=?

A) –9 B) 9 C) 0 D) Cannotdetermine

2)

xy z

uv w

12–4

=112

xyz

uvw

–2–48

=?

A) –224 B) –112 C) 224 D) 112

3)

xy z

uvw

11 3

=–46

uvw

33 9

xy z

=?

A) –138 B) 138 C) 46 D) –46

4)

xyz

uvw

1–33

=79

1–33

–3u –3v –3w

x–1y+3z–3

=?

A) 237 B) –79 C) –237 D) 79

5)

xyz

uvw

1–23

=41

xy z–x

uvw–u

1–22

=?

A) 41 B) –41 C) 0 D) Cannotdetermine

6)

xy z

uvw

13 3

=123

x+3y+9z+9

3u–13v–33w–3

133

=?

A) 369 B) –123 C) –369 D) 123

7) Given

stu

vwx

428

=3,findthevalueof

32–s 16–t 64–u

vw x

42 8

.

A) 3 B) –24 C) 24 D) –3

Page29

Solvetheproblem.

8) Giventhat

xyz

abc

245

=3,findthevalueofthedeterminant

245

3a 3b 3c

x–2y–4z–5

.

A) –9B)9 C)0 D)6

8.4 MatrixAlgebra

1 FindtheSumandDifferenceofTwoMatrices

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Performtheindicatedoperation(s),wheneverpossible.

1)

LetA=[24]andB= –3

8.FindA+B.

A)

[–112]

B)

–1

12

C)

2–3

48

D) notdefined

2)

LetA=

3

–2

–4

andB=

–5

4

9

.FindA+B.

A)

–2

2

5

B)

[–225]

C)

3–5

–24

–49

D)

2

5

6

3)

LetA=

8–6

62

61

andB=

–1–9

–61

9–6

.FindA+B.

A)

7–15

03

15 –5

B)

93

12 1

–310

C)

7–15

02

15 5

D)

72

03

15 –5

4)

LetA=

–15

04

9–4

andB=

72

17 4

22

.FindA–B.

A)

–83

–17 0

7–6

B)

16

78

11 –1

C)

13

70

7–2

D)

3–4

70

–76

Page30

5)

LetA=–11

25

andB=62

23 .FindA+B.

A)

53

48

B)

34

48

C)

51

12

D)

20

6)

LetA=–10

61

andB=–16

31 .FindA–B.

A)

0–6

30

B)

–26

92

C)

06

–30

D)

[–3]

7)

LetA=–6–8

8–5,B=–59

7–4

andC=–16

–59 .FindA+B–C.

A)

–10 –5

20 –18

B)

–12 7

10 0

C)

0–23

6–10

D)

–2–11

–48

8) LetA=

7–48

–65

–1

06

–3

andB=

–2–6–1

–7–43

–3–9–5

. FindA–B.

A)

929

19

–4

315 2

B)

929

192

315–4

C)

5–10 7

–13 1 –8

–3–32

D)

5–10 7

–13 1 2

–3–3–8

9) LetA=

–467

3–512

7–11 14

andB=

610 –4

–56–8

311 7

.FindA–B.

A)

–10 –411

8–11 20

4–22 7

B)

216 3

–214

10 021

C)

–10 –43

–2–11 4

4–22 7

D)

10 4–11

–811 –20

–422 –7

10)

LetA=

461

84–2

336

andB=

1–62

303

–45–3

.FindA+B.

A)

503

11 4 1

–183

B)

11 –12 3

541

–183

C)

11 –12 3

11 4 –5

183

D)

50 3

54–5

18 3

Page31

11)

LetA=

325

60–1

–163

andB=

4–21

–101

07–6

.FindA–B.

A)

–144

70

–2

–1–19

B)

–10 4

70–2

1–1–3

C)

–1–44

70

–2

–119

D)

70 6

500

–113–3

2 FindScalarMultiplesofaMatrix

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethegivenmatricestocomputethegivenexpression.

1) LetA=–31

02 .Find2A.

A)

–62

04

B)

–62

02

C)

–61

02

D)

–13

24

2) LetB=–135–3.Find–4B.

A) 4–12 –20 12 B) 435–3C) –41220–12 D) –313–5

3) LetC=

2

–2

10

.Find1/2C.

A)

1

–1

5

B)

4

–4

20

C)

1

–2

10

D)

2

–1

10

4) LetA=23

26

andB=04

–16 .Find3A+B.

A)

613

524

B)

621

336

C)

613

112

D)

67

512

5) LetC=

1

–3

2

andD=

–1

3

–2

.FindC–2D.

A)

3

–9

6

B)

–1

3

4

C)

–3

9

–6

D)

3

–6

4

6) LetA=–12 andB=10 .Find2A+3B.

A) 14 B) –24 C) –14 D) 22

Page32

7) IfA=2–1

79

andB=5–3

47

,find–2A+4B.

A)

–24 –18

–30 –34

B)

16 –10

210

C)

74

11 13

D)

–3–6

35

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

8) IfA=7–5

–81

andB=–9–3

2–8,find–5A+4B.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

9) LetA=

79

–4

492

–2–8–5

andB=

–191

4–83

–41

–6

.Find–3A+4B.

A)

–25 9 16

4–59 6

–10 28 –9

B)

–22 –18 13

–8–35 –3

2259

C)

618 –3

81 5

–6–7–11

D)

68 –6

18 1 –7

–35–11

3 FindtheProductofTwoMatrices

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Computetheproduct.

1)

–13

32

–20

–14

A)

–112

–88

B)

20

–38

C)

2–6

–25

D)

12 –1

8–8

2)

13–1

40 5

30

–11

05

A)

0–2

12 25

B)

–20

25 12

C)

3–30

0025

D) notdefined

3)

0–31

5–10

12

01

1–1

A)

1–4

59

B)

15

–49

C)

010

0–1

00

D)

10 –51

5–10

–5–20

Page33

4)

7–34

18

–7

–3

1

A)

–8

–34

B)

7–34

18–7

–3–31

C)

–8–34

D) notdefined

5)

–622

5

0

–3

A) –36 B) 54 C) –30 0 –6D)

–30

0

–6

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

6)

–2

1

–1

1–10

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

7)

–23 6

39–7

4–5–2

–51

–1

–31

–4

A)

–41 19 –23

–12 –13 13

B)

–41 –12

19 –13

–23 13

C)

–236

39–7

49–7

–51

–1

–3–3–4

D)

–8–15 –12

–15 9 7

–9928

8)

3–6–1

2–65

988

344

–8–57

9–56

A)

48 47 –36

99 13 –4

35 –44 140

B)

48 99 35

47 99 –44

–36 –4140

C)

3–6–1

2–65

98 8

34 4

–8–57

9–56

D)

9–24 –4

–16 30 35

81 –40 48

Page34

9)

2–8

16

01

–32

1–1

61

A)

410 –24

12 –59

1–12

B)

–24 10 4

9–512

2–11

C)

9–14

–24 –51

210 12

D) notdefined

10)

3–21

04

–3

30

–22

A)

90

08

B)

9–63

–612–8

C)

9–6

–612

3–8

D) notdefined

11)

–13

52

0–24

1–32

A)

3–72

2–16 24

B) notdefined C)

32

–7–16

224

D)

0–612

5–64

12)

[9–31]

–7–86

8–5–6

–5–47

A) –92 –61 79 B)

–92

–61

79

C)

9–31

–7–86

8–5–6

–5–47

D)

–63 24 6

72 15 –6

–45 12 7

Performtheindicatedoperationsandsimplify.

13) LetA=3–4

–25

,B=5–28

10–3

,andC=

7–90

3–51

–162

.FindAB+BC.

A) 32 750

5–23 –37 B) 68 331

8–2–5C) 32 19 40

–15 31 –37 D) –10 –19 12

–15 31 –25

14)

LetA=

0–1

40

34

,B=

–40

11

44

,andC=–314

0–4–1.FindC(A–B).

A) –13 2

–11 4 B) 12 3

–5–4C) –5–4

12 3 D) –24 5

14 0

Page35

15)

LetA=

24

10

6–1

,B=

–51

06

–1–4

,andC=465

1–20.FindBC+3ℐ3.

A)

–16 –32 –25

6–90

–82

–2

B)

–19 –32 –25

6–12 0

–82

–5

C)

24 28 25

6150

814 8

D)

21 28 25

6120

814 5

Solvetheproblem.

16) StateUniversityhasaCollegeofArts&Sciences,aCollegeofBusiness,andaCollegeofEngineering.The

percentageofstudentsineachcategoryaregivenbythefollowingmatrix.

Freshman Sophomore Junior Senior

Arts&Sciences

Business

Engineering

60% 50% 40% 70%

20% 40% 30% 10%

20% 10% 30% 20%

Thestudentpopulationisdistributedbyclassandageasgiveninthefollowingmatrix.

Female Male

Freshman

Sophomore

Junior

Senior

430 780

550 750

840 610

630 480

HowmanyfemalestudentsareintheCollegeofBusiness?HowmanymalestudentsareintheCollegeof

Arts&Sciences?

A) 621students;1423students B) 519 students;687students

C) 687students;510students D) 1310 students;621students

17) Thefinalgradeforanalgebracourseisdeterminedbygradesonthemidtermandfinalexam.Thegrades

forfourstudentsandtwopossiblegradingsystemsaremodeledbythefollowingmatrices.

Midterm Final

Student1

Student2

Student3

Student4



73 79

44 62

78 86

98 96

System

1

System

2

Midterm

Final

0.4 0.5

0.6 0.5

FindthefinalcoursescoreforStudent3forbothgradingSystem1andSystem2.

A) System1:82.8;System2:82 B) System1:74.2;System2:89.8

C) System1:76.6;System2:76 D) System1:48.6;System2:53

Page36

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

18) Acompanymanufacturesthreetypesofwoodenchairsattwodifferentlocations.Inoneweek,themain

locationproduces18Kituichairs,12Goachairs,and6SantaFechairswhilethesecondarylocation

produces10Kituichairs,3Goachairs,and5SantaFechairs.

(a)Finda2by3matrixrepresentingtheabovedata.

(b)IfeachKituichairrequires25board–feetofwood,aGoachairrequires29board–feetofwood,anda

SantaFerequires30board–feetofwood,finda3by1matrixrepresentingtheamountofmaterial.

(c)Multiplythe2by3matrixfoundinpartaandthe3by1matrixfoundinpartbtogeta2by1matrix

showingtheweekʹsusageofmaterialatbothlocations.

19) Thematrix922

435

representsoneweekʹsusageofwood(inboard–feet)attwolocationsofacompanythat

manufactureschairs.Duetodifferencesintransportationcosts,thefirstlocationpays$0.89perboard–foot

andthesecondlocationpays$0.94perboard–foot.

(a)Finda1by2matrixrepresentingthecostofthewood.

(b)Multiplythegivenmatrixandthematrixfoundinpartatodeterminethetotalcostofwoodforthat

week.

4 FindtheInverseofaMatrix

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Eachmatrixisnonsingular.Findtheinverseofthematrix.Besuretocheckyouranswer.

1)

53

32

A) 2–3

–35 B) 23

35 C)

1

2

–3

–31

5

D)

1

23

31

5

2)

10 1

–10

A) 0–1

110 B) 01

–110 C) 01

110 D) 0–1

–110

3)

–24

4–4

A)

1

2

1

2

1

2

1

4

B)

1

2

1

4

1

2

1

4

C)

–1

2

1

4

1

2

–1

4

D)

1

2

–1

2

–1

2

1

4

Page37

4)

–5–1

60

A)

01

6

–1–5

6

B)

01

6

–15

6

C)

0–1

6

1–5

6

D)

05

6

–1–1

6

5)

15

1–5

A)

1

2

1

2

1

10

–1

10

B)

1

2

–1

2

1

10

1

10

C)

–1

2

–1

2

–1

10

1

10

D)

1

10

1

2

1

10

–1

2

6)

5–3

61

A)

1

23

3

23

–6

23

5

23

B)

1

23

–3

23

6

23

5

23

C)

–6

23

5

23

1

23

3

23

D)

5

23

3

23

–6

23

1

23

7)

0–3

–6–1

A)

1

18

–1

6

–1

30

B)

1

18

1

6

1

30

C)

–1

30

1

18

–1

6

D)

0–1

6

–1

3

1

18

8)

–6–5

06

A)

–1

6

–5

36

01

6

B)

–1

6

5

36

01

6

C)

01

6

–1

6

–5

36

D)

1

6

–5

36

0–1

6

Page38

9)

33

30

A)

01

3

1

3

–1

3

B)

0–1

3

–1

3

–1

3

C)

1

3

–1

3

01

3

D)

–1

3

1

3

1

30

10)

–5a

–7a

A)

a

2

–a

2

7

2

–5

2

B)

a

2

1

2

–7

2

a

2

C)

a

2

–7

2

a

2

5

2

D)

a

2

1

2

7

2

a

2

11)

2–10

–11

–2

10

–1

A)

1–12

–3–2–4

–1–11

B)

1–12

–3–24

–111

C)

1–22

3–2–4

–1–12

D)

1–12

–2–1–4

–1–11

12)

100

110

111

A)

100

–110

0–11

B)

100

111

0–11

C)

10–1

–110

0–11

D)

100

–1–10

011

13)

3–31

–22–1

–45

–2

A)

1–11

0–21

–2–30

B)

1–31

0–21

–2–30

C)

1–11

–2–30

0–21

D)

1–10

–2–30

0–2–1

14)

10 0

61 0

08 1

A)

100

–610

48 –81

B)

100

610

008

C)

18–48

01 1

00 1

D)

100

8–10

–48 6 1

Page39

15)

100

–110

111

A)

100

110

–2–11

B)

111

011

001

C)

1–11

01

–1

001

D)

–100

–1–10

–1–1–1

16)

111

211

223

A)

–110

4–1–1

–201

B)

–1–1–1

–2–1–1

–2–2–3

C)

11 1

1

211

1

2

 1

2

1

3

D)

1–1 1

1

211

1

2

 1

2

–1

3

17)

132

133

278

A)

–310–3

2–41

–110

B)

–1–3–2

–1–3–3

–2–7–8

C)

1 1

3

 1

2

11

3

1

3

1

2

1

7

1

8

D)

1–1

3

 1

2

1–1

3

1

3

1

7

1

2

1

8

18)

108

123

253

A)

9–40 16

–313 –5

–15–2

B)

112

025

833

C)

–1 0 –8

–1–2–3

–2–5–3

D)

–1–12

02

–5

823

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Showthatthematrixhasnoinverse.

19)

14 –4

–72

20)

2104

–3–11

–174

Page40

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Useagraphingutilitytofindtheinverse,ifitexists,ofthematrix.Roundanswerstotwodecimalplaces.

21)

–16 3 28

5–14 15

34 25 2

A)

–0.02 0.03 0.02

0.02 –0.04 0.02

0.02 0.02 0.01

B)

–0.01 0.02 0.01

0.02 –0.03 0.01

0.02 0.02 0.01

C)

–0.01 0.03 0.02

0.02 –0.03 0.01

0.02 0.02 0.01

D)

–0.02 0.03 0.02

0.02 –0.04 0.01

0.02 0.02 0.00

5 SolveaSystemofLinearEquationsUsinganInverseMatrix

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemusingtheinversematrixmethod.

1)

–2x–6y=-

2

2x–y=-

5

A) x=–2,y=1;(–2,1) B) x=1,y= –2;(1,–2)

C) x=2,y=–1;(2,–1) D) x= –1,y=2;(–1,2)

2)

x+3y=-

8

21x+6y=3

A) x=1,y=–3;(1,–3) B) x= –3,y=1;(–3,1)

C) x=–1,y=3;(–1,3) D) x=3,y= –1;(3,–1)

3)

–2x+6y=6

3x+2y=13

A) x=3,y=2;(3,2) B) x=2,y=3;(2,3)

C) x=–3,y=–2;(–3,–2) D) x= –2,y= –3;(–2,–3)

4)

–5x+3y=8

3x–6y=-

30

A) x=2,y=6;(2,6) B) x=6,y=2;(6,2)

C) x=–2,y=–6;(–2,–6) D) x= –6,y= –2;(–6,–2)

5)

bx+4y =6

bx+5y =9b≠0

A) x=–6

b,y=3;–6

b,3 B) x=3,y=–6

b;3,–6

b

C) x=6

b,y=–3;6

b,–3 D) x=6

b,y=3;6

b,3

Page41

6)

x+2y+3z =-

3

x+y+z=-

10

2x+2y+z=-

11

Theinverseof

123

111

221

is

–14–1

1–52

02–1

.

A) x=–26

,

y=25

,

z=–9;(–26

,

25

,

–9) B) x= –7

,

y=16

,

z=–12;(–7

,

16

,

–12)

C) x=–12

,

y=20

,

z=–11;(–12

,

20

,

–11) D) x= –54

,

y= –75

,

z=–31;(–54

,

–75

,

–31)

7)

x+2y+3z =1

x+y+z=-

6

x–2z =5

Theinverseof

123

111

10–2

is

–24–1

3–52

–12–1

.

A) x=–31

,

y=43

,

z=–18;(–31

,

43

,

–18) B) x= –25

,

y=44

,

z=–18;(–25

,

44

,

–18)

C) x=1

,

y=0,z=0;(1

,

0,0) D) x= –17

,

y= –17

,

z=–6;(–17

,

–17

,

–6)

8)

x+2y+3z =-

11

x+y+z=12

–x+y+2z =-

9

Theinverseof

123

111

–112

is

1–1–1

–352

2–3–1

.

A) x=–14

,

y=75

,

z=–49;(–14

,

75

,

–49) B) x= –65

,

y=98

,

z=44;(–65

,

98

,

44)

C) x=–11

,

y=48

,

z=18;(–11

,

48

,

18) D) x= –8

,

y=9

,

z=5;(–8

,

9

,

5)

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

9)

4x +2z =-

20

x–y–5z =5

–3x –2y –z=12

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

10)

2x +4y –5z =-

8

x+5y +2z =-

1

3x +3y +3z =15

A) x=5,y=–2,z=2;(5,–2,2) B) x=2,y=5,z=2;(2,5,2)

C) x=5,y=2,z=–2;(5,2,–2) D) x= –5,y= –2,z=–2;(–5,–2,–2)

Page42

Useagraphingutilitytosolvethesystemofequations.Roundanswerstotwodecimalplaces.

11)

25x+61y–12z=15

18x–12y+7z=-

3

3x+4y–z=12

A) x=4.56,y=–6.06,z=–22.55;(4.56,–6.06,–22.55)

B) x=-22.55,y=4.56,z=–6.06;(–22.55,4.56,–6.06)

C) x=4.56,y=–22.55,z=–6.06;(4.56,–22.55,–6.06)

D) x=–22.55,y=–6.06,z=4.56;(–22.55,–6.06,4.56)

Encodeordecodethegivenmessage,asrequested,numberingthelettersofthealphabet1through26intheirusual

order.

12) UsethecodingmatrixA=37

25

toencodethemessageLIFE.

A) 99 53

69 37 B) 78 62

54 43 C) 54 28

129 67 D) –3–5

33

13) UsethecodingmatrixA=–1–3

25

toencodethemessageCARE.

A) –6–33

11 61 B) –57 –16

96 27 C) 18 105

–7–4D) –1–8

–4–29

14) UsethecodingmatrixA=

10–2

123

111

toencodethemessageCOME_HERE.

A)

–23 –11 –5

72 29 56

31 13 28

B)

35 5

15 0 18

13 8 5

C)

19 27 –21

–14 –30 39

811 –13

D)

–7–21 3

28 69 44

13 33 26

15) UsethecodingmatrixA=1–4

–29

anditsinverseA–1=94

21

todecodethecryptogram–7–8

16 21

.

A) ABLE B) ACTS C) ARMS D) ALAS

16) UsethecodingmatrixA=21

53

anditsinverseA–1=3–1

–52

todecodethecryptogram96

25 17

.

A) BEAD B) CARE C) DARE D) CURB

17) UsethecodingmatrixA=

111

–112

123

anditsinverseA–1=

–1–11

52–3

–3–12

todecodethecryptogram

37 16 35

38 20 4

82 40 60

.

A) GOOD_LUCK B) STAY_CALM C) LOOK_DOWN D) HELP_THEM

8.5 PartialFractionDecompositio

n

1 DecomposeP/Q,WhereQHasOnlyNonrepeatedLinearFactors

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Tellwhetherthegivenrationalexpressionisproperorimproper.

1) x2–11x+28

(x2–13x+42)(x+7)

A) proper B) improper

Page43

2) x3–9x+20

x2–13x+40

A) improper B) proper

Tellwhetherthegivenrationalexpressionisproperorimproper.Ifimproper,rewriteitasthesumofapolynomial

andaproperrationalexpression.

3) x

2–x

A) improper;–1+2

2–xB) proper

C) improper;1+2

2–xD) improper;1–2

2–x

4) x–2

x2+6

A) proper B) improper; x–2

(x+6)(x–6)

C) improper; 2

x+6

+2

x–6D) improper; 2

x+6

–2

x–6

Writethepartialfractiondecompositionoftherationalexpression.

5) x

(x–2)(x–3)

A) –2

x–2

+3

x–3B) 2

x–2

+–3

x–3C) –3

x–2

+2

x–3D) –2

x–2

+–3

x–3

6) x

x2–11x+30

A) –5

x–5

+6

x–6B) 5

x–5

+–6

x–6C) –6

x–5

+5

x–6D) –5

x–5

+–6

x–6

7) x–14

(x–2)(x–5)

A) 4

x–2

+–3

x–5B) –3

x–2

+4

x–5C) 4

x–2

+3

x–5D) 3

x–2

+–4

x–5

8) 3x–24

(x+4)(x–5)

A) 4

x+4

–1

x–5B) 4

x+4

+1

x–5C) 1

x–5

–4

x+4D) 3

x+4

–24

x–5

9) 3x2–x–20

x(x+1)(x–1)

A) 20

x

+–8

x+1

+–9

x–1B) 20

x

+8

x+1

+–9

x–1C) 20

x

+–8

x+1

+9

x–1D) 20

x

+–9

x+1

+8

x–1

Page44

10) 12x2+162x+384

(x+8)(x+2)(x+11)

A) 8

x+8

+2

x+2

+2

x+11 B) –8

x+8

–2

x+2

–2

x+11

C) –8

x+8

+2

x+2

+2

x+11 D) 8

x+8

+2

x+2

–2

x+11

11) 2x–5

x2–5x–6

A) 1

x–6

+1

x+1B) 1

x–3

+1

x–2C) 9

x+2

+1

x–3D) 17

x+6

–3

x–1

12) 10x+2

x2–2x–3

A) 2

x+1

+8

x–3B) 2

x+1

+–8

x–3C) 3

x+1

+7

x–3D) 2

x–1

+8

x+3

13) 10x2–30x–12

x(x–3)(x–4)

A) –1

x

+4

x–3

+7

x–4B) –1

x

+5

x–3

+6

x–4C) –1

x

+3

x–3

+8

x–4D) x+4

x2–3

+7

x–4

2 DecomposeP/Q,WhereQHasRepeatedLinearFactors

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethepartialfractiondecompositionoftherationalexpression.

1) –5x2–11x–5

(x+2)(x+1)2

A) –3

x+2

+–2

x+1

+1

(x+1)2B) 3

x+2

+–2

x+1

+–1

(x+1)2

C) –3

x+2

+2

x+1

+1

(x+1)2D) 1

x+2

+–2

x+1

+3

(x+1)2

2) x+5

x3–2x2+x

A) 5

x

+–5

x–1

+6

(x–1)2B) 5

x

+6

x–1

+–5

(x–1)2

C) 5

x

+–5

x–1

+11

(x–1)2D) –5

x

+5

x–1

+6

(x–1)2

Page45

3) 125–19x

x3–10x2+25x

A) 5

x

+–5

x–5

+6

(x–5)2B) 5

x

+6

x–5

+–5

(x–5)2

C) 5

x

+–5

x–5

+12

(x–5)2D) –5

x

+5

x–5

+6

(x–5)2

4) x+1

(x–2)2(x+4)

A)

1

12

x–2

+

1

2

(x–2)2

+

–1

12

x+4

B)

1

2

(x–2)2

+

–1

12

x+4

C)

–1

x–2

+

1

4x

(x–2)2

+

–1

4

x+4

D) 12

x–2

+2

(x–2)2

+–12

x+4

5) 7x3–2

x2(x+1)3

A) 6

x

–2

x2

–6

x+1

+3

(x+1)2

–9

(x+1)3B) 2

x2

–6

x+1

+3

(x+1)2

–9

(x+1)3

C) 6

x

–6

x+1

+3

(x+1)2

–9

(x+1)3D) 6

x

+2

x2

+6

x+1

–3

(x+1)2

+9

(x+1)3

6) 6x+7

(x–4)2

A) 6

x–4

+31

(x–4)2B) 6

x–4

+48

(x–4)2C) 6

x–4

+x+31

(x–4)2D) 1

x–4

+x+31

(x–4)2

7) 2x2–2x+4

(x–1)3

A) 2

x–1

+2

(x–1)2

+4

(x–1)3B) 2

x–1

+2

(x–1)2

C) 2

x–1

+x+2

(x–1)2

+x2+4

(x–1)3D) 2

x–1

–2

(x–1)2

+4

(x–1)3

Page46

3 DecomposeP/Q,WhereQHasaNonrepeatedIrreducibleQuadraticFactor

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethepartialfractiondecompositionoftherationalexpression.

1) 14x+1

(x–1)(x2+x+1)

A) 5

x–1

+–5x+4

x2+x+1

B) 5

x–1

+–5

x+1

+4

x–1

C) –5

x–1

+5x+4

x2+x+1

D) 5

x–1

+4x–5

x2+x+1

2) x2–111

x4–x2–72

A) 1

x+3

–1

x–3

+7

x2+8

B) 1

x+3

+1

x–3

–7

x2+8

C) 1

x+3

–1

x–3

–7

x2+8

D) 1

x+3

+1

x–3

+7

x2+8

3) 3x–2

x3–1

A)

1

3

x–1

+

–1

3x+7

3

(x2+x+1)

B) 3

(x–1)2

+1

(x–1)3

C) 3

x–1

+–3(x–7)

x2+x+1

D)

1

2

x–1

+

5

2

x+1

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

4) x2–56

x4+5x2–36

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

5) 12x+3

x3–1

A) 5

x–1

+–5x+2

x2+x+1

B) 5

x–1

+–5

x+1

+2

x–1

C) –5

x–1

+5x+2

x2+x+1

D) 5

x–1

+2x–5

x2+x+1

Page47

6) 8x2+16x+15

(x+5)(x2+2)

A) 5

x+5

+3x+1

x2+2

B) 5

x+5

+3

x2+2

C) 5

x+5

+3x–1

x2+2

D) 5

x+5

+3

x+2

+1

(x+2)2

7) 6x2+10x+24

x3+2x2+3x+6

A) 4

x+2

+2x+6

x2+3

B) 4

x+2

+2

x2+3

C) 4

x+3

+2x+6

x2+2

D) 4

x+2

+2

x+3

+6

(x+3)2

8) 10x+52

(x+3)2(x2+2)

A) 2

x+3

+2

(x+3)2

+–2x+4

x2+2

B) 2

x+3

+2

(x+3)2

+–2

x2+2

C) 2

x+3

+6

(x+3)2

+–2x–4

x2+2

D) 6x–4

(x+3)2

+–2x+4

x2+2

4 DecomposeP/Q,WhereQHasaRepeatedIrreducibleQuadraticFactor

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethepartialfractiondecompositionoftherationalexpression.

1) 2x3+4x2

(x2+5)2

A) 2x+4

x2+5

+–10x–20

(x2+5)2B) 2x–4

x2+5

+–10x+20

(x2+5)2C) 2x+4

x2+5

+10x+20

(x2+5)2D) 2x+4

x2+5

+10x–20

(x2+5)2

2) x2+2x+3

(x2+5)2

A) 1

x2+5

+2x–2

(x2+5)2B) x+1

x2+5

+2x–2

(x2+5)2C) 1

x2+5

+–x–2

(x2+5)2D) –1

x2+5

+2x–2

(x2+5)2

3) 5x3+14x–5

(x2+2)2

A) 5x

x2+2

+4x–5

(x2+2)2B) 5x+1

x2+2

+4x–5

(x2+2)2C) 5x

x2+2

+–4x–5

(x2+2)2D) 5x

x2+2

+4x+5

(x2+2)2

Page48

4) 2x3+2x2+3x+4

(x2+3)3

A) 2x+2

(x2+3)2

+–3x–2

(x2+3)3B) x+1

x2+3

+2x+2

(x2+3)2

+–3x–2

(x2+3)3

C) 2x–2

(x2+3)2

+–3x+2

(x2+3)3D) x

x2+3

+2x+2

(x2+3)2

+–3x–2

(x2+3)3

8.6 SystemsofNonlinearEquations

1 SolveaSystemofNonlinearEquationsUsingSubstitution

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Thegraphoftwoequationsalongwiththepointsofintersectionaregiven.Substitutethepointsofintersection

intothesystemsofequations.Arethepointsofintersectionsolutionstothesystemofequations(Y/N)?

1)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

35

30

25

20

15

10

5

-5

(1, 2)

(-5, 26)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

35

30

25

20

15

10

5

-5

(1, 2)

(-5, 26)

x2=y–1

y=–4x+6

A) Yes B) No

2)

x

–10–8-6-4-2 2 4 6 810

y

100

50

-50

(0.8, –3.4)

(-4.8, 41.4)

x

–10–8-6-4-2 2 4 6 810

y

100

50

-50

(0.8, –3.4)

(-4.8, 41.4)

2x2=y+5

y=–8x+3

A) No B) Yes

Page49

3)

x

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

y

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

(3, 4)

(-5, 0)

x

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

y

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

(3, 4)

(-5, 0)

x2+y2=25

2y+x=5

A) No B) Yes

4)

x

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

y

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7 (4, –6)

(-4, 6)

x

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

y

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7 (4, –6)

(-4, 6)

x2+y2=52

2y+3x=0

A) Yes B) No

Page50

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Graphtheequationsofthesystem.Thensolvethesystemtofindthepointsofintersection.

5)

y=x2–8x+16

y=–x+6

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

6)

x2+y2=81

x2–y2=81

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page51

7)

x2+y2=49

y=x2–7

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

8)

y=x

y=2–x

x

510

y

10

5

x

510

y

10

5

Page52

9)

(x+2)2+(y+4)2=4

y2+8y–x+10 =0

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvethesystemofequationsusingsubstitution.

10)

x2+y2=5

x+y=3

A) x=2

,

y=1;x=1

,

y=2

or(2,1),(1,2)

B) x= –2

,

y=1;x=–1

,

y=2

or(–2,1),(–1,2)

C) x=2

,

y=–1;x=1

,

y=–2

or(2,–1),(1,–2)

D) x= –2

,

y= –1;x=–1

,

y=–2

or(–2,–1),(–1,–2)

11)

xy=90

x+y=–19

A) x=–9

,

y=–10;x=–10

,

y=–9

or(–9,–10),(–10,–9)

B) x=9

,

y= –10;x=10

,

y=–9

or(9,–10),(10,–9)

C) x=–9

,

y=10;x= –10

,

y=9

or(–9,10),(–10,9)

D) x=9

,

y=10;x=10

,

y=9

or(9,10),(10,9)

12)

x2+y2=113

x–y=1

A) x=–7

,

y=–8;x=8

,

y=7

or(–7,–8),(8,7)

B) x=7

,

y= –8;x=8

,

y=–7

or(7,–8),(8,–7)

C) x=–7

,

y=8;x=–8

,

y=7

or(–7,8),(–8,7)

D) x=7

,

y=8;x=8

,

y=7

or(7,8),(8,7)

Page53

13)

y=x2–12x+36

x+y=12

A) x=3

,

y=9;x=8

,

y=4

or(3,9),(8,4)

B) x= –3

,

y=15;x=–8

,

y=20

or(–3,15),(–8,20)

C) x=6

,

y=6or(6

,

6) D) x=3

,

y=15;x=8

,

y=4

or(3,15),(8,4)

14)

y=–x2+3

x2+y2=5

A) x=2

,

y=–1;x=1

,

y=2;x=–1

,

y=2;x= –2

,

y= –1

or(2,–1),(1,2),(–1,2),(–2,–1)

B) x=1

,

y=2;x=–1

,

y=2

or(1,2),(–1,2)

C) x=2

,

y=–1;x=–2

,

y=–1

or(2,–1),(–2,–1)

D) x=1

,

y=4;x=4

,

y=19

or(1,4),(4,19)

15)

x2+y2=4

x+y=2

A) x=0,y=2;x=2,y=0

or(0,2),(2,0)

B) x=0,y=0;x=2,y=–2

or(0,0),(2,–2)

C) x=0,y=–2;x=–2,y=0

or(0,–2),(–2,0)

D) x=2,y= –2;x=–2,y=–2

or(2,–2),(–2,–2)

16)

xy=20

x+y=9

A) x=5,y=4;x=4,y=5

or(5,4),(4,5)

B) x=6,y=3;x=3,y=6

or(6,3),(3,6)

C) x=10,y=2;x=2,y=10

or(10,2),(2,10)

D) x=20,y=1;x=1,y=20

or(20,1),(1,20)

17)

x2+y2=169

x+y=17

A) x=12,y=5;x=5,y=12

or(12,5),(5,12)

B) x= –12,y=5;x=–5,y=12

or(–12,5),(–5,12)

C) x=12,y=–5;x=5,y=–12

or(12,–5),(5,–12)

D) x= –12,y= –5;x=–5,y=–12

or(–12,–5),(–5,–12)

Page54

18)

xy–x2=–20

x–2y=3

A) x=5,y=1;x=–8,y=–11

2

or(5,1),–8,–11

2

B) x=–5,y=–1;x=8,y=11

2

or(–5,–1),8,11

2

C) x=5,y=1;x=–11

2,y=–8

or(5,1),–11

2,–8

D) x=–5,y=–1;x=11

2,y=8

or(–5,–1),–11

2,8

19)

x2–y2=39

x–y=3

A) x=8,y=5or(8,5) B) x=8,y= –5or(8,–5)

C) x=–8,y=5or(–8,5) D) x= –8,y= –5or(–8,–5)

20)

y=6x2–5x

y=2x+3

A) x=3

2,y=6;x=–1

3,y=7

3

or3

2,6 ),–1

3,–7

3

B) x=1

3,y=11

3;x=–3

2,y=0

or1

3,11

3,–3

2,0

C) x=1

6,y=10

3;x=1,y=5

or1

6,10

3,(1,5)

D) x=–1

2,y=2;x=1,y=5

or–1

2,2 ,(1,5)

21)

lnx=3lny

3x=27y

A) x=33,y=3or(3 3,3)B)x=3,y=33or(3,33)

C) x=9,y=3or(9,3)D)x=3,y=9or(3,9)

22)

–4x–y=–35

y=x2+3

A) x=4

,

y=19;x=–8

,

y=67

or(4,19),(–8,67)

B) x= –4

,

y=19;x=8

,

y=67

or(–4,19),(8,67)

C) x=4

,

y=13;x=–8

,

y=61

or(4,13),(–8,61)

D) x= –4

,

y=19;x=–8

,

y=67

or(–4,19),(–8,67)

Page55

23)

y=x+3

y2=12x

A) x=3

,

y=6or(3

,

6) B) x=3

,

y=6;x= –3

,

y=0

or(3,6),(–3,0)

C) x=3

,

y=6;x=3

,

y=–6

or(3,6),(3,–6)

D) x=3

,

y=6;x=3

,

y=–6;x=–3

,

y=0

or(3,6),(3,–6),(–3,0)

24)

xy=1

–2x–y=3

A) x=–1

2

,y=–2;x=–1,y=–1

or–1

2,–2 ,–1,–1

B) x=–2,y=–1

2;x=–1,y=–1

or–2,–1

2,–1,–1

C) x=2,y=–2;x=1

,

y=–1

or(2,–2),(1,–1)

D) x= – 1 ,y= –1 or–1

,

–1

25)

11x2+6y2=24

y=x+2

A) x=0,y=2;x=–24

17,y=10

17

or0,2 ,–24

17,10

17

B) x=0,y=–2;x=–24

17,y=10

17

or0,–2 ,–24

17,10

17

C) x=0,y=2;x=24

17,y=58

17

or0,2 ,24

17,58

17

D) x=0,y=–2;x=24

17,y=58

17

or0,–2 ,24

17,58

17

26)

xy=32

x2+y2=80

A) x=4

,

y=–4;x=–4

,

y=–8;x=8

,

y=4;x= –8

,

y= –4

or(4,–4),(–4,–8),(8,4),(–8,–4)

B) x=4

,

y=8;x=–4

,

y=–8;x=4

,

y= –8;x= –4

,

y=8

or(4,8),(–4,–8),(4,–8),(–4,8)

C) x=4

,

y=8;x=8

,

y=4;x=4

,

y= –8;x=8

,

y= –4

or(4,8),(8,4),(4,–8),(8,–4)

D) x=–4

,

y=–8;x=–8

,

y=–4;x= –4

,

y=8;x= –8

,

y=4

or(–4,–8),(–8,–4),(–4,8),(–8,4)

Page56

27)

y=(x+5)2+1

2x–y+10=0

A) x=–4,y=2or(–4,2) B) x= –4,y=2;x=4,y=18

or(–4,2),(4,18)

C) x=0,y=10;x=0,y=26

or(0,10),(0,26)

D) x= –5,y=0or(–5,0)

28)

2y–x=10

x2+y2–100=0

A) x=–10

,

y=0;x=6

,

y=8

or(–10,0),(6,8)

B) x=10

,

y=0;x= –10

,

y=0;x=6

,

y=8

or(10,0),(–10,0),(6,8)

C) x=0,y=5;x=8

,

y=9

or0,5,8,9

D) x=0,y=5;x=0,y=–5;x=8

,

y=9

or0,5,0,–5 ,8,9

29)

y=x2–3x–6

y=–x2+3x+2

A) x=4

,

y=–2;x=–1

,

y=–2

or(4,–2),(–1,–2)

B) x= –4

,

y=22;x=1

,

y=–8

or(–4,22),(1,–8)

C) x=4

,

y=–2;x=0,y=–6

or(4,–2),(0,–6)

D) x= –4

,

y=22;x=0,y=–6

or(–4,22),(0,–6)

30)

x+y=11

x2+y2=–4y+85

A) x=5

,

y=6;x=8

,

y=3

or(5,6),(8,3)

B) x=6

,

y=5;x=3

,

y=8

or(6,5),(3,8)

C) x=17

,

y=–6;x=14

,

y=–3

or(17,–6),(14,–3)

D) x= –6

,

y=17;x=–3

,

y=14

or(–6,17),(–3,14)

2 SolveaSystemofNonlinearEquationsUsingElimination

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solveusingelimination.

1)

x2+y2=65

x2–y2=33

A) x=7

,

y=4;x=–7

,

y=4;x=7

,

y= –4;x= –7

,

y= –4

or(7,4),(–7,4),(7,–4),(–7,–4)

B) x=7

,

y=4;x=4

,

y=7;x=–7

,

y= –4;x= –4

,

y= –7

or(7,4),(4,–7),(–7,–4),(–4,–7)

C) x=7

,

y=–4;x=7

,

y=4or(7

,

–4),(7

,

4)

D) x=–7

,

y=–4;x=–4

,

y=–7or(–7

,

–4),(–4

,

–7)

Page57

2)

4x2–4y2=–48

5x2+3y2=68

A) x=2

,

y=4;x=–2

,

y=4;x=2

,

y= –4;x= –2

,

y= –4

or(2,4),(–2,4),(2,–4),(–2,–4)

B) x=2

,

y=4;x=4

,

y=2;x=–2

,

y= –4;x= –4

,

y= –2

or(2,4),(4,2),(–2,–4),(–4,–2)

C) x=2

,

y=–4;x=2

,

y=4or(2

,

–4),(2

,

4)

D) x=–2

,

y=–4;x=–4

,

y=–2or(–2

,

–4),(–4

,

–2)

3)

x2+y2=64

x2–y2=64

A) x=8

,

y=0;x=–8

,

y=0or(8

,

0),(–8

,

0) B) x=8

,

y=8;x= –8

,

y=8or(8

,

8),(–8

,

8)

C) x=8

,

y=0;x=8

,

y=8or(8

,

0),(8

,

8) D) x= –8

,

y=0;x=–8

,

y=8or(–8

,

0),(–8

,

8)

4)

x2+y2=64

x2

64

+y2

9

=1

A) x=–8,y=0;x=8,y=0or(–8,0),(8,0) B) x=0,y= –8;x=0,y=8or(0,–8),(0,8)

C) x=0,y=–3;x=0,y=3or(0,–3),(0,3) D) Norealsolutionexists.

5)

3x2+2y2=89

x2–2y2=–21

A) x=17,y=19;x=–17,y=19;x=,17,y=–19;x=–17,y=–19

or(17,19),(–17,19),(17,–19),(–17,–19)

B) x=19,y=17;x=–19,y=17;x=,19,y=–17;x=–19,y=–17

or(17,19),(–17,19),(17,–19),(–17,–19)

C) x=17,y=19;x=,17,y=–19;

or(17

,19),(17,–19)

D) x=–17,y=19;x=–17,y=–19

or(–17,19),(–17,–19)

6)

2x2+y2=17

3x2–2y2=–6

A) x=2,y=3;x=2,y=–3;x=–2,y=3;x= –2,y= –3

or(2,3),(2,–3),(–2,3),(–2,–3)

B) x=1,y=3;x=1,y=–3;x=–1,y=3;x= –1,y= –3

or(1,3),(1,–3),(–2,3),(–2,–3)

C) x=2,y=–3;x=–2,y=3or(2,–3),(–2,3)

D) x=1,y=3;x=–1,y=–3or(1,3),(–1,–3)

Page58

7)

2x2+xy–y2=3

x2+2xy+y2=3

A) x=23

3,y=3

3;x=–23

3,y=–3

3

or23

3,3

3,–23

3,–3

3

B) x=23

3,y=–3

3;x=–23

3,y=3

3

or23

3,–3

3,–23

3,3

3

C) x=–2

3,y=1

3;x=23

3,y=–1

3

or–2

3,1

3,23

3,–1

3

D) x=–2

3,y=–1

3;x=23

3,y=1

3

or–2

3,–1

3,23

3,1

3

Useagraphingutilitytosolvethesystemofequations.Expressthesolutionroundedtotwodecimalplaces.

8)

3x3+y2=6

x4y=2

A) x=1.18,y=1.03;x=1.07,y=1.52;x= –0.91,y=2.88

or(1.18,1.03),(1.07,1.52),(–0.91,2.88)

B) x=1.99,y=0.13;x=1.04,y=1.70;x= –0.91,y=2.95

or(1.99,0.13),(1.04,1.70),(–0.91,2.95)

C) x=1.81,y=0.28;x=0.20,y=2.45;x= –0.20,y= –2.45

or(1.81,0.28),(0.20,2.45),(–0.20,–2.45)

D) Norealsolutionexists.

9)

x3+y2=2

x2y=4

A) x=–1.37,y=2.14or(–1.37,2.14) B) x=2.14,y=1.37or(2.14,1.37)

C) x=1.37,y=2.14or(1.37,2.14) D) x=2.14,y= –1.37or(2.14,–1.37)

10)

y=x–4/3

y=ex

A) x=0.64,y=1.89or(0.64,1.89) B) x=0.48,y=1.63or(0.48,1.63)

C) x=0.22,y=0.98or(0.22,0.98) D) x=1.14,y=0.81or(1.14,0.81)

11)

x2+y2=25

y=–lnx

A) x=4.76,y=–1.52;x=0.01,y=5.00

or(4.76,–1.52),(0.01,5.00)

B) x= –4.76,y=1.52;x=–0.01,y=5.00

or(–4.76,1.52),(–0.01,5.00)

C) x=5.12,y=–3.67;x=0.76,y=2.05

or(5.12,–3.67),(0.76,2.05)

D) x=5.12,y=3.67;x=–0.76,y=2.05

or(5.12,3.67),(–0.76,2.05)

Solvetheproblem.

12) Thesumofthesquaresoftwonumbersis145.Thesumofthetwonumbersis17.Findthetwonumbers.

A) 8and9B)8and9 or–9 and–8

C) –9and–8D)

–8 and9 or–9 and8

Page59

13) Thesumofthesquaresoftwonumbersis80.Thedifferenceofthetwonumbersis–12.Findthetwo

numbers.

A) –4and8or–8and4B)

–4 and8

C) –8and4D)

–8 and–4 or4 and8

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

14) Thedifferenceoftwonumbersis5andthedifferenceoftheirsquaresis55.Findthenumbers.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

15) Arighttrianglehasanareaof36squareinches.Thesquareofthehypotenuseis145.Findthelengthsof

thelegsofthetriangle.Roundyouranswertothenearestinch.

A) 8in.and9in. B) 64in.and81 in. C) 4 in.and18 in. D) 16in.and4.5 in.

16) Theperimeterofarectangleis32inchesanditsareais55 squareinches.Whatareitsdimensions?

A) 5in.by11in. B) 4in.by12 in. C) 6 in.by10 in. D) 4in.by10 in.

17) Asystemfortrackingshipsindicatedthatashipliesonahyperbolicpathdescribedby6x2–y2=50.The

processisrepeatedandtheshipisfoundtolieonahyperbolicpathdescribedbyy2–2x2=50.Ifitis

knownthattheshipislocatedinthefirstquadrantofthecoordinatesystem,determineitsexactlocation.

A) (5

,

10) B) (–5

,

–10) C) (10

,

5) D) (–10

,

–5)

18) Findthedimensionsofarectanglewhoseperimeteris24 feetandwhoseareais32squarefeet.

A) 4ftby8ft B) 3ftby9 ft C) 5 ftby7 ft D) 3ftby7 ft

19) Theareaofagardenis2940squarefeet,andthelengthofitsdiagonalis91 feet.Findthedimensionsofthe

garden.

A) 35ftby84ft B) 5ftby588 ft C) 245 ftby12 ft D) 60ftby49 ft

20) Theareaofarectangularpieceofcardboardshownis945 squareinches.Thecardboardisusedtomakean

openboxbycuttinga2–inchsquarefromeachcornerandturningupthesides.Iftheboxistohavea

volumeof1394cubicinches,findthedimensionsofthecardboardthatmustbeused.

A) 21in.by45in. B) 23in.by47 in. C) 19 in.by43 in. D) 17in.by39 in.

21) Arectangularpieceoftinhasanareaof792 squareinches.Asquareof3 inchesiscutfromeachcorner,

andanopenboxismadebyturninguptheendsandsides.Ifthevolumeoftheboxis1440cubicinches,

whatweretheoriginaldimensionsofthepieceoftin?

A) 22in.by36in. B) 25in.by39 in. C) 19 in.by33 in. D) 16in.by27 in.

22) Thediagonalofthefloorofarectangularofficecubicleis2ftlongerthanthelengthofthecubicleand5ft

longerthantwicethewidth.Findthedimensionsofthecubicle.Roundtothenearesttenth,ifnecessary.

A) width=9.7ft,length=22.4ft B) width=4ft,length=11ft

C) width=2ft,length=9ft D) width=3.9ft,length=9.7ft

Page60

23) Ina1–milerace,thewinnercrossesthefinishline13 feetaheadofthesecond–placerunnerand24 feet

aheadofthethird–placerunner.Assumingthateachrunnermaintainsaconstantspeedthroughoutthe

race,byhowmanyfeetdoesthesecond–placerunnerbeatthethird–placerunner?(5280feetin1mile.)

A) 11.03ft B) –13.06 ft C) –11.05 ft D) –2ft

24) Apersonatthetopofa600foottallbuildingdropsayellowball.Theheightoftheyellowballisgivenby

theequationh=–16t2+600wherehismeasuredinfeetandtisthenumberofsecondssincetheyellow

ballwasdropped.Asecondperson,inthesamebuildingbutonalowerfloorthatis180feetfromthe

ground,dropsawhiteball3.5secondsaftertheyellowballwasdropped.Theheightofthewhiteballis

givenbytheequationh=–16(t–3.5)2+180wherehismeasuredinfeetandtisthenumberofseconds

sincetheyellowballwasdropped.Findthetimethattheballsarethesamedistanceabovethegroundand

findthisdistance.

A) 5.5sec;116ft B) 4.5sec;276 ft C) 5 sec;200 ft D) 6sec;24 ft

Page61

8.7 SystemsofInequalities

1 GraphanInequality

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Graphtheinequality.

1) x>8

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page62

2) y≤–1

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page63

3) 2x+y≤–1

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page64

4) x+5y≥3

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page65

5) y≥x–2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page66

6) y+4

<

x

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page67

7) y≤–5x–3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page68

8) –2x–4y≤8

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page69

9) 2x+5y≤10

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page70

10) –2x–4y≤–8

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page71

11) x–y>–5

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page72

12) x+y

<

–3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page73

13) x–y

<

–4

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page74

14) x2+y2≤36

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page75

15) x2+y2>16

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page76

16) y>x2+3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page77

17) y≤x2+1

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page78

18) xy≤2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page79

2 GraphaSystemofInequalities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Graphthesystemofinequalities.

1) y>5

x≥2

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page80

2) –3x+y>6

–3x+y<–1

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

A) Nosolution

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

B)

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

C)

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

D)

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

Page81

3) –2x+y

<

2

–2x+y>1

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D) Nosolution

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page82

4) 3x–y≤–3

x+4y≥–12

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page83

5) y

<

–x+4

y>4x–3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

–20 –10 10 20

y

20

10

-10

-20

x

–20 –10 10 20

y

20

10

-10

-20

D)

x

–20 –10 10 20

y

20

10

-10

-20

x

–20 –10 10 20

y

20

10

-10

-20

Page84

6) x+2y≥2

x–y≤0

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page85

7) –x+4y

<

–12

x≥–3

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page86

8) y

<

x+1

4x+7y>7

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page87

9) x+7y≤7

x+7y≥0

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D) Nosolution

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page88

10) 7x+y≥7

7x+y≥0

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D) Nosolution

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page89

11) 2x+3y≤6

x–y≤3

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page90

12) 4x+3y≥12

x≥y

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

B)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

C)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

D)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

Page91

13) 2x+3y≥6

x–y≤3

y≤2

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

A)

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

B)

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

C)

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

D)

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

Page92

14) 2x+3y≥6

x–y≥3

y≤2

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

A)

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

B)

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

C)

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

D)

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

x

-8 -6 -4 -2 2 4 6 8

y

8

6

4

2

-2

-4

-6

-8

Page93

15) –x+2y≤–6

3x+2y>–18

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page94

16) x2+y2≤64

–10x+5y≤–50

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page95

17) y≥x2

x+y>3

x

y

x

y

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

D)

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

x

-12 -8 -4 4 8 12

y

12

8

4

-4

-8

-12

Page96

18) x2+y≤4

x2–y≤3

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

A)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

B)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

C)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

D)

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

x

-5 -4 -3 -2 -1 1 2 3 4 5

y

5

4

3

2

1

-1

-2

-3

-4

-5

Page97

19) y>x2

9x+4y≤36

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page98

20) y>x2

2x+6y≤12

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page99

21) x2+y2≤49

8x+6y≤48

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page100

22) x2+y2≤81

y–x2>0

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page101

23) x2+y2≤4

x+y>1

x

–10–8-6-4-2 2 4 6 810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x

–10–8-6-4-2 2 4 6 810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

A)

x

–10–8–6–4–2 246810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x

–10–8–6–4–2 246810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

B)

x

–10–8-6-4-2 2 4 6 810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x

–10–8-6-4-2 2 4 6 810

y

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

C)

x

–10–8-6-4-2 2 4 6 8

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 8

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8-6-4-2 2 4 6 810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Page102

24) xy

<

7

y≤x2–1

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

B)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

C)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

D)

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

Page103

25) x2+y2≤64

x2+y2≥49

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

A)

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

B) nosolution

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

C)

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

D)

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

x

–8–6–4–2 2468

y

8

6

4

2

-2

-4

-6

-8

Page104

26) 2x+3y≥6

x–y≥3

y≤2

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page105

27) 2x+3y≥6

x–y≤3

x≥1

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page106

28) 2x+3y≤6

x–y≤3

x≥1

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page107

29) 2x+3y≤6

x–y≥3

x≥1

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

A)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

B)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

C)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

D)

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

x

-6 -4 -2 2 4 6

y

6

4

2

-2

-4

-6

Page108

30) y–x≤5

x+y≥3

y–3x≥–1

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

A)

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

B)

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

C)

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

D)

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

x

–10–8–6–4–2 246810

y

10

8

6

4

2

-2

-4

-6

-8

-10

Graphthesystemoflinearinequalities.Tellwhetherthegraphisboundedorunbounded,andlabelthecorner

points.

31)

x≥0

y≥0

x+y≤9

x+y≥1

Page109

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A) bounded;

cornerpoints(9,0),(0,9),(0,1),(1,0)

x

y

12

-12

(0, 9)

(9, 0)

(1, 0)

(0, 1)

x

y

12

-12

(0, 9)

(9, 0)

(1, 0)

(0, 1)

B) unbounded;

cornerpoints(0,0),(0,1),(1,0)

x

y

12

-12

(0, 9)

(9, 0)

(1, 0)

(0, 1)

x

y

12

-12

(0, 9)

(9, 0)

(1, 0)

(0, 1)

C) unbounded;

cornerpoints(9,0),(0,9)

x

y

12

-12

(0, 9)

(9, 0)

x

y

12

-12

(0, 9)

(9, 0)

D) nosolution

x

y

12

-12

(0, 9)

(9, 0)

(1, 0)

(0, 1)

x

y

12

-12

(0, 9)

(9, 0)

(1, 0)

(0, 1)

Page110

32)

x≥0

y≥0

x+y≥2

x+2y≥4

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A) unbounded;cornerpoints(0,2),(4

,

0)

x

-5 5 10

y

10

5

-5

-10

(0, 2) (4, 0)

x

-5 5 10

y

10

5

-5

-10

(0, 2) (4, 0)

B) bounded;cornerpoints(0,0),(0,2),(2

,

0)

x

-10 -5 5 10

y

10

5

-5

-10

(0, 2)

(2, 0)

(0, 0)

x

-10 -5 5 10

y

10

5

-5

-10

(0, 2)

(2, 0)

(0, 0)

C) bounded;cornerpoints(2

,

0),(0,2),(4

,

0)

x

-10 -5 5 10

y

10

5

-5

-10

(2, 0)

(0, 2)

(4, 0)

x

-10 -5 5 10

y

10

5

-5

-10

(2, 0)

(0, 2)

(4, 0)

D) unbounded;cornerpoints(0,2),(2

,

0)

x

-10 -5 5 10

y

10

5

-5

-10

(0, 2)

(2, 0)

x

-10 -5 5 10

y

10

5

-5

-10

(0, 2)

(2, 0)

Page111

Graphtheregionindicatedbygraphingthesystemofinequalities.Labelallpointsofintersection.

33)

y≥x2–8

y≤–x2

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

A) O(0,0),A(0,–8)

x

-10 -5 5 10

y

10

5

-5

-10

O

A

(-2, –4) (2, –4)

x

-10 -5 5 10

y

10

5

-5

-10

O

A

(-2, –4) (2, –4)

B) O(0,0),A(0,–8)

x

-10 -5 5 10

y

10

5

-5

-10

O

A

(-2, –4) (2, –4)

x

-10 -5 5 10

y

10

5

-5

-10

O

A

(-2, –4) (2, –4)

C) O(0,0),A(0,–8)

x

-10 -5 5 10

y

10

5

-5

-10

O

A

(-2, –4) (2, –4)

x

-10 -5 5 10

y

10

5

-5

-10

O

A

(-2, –4) (2, –4)

D) O(0,0),A(0,–8)

x

-10 -5 5 10

y

10

5

-5

-10

O

A

(-2, –4) (2, –4)

x

-10 -5 5 10

y

10

5

-5

-10

O

A

(-2, –4) (2, –4)

Page112

Writeasystemoflinearinequalitiesthathasthegivengraph.

34)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

y≥0

x≥0

x≤3

y+x≤5

B)

y≥0

x≥0

x≤5

y+x≤3

C)

x≤3

y+x≤5

D)

y≥0

x≥0

x≤3

y+x≥5

35)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

y≥0

x≥0

x≤7

y≤8

x+y≥2

B)

y≥0

x≥0

y≤8

x+y≥2

C)

y≥0

x≥0

x≤8

x+y≥2

D)

x≤7

y≤8

x+y≥2

Page113

36)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

x≥0

y≥2

y≤x

y+x≤8

B)

y≥0

x≥0

y≤2

y≤x

y+x≤8

C)

y≥2

y≥x

y+x≤8

D)

y≥0

x≥0

y≤2

y+x≤8

37)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

y≥0

y≤9

y≤x–4

y≥4–x

B)

x≥0

y≥0

y≤9

y≤x–4

y≤4–x

C)

x≥0

y≤9

y≤x–4

y≥4–x

D)

y≥0

x≤9

y≤x–4

y≥4–x

Page114

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Solvetheproblem.

38) Acoffeestorehasavailable75poundsofAgradecoffeeand120poundsofBgradecoffee.Thesewillbe

blendedinto1poundpackagesasfollow:aneconomyblendthatcontains4ouncesofAgradecoffeeand

12ouncesofBgradecoffeeandasuperiorblendthatcontains8ouncesofAgradecoffeeand8ouncesof

Bgradecoffee.Usingxtodenotethenumberofpackagesoftheeconomyblendandytodenotethe

numberofpackagesofthesuperiorblend,writeasystemoflinearinequalitiesthatdescribesthepossible

numberofpackagesofeachblend.Graphthesystemandlabelthecornerpoints.

x

y

x

y

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

39) Apersonwithnomorethan$1000toinvestplanstoplacethemoneyintwoinvestments,

telecommunicationsandpharmaceuticals.Thetelecommunicationsinvestmentistobenomorethan3

timesthepharmaceuticalsinvestment.Writeasystemofinequalitiestodescribethesituation.Let

x=amounttobeinvestedintelecommunicationsandy=amounttobeinvestedinpharmaceuticals.

A) x+y≤1000

x≤3y

x≥0

y≥0

B) x+y=1000

x≤3y

x≥0

y≥0

C) x+y=1000

y≥3x

x≥0

y≥0

D) x+y≤1000

3x≤y

x≥0

y≥0

40) Amanisplantingasectionofgardenwithtomatoesandcucumbers.Theavailableareaofthesectionis

100squarefeet.Hewantstheareaplantedwithtomatoestobemorethan20%oftheareaplantedwith

cucumbers.Writeasystemofinequalitiestodescribethesituation.Letx=amounttobeplantedin

tomatoesandy=amounttobeplantedincucumbers.

A) x+y≤100

x>0.20y

x≥0

y≥0

B) x+y=100

x≥0.20y

x≥0

y≥0

C) x+y≤100

x>20y

x≥0

y≥0

D) x+y≤100

x<0.20y

x≥0

y≥0

Page115

8.8 LinearProgramming

1 SetupaLinearProgrammingProblem

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Setupthelinearprogrammingproblem.

1) Asteelcompanyproducestwotypesofmachinedies,partAandpartB.Thecompanymakesa$3.00

profitoneachpartAthatitproducesanda$6.00profitoneachpartBthatitproduces.Letx=thenumber

ofpartAproducedinaweekandy=thenumberofpartBproducedinaweek.Writetheobjective

functionthatdescribesthetotalweeklyprofit.

A) z=3x+6y B) z=6x+3y

C) z=9(x+y) D) z=3(x–6)+6(y–3)

2) Adietitianneedstopurchasefoodforpatients.Shecanpurchaseanounceofchickenfor$0.25 andan

ounceofpotatoesfor$0.03.Letx=thenumberofouncesofchickenandy=thenumberofouncesof

potatoespurchasedperpatient.Writetheobjectivefunctionthatdescribesthetotalcostperpatientper

meal.

A) z=0.25x+0.03y B) z=0.03x+0.25y C) z=25x+3y D) z=3y+25y

3) Asteelcompanyproducestwotypesofmachinedies,partAandpartBandisboundbythefollowing

constraints:

·PartArequires1hourofcastingtimeand10hoursoffiringtime.

·PartBrequires4hoursofcastingtimeand3hoursoffiringtime.

·Themaximumnumberofhoursperweekavailableforcastingandfiringare100and70,respectively.

·Thecosttothecompanyis$0.75perpartAand$3.00perpartB.Totalweeklycostscannotexceed

$45.00.

Letx=thenumberofpartAproducedinaweekandy=thenumberofpartBproducedinaweek.Write

asystemofthreeinequalitiesthatdescribestheseconstraints.

A)

x+4y ≤100

10x +3y ≤70

0.75x +3y ≤45

B)

x+10y ≤100

4x +3y ≤70

0.75x +3y ≤45

C)

x+10y ≥100

4x +3y ≥70

0.75x +3y ≤45

D)

x+4y ≤100

10x +3y ≤70

3x +0.75y ≤45

4) Adietitianneedstopurchasefoodforpatients.Shecanpurchaseanounceofchickenfor$0.25andan

ounceofpotatoesfor$0.02.Thedieticianisboundbythefollowingconstraints.

·Eachounceofchickencontains13gramsofproteinand24gramsofcarbohydrates.

·Eachounceofpotatoescontains5gramsofproteinand35gramsofcarbohydrates.

·Theminimumdailyrequirementsforthepatientsunderthedietitianʹscareare45gramsofproteinand

58gramsofcarbohydrates.

Letx=thenumberofouncesofchickenandy=thenumberofouncesofpotatoespurchasedperpatient.

Writeasystemofinequalitiesthatdescribestheseconstraints.

A)

13x +5y ≥45

24x +35y ≥58

B)

13x +24y ≥45

5x +35y ≥58

C)

13x +5y ≥58

24x +35y ≥45

D)

13x +24x ≥45

5y +35y ≥58

Page116

5) Mrs.Whitewantstocrochethatsandafghansforachurchfundraisingbazaar.Sheneeds8hourstomake

ahatand3hourstomakeanafghan,andshehasnomorethan54hoursavailable.Shehasmaterialforno

morethan13items,andshewantstomakeatleasttwoafghans.Letx=thenumberofhatsshemakesand

y=thenumberofafghansshemakes.Writeasystemofinequalitiesthatdescribestheseconstraints.

A) 8x+3y≤54

x+y≤13

y≥2

B) 8x+3y≤54

x+y≤13

x≥2

C) 3x+8y≤54

x+y≤13

x≤2

D) 8x+3y≥54

x+y≤13

y≥2

6) Anofficemanagerisbuyingusedfilingcabinets.Smallfilecabinetscost$6 eachandlargefilecabinetscos

t

$8each,andthemanagercannotspendmorethan$74onfilecabinets.Asmallcabinettakesup7square

feetoffloorspaceandalargecabinettakesup8squarefeet,andtheofficehasnomorethan81squarefeet

offloorspaceavailableforfilecabinets.Themanagermustbuyatleast7filecabinetsinordertogetfree

delivery.Letx=thenumberofsmallfilecabinetsboughtandy=thenumberoflargefilecabinetsbought.

Writeasystemofinequalitiesthatdescribestheseconstraints.

A) 6x+8y≤74

7x+8y≤81

x+y≥7

B) 6x+8y≤74

8x+7y≤81

x≥7

C) 6x+8y≤74

7x+8y≤81

x+y≤7

D) 6x+8y≤74

7x+8y≤81

y≥7

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

7) TheJillsonʹshaveupto$75,000toinvest.Theydecidethattheywanttohaveatleast$40,000investedin

stablebondsyielding6%andthatnomorethan$20,000shouldbeinvestedinmorevolatilebonds

yielding12%.

(a)Usingxtodenotetheamountofmoneyinvestedinthestablebondsandytheamountinvestedinthe

morevolatilebonds,writeasystemoflinearinequalitiesthatdescribesthepossibleamountsofeach

investment.

(b)Graphthesystemandlabelthecornerpoints.

x

y

x

y

Page117

8) Charlenebaby–sitsfor$4perhour.Shealsoworksasatutorfor$7perhour.Becauseofschool,her

parentsonlyallowhertowork13hoursperweek.HowmanyhourscanCharlenetutorandbaby–sitand

stillmakeatleast$50perweek?

(a)Letx=hoursspentbaby–sittingandlety=hoursspenttutoring.Writeasystemofinequalitiesforthis

situation.

(b)Graphthesolutionset.

x

4 8 12 16 20

y

16

12

8

4

x

4 8 12 16 20

y

16

12

8

4

9) EricʹsCarpentrymanufacturestwotypesofbookshelvesthatare4feettalland3feetwide,abasicmodel

andadeluxemodel.Eachbasicbookshelfrequires1.5hoursforassemblyand1hourforfinishing;each

deluxemodelrequires2.5hoursforassemblyand1hourforfinishing.Twoassemblersandonefinisherar

e

employedbythecompany,andeachworks40hoursperweek.

(a)Usingxtodenotethenumberofbasicbookcasesandytodenotethenumberofdeluxebookcases,

writeasystemoflinearinequalitiesthatdescribesthepossiblenumberofeachmodelofbookcasethatcan

bemanufacturedinaweek.

(b)Graphthesystemandlabelthecornerpoints.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

10) Theliquidportionofadietistoprovideatleast300calories,36unitsofvitaminA,and90unitsofvitamin

Cdaily.AcupofdietarydrinkXprovides60calories,12unitsofvitaminA,and10unitsofvitaminC.A

cupofdietarydrinkYprovides60calories,6unitsofvitaminA,and30unitsofvitaminC.Setupasystem

oflinearinequalitiesthatdescribestheminimumdailyrequirementsforcaloriesandvitamins.Let

x=numberofcupsofdietarydrinkX,andy=numberofcupsofdietarydrinkY.Writealltheconstraints

asasystemoflinearinequalities.

A)

60x+60y≥300

12x+6y≥36

10x+30y≥90

x≥0

y≥0

B)

60x+60y>300

12x+6y>36

10x+30y>90

x>0

y>0

C)

60x+60y≥300

12x+6y>36

10x+30y≥90

D)

60x+60y≤300

12x+6y≤36

10x+30y≤90

Page118

2 SolveaLinearProgrammingProblem

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthemaximumorminimumvalueofthegivenobjectivefunctionofalinearprogrammingproblem.Thefigure

illustratesthegraphofthefeasiblepoints.

1) z=2x+8y.Findmaximumandminimum.

x

y

(0, 8) (7, 8)

(7, 2)

(

2

,

0

)

(0, 2)

x

y

(0, 8) (7, 8)

(7, 2)

(

2

,

0

)

(0, 2)

A) maximumvalue:78;minimumvalue:4 B) maximumvalue:78;minimumvalue:16

C) maximumvalue:30;minimumvalue:4 D) maximumvalue:30;minimumvalue:16

Findthemaximumorminimumvalueofthegivenobjectivefunctionofalinearprogrammingproblem.Thefigure

illustratesthegraphoffeasiblepoints.

2) z=x+7y.Findmaximum.

A) maximum:37 B) maximum:24 C) maximum:18 D) maximum:30

3) z=–x–8y.Findmaximum.

A) maximum:–20 B) maximum:–27 C) maximum:–42 D) maximum:–34

4) z=6x+7y.Findminimum.

A) minimum:38 B) minimum:39 C) minimum:47 D) nominimum

5) z=x+5y+9.Findminimum.

A) minimum:23 B) minimum:36 C) minimum:28 D) nominimum

6) z=–6x–y.Findmaximum.

A) maximum:–17 B) maximum:–21 C) maximum:–26 D) nomaximum

Solvethelinearprogrammingproblem.

7) Maximizeandminimizez=25x+22ysubjectto:

x≥0, y≥0, 2x+3y≥6, x≤10, y≤5.

A) maximum:360;minimum:44 B) maximum:250;minimum:66

C) maximum:110;minimum:66 D) maximum:360;minimum:250

Page119

8) Minimizez=13x+10y+18subjectto:

x≥0, y≥0, x+y≥1.

A) minimum:28 B) minimum:31 C) minimum:41 D) minimum:18

9) Maximizeandminimizez=13x–14ysubjectto:

x≥0, y≥0, 4x+5y≤30, 4x+3y≤20, x≤5, y≤8.

A) maximum:65;minimum:–84 B) maximum:65;minimum:0

C) maximum:–53.75;minimum:–84 D) maximum:–84;minimum:0

Anobjectivefunctionandasystemoflinearinequalitiesrepresentingconstraintsaregiven.Graphthesystemof

inequalitiesrepresentingtheconstraints.Findthevalueoftheobjectivefunctionateachcornerofthegraphed

region.Usethesevaluestodeterminethemaximumvalueoftheobjectivefunctionandthevaluesofxandyfor

whichthemaximumoccurs.

10) ObjectiveFunction z=17x+9y

Constraints 0≤x≤10

0≤y≤5

3x+2y≥6

A) maximum:215;at(10,5) B) maximum:170;at(10,0)

C) maximum:45;at(0,5) D) maximum:27;at(0,3)

11) ObjectiveFunction z=6x+7y

Constraints x≥0

y≥0

2x+3y≤12

2x+y≤8

A) maximum32;at(3,2) B) maximum32;at(2,3)

C) maximum52;at(4,4) D) maximum24;at(4,0)

12) ObjectiveFunction z=3x+5y

Constraints x≥0

y≥0

2x+y≤15

x–3y≥–3

A) maximum33;at(6,3) B) maximum75;at(0,15)

C) maximum22.5;at(7.5,0) D) maximum38;at(6,4)

13) ObjectiveFunction z=7x–10y

Constraints 0≤x≤5

0≤y≤8

4x+5y≤30

4x+3y≤20

A) maximum:35;at(5,0) B) maximum:0;at(0,0)

C) maximum:–41.25;at(1.25,5) D) maximum:–60;at(0,6)

14) ObjectiveFunction z=5x+7y

Constraints x≥0

0≤y≤5

2x+3y≥12

2x+3y≤20

A) maximum:50;at(10,0) B) maximum:45;at(2,5)

C) maximum:30;at(6,0) D) maximum:–20;at(4,0)

Page120

15) ObjectiveFunction z=7x–3y

Constraints x≥0

0≤y≤5

x–y≤3

x+2y≤12

A) maximum:33;at(6

,

3) B) maximum:21;at(3

,

0)

C) maximum:–1;at(2,5) D) maximum:–15;at(0,5)

16) ObjectiveFunction z=7x+6y

Constraints x≥0

y≥0

3x+y≤21

x+y≤10

x+2y≥12

A) maximum65.5;at(5.5,4.5) B) maximum60;at(6,3)

C) maximum68;at(8,2) D) maximum66;at(6,4)

Solvethelinearprogrammingproblem.

17) Twokindsofcratedcargo,AandB,aretobeshippedbytruck.Theweightandvolumeofeachtypeare

giveninthefollowingtable:

AB

Volume 50cubicfeet 10cubicfeet

Weight 200pounds 360pounds

Theshippingcompanycharges$75percrateforcargoAand$100percrateforcargoB.Thetruckhasa

maximumloadlimitof7,200poundsand1,000cubicfeet.Howmanyofeachtypeofcargoshouldbe

shippedtomaximizeprofitfortheshippingcompany?

A) 18cratesofcargoAand10cratesofcargoBB)20cratesofcargoAand0cratesofcargoB

C) 0cratesofcargoAand20cratesofcargoBD)10cratesofcargoAand18cratesofcargoB

18) Avineyardproducestwospecialwines,awhiteandared.Abottleofthewhitewinerequires14pounds

ofgrapesand1hourofprocessingtime.Abottleofredwinerequires25poundsofgrapesand2hoursof

processingtime.Thevineyardhasonhand2,198poundsofgrapesandcanallot160hoursofprocessing

timetotheproductionofthesewines.Abottleofthewhitewinesellsfor$11.00,whileabottleofthered

winesellsfor$20.00.Howmanybottlesofeachtypeshouldthevineyardproduceinordertomaximize

grosssales?

A) 132bottlesofwhiteand14bottlesofred B) 14bottlesofwhiteand132bottlesofred

C) 76bottlesofwhiteand42bottlesofred D) 42bottlesofwhiteand59bottlesofred

19) Mrs.Whitewantstocrochetbeachhatsandbabyafghansforachurchfund–raisingbazaar.Sheneeds8

hourstomakeahatand2hourstomakeanafghanandshehas46hoursavailable.Shewantstomakeno

morethan11itemsandnomorethan9afghans.Thebazaarwillsellthehatsfor$14eachandtheafghans

for$7each.Howmanyofeachshouldshemaketomaximizetheincomeforthebazaar?Whatisthe

maximumincome?

A) 4hats,7afghans,$105 B) 7 hats,4 afghans,$126

C) 9hats,2afghans,$140 D) 6 hats,5 afghans,$119

Page121

20) Acandycompanyhas130poundsofcashewsand175 poundsofpeanutswhichtheycombineintotwo

differentmixes.Thedeluxemixhashalfcashewsandhalfpeanutsandsellsfor$6perpound.The

economymixhasonethirdcashewsandtwothirdspeanutsandsellsfor$5.60perpound.Howmany

poundsofeachmixshouldbepreparedformaximumrevenue?

A) 170deluxe,135economy B) 255 deluxe,90 economy

C) 85deluxe,45economy D) 130 deluxe,0economy

21) Adoctorhastoldasickpatienttotakevitaminpills.Thepatientneedsatleast6unitsofvitaminAanda

t

least6unitsofvitaminB.Theredvitaminpillscost10¢eachandcontain1unitofAand2unitsofB.The

bluevitaminpillscost25¢eachandcontain2unitsofAand1unitofB.Toavoidindigestion,thepatient

shouldtakenomorethan4redpillsandnomorethan7bluepills.Howmanypillsshouldthepatienttake

eachdaytominimizecosts?

A) 4redand1blue B) 2redand2blue C) 0redand6blue D) 4redand7blue

22) Adoctorhastoldapatienttotakevitaminpills.Thepatientneedsatleast12unitsofvitaminAanda

t

least8unitsofvitaminD.Theredvitaminpillscost20¢eachandcontain3unitsofAand1unitofD.The

bluevitaminpillscost35¢eachandcontain2unitsofAand2unitsofD.Toavoidindigestion,thepatient

shouldtakenomorethan6redpillsandnomorethan9bluepills.Howmanypillsshouldthepatienttake

eachdaytominimizecosts?

A) 2redand3blue B) 6redand1blue C) 0redand9blue D) 6redand9blue

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

23) Yourcomputersupplystoresellstwotypesoflaserprinters.Thefirsttype,A,hasacostof$86andyou

makea$45profitoneachone.Thesecondtype,B,hasacostof$130andyoumakea$35profitoneach

one.Youexpecttosellatleast100laserprintersthismonthandyouneedtomakeatleast$3850profiton

them.Howmanyofwhattypeofprintershouldyouorderifyouwanttominimizeyourcost?

24) TheJillsonʹshaveupto$75,000toinvest.Theydecidethattheywanttohaveatleast$25,000investedin

stablebondsyielding6%andthatnomorethan$45,000shouldbeinvestedinmorevolatilebonds

yielding12%.Howmuchshouldtheyinvestineachtypeofbondtomaximizeincomeiftheamountinthe

morevolatilebondshouldnotexceedtheamountinthemorestablebond?Whatisthemaximumincome?

25) TheFiedlerfamilyhasupto$130,000toinvest.Theydecidethattheywanttohaveatleast$40,000

investedinstablebondsyielding5.5%andthatnomorethan$60,000shouldbeinvestedinmorevolatile

bondsyielding11%.Howmuchshouldtheyinvestineachtypeofbondtomaximizeincomeiftheamount

inthestablebondshouldnotexceedtheamountinthemorevolatilebond?Whatisthemaximum

income?

26) EricʹsCarpentrymanufacturestwotypesofbookshelvesthatare4feettalland3feetwide,abasicmodel

andadeluxemodel.Eachbasicbookshelfrequires1.5hoursforassemblyand1hourforfinishing;each

deluxemodelrequires2.5hoursforassemblyand1hourforfinishing.Twoassemblersandonefinisher

areemployedbythecompany,andeachworks40hoursperweek.Ericcansellmorebasicmodelsthan

deluxemodels,sohewantsthenumberofbasicmodelsproducedtobe50%morethanthenumberof

deluxemodelsproduced.Ifhemakes$50profitonthebasicmodelsand$65profitonthedeluxemodels,

howmanyshouldhemaketomaximizetheprofit?Whatisthemaximumprofit?

27)

J

oelyʹsTeaShop,astorethatspecializesinteablends,hasavailable45poundsofAgradeteaand70

poundsofBgradetea.Thesewillbeblendedinto1poundpackagesasfollows:Abreakfastblendthat

containsonethirdofapoundofAgradeteaandtwothirdsofapoundofBgradeteaandanafternoon

teathatcontainsonehalfpoundofAgradeteaandonehalfpoundofBgradetea.IfJoelymakesaprofit

of$1.50oneachpoundofthebreakfastblendand$2.00profitoneachpoundoftheafternoonblend,how

manypoundsofeachblendshouldshemaketomaximizeprofits?Whatisthemaximumprofit?

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28) Anartistiscreatingamosaicthatcannotbelargerthanthespaceallottedwhichis4feettalland6fee

t

wide.Themosaicmustbeatleast3feettalland5feetwide.Thetilesinthemosaichavewordswrittenon

themandtheartistwantsthewordstoallbehorizontalinthefinalmosaic.Thewordtilescomeintwo

sizes:Thesmallertilesare4inchestalland4incheswide,whilethelargetilesare6inchestalland12

incheswide.Ifthesmalltilescost$3.50eachandthelargertilescost$4.50each,howmanyofeachshould

beusedtominimizethecost?Whatistheminimumcost?

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

29) TheAcmeClassRingCompanydesignsandsellstwotypesofrings:theVIPandtheSST.Theycan

produceupto24ringseachdayusingupto60totalman–hoursoflabor.Ittakes3man–hourstomake

oneVIPringand2man–hourstomakeoneSSTring.Howmanyofeachtypeofringshouldbemade

dailytomaximizethecompanyʹsprofit,iftheprofitonaVIPringis$40andonanSSTringis$35?

A) 12VIPand12SST B) 14VIPand10SST C) 16VIPand8SST D) 18VIPand6SST

30) Anairlinewithtwotypesofairplanes,P1andP2

,

hascontractedwithatourgrouptoprovide

transportationforaminimumof400firstclass,750touristclass,and1500economyclasspassengers.Fora

certaintrip,airplaneP1costs$10,000tooperateandcanaccommodate20firstclass,50touristclass,and

110economyclasspassengers.AirplaneP2costs$8500tooperateandcanaccommodate18firstclass,30

touristclass,and44economyclasspassengers.Howmanyofeachtypeofairplaneshouldbeusedin

ordertominimizetheoperatingcost?

A) 9P1planesand13P2planes B) 5P1planesand17P2planes

C) 11P1planesand7P2planes D) 7P1planesand11P2planes

31) Asummercampwantstohirecounselorsandaidestofillitsstaffingneedsatminimumcost.Theaverage

monthlysalaryofacounseloris$2400andtheaveragemonthlysalaryofanaideis$1100.Thecampcan

accommodateupto45staffmembersandneedsatleast30torunproperly.Theymusthaveatleast10

aides,andmayhaveupto3aidesforevery2counselors.Howmanycounselorsandhowmanyaides

shouldthecamphiretominimizecost?

A) 12counselorsand18aides B) 27counselorsand18aides

C) 35counselorsand10aides D) 18counselorsand12aides

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Ch.8 SystemsofEquationsandInequalities

AnswerKey

8.1 SystemsofLinearEquations:SubstitutionandEliminatio

n

1 SolveSystemsofEquationsbySubstitution

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