Elementary Education Chapter 12 Systems Equations And Inequalities 121 Systems

Ch.12 SystemsofEquationsandInequalities

12.1 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=4

x=3,y=–1

A) solution B) notasolution

2)

x+y=8

x–y=2

x=–5,y=3

A) solution B) notasolution

3)

3x+y=–12

2x+3y=–22

x=–2,y=–6

A) solution B) notasolution

4)

4x+y=18

3x+4y=20

x=4,y=–2

A) solution B) notasolution

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

5)

x+y=–8

x–y=18

A) x=5

,

y=–13;(5

,

–13) B) x=5

,

y=13;(5

,

13)

C) x=8

,

y=–13;(8

,

–13) D) x=8

,

y=5;(8

,

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+2y=2

7x–8y=-

8

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)

2x+4y =10

2x =-

6

A) x=–3

,

y=4;(–3

,

4) B) x=4

,

y= –3;(4

,

–3)

C) x=–3

,

y=–4;(–3

,

–4) D) x= –4

,

y=0;(–3

,

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)

5x+36y=36

3x–9y=-

9

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=-

5

x–y=17

A) x=6

,

y=–11;(6

,

–11) B) x=6

,

y=11;(6

,

11)

C) x=5

,

y=–11;(5

,

–11) D) x=5

,

y=6;(5

,

6)

4)

5x+4y=74

5x+2y=82

A) x=18

,

y=–4;(18

,

–4) B) x= –18

,

y=5;(–18

,

5)

C) x=–18

,

y=4;(–18

,

4) D) x= –4

,

y=18;(–4

,

18)

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+6y=-

36

12x+2y=90

A) x=9

,

y=–9;(9

,

–9) B) x= –9

,

y=9;(–9

,

9)

C) x=12

,

y=–12;(12

,

–12) D) x= –2

,

y=9;(–2

,

9)

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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)

3

5x+2

5y=51

5

6x+2y=108

A) x=19

,

y=–3;(19

,

–3) B) x= –19

,

y=6;(–19

,

6)

C) x=–19

,

y=4;(–19

,

4) D) x= –3

,

y=19;(–3

,

19)

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68 inches.Thelengthis10incheslongerthanthe

width.Findthewidth.

A) 12in. B) 22in. C) 32 in. D) 34in.

13) TheFamilyFineArtsCentercharges$25 peradultand$10 perseniorcitizenforitsperformances.Ona

recentweekendeveningwhen534peoplepaidadmission,thetotalreceiptswere$7980.Howmanywho

paidwereseniorcitizens?

A) 358seniorcitizens B) 176seniorcitizens C) 268 seniorcitizens D) 266 seniorcitizens

14) Aretiredcouplehas$140,000toinvesttoobtainannualincome.Theywantsomeofitinvestedinsafe

CertificatesofDeposityielding7%.TheresttheywanttoinvestinAAbondsyielding10%peryear.How

muchshouldtheyinvestineachtorealizeexactly$12,800peryear?

A) $100,000at10%and$40,000at7% B) $100,000 at7%and$40,000at10%

C) $90,000at7%and$50,000at10% D) $110,000 at10%and$30,000at7%

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4softdrinksfor$36.60.Thesecondgroupbought7slicesofpizzaand4soft

drinksfor$32.67.Howmuchdoesonesliceofpizzacost?

A) $3.93persliceofpizza B) $1.29 persliceofpizza

C) $3.43persliceofpizza D) $1.79 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17paperbackbookstotradeandgets$18.83

credit.Someofthebookshadacoverpriceof$6.95,therest$4.95.ShewantstogetsomeTomClancy

bookshavingacoverpriceof$6.95.Howmany$6.95booksdidshebringinandhowmanyClancybooks

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

A) 5$6.95books,5Clancybooks B) 12 $6.95 books,2Clancybooks

C) 5$6.95books,6Clancybooks D) 5 $6.95 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=–6

x+y=2

A) x=0,y=0;(0,0) B) x= –6

,

y=2;(–6

,

2)

C) x=0,y=–4;(0,–4) 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)

4x–7y=1

4x–7y=–9

A) x=4

,

y=7;(4

,

7) B) x=1

,

y= –9;(1

,

–9)

C) x=1

7,y=–9

4;1

7,–9

4D) inconsisten

t

4)

4x–6y=6

8x–12y=30

A) x=2

,

y=5;(2

,

5) B) x=3

,

y= – 2;3

,

2

C) x=6

,

y=30;(6

,

30) D) inconsisten

t

5)

5x–9y=6

–20x+36y=–18

A) x=5

,

y=6;(5

,

6) B) x=5

4,y=–9

4;(5

4,–9

4)

C) x=4

,

y=3;(4

,

3) 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+7y=5

2x+14y=10

A) y=–x

7

+5

7,wherexisanyrealnumber

or{(x,y)|y=–x

7

+5

7,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=8

–45x –5y =-

40

A) y=–9x+8

,

wherexisanyrealnumber

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

B) x=–9y+8

,

whereyisanyrealnumber

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

C) y=9x+8

,

wherexisanyrealnumber

or{(x,y)|y=9x+8,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=0

x–y+3z=14

5x+y+z=–4

x=4,y=–3,z=–1

A) solution B) notasolution

2)

x–y+2z=1

4x+z=1

x+5y+z=6

x=0,y=1,z=1

A) solution B) notasolution

Page7

Solvethesystemofequations.

3)

x+5y+2z=-

4

5y+4z=7

z=3

A) x=–5

,

y=–1

,

z=3;(–5

,

–1

,

3) B) x= –5

,

y=3

,

z= –1;(–5

,

3

,

–1)

C) x=3

,

y=–1

,

z=–5;(3

,

–1

,

–5) D) inconsisten

t

4)

x–y+3z=-

2

3x+z=-

2

x+3y+z=-

14

A) x=0

,

y=–4

,

z=–2;(0

,

–4

,

–2) B) x= –2

,

y=0

,

z= –4;(–2

,

0

,

–4)

C) x=–2

,

y=–4

,

z=0;(–2

,

–4

,

0) D) inconsisten

t

5)

x–y+5z=-

3

5x+z=0

x+5y+z=15

A) x=0

,

y=3

,

z=0;(0

,

3

,

0) B) x=0

,

y=0

,

z=3;(0

,

0

,

3)

C) x=0

,

y=3

,

z=–3;(0

,

3

,

–3) D) inconsisten

t

6)

5x+4y+z=–29

2x–4y–z=1

5x+y+2z=–24

A) x=–4

,

y=–2

,

z=–1;(–4

,

–2

,

–1) B) x= –4

,

y= –1

,

z=–2;(–4

,

–1

,

–2)

C) x=–1

,

y=–2

,

z=–4;(–1

,

–2

,

–4) 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=8

x–y+3z=6

5x+y+z=28

A) x=5

,

y=2

,

z=1;(5

,

2

,

1) B) x=1

,

y=5

,

z=2;(1

,

5

,

2)

C) x=1

,

y=2

,

z=5;(1

,

2

,

5) D) inconsisten

t

9)

x–y+z=1

x+y+z=–5

x+y–z=3

A) x=2

,

y=–3

,

z=–4;(2

,

–3

,

–4) B) x=2

,

y= –4

,

z= –3;(2

,

–4

,

–3)

C) x=–3

,

y=–4

,

z=2;(–3

,

–4

,

2) D) inconsisten

t

Page8

10)

x+y+z=-

1

x–y+4z=7

5x+y+z=-

9

A) x=–2

,

y=–1

,

z=2;(–2

,

–1

,

2) B) x=2

,

y= –2

,

z= –1;(2

,

–2

,

–1)

C) x=2

,

y=–1

,

z=–2;(2

,

–1

,

–2) D) inconsisten

t

11)

4x+4y+z=-

25

3x–4y–z=-

3

2x+y+2z=-

19

A) x=–4

,

y=–1

,

z=–5;(–4

,

–1

,

–5) B) x= –4

,

y= –5

,

z=–1;(–4

,

–5

,

–1)

C) x=–5

,

y=–1

,

z=–4;(–5

,

–1

,

–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$23foradults,$15 forseniorcitizens,and$7 forchildrenunder12for

theirliveperformancesonSundayafternoon.ThispastSunday,thepaidrevenuewas$11,099for765

ticketssold.Therewere47morechildrenthanadults.Howmanychildrenattended?

A) 259children B) 212children C) 294 children D) 249 children

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

to122gprotein,118gfat,and159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,4meatballs,3eggs B) 4 mushrooms,3 meatballs,9eggs

C) 3mushrooms,9meatballs,4eggs D) 10 mushrooms,5meatballs,4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9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,15hourstopaint,and7hourstofire.Iftheworkshophas

93hoursforpreparationtime,56hoursforpainting,and93hoursforfiring,howmanyofeachcanbe

made?

A) 7wreaths,4trees,2sleighs B) 4 wreaths,2 trees,7sleighs

C) 2wreaths,7trees,4sleighs D) 8 wreaths,5 trees,3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248.1poundsofshrimp,CaptainJʹs

received218.4pounds,andMainstreetreceived162.5pounds.Howmanypoundsofshrimpdideachboat

catch?

A) Annabelle166lbs,CurlyQ262lbs,SloJoe201 lbs

B) Annabelle262lbs,CurlyQ201lbs,SloJoe166 lbs

C) Annabelle201lbs,CurlyQ262lbs,SloJoe166 lbs

D) Annabelle201lbs,CurlyQ166lbs,SloJoe262 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

28+56–14I1–14I2=0

56–14I1–7I3=0

FindthecurrentsI1,I2,andI3.

28V56V

11Ω

7Ω 4Ω

3Ω 10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=-

1

x–y+3z=-

5

2x+2y+2z=4

A) x=5

,

y=–2

,

z=–4;(5

,

–2

,

–4) B) x= –4

,

y= –2

,

z=5;(–4

,

–2

,

5)

C) x=–4

,

y=5

,

z=–2;(–4

,

5

,

–2) D) inconsisten

t

2)

x–y+4z=1

4x+z=0

–x+y–4z=-

4

A) x=0

,

y=–1

,

z=0;(0

,

–1

,

0) B) x=4

,

y= –1

,

z=0;(4

,

–1

,

0)

C) x=0

,

y=0

,

z=–1;(0

,

0

,

–1) 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

12.2 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) 7x+2y=53

8x–2y=22

A) 7253

8–222 B) 53 2 7

22 8 –2C) 7853

2–222 D) 7222

–2853

2) –2x+4y=6

8y=48

A) –24 6

0848 B) –246

8480 C) 8048

–24 4 D) 64–2

48 0 8

3)

7x–y=0

–7x+y–14=0

A)

7–10

–7114

B)

7–10

–71–14

C)

7–70

–1114

D)

7–714

–110

4)

2.5x+0.3y=11.9

0.5x–0.9y=4.3

A)

2.5 0.3 11.9

0.5 –0.9 4.3

B)

2.5 11.9 0.3

0.5 4.3 –0.9

C)

2.5 0.5 11.9

–0.9 0.3 4.3

D)

11.9 0.3 2.5

4.3 –0.9 0.5

5)

1

2x+4

11y=–1

8

4

11x–1

2y=1

2

A)

1

2

 4

11

–1

8

4

11

–1

2

 1

2

B)

1

2

4

11

1

8

4

11

1

2

 1

2

C)

1

2

4

11

–1

8

4

11

1

2

–1

2

D)

1

2

4

11

–1

80

4

11

–1

2

1

20

6)

3x+6y+4z=73

7x+5y+9z=89

7x+7y–2z=96

A)

36 473

75 989

77–296

B)

36 4

75 9

77–2

C)

37 773

65 789

49–296

D)

73 463

89 957

96 –277

7)

–2x+4z=18

2y+7z=41

9x+3y+9z=63

A)

–20418

02741

93963

B)

–20918

02341

47963

C)

–24018

27041

93963

D)

–204

027

939

8)

9x+4y+5z=74

2x+5y+8z=59

2x+2y+8z=38

4x+2y+2z=2

A)

94574

25859

22838

4222

B)

9224

4522

5882

74 59 38 2

C)

9452

25838

22859

42274

D)

9222

4522

5882

74 59 38 4

Page14

9)

12x+2y–7z+w=–7

3y+z=–6

x–y–11z=–4

6x–6y+8z=1

A)

12 2 –71

–7

0310

–6

1–1–11 0 –4

6–6801

B)

12 2 –7–7

0 316

1–1–11 –4

6–681

C)

12019

23

–1–6

–71

–11 8

1 000

–7–6–41

D)

12 2 7 1 –7

0310

–6

1 1 11 0 –4

66801

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) –53–4

63 3

A)

–5x+3y=–4

6x+3y=3 B)

–5x+3y=0

6x+3y=0 C)

3x–5y= –4

6x+3y=3 D)

–5x+3y= –4

3x+6y=3

2) 45 6

10 3 12

A)

4x+5y=6

10x+3y=12 B)

4x+5y= – 6

10x+3y=–12 C)

5x+4y=6

10x+3y=12 D)

4x+5y=6

3x+10y=12

3)

4723

4238

A)

4x+7y=23

4x+2y=38

B)

4x+4y=23

7x+2y=38

C)

23x+7y=–4

38x+2y=–4

D)

4x+38y=–7

7x+23y=–2

4)

1004

0108

0015

A)

x=4

y=8

z=5

B)

x= –4

y=–8

z=–5

C)

x=0

y=12

z=9

D)

x= –1

y=3

z=0

Page15

5)

2–4–3–7

–360–7

80–6–8

A)

2x–4y–3z=–7

–3x+6y=–7

8x–6z=–8

B)

2x–4y–3z=–7

–3x+6y–7z=0

8x–6y–8z=0

C)

2x–4y–3z=–7

–3x+6z=–7

8x–6y=–8

D)

2x–4y–3z=–7

–3x+6y=–7

8x–6y=–8

6)

766–2

8054

790 2

A)

7x +6y +6z =–2

8x +5z =4

7x +9y =2

B)

7x –6y +6z = –2

8x +5z =–4

7x +9y =–2

C)

7x +6y +6z = –2

8x +5z =4

7x +9z =2

7)

1000–8

0100–2

001010

00010

A)

x1=–8

x2=–2

x3=10

x4=0

B)

x1=8

x2=2

x3=–10

x4=0

C)

x1=0

x2=–10

x3=2

x4=8

D)

x1=–18

x2=–12

x3=0

x4=–12

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7

010 7

000–8

A) consistent;x=7

,

y=7;(7

,

7) B) consistent;x= –7

,

y=–7;(–7

,

–7)

C) consistent;x=7

,

y=7

,

z=–8;(7

,

7

,

–8) D) inconsisten

t

9)

1000

010 0

000–1

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

C) consistent;z=–1;(–9) D) inconsisten

t

Page16

10)

1559

0555

0013

A) consistent;x=4

,

y=–2

,

z=3;(4

,

–2

,

3) B) consistent;x=4

,

y=3

,

z=–2;(4

,

3

,

–2)

C) consistent;x=3

,

y=–2

,

z=4;(3

,

–2

,

4) D) inconsisten

t

11)

10–58

017 –6

000 0

A) consistent;x=8+5z,y=–6–7z,zanyrealnumber

or{(x,y,z)|x=8+5z,y=–6–7z,zanyrealnumber}

B) consistent;x=8–5z,y=–6+7z,zanyrealnumber

or{(x,y,z)|x=8–5z,y=–6+7z,zanyrealnumber}

C) consistent;x=8

,

y=–6

,

z=–5;(8

,

–6

,

–5)

D) inconsisten

t

12)

1001 2

0104 –4

001–70

0000 0

A) consistent;x1=2–x4

,

x2=–4–4x4

,

x3=7x4

,

x4anyrealnumber

or{(x,y,z)|x1=2–x4,x2=–4–4x4,x3=7x4,x4anyrealnumber}

B) consistent;x1=2+x4

,

x2=–4+4x4

,

x3= –7x4

,

x4anyrealnumber

or{(x,y,z)|x1=2+x4,x2=–4+4x4,x3=–7x4,x4anyrealnumber}

C) consistent;x1=4

,

x2=–7

,

x3=–2

,

x4= –7;(4

,

–7

,

–2

,

–7)

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

4r1

420 16

42

–4

A) 154

42–4B)

15 4

11

2

–1C) 15 4

570D) 15 16

42–4

2) R2=–1r1+r2

1210

15–3

A) 1210

03–13 B) 12 10

077C) –1–2–10

03–13 D) –1–2–10

15–3

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=–5r1+r2

4–415

–50 5–2

–12

–3–1

A)

4–41 5

–25 20 0 –27

–12

–3–1

B)

4–415

15 –20 10 23

–12

–3–1

C)

29 –4–24 15

–50 5

–2

–12

–3–1

D)

–25 20 0 –27

–505

–2

–12

–3–1

6) (a)R2=–3r1+r2

(b)R3=–3r1+r3

(c)R3=5r2+r3

1–3–5–2

3–5–45

3546

A)

1–3–5–2

0 4 11 11

0347467

B)

1–3–5–2

0–8–93

0–26 –26 –3

C)

1–3–5–2

0 4 11 11

0183023

D)

1–3–5–2

014194

0 84 114 32

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

0417

–1

01091 6

B)

1–3–52

0417

–1

0–98 –89 6

C)

1–3–52

0417

–1

0 28 121 6

D)

1–3–52

01417–1

00611

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=–4r1+r3

(b)R4=2r1+r4

11

–11 5

0–24

–30

40

–1–44

–110 5

–4

A)

11

–11 5

0–24

–30

0–43

–8–16

13

–27 6

B)

11

–115

0–24

–30

0–43

–8–16

–11 0 5

–4

C)

11

–115

0–24

–30

84

–5024

13

–276

D)

11

–11 5

0–24

–30

0–43

–80

13

–273

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) 6x+6y=12

3x+2y=1

A) x=–3

,

y=5;(–3

,

5) B) x=5

,

y= –3;(5

,

–3)

C) x=–3

,

y=–5;(–3

,

–5) D) inconsisten

t

2) 9x–5y=-

3

–9x+5y=8

A) inconsisten

t

B) x=9

,

y=8;(9

,

8) C) x=-3

,

y=8;(–3

,

8) D) x=9

,

y=9;(9

,

9)

3)

x–y=5

x+y=3

A) x=4

,

y=–1;4

,

–1 B) x=1

,

y=2;1

,

2

C) x=8

,

y=–2;8

,

–2 D) inconsisten

t

Page19

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) 3x+5y =1

2x =-

6

A) x=–3

,

y=2;(–3

,

2) B) x=2

,

y= –3;(2

,

–3)

C) x=–3

,

y=–2;(–3

,

–2) D) inconsisten

t

6) 4x+y=3

5x+3y=-

5

A) x=2

,

y=–5;(2

,

–5) B) x= –5

,

y=2;(–5

,

2)

C) x=–5

,

y=–2;(–5

,

–2) D) inconsisten

t

7)

8x+5y–z=96

x–4y+6z=-

5

–6x+y+z=-

48

A) x=9

,

y=5

,

z=1;(9

,

5

,

1) B) x=9

,

y=1

,

z=5;(9

,

1

,

5)

C) x=–9

,

y=5

,

z=18;(–9

,

5

,

18) D) inconsisten

t

8)

7x–y–8z=-

16

–7x–7z=-

105

3y+z=11

A) x=7

,

y=1

,

z=8;(7

,

1

,

8) B) x=7

,

y=8

,

z=1;(7

,

8

,

1)

C) x=–7

,

y=1

,

z=14;(–7

,

1

,

14) D) inconsisten

t

9)

–5x+8y–z=26

x–7y+9z=-

22

5x+y+z=46

A) x=7

,

y=8

,

z=3;(7

,

8

,

3) B) x=7

,

y=3

,

z=8;(7

,

3

,

8)

C) x=–7

,

y=8

,

z=14;(–7

,

8

,

14) D) inconsisten

t

10)

8x–9y+5z=-

78

–24x+27y–15z=234

24x–27y+15z=-

234

A) x=9

8y–5

8z–39

4,yisanyrealnumber,zisanyrealnumber

or(x,y,z)|x=9

8y–5

8z–39

4,yisanyrealnumber,zisanyrealnumber

B) x=24

,

y=27

,

z=15;(24

,

27

,

15)

C) x=–2

,

y=8

,

z=2;(–2

,

8

,

2)

D) inconsisten

t

Page20

11)

x–y+3z=–1

5x+z=1

x+4y+z=17

A) x=0

,

y=4

,

z=1;(0

,

4

,

1) B) x=0

,

y=1

,

z=4;(0

,

1

,

4)

C) x=1

,

y=4

,

z=0;(1

,

4

,

0) D) x=1

,

y=0

,

z=4;(1

,

0

,

4)

12)

x+y+z=1

x–y+4z=–15

4x+y+z=10

A) x=3

,

y=2

,

z=–4;(3

,

2

,

–4) B) x=3

,

y= –4

,

z=2;(3

,

–4

,

2)

C) x=–4

,

y=2

,

z=3;(–4

,

2

,

3) D) x= –4

,

y=3

,

z=2;(–4

,

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=3

y+z=1

z–w=4

A) x=–8–4w,y=–3–w,z=4+w,wherewisanyrealnumber

B) inconsisten

t

C) x=–8

,

y=–3

,

z=4

,

w=0;(–8

,

–3

,

4

,

0)

D) x=–12

,

y=–4

,

z=5

,

w=1;(–12

,

–4

,

5

,

1)

16)

x+3y–2z–w=7

4x+y+z+2w=16

–3x–y–3z–2w=–13

x–y–3z–2w=–9

A) x=1

,

y=3

,

z=–1

,

w=5;(1

,

3

,

–1

,

5)

B) x=7

,

y=16

,

z=–13

,

w=–9;(7

,

16

,

–13

,

–9)

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

D) x=0

,

y=5

,

z=–5

,

w=7;(0

,

5

,

–5

,

7)

Page21

17)

5x +y=6

3x –y+z–w=6

z+w=4

A) x=1+1

4w,y=1–5

4w,z=4–w,wherewisanyrealnumber

B) x=1–5

4w,y=1+1

4w,z=4–w,wherewisanyrealnumber

C) x=1–1

4w,y=1+5

4w,z=4–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)

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

Page22

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

28+56–14I1–14I2=0

56–14I1–7I3=0

FindthecurrentsI1,I2,andI3.

28V56V

11Ω

7Ω 4Ω

3Ω 10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

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?

Page23

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  510

y(Numberofbirds) 40 160 445

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=3x2+12x+25 B) y=4x2+24x+20 C) y=5x2–12x+28 D) y=6x2–36x+21

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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

125hoursforpreptime,95hoursforpainting,and128hoursforfiring,howmanyofeachcanbemade?

A) 8wreaths;5trees;4sleighs B) 5 wreaths;4 trees;8sleighs

C) 4wreaths;8trees;5sleighs D) 9 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28.Ifthenumberofbuyersinthe

oldestgroupisdoubled,itis25lessthanthenumberofusersinthemiddlegroup.Findthenumberof

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

A) 36

,

35andyounger;51

,

36–59yearolds;13

,

60andolder

B) 38

,

35andyounger;48

,

36–59yearolds;14

,

60andolder

C) 13

,

35andyounger;51

,

36–59yearolds;36

,

60andolder

D) 30

,

35andyounger;53

,

36–59yearolds;17

,

60andolder

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

to157gprotein,137gfat,and177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,4eggs B) 6 mushrooms,4 meatballs,7eggs

C) 4mushrooms,7meatballs,6eggs D) 8 mushrooms,7 meatballs,5eggs

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

Page24

12.3 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) 27

92

A) –59 B) 67 C) 59 D) –4

2) 3–4

–78

A) –4 B) 52 C) 4 D) 44

3) 22

–13

A) 8 B) 4 C) –8D)7

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+5y=19

3x+y=-

4

A) x=–3

,

y=5;(–3

,

5) B) x=5

,

y= –3;(5

,

–3)

C) x=–5

,

y=–3;(–5

,

–3) D) x=3

,

y= –5;(3

,

–5)

2) 4x+4y=48

2x+5y=36

A) x=8

,

y=4;(8

,

4) B) x=4

,

y=8;(4

,

8)

C) x=–4

,

y=8;(–4

,

8) D) x= –8

,

y= –4;(–8

,

–4)

3) 2x+2y=26

2x–2y=-

6

A) x=5

,

y=8;(5

,

8) B) x=8

,

y=5;(8

,

5)

C) x=–8

,

y=5;(–8

,

5) D) x= –5

,

y= –8;(–5

,

–8)

Page25

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)

2x–3y=6

6x–9y=24

A) notapplicable B) x=3

,

y=4;(3

,

4)

C) x=15

4,y=–5

2;15

4,–5

2D) x=6

,

y=24;(6

,

24)

6)

4x+2y=8

7

7x–7y=7

A) x=11

21,y=–10

21;11

21,–10

21 B) x=11

21,y=32

21;11

21,32

21

C) x=–11

21,y=–10

21;–11

21,–10

21 D) notapplicable

7)

x+2y=10

1

4x–y=–20

A) x=–20

,

y=15;–20

,

15 B) x= – 60

,

y=5;–60

,

5

C) x=100

3,y=85

3;100

3,85

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)

122

–115

215

A) 24 B) –16 C) –24 D) 22

2)

355

213

513

A) 30 B) 158 C) –30 D) –120

3)

–254

3–21

16–3

A) 130 B) 80 C) –90 D) –12

Page26

4)

300

765

393

A) –81 B) 189 C) 81 D) –76

5)

123

215

123

A) 0 B) 50 C) 1 D) –20

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4

1x–6

014

=7x

A) 0or1

4B) 0or–1

4C) 1

4D) –1

4

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

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(9,7)and

(–3,–9).Expressthelineʹsequationinstandardform.

A) 16x–12y–60=0B)7x–3y–81 =0

C) –16x+12y–60=0D)

–9x+9y–21 =0

Page27

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(–4,–7),(0,8),and(–20,–67)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(–8,9),(0,–2),and(–24,34)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(10,7),(3,–7),and(–10,–5).

A) 98 B) 196 C) 7 D) 14

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)

9x+4y–z=22

x–8y–4z=-

41

–7x+y+z=-

11

A) x=3

,

y=1

,

z=9;(3

,

1

,

9) B) x=1

,

y=9

,

z=1;(1

,

9

,

1)

C) x=3

,

y=–1

,

z=–9;(3

,

–1

,

–9) D) x=4

,

y= –1

,

z=9;(4

,

–1

,

9)

2)

–5x–6y–7z=-

110

–9x+9y+8z=27

–9x–3y+7z=-

6

A) x=7

,

y=2

,

z=9;(7

,

2

,

9) B) x=2

,

y=9

,

z=2;(2

,

9

,

2)

C) x=7

,

y=–2

,

z=–9;(7

,

–2

,

–9) D) x=8

,

y=0

,

z=9;(8

,

0

,

9)

3)

–5x+9z=47

9x+6y–5z=41

5x–3y=7

A) x=5

,

y=6

,

z=8;(5

,

6

,

8) B) x=6

,

y=8

,

z=6;(6

,

8

,

6)

C) x=5

,

y=–6

,

z=–8;(5

,

–6

,

–8) D) x=6

,

y=4

,

z=8;(6

,

4

,

8)

Page28

4)

x–y+2z=-

1

3x+z=0

–x+y–2z=5

A) notapplicable B) x=0

,

y=1

,

z=0;(0

,

1

,

0)

C) x=2

,

y=1

,

z=0;(2

,

1

,

0) D) x=0

,

y=0

,

z=1;(0

,

0

,

1)

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)

xy z

uv w

12–4

=43

12–4

uv w

xy z

=?

A) –43 B) 43 C) 0 D) Cannotdetermine

2)

xyz

uvw

1–2–4

=90

xy z

uv w

3–6–12

=?

A) 270 B) –90 C) –270 D) 90

3)

xy z

uvw

12 3

=40

uvw

36 9

xy z

=?

A) 120 B) –120 C) –40 D) 40

4)

xyz

uvw

1–3–1

=10

1–3–1

2u 2v 2w

x–1y+3z+1

=?

A) –20 B) –10 C) 20 D) 10

5)

xyz

uvw

1–2–4

=27

xy z–x

uvw–u

1–2–5

=?

A) 27 B) –27 C) 0 D) Cannotdetermine

6)

xy z

uvw

11 4

=–56

x–2y–2z–8

–2u–1–2v–1–2w–4

11 4

=?

A) 112 B) 56 C) –112 D) –56

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

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

Page29

12.4 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= –4

8.FindA+B.

A)

[–212]

B)

–2

12

C)

2–4

48

D) notdefined

2)

LetA=

3

–6

–4

andB=

–5

4

7

.FindA+B.

A)

–2

3

B)

[–2–23]

C)

3–5

–64

–47

D)

2

3

4

3)

LetA=

–46

–56

–6–5

andB=

–32

21

–36

.FindA+B.

A)

–78

–37

–91

B)

–14

–75

–3–14

C)

–78

36

–9–1

D)

–76

–37

–91

4)

LetA=

–12

04

7–4

andB=

72

17 4

32

.FindA–B.

A)

–80

–17 0

4–6

B)

13

78

10 2

C)

10

70

4–2

D)

3–1

70

–46

5)

LetA=–91

25

andB=62

32 .FindA+B.

A)

–33

57

B)

34

37

C)

–3–7

–6–7

D)

12

Page30

6)

LetA=–10

43

andB=–14

31 .FindA–B.

A)

0–4

12

B)

–24

74

C)

04

–1–2

D)

[–1]

7)

LetA=2–4

–6–6,B=9–3

–4–6

andC=–57

–7–8.FindA+B–C.

A)

16 –14

–3–4

B)

60

–17 –20

C)

–2–8

58

D)

–12 6

–9–8

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=

222

40–1

–333

andB=

5–21

–301

24–3

.FindA+B.

A)

703

100

–170

B)

1–43

700

–170

C)

1–43

10

–2

170

D)

70 3

70–2

17 0

11)

LetA=

2–81

4–10 –2

536

andB=

–382

50–4

–65–3

.FindA–B.

A)

5–16 –1

–1–10 2

11 –29

B)

50

–1

–1–10 2

1–23

C)

516

–1

–1–10 2

11 2 9

D)

–103

9–10 –6

–183

Page31

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=–34

02 .Find2A.

A)

–68

04

B)

–68

02

C)

–64

02

D)

–16

24

2) LetB=–157–3.Find–4B.

A) 4–20 –28 12 B) 457–3C) –42028–12 D) –335–5

3) LetC=

2

–2

12

.Find1/2C.

A)

1

–1

6

B)

4

–4

24

C)

1

–2

12

D)

2

–1

12

4) LetA=23

26

andB=04

–16 .Find2A+B.

A)

410

318

B)

414

224

C)

410

112

D)

47

312

5) LetC=

1

–3

2

andD=

–1

3

–2

.FindC–4D.

A)

5

–15

10

B)

–3

9

4

C)

–5

15

–10

D)

5

–6

4

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

A) 14 B) –24 C) –14 D) 22

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.

Page32

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

9) LetA=

–2–46

44

–8

62

–3

andB=

–233

–9–9–9

5–1–8

.Find2A+4B.

A)

–12 4 24

–28 –28 –52

32 0 –38

B)

–6–515

–1–1–25

17 3 –14

C)

–4–19

–5–5–17

11 1 –11

D)

–4–511

–1–51

9–17 –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

42

–20

–15

A)

–115

–10 10

B)

20

–410

C)

2–6

–37

D)

15 –1

10 –10

2)

13–1

40 3

30

–11

03

A)

00

12 9

B)

00

912

C)

3–30

009

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

4)

–619

–2–3–8

5

–9

–2

A)

–57

33

B)

–619

–2–3–8

5–9–2

C)

–57 33

D) notdefined

Page33

5)

–825

8

0

–3

A) –79 B) 314 C) –64 0 –15 D)

–64

0

–15

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)

26

–8

9–49

–3–46

9–5–4

829

A)

–16 –54 –84

9 2 151

B)

–16 9

–54 2

–84 151

C)

26–8

9–49

–3–49

9–5–4

889

D)

–6–24 –48

81 20 –36

72 –881

8)

–715

–296

5–69

513

–799

–795

A)

–77 47 13

–115 133 105

432 6

B)

–77 –115 4

47 –115 32

13 105 6

C)

–715

–296

5–69

513

–799

–795

D)

–35 1 15

14 81 54

–35 –54 45

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

Page34

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

12

0–24

1–32

A)

3–72

2–88

B) notdefined C)

32

–7–8

28

D)

0–612

1–64

12)

[172]

–66

–5

3–9–4

–4–9–1

A) 7–75 –35 B)

7

–75

–35

C)

1702

–66–5

3–9–4

–4–9–1

D)

–642

–10

3–63 –8

–4–63 –2

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

60

24

,B=

–40

11

56

,andC=–216

0–5–1.FindC(A–B).

A) –21 –11

–22 7 B) –4–10

–18 –3C) –18 –3

–4–10 D) –32 8

19 1

15)

LetA=

33

10

3–1

,B=

–41

03

–1–3

,andC=334

1–30.FindBC+6ℐ3.

A)

–5–15 –16

3–30

–66 2

B)

–11 –15 –16

3–90

–66

–4

C)

19 9 16

3150

612 10

D)

13 9 16

390

612 4

Page35

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

410 780

550 750

820 670

630 480

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

Arts&Sciences?

A) 611students;1447students B) 509 students;705students

C) 705students;528students D) 1290 students;611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

83 83

98 96

System

1

System

2

Midterm

Final

0.2 0.5

0.8 0.5

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

A) System1:83;System2:83 B) System1:58.1;System2:107.9

C) System1:77.8;System2:76 D) System1:39.8;System2:53

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.

Page36

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

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)

45

1–5

A)

1

5

1

5

1

25

–4

25

B)

1

5

–1

5

1

25

4

25

C)

–1

5

–1

5

–1

25

4

25

D)

4

25

1

5

1

25

–1

5

Page37

6)

6–4

–3–4

A)

1

9

–1

9

–1

12

–1

6

B)

1

9

1

9

1

12

–1

6

C)

–1

12

–1

6

1

9

–1

9

D)

–1

6

–1

9

–1

12

1

9

7)

05

13

A)

–3

51

1

50

B)

–3

5

–1

50

C)

1

50

–3

51

D)

01

1

5

–3

5

8)

55

05

A)

1

5

–1

5

01

5

B)

1

5

1

5

01

5

C)

01

5

1

5

–1

5

D)

1

5

–1

5

01

5

9)

4–2

30

A)

01

3

–1

2

3

B)

0–1

3

1

2

3

C)

–1

2

3

01

3

D)

2

3

1

3

–1

20

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

Page38

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

–11 0

08 1

A)

100

110

–8–81

B)

100

–110

008

C)

188

011

001

D)

100

8–10

8–11

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

Page39

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)

4–6

–23

20)

2104

–3–11

–174

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)

–3x+9y=9

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

2x–4y=-

20

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+2y =10

bx+3y =7b≠0

A) x=16

b,y=–3;16

b,–3 B) x=–3,y=16

b;–3,16

b

C) x=–16

b,y=3;–16

b,3 D) x=–16

b,y=–3;–16

b,–3

6)

x+2y+3z =-

9

x+y+z=7

2x+2y+z=-

6

Theinverseof

123

111

221

is

–14–1

1–52

02–1

.

A) x=43

,

y=–56

,

z=20;(43

,

–56

,

20) B) x=16

,

y= –83

,

z=11;(16

,

–83

,

11)

C) x=–36

,

y=–14

,

z=–6;(–36

,

–14

,

–6) D) x=13

,

y=14

,

z=8;(13

,

14

,

8)

Page41

7)

x+2y+3z =-

2

x+y+z=-

8

x–2z =8

Theinverseof

123

111

10–2

is

–24–1

3–52

–12–1

.

A) x=–36

,

y=50

,

z=–22;(–36

,

50

,

–22) B) x= –28

,

y=48

,

z=–22;(–28

,

48

,

–22)

C) x=–2

,

y=0,z=0;(–2

,

0,0) D) x= –28

,

y= –30

,

z=–10;(–28

,

–30

,

–10)

8)

x+2y+3z =12

x+y+z=-

2

–x+y+2z =-

10

Theinverseof

123

111

–112

is

1–1–1

–352

2–3–1

.

A) x=24

,

y=–66

,

z=40;(24

,

–66

,

40) B) x= –2

,

y=8

,

z= –6;(–2

,

8

,

–6)

C) x=12

,

y=–8

,

z=20;(12

,

–8

,

20) D) x=0

,

y=6

,

z=8;(0

,

6

,

8)

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)

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)

Page42

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

12.5 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–9x+20

(x2–7x+12)(x+4)

A) proper B) improper

2) x3–13x+42

x2–9x+20

A) improper B) proper

Page43

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

9–x

A) improper;–1+9

9–xB) proper

C) improper;1+9

9–xD) improper;1–9

9–x

4) x–4

x2+6

A) proper B) improper; x–4

(x+6)(x–6)

C) improper; 4

x+6

+4

x–6D) improper; 4

x+6

–4

x–6

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

5) x

(x–5)(x–6)

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

6) x

x2–5x+6

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

7) x–13

(x–3)(x–5)

A) 5

x–3

+–4

x–5B) –4

x–3

+5

x–5C) 5

x–3

+4

x–5D) 4

x–3

+–5

x–5

8) 7x–37

(x+5)(x–4)

A) 8

x+5

–1

x–4B) 8

x+5

+1

x–4C) 1

x–4

–8

x+5D) 7

x+5

–37

x–4

9) 5x2–x–16

x(x+1)(x–1)

A) 16

x

+–5

x+1

+–6

x–1B) 16

x

+5

x+1

+–6

x–1C) 16

x

+–5

x+1

+6

x–1D) 16

x

+–6

x+1

+5

x–1

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

Page44

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) 17x–95

x2–11x+30

A) 10

x–5

+7

x–6B) 10

x–5

+–7

x–6C) 11

x–5

+6

x–6D) 10

x+5

+7

x+6

13) 16x2–21x–27

x(x+3)(x–3)

A) 3

x

+10

x+3

+3

x–3B) 3

x

+11

x+3

+2

x–3C) 3

x

+9

x+3

+4

x–3D) x+10

x2+3

+3

x–3

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) 4x2+13x+11

(x+2)(x+1)2

A) 1

x+2

+3

x+1

+2

(x+1)2B) –1

x+2

+3

x+1

+–2

(x+1)2

C) 1

x+2

+–3

x+1

+2

(x+1)2D) 2

x+2

+3

x+1

+–1

(x+1)2

2) x+3

x3–2x2+x

A) 3

x

+–3

x–1

+4

(x–1)2B) 3

x

+4

x–1

+–3

(x–1)2

C) 3

x

+–3

x–1

+7

(x–1)2D) –3

x

+3

x–1

+4

(x–1)2

3) 100–15x

x3–10x2+25x

A) 4

x

+–4

x–5

+5

(x–5)2B) 4

x

+5

x–5

+–4

(x–5)2

C) 4

x

+–4

x–5

+10

(x–5)2D) –4

x

+4

x–5

+5

(x–5)2

Page45

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) 5x+9

(x–7)2

A) 5

x–7

+44

(x–7)2B) 5

x–7

+70

(x–7)2C) 5

x–7

+x+44

(x–7)2D) 1

x–7

+x+44

(x–7)2

7) 9x2–9x+9

(x–1)3

A) 9

x–1

+9

(x–1)2

+9

(x–1)3B) 9

x–1

+9

(x–1)2

C) 9

x–1

+x+9

(x–1)2

+x2+9

(x–1)3D) 9

x–1

–9

(x–1)2

+9

(x–1)3

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

Page46

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) 13x+2

x3–1

A) 5

x–1

+–5x+3

x2+x+1

B) 5

x–1

+–5

x+1

+3

x–1

C) –5

x–1

+5x+3

x2+x+1

D) 5

x–1

+3x–5

x2+x+1

6) 6x2–2x–16

(x–4)(x2+2)

A) 4

x–4

+2x+6

x2+2

B) 4

x–4

+2

x2+2

C) 4

x–4

+2x–6

x2+2

D) 4

x–4

+2

x+2

+6

(x+2)2

7) 8x2+8x+34

x3+2x2+6x+12

A) 5

x+2

+3x+2

x2+6

B) 5

x+2

+3

x2+6

C) 5

x+6

+3x+2

x2+2

D) 5

x+2

+3

x+6

+2

(x+6)2

Page47

8) –8x–56

(x–3)2(x2+1)

A) 4

x–3

+–8

(x–3)2

+–4x–4

x2+1

B) 4

x–3

+–8

(x–3)2

+–4

x2+1

C) 4

x–3

+–12

(x–3)2

+–4x+4

x2+1

D) –12x+4

(x–3)2

+–4x–4

x2+1

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+1

(x2+4)2

A) 1

x2+4

+2x–3

(x2+4)2B) x+1

x2+4

+2x–3

(x2+4)2C) 1

x2+4

+–x–3

(x2+4)2D) –1

x2+4

+2x–3

(x2+4)2

3) 4x3+26x–4

(x2+6)2

A) 4x

x2+6

+2x–4

(x2+6)2B) 4x+1

x2+6

+2x–4

(x2+6)2C) 4x

x2+6

+–2x–4

(x2+6)2D) 4x

x2+6

+2x+4

(x2+6)2

4) 2x3+4x2+12x+12

(x2+4)3

A) 2x+4

(x2+4)2

+4x–4

(x2+4)3B) x+1

x2+4

+2x+4

(x2+4)2

+4x–4

(x2+4)3

C) 2x–4

(x2+4)2

+4x+4

(x2+4)3D) x

x2+4

+2x+4

(x2+4)2

+4x–4

(x2+4)3

12.6 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

(5, 26)

(-3, 10)

x

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

y

35

30

25

20

15

10

5

-5

(5, 26)

(-3, 10)

x2=y–1

y=2x+16

A) Yes B) No

2)

x

–10–8-6-4-2 2 4 6 810

y

100

50

-50

(0.6, -3.8)

(-4.6, 37.8)

x

–10–8-6-4-2 2 4 6 810

y

100

50

-50

(0.6, -3.8)

(-4.6, 37.8)

2x2=y+5

y=–8x+1

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–4x+4

y=–x+8

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=81

y=x2–9

x

-10 -5 5 10

y

10

5

-5

-10

x

-10 -5 5 10

y

10

5

-5

-10

8)

y=x

y=6–x

x

510

y

10

5

x

510

y

10

5

Page52

9)

(x+4)2+(y–1)2=4

y2–2y–x–7=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=13

x+y=5

A) x=3

,

y=2;x=2

,

y=3

or(3,2),(2,3)

B) x= –3

,

y=2;x=–2

,

y=3

or(–3,2),(–2,3)

C) x=3

,

y=–2;x=2

,

y=–3

or(3,–2),(2,–3)

D) x= –3

,

y= –2;x=–2

,

y=–3

or(–3,–2),(–2,–3)

11)

xy=20

x+y=9

A) x=5

,

y=4;x=4

,

y=5

or(5,4),(4,5)

B) x= –5

,

y=4;x=–4

,

y=5

or(–5,4),(–4,5)

C) x=5

,

y=–4;x=4

,

y=–5

or(5,–4),(4,–5)

D) x= –5

,

y= –4;x=–4

,

y=–5

or(–5,–4),(–4,–5)

12)

x2+y2=181

x–y=1

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)

Page53

13)

y=x2–4x+4

x+y=8

A) x=4

,

y=4;x=–1

,

y=9

or(4,4),(–1,9)

B) x= –4

,

y=12;x=1

,

y=7

or(–4,12),(1,7)

C) x=2

,

y=6or(2

,

6) D) x=4

,

y=12;x=–1

,

y=9

or(4,12),(–1,9)

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)

7x–y=–3

y=x2–5

A) x=8

,

y=59;x=–1

,

y=–4

or(8,59),(–1,–4)

B) x= –8

,

y=59;x=1

,

y=–4

or(–8,59),(1,–4)

C) x=8

,

y=69;x=–1

,

y=6

or(8,69),(–1,6)

D) x= –8

,

y=59;x=–1

,

y=–4

or(–8,59),(–1,–4)

Page55

23)

y=x+2

y2=8x

A) x=2

,

y=4or(2

,

4) B) x=2

,

y=4;x= –2

,

y=0

or(2,4),(–2,0)

C) x=2

,

y=4;x=2

,

y=–4

or(2,4),(2,–4)

D) x=2

,

y=4;x=2

,

y=–4;x=–2

,

y=0

or(2,4),(2,–4),(–2,0)

24)

xy=1

–10x–y=7

A) x=–1

2

,y=–2;x=–1

5,y=–5

or–1

2,–2 ,–1

5,–5

B) x=–2,y=–1

2;x=–5,y=–1

5

or–2,–1

2,–5,–1

5

C) x=2,y=–2;x=5

,

y=–5

or(2,–2),(5,–5)

D) x=–1

5

,y=–5or–1

5,–5

25)

5x2+4y2=36

y=x+3

A) x=0,y=3;x=–8

3,y=1

3

or0,3 ,–8

3,1

3

B) x=0,y=–3;x=–8

3,y=1

3

or0,–3 ,–8

3,1

3

C) x=0,y=3;x=8

3,y=17

3

or0,3 ,8

3,17

3

D) x=0,y=–3;x=8

3,y=17

3

or0,–3 ,8

3,17

3

26)

xy=40

x2+y2=89

A) x=8

,

y=–8;x=–8

,

y=–5;x=5

,

y=8;x= –5

,

y= –8

or(8,–8),(–8,–5),(5,8),(–5,–8)

B) x=8

,

y=5;x=–8

,

y=–5;x=8

,

y= –5;x= –8

,

y=5

or(8,5),(–8,–5),(8,–5),(–8,5)

C) x=8

,

y=5;x=5

,

y=8;x=8

,

y= –5;x=5

,

y= –8

or(8,5),(5,8),(8,–5),(5,–8)

D) x=–8

,

y=–5;x=–5

,

y=–8;x= –8

,

y=5;x= –5

,

y=8

or(–8,–5),(–5,–8),(–8,5),(–5,8)

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=5

x2+y2–25=0

A) x=–5

,

y=0;x=3

,

y=4

or(–5,0),(3,4)

B) x=5

,

y=0;x= –5

,

y=0;x=3

,

y=4

or(5,0),(–5,0),(3,4)

C) x=0,y=5

2;x=4,y=9

2

or0,5

2,4,9

2

D) x=0,y=5

2;x=0,y=–5

2;x=4,y=9

2

or0,5

2,0,–5

2,4,9

2

29)

y=x2–2x–6

y=–x2–8x+2

A) x=–4

,

y=18;x=1

,

y=–7

or(–4,18),(1,–7)

B) x=4

,

y=2;x= –1

,

y=–3

or(4,2),(–1,–3)

C) x=–4

,

y=18;x=0,y=–6

or(–4,18),(0,–6)

D) x=4

,

y=2;x=0,y=–6

or(4,2),(0,–6)

30)

x+y=–7

x2+y2=6y+43

A) x=–6

,

y=–1;x=–4

,

y=–3

or(–6,–1),(–4,–3)

B) x= –1

,

y= –6;x=–3

,

y=–4

or(–1,–6),(–3,–4)

C) x=–8

,

y=1;x=–10

,

y=3

or(–8,1),(–10,3)

D) x=1

,

y= –8;x=3

,

y=–10

or(1,–8),(3,–10)

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=106

x2–y2=56

A) x=9

,

y=5;x=–9

,

y=5;x=9

,

y= –5;x= –9

,

y= –5

or(9,5),(–9,5),(9,–5),(–9,–5)

B) x=9

,

y=5;x=5

,

y=9;x=–9

,

y= –5;x= –5

,

y= –9

or(9,5),(5,–9),(–9,–5),(–5,–9)

C) x=9

,

y=–5;x=9

,

y=5or(9

,

–5),(9

,

5)

D) x=–9

,

y=–5;x=–5

,

y=–9or(–9

,

–5),(–5

,

–9)

Page57

2)

3x2–3y2=63

3x2+4y2=91

A) x=5

,

y=2;x=–5

,

y=2;x=5

,

y= –2;x= –5

,

y= –2

or(5,2),(–5,2),(5,–2),(–5,–2)

B) x=5

,

y=2;x=2

,

y=5;x=–5

,

y= –2;x= –2

,

y= –5

or(5,2),(2,5),(–5,–2),(–2,–5)

C) x=5

,

y=–2;x=5

,

y=2or(5

,

–2),(5

,

2)

D) x=–5

,

y=–2;x=–2

,

y=–5or(–5

,

–2),(–2

,

–5)

3)

x2+y2=16

x2–y2=16

A) x=4

,

y=0;x=–4

,

y=0or(4

,

0),(–4

,

0) B) x=4

,

y=4;x= –4

,

y=4or(4

,

4),(–4

,

4)

C) x=4

,

y=0;x=4

,

y=4or(4

,

0),(4

,

4) D) x= –4

,

y=0;x=–4

,

y=4or(–4

,

0),(–4

,

4)

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68.Thesumofthetwonumbersis10.Findthetwonumbers.

A) 2and8B)2and8 or–8 and–2

C) –8and–2D)

–2 and8 or–8 and2

Page59

13) Thesumofthesquaresoftwonumbersis58.Thedifferenceofthetwonumbersis–10.Findthetwo

numbers.

A) –7and3or–3and7B)

–7 and3

C) –3and7D)3and7 or–7 and–3

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16squareinches.Thesquareofthehypotenuseis80.Findthelengthsofthe

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

A) 4in.and8in. B) 16in.and64 in. C) 2 in.and16 in. D) 8in.and4 in.

16) Theperimeterofarectangleis46inchesanditsareais130 squareinches.Whatareitsdimensions?

A) 10in.by13in. B) 9in.by14 in. C) 11 in.by12 in. D) 9in.by12 in.

17) Asystemfortrackingshipsindicatedthatashipliesonahyperbolicpathdescribedby5x2–y2=4.The

processisrepeatedandtheshipisfoundtolieonahyperbolicpathdescribedbyy2–2x2=8.Ifitis

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

A) (2

,

4) B) (–2

,

–4) C) (4

,

2) D) (–4

,

–2)

18) Findthedimensionsofarectanglewhoseperimeteris36 feetandwhoseareais56squarefeet.

A) 4ftby14ft B) 3ftby15 ft C) 5 ftby13 ft D) 3ftby13 ft

19) Theareaofagardenis5880squarefeet,andthelengthofitsdiagonalis119 feet.Findthedimensionsof

thegarden.

A) 56ftby105ft B) 8ftby735 ft C) 392 ftby15 ft D) 120 ftby49 ft

20) Theareaofarectangularpieceofcardboardshownis576 squareinches.Thecardboardisusedtomakean

openboxbycuttinga5–inchsquarefromeachcornerandturningupthesides.Iftheboxistohavea

volumeof780cubicinches,findthedimensionsofthecardboardthatmustbeused.

A) 16in.by36in. B) 21in.by41 in. C) 11 in.by31 in. D) 6in.by21 in.

21) Arectangularpieceoftinhasanareaof810 squareinches.Asquareof2 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1148cubicinches,

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

A) 18in.by45in. B) 20in.by47 in. C) 16 in.by43 in. D) 14in.by39 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15 feetaheadofthesecond–placerunnerand34 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) 19.05ft B) –15.1 ft C) –19.12 ft D) 4.01 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440feetfromthe

ground,dropsawhiteball2secondsaftertheyellowballwasdropped.Theheightofthewhiteballis

givenbytheequationh=–16(t–2)2+440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) 3.5sec;404ft B) 2.5sec;500 ft C) 3 sec;456 ft D) 4sec;344 ft

Page61

12.7 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>–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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≤–6

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≥2

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–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

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+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

8) –3x–5y≤15

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+3y≤6

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) –3x–5y≤–15

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

<

–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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>49

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

16) y>x2–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77

17) y≤x2+6

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≤6

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) 2x+y>4

2x+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

<

8

–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) 2x–y≤–8

x+4y≥–16

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+3

y>3x–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

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

7) –x+3y

<

9

x≥–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86

8) y

<

x+1

4x+10y>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87

9) x+5y≤5

x+5y≥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

10) 2x+y≥2

2x+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≤49

10x+6y≤60

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

18) x2+y≤4

x2–y≤1

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

19) y>x2

3x+4y≤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98

20) y>x2

8x+2y≤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99

21) x2+y2≤36

–8x+3y≤–24

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≤36

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≤16

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

<

9

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≥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)

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≥5

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,5),(5,0)

x

y

12

-12

(0, 9)

(9, 0)

(5, 0)

(0, 5)

x

y

12

-12

(0, 9)

(9, 0)

(5, 0)

(0, 5)

B) unbounded;

cornerpoints(0,0),(0,5),(5,0)

x

y

12

-12

(0, 9)

(9, 0)

(5, 0)

(0, 5)

x

y

12

-12

(0, 9)

(9, 0)

(5, 0)

(0, 5)

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)

(5, 0)

(0, 5)

x

y

12

-12

(0, 9)

(9, 0)

(5, 0)

(0, 5)

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≤4

y+x≤8

B)

y≥0

x≥0

x≤8

y+x≤4

C)

x≤4

y+x≤8

D)

y≥0

x≥0

x≤4

y+x≥8

35)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

y≥0

x≥0

x≤8

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≤8

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≤9

B)

y≥0

x≥0

y≤2

y≤x

y+x≤9

C)

y≥2

y≥x

y+x≤9

D)

y≥0

x≥0

y≤2

y+x≤9

37)

x

-10 10

y

10

-10

x

-10 10

y

10

-10

A)

y≥0

y≤9

y≤x–3

y≥3–x

B)

x≥0

y≥0

y≤9

y≤x–3

y≤3–x

C)

x≥0

y≤9

y≤x–3

y≥3–x

D)

y≥0

x≤9

y≤x–3

y≥3–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2

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≤2y

x≥0

y≥0

B) x+y=1000

x≤2y

x≥0

y≥0

C) x+y=1000

y≥2x

x≥0

y≥0

D) x+y≤1000

2x≤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

140squarefeet.Hewantstheareaplantedwithtomatoestobemorethan25%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≤140

x>0.25y

x≥0

y≥0

B) x+y=140

x≥0.25y

x≥0

y≥0

C) x+y≤140

x>25y

x≥0

y≥0

D) x+y≤140

x<0.25y

x≥0

y≥0

Page115

12.8 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$2.00

profitoneachpartAthatitproducesanda$5.00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=2x+5y B) z=5x+2y

C) z=7(x+y) D) z=2(x–5)+5(y–2)

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2hourstomakeanafghan,andshehasnomorethan58hoursavailable.Shehasmaterialforno

morethan14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+2y≤58

x+y≤14

y≥2

B) 8x+2y≤58

x+y≤14

x≥2

C) 2x+8y≤58

x+y≤14

x≤2

D) 8x+2y≥58

x+y≤14

y≥2

6) Anofficemanagerisbuyingusedfilingcabinets.Smallfilecabinetscost$5 eachandlargefilecabinetscos

t

$12each,andthemanagercannotspendmorethan$102onfilecabinets.Asmallcabinettakesup6square

feetoffloorspaceandalargecabinettakesup9squarefeet,andtheofficehasnomorethan90squarefeet

offloorspaceavailableforfilecabinets.Themanagermustbuyatleast6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) 5x+12y≤102

6x+9y≤90

x+y≥6

B) 5x+12y≤102

9x+6y≤90

x≥6

C) 5x+12y≤102

6x+9y≤90

x+y≤6

D) 5x+12y≤102

6x+9y≤90

y≥6

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+6y.Findmaximumandminimum.

x

y

(0, 8) (6, 8)

(6, 1)

(

2

,

0

)

(0, 2)

x

y

(0, 8) (6, 8)

(6, 1)

(

2

,

0

)

(0, 2)

A) maximumvalue:60;minimumvalue:4 B) maximumvalue:60;minimumvalue:12

C) maximumvalue:18;minimumvalue:4 D) maximumvalue:18;minimumvalue:12

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–9y.Findmaximum.

A) maximum:–22 B) maximum:–30 C) maximum:–47 D) maximum:–38

4) z=6x+7y.Findminimum.

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

5) z=x+7y+9.Findminimum.

A) minimum:27 B) minimum:46 C) minimum:34 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=8x+6ysubjectto:

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

A) maximum:110;minimum:12 B) maximum:80;minimum:18

C) maximum:30;minimum:18 D) maximum:110;minimum:80

Page119

8) Minimizez=13x+8y+17subjectto:

x≥0, y≥0, x+y≥1.

A) minimum:25 B) minimum:30 C) minimum:38 D) minimum:17

9) Maximizeandminimizez=5x–11ysubjectto:

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

A) maximum:25;minimum:–66 B) maximum:25;minimum:0

C) maximum:–48.75;minimum:–66 D) maximum:–66;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=20x+15y

Constraints 0≤x≤10

0≤y≤5

3x+2y≥6

A) maximum:275;at(10,5) B) maximum:200;at(10,0)

C) maximum:75;at(0,5) D) maximum:45;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–18y

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:–81.25;at(1.25,5) D) maximum:–108;at(0,6)

14) ObjectiveFunction z=5x+6y

Constraints x≥0

0≤y≤5

2x+3y≥12

2x+3y≤20

A) maximum:50;at(10,0) B) maximum:40;at(2,5)

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

Page120

15) ObjectiveFunction z=6x–4y

Constraints x≥0

0≤y≤2

x–y≤3

x+2y≤6

A) maximum:20;at(4

,

1) B) maximum:18;at(3

,

0)

C) maximum:4;at(2,2) D) maximum:–8;at(0,2)

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)2

0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3hourstomakeanafghanandshehas59hoursavailable.Shewantstomakeno

morethan13itemsandnomorethan12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,9afghans,$119 B) 9 hats,4 afghans,$154

C) 12hats,1afghans,$175 D) 6 hats,7 afghans,$133

Page121

20) Acandycompanyhas120poundsofcashewsand150 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$5.30perpound.Howmany

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

A) 180deluxe,90economy B) 270 deluxe,60 economy

C) 90deluxe,30economy D) 120 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

Ch.12 SystemsofEquationsandInequalities

AnswerKey

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

n

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

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