6-4:APPLICATIONSOFLINEARSYSTEMSLessonObjectives:

• Understand,write,andsolvesystemsoflinearequationsfromwordproblems

EXAMPLE1:MIXTUREPROBLEMS1.Toproducea50%mixtureofantifreeze,a30%mixtureofantifreezeismixedwithpureantifreeze.Howmuchofeachtypeisneededtoproduce7quartsofthe50%mixture?2.Ametalworkerhassomeingotsofmetalalloythatare20%copperandothersthatare60%copper.Howmanykilogramsofeachtypeofingotshouldthemetalworkercombinetocreate80kgofa52%copperalloy?3.Supposeyoucombineingotsof25%copperalloyand50%copperalloytocreate40kgof45%copperalloy.Howmankilogramsofeachdoyouneed?4.A20ozalloyofplatinumthatcosts$220perounceismixedwithanalloythatcosts$400perounce.Howmanyouncesofthe$400alloyshouldbeusedtomakeanalloythatcosts$300perounce?

100

let qtsof30 antifreeze

gqtsof1oo afreeze z y g 3s

t Eyt L

let kgof20 copperXt Yi 80 o2X 2y 16

y kgof60copper 2Xt.by 5280

t 2xt.ky41.6ly 25.6

i xt.eii.se

L

let kgof25 copper

y kgof50 copper

o2 Xty 40 2Sxo2sy to

25Xtosy45610 t 2sxt.su

oi

Let y oz of 400alloy 2 ty y Zo22ffoo2oYI4o9fE.E.ioItoDoy22oC2ot4ooy 30oz sooy sooyOz of 300alloy 440010026000woo woo

toy1640zaky y

EXAMPLE2:FINDINGABREAK-EVENPOINT5.Supposeamodelairplaneclubpublishesanewsletter.Expensesare$0.90forprintingandmailingeachcopy,plus$600totalforresearchandwriting.Thepriceofthenewsletteris$1.50percopy.Howmanycopiesofthenewslettermusttheclubselltobreakeven?6.Supposeyouarestartinganoffice-cleaningservice.Youhavespent$315onequipment.Tocleananoffice,youuse$4worthofsupplies.Youcharge$25peroffice.Howmanyofficesmustyoucleantobreakeven?7.Supposeyouinvest$1500inequipmenttoputpicturesonT-shirts.YoubuyeachT-shirtfor$3.Afteryouhaveplacedthepictureontheshirt,yousellitfor$20.HowmanyT-shirtsmustyouselltobreakeven?EXAMPLE3:AIRSPEED/WINDSPEEDPROBLEMS8.SupposeyouflyfromMiami,FloridatoSanFrancisco,California.Ittakes6.5hourstofly2600milesagainstaheadwind.Atthesametime,yourfriendfliesfromSanFranciscotoMiami.Herplanetravelsatthesameaverageairspeed,butherflightonlytakes5.2hours.Findtheaveragewindspeed.

Alway itemsorservices

g amount I

rateinitial

Let X newsletters go.gftaooho.gg

y amount oo

Expenses y yytlooo

Income 1ooonewsle

3154 2sxLet offices 4 4Xy amount

315211y 3is

2T 21

2sx 15015offices

15oot3x 2oLet t shirts sx K

y amount 1,50,0474

g 1500 30 88.23 Xsox 89Lshirt

RoTDCAtwJ.TDA w 1 D

Aairspeed Atg.q26 CAg52.600wwindspeedAtw 500 A w 400plane450mph tAtw5oowiI I Iw

9.Aboattravels60kmupstream(againstthecurrent)in5hours.Theboattravelsthesamedistancedownstreamin3hours.Whatistherateoftheboatinstillwater?Whatistherateofthecurrent?10.WhenLucyswimswiththecurrent,sheswims18kmin2hours.Againstthecurrent,shecanswimonly14kminthesametime.HowfastcanLucyswiminstillwater?Whatistherateofthecurrent?11.Onanupstreamtrip,acanoetravels40kmin5hours.Downstream,ittravelsthesamedistanceinhalfthetime.Whatistherateofthecanoeinstillwaterandtherateofthecurrent?12.Aduckcanfly2400min10minwiththewind.Againstthewind,itcanflyonlytwothirdsofthisdistancein10min.Howfastcouldtheduckflyinstillair?Whatistherateofthewind?

TrentetBIdeed B E 360 C E 6gBtc 20 B c 12

Boat 16kmh Btc 20

ar Wh E

dLte 2 18 L c 2 14

Let f Lucy'sspeed 21 2 21 2currentspeed

c q L C 7

Lucy 8km h TfLtc9curmtssh.Ee

let

Btc _16 B C 8BtC

O0

LetD duck'sspeed Dt 240 C

w windspeedsD w 24o D w 160

tD w 24o

in i

EXAMPLE4:RATIOPROBLEMS13.Kayspends250min/weekexercising.Herratiooftimespentonaerobicstotimespentweighttrainingis3to2.Howmanyminutesperweekdoesshespendonaerobics?Howmanyminutesperweekdoesshespendonweighttraining?14.Supposetheratioofgirlstoboysinyourschoolis19:17.Thereare1908studentsaltogether.Writeandsolveasystemofequationstofindthetotalnumberofboysandgirlsintheschool.EXAMPLE5:TWONUMBERSPROBLEMS15.Threetimesthelargeroftwonumbersisequaltofourtimesthesmaller.Thesumofthenumbersis21.Findthenumbers.16.Thedifferencebetweentwonumbersis16.Fivetimesthesmalleristhesameas8lessthantwicethelarger.Findthenumbers.

3to2323

OLet X timeonaerobics Xty 25Ogtimeonweights zyy

t

Ify z

x 3 I

C

Let b boysggirls

btg 1908

xE

bbg g F gg seiosigloo7f

Tya ss

90

tx largert t't

sy 2T s

g smallerX F 16

5y 2Xg8 Sy22 22 8

x i6252

24Izz

E

Name__________________________________________ 6-4PracticeWorksheet Period________________Writeandsolveasystemofequationsforeachproblem.1.A4%solutionofboraxmustbeaddedtoa12%solutionofboraxtoobtain32poundsofa10%solutionofborax.Howmanypoundsofeachtypeofsolutionshouldbeused?2.Onemixtureofsaltwateris10%salt.Anothermixtureis3%salt.Howmanypoundsofsaltwaterofeachtypemustbemixedtoproduce42poundsof5%saltwater?3.50gramsofasolutionwhichis36%acidisneededbymixingtogethera20%acidsolutionwithpure(100%)acid.Howmanygramsofeachtypeshouldbeused?4.Whenaplanefliesintothewind,itcantravel3000kmin6hours.Whenitflieswiththewind,itcantravelthesamedistancein5hours.Findtherateoftheplaneinstillairandtherateofthewind.5.Withthewind,ajetcanfly2500kmin2hoursand30minutes.Againstthewind,itcanflyonly2000kminthesameamountoftime.Findtherateofthejetinstillairandtherateofthewind.

6.Withthewind,aplaneflew1400kmin4hours.Onthereturntrip,thepilotwasforcedtolandafter1hourand30minutes,havingtraveledonly450km.Findtherateoftheplaneinstillairandtherateofthewind.7.Asalmonswims100min8mindownstream.Upstream,itwouldtakethefish20mintoswimthesamedistance.Whatistherateofthesalmoninstillwater?Whatistherateofthecurrent?8.Anewparkinglothasspacesfor450cars.Theratioofspacesforfull-sizedcarstocompactcarsis11to4.Howmanyspacesareforfull-sizedcars?Howmanyspacesareforcompactcars?9.Thelargeroftwonumbersis1morethantwicethesmaller.Thesumofthenumbersis20lessthanthreetimesthelarger.Findthenumbers.10.Thesumoftwonumbersisthesameasfourtimesthesmallernumber.Iftwicethelargerisdecreasedbythesmaller,theresultis30.Findthenumbers

11.Youstartacompanythatsellswidgetsonline.Theinitialequipmentandsetupcost$1200.Itcostsanadditional$10foreverywidgetproduced.Ifthewidgetsaresoldfor$35,howmanymustbesoldtobreakeven?12.Thefreshmanclassissellingt-shirtsasafundraiser.Theyfoundacompanythatwillcharge$125tomaketheprintingscreenplus$8.75pershirt.Iftheclasssellstheshirtsfor$15each,howmanymusttheyselltobreakeven?13.Thepriceofasweateris$5lessthantwicethepriceofashirt.Iffoursweatersandthreeshirtscost$200,findthepriceofeachshirtandeachsweater.14.AshipmentofTVsets,someweighing30poundseachandothersweighing50poundseachhasatotalweightof880pounds.Ifthereare20TVsetsaltogether,howmanyweigh50pounds?15.Yourteacherisgivingyouatestworth100pointscontaining40questions.Therearetwo-pointquestionsandfour-pointquestionsonthetest.Howmanyofeachtypeofquestionareonthetest?