Day 3 Linear Equations, Inequalities & Systems

NAME DATE SECTION

Day3LinearEquations,Inequalities&Systems

EquationsandTheirSolutions

WhatisaSolution?

Agranolabitecontains27calories.Mostofthecaloriescomefrom𝑐gramsofcarbohydrates.Therestcomefromotheringredients.Onegramofcarbohydratecontains4calories.

Theequation4𝑐 + 5 = 27representstherelationshipbetweenthesequantities.

1. Whatcouldthe5representinthissituation?2. Priyasaidthatneither8nor3couldbethesolutiontotheequation.Explainwhysheis

correct.3. Findthesolutiontotheequation.

WeekendEarnings

Jadahastimeontheweekendstoearnsomemoney.Alocalbookstoreislookingforsomeonetohelpsortbooksandwillpay$12.20anhour.Togettoandfromthebookstoreonaworkday,however,Jadawouldhavetospend$7.15onbusfare.

1. WriteanequationthatrepresentsJada’stake-homeearningsindollars,𝐸,ifsheworksatthebookstoreforℎhoursinoneday.

2. Oneday,Jadatakeshome$90.45afterworkingℎhoursandafterpayingthebusfare.Writeanequationtorepresentthissituation.

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3. Is4asolutiontothelastequationyouwrote?Whatabout7?

– Ifso,bepreparedtoexplainhowyouknowoneorbothofthemaresolutions.– Ifnot,bepreparedtoexplainwhytheyarenotsolutions.Then,findthe

solution.4. Inthissituation,whatdoesthesolutiontotheequationtellus?

CaloriesfromProteinandFat

Onegramofproteincontains4calories.Onegramoffatcontains9calories.Asnackhas60caloriesfrom𝑝gramsofproteinand𝑓gramsoffat.

Theequation4𝑝 + 9𝑓 = 60representstherelationshipbetweenthesequantities.

1. Determineifeachpairofvaluescouldbethenumberofgramsofproteinandfatinthesnack.Bepreparedtoexplainyourreasoning.

a. 5gramsofproteinand2gramsoffat

b. 10.5gramsofproteinand2gramsoffat

c. 8gramsofproteinand4gramsoffat2. Ifthereare6gramsoffatinthesnack,howmanygramsofproteinarethere?Show

yourreasoning.3. Inthissituation,whatdoesasolutiontotheequation4𝑝 + 9𝑓 = 60tellus?Givean

exampleofasolution.

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SnacksinBulk

Togetsnacksforaclasstrip,Clarewenttothe“bulk”sectionofthegrocerystore,whereshecouldbuyanyquantityofaproductandthepricesareusuallygood.

Clarepurchasedsomesaltedalmondsat$6apoundandsomedriedfigsat$9perpound.Shespent$75beforetax.

4. Ifshebought2poundsofalmonds,howmanypoundsoffigsdidshebuy?5. Ifshebought1poundoffigs,howmanypoundsofalmondsdidshebuy?6. Writeanequationthatdescribestherelationshipbetweenpoundsoffigsandpounds

ofalmondsthatClarebought,andthedollaramountthatshepaid.Besuretospecifywhatthevariablesrepresent.

7. Hereisagraphthatrepresentsthequantitiesinthissituation.

a. Chooseanypointontheline,stateitscoordinates,andexplainwhatittellsus.

b. Chooseanypointthatisnotontheline,stateitscoordinates,andexplainwhatittellsus.

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GraphIt!8. Astudenthasasavingsaccountwith$475init.Shedeposits$125ofherpaycheck

intotheaccounteveryweek.Hergoalistosave$7,000forcollege.

a. Howmuchwillbeintheaccountafter3weeks?

b. Howlongwillittakebeforeshehas$1,350?

c. Writeanequationthatrepresentstherelationshipbetweenthedollaramountinheraccountandthenumberofweeksofsaving.

d. Graphyourequationusinggraphingtechnology.Markthepointsonthegraphthatrepresenttheamountafter3weeksandtheweekshehas$1,350.Writedownthecoordinates.

e. Howlongwillittakehertoreachhergoal?9. A450-gallontankfullofwaterisdrainingatarateof20gallonsperminute.

a. Howmanygallonswillbeinthetankafter7minutes?

b. Howlongwillittakeforthetanktohave200gallons?

c. Writeanequationthatrepresentstherelationshipbetweenthegallonsofwaterinthetankandminutesthetankhasbeendraining.

d. Graphyourequationusinggraphingtechnology.Markthepointsonthegraphthatrepresentthegallonsafter7minutesandthetimewhenthetankhas200gallons.Writedownthecoordinates.

e. Howlongwillittakeuntilthetankisempty?

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Day3Summary

Anequationthatcontainsonlyoneunknownquantityoronequantitythatcanvaryiscalledanequationinonevariable.

Forexample,theequation2ℓ+ 2𝑤 = 72representstherelationshipbetweenthelength,ℓ,andthewidth,𝑤,ofarectanglethathasaperimeterof72units.Ifweknowthatthelengthis15units,wecanrewritetheequationas:

2(15)+ 2𝑤 = 72.

Thisisanequationinonevariable,because𝑤istheonlyquantitythatwedon'tknow.Tosolvethisequationmeanstofindavalueof𝑤thatmakestheequationtrue.

Inthiscase,21isthesolutionbecausesubstituting21for𝑤intheequationresultsinatruestatement.

2(15)+ 2𝑤 = 722(15)+ 2(21) = 72

30+ 42 = 7272 = 72

Anequationthatcontainstwounknownquantitiesortwoquantitiesthatvaryiscalledanequationintwovariables.Asolutiontosuchanequationisapairofnumbersthatmakestheequationtrue.

SupposeTylerspends45𝑜𝑛𝑇 − 𝑠ℎ𝑖𝑟𝑡𝑠𝑎𝑛𝑑𝑠𝑜𝑐𝑘𝑠.𝐴𝑇 − 𝑠ℎ𝑖𝑟𝑡𝑐𝑜𝑠𝑡𝑠10andapairofsockscosts$2.50.If𝑡representsthenumberofT-shirtsand𝑝representsthenumberofpairsofsocksthatTylerbuys,wecancanrepresentthissituationwiththeequation:

10𝑡 + 2.50𝑝 = 45

Thisisanequationintwovariables.Morethanonepairofvaluesfor𝑡and𝑝maketheequationtrue.

𝑡 = 3and𝑝 = 6

10(3)+ 2.50(6) = 4530+ 15 = 4545 = 45

𝑡 = 4and𝑝 = 2

10(4)+ 2.50(2) = 4540+ 5 = 4545 = 45

𝑡 = 2and𝑝 = 10

10(2)+ 2.50(10) = 4520+ 25 = 4545 = 45

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Inthissituation,oneconstraintisthatthecombinedcostofshirtsandsocksmustequal$45.Solutionstotheequationarepairsof𝑡and𝑝valuesthatsatisfythisconstraint.

Combinationssuchas𝑡 = 1and𝑝 = 10or𝑡 = 2and𝑝 = 7arenotsolutionsbecausetheydon’tmeettheconstraint.Whenthesepairsofvaluesaresubstitutedintotheequation,theyresultinstatementsthatarefalse.

GraphingSummary

Likeanequation,agraphcangiveusinformationabouttherelationshipbetweenquantitiesandtheconstraintsonthem.

Supposewearebuyingbeansandricetofeedalargegatheringofpeople,andweplantospend120𝑜𝑛𝑡ℎ𝑒𝑡𝑤𝑜𝑖𝑛𝑔𝑟𝑒𝑑𝑖𝑒𝑛𝑡𝑠.𝐵𝑒𝑎𝑛𝑠𝑐𝑜𝑠𝑡2apoundandricecosts$0.50apound.

If𝑥representspoundsofbeansand𝑦poundsofrice,theequation2𝑥 + 0.50𝑦 = 120canrepresenttheconstraintsinthissituation.

Thegraphof2𝑥 + 0.50𝑦 = 120showsastraightline.

Eachpointonthelineisapairof𝑥-and𝑦-valuesthatmaketheequationtrueandisthusasolution.Itisalsoapairofvaluesthatsatisfytheconstraintsinthesituation.

• Thepoint(10,200)isontheline.Ifwebuy10poundsofbeansand200poundsofrice,thecostwillbe2(10)+ 0.50(200),whichequals120.

• Thepoints(60,0)and(45,60)arealsoontheline.Ifwebuyonlybeans—60poundsofthem—andnorice,wewillspend120. 𝐼𝑓𝑤𝑒𝑏𝑢𝑦45𝑝𝑜𝑢𝑛𝑑𝑠𝑜𝑓𝑏𝑒𝑎𝑛𝑠𝑎𝑛𝑑60𝑝𝑜𝑢𝑛𝑑𝑠𝑜𝑓𝑟𝑖𝑐𝑒,𝑤𝑒𝑤𝑖𝑙𝑙𝑎𝑙𝑠𝑜𝑠𝑝𝑒𝑛𝑑120.

Whataboutpointsthatarenotontheline?Theyarenotsolutionsbecausetheydon'tsatisfytheconstraints,buttheystillhavemeaninginthesituation.

• Thepoint(20,80)isnotontheline.Buying20poundsofbeansand80poundsofricecosts2(20)+ 0.50(80)or80,whichdoesnotequal120.Thiscombinationcostslessthanwhatweintendtospend.

• Thepoint(70,180)meansthatwebuy70poundsofbeansand180poundsofrice.Itwillcost2(70)+ 0.50(180)or230,whichisoverourbudgetof120.