day 3 linear equations, inequalities & systems
TRANSCRIPT
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.