SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.3

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

5. 3 Some of One, None of the Other

A Solidify Understanding Task

CarlosandClaritaarecomparingstrategiesforwritingequationsoftheboundarylinesfor

the“PetSitter”constraints.Theyarediscussingtheirworkonthespaceconstraint.

• Space:Catpenswillrequire6ft2ofspace,whiledogrunsrequire24ft2.Carlosand

Claritahaveupto360ft2availableinthestorageshedforpensandruns,whilestillleavingenoughroomtomovearoundthecages.

Carlos’Method:“Imadeatable.IfIdon’thaveanycats,thenIhaveroomfor15dogs.IfIusesome

ofthespacefor4cats,thenIcanhave14dogs.With8cats,Ihaveroomfor13dogs.Foreach

additionaldogrunthatIdon’tbuy,Icanbuy4morecatpens.FrommytableIknowthey-intercept

ofmylineis15andtheslopeis− !!,somyequationis! = − !

! ! + 15.”

Clarita’sMethod:“Iletxrepresentthenumberofcats,andythenumberofdogs.Sincecatpens

require6ft2,6xrepresentsthespaceusedbycats.Sincedogrunsrequire24ft2,24yrepresentsthe

amountofspaceusedbydogs.Somyequationis6x+24y=360.”

1. Sincebothequationsrepresentthesameinformation,theymustbeequivalenttoeachother.

a. ShowthestepsyoucouldusetoturnClarita’sequationintoCarlos’equation.Explainwhyyoucandoeachstep.

b. ShowthestepsyoucouldusetoturnCarlos’equationintoClarita’s.Explainwhy

youcandoeachstep.

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SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.3

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

2. UsebothCarlos’andClarita’smethodstowritetheequationoftheboundarylineforthestart-upcostsconstraint.

3. ShowthestepsyoucouldusetoturnClarita’sstart-upcostsequationintoCarlos’equation.Explainwhyyoucandoeachstep.

4. ShowthestepsyoucouldusetoturnCarlos’start-upcostsequationintoClarita’s.Explainwhyyoucandoeachstep.

Inadditiontowritinganequationoftheboundarylines,CarlosandClaritaneedtograph

theirlinesonacoordinategrid.

Carlos’equationsarewritteninslope-interceptform.Clarita’sequationsarewrittenin

standardform.Bothformsarewaysofwritinglinearequations.

BothCarlosandClaritaknowtheyonlyneedtoplottwopointsinordertographaline.

5. Carlos’strategy:HowmightCarlosusehisslope-interceptform,! = − !! ! + 15,toplottwo

pointsonhisline?

6. Clarita’sstrategy:HowmightClaritauseherstandardform,6x+24y=360,toplottwopointsonherline?(Claritaisreallyclever,soshelooksforthetwoeasiestpointsshecanfind.)

• Start-upCosts:CarlosandClaritaplantoinvestmuchofthe$1280theyearnedfromtheirlastbusinessventuretopurchasecatpensanddogruns.Itwillcost$32foreachcatpenand$80foreachdogrun.

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