NAME DATE SECTION

Day11LinearEquations,Inequalities&Systems

SystemsofLinearEquationsandTheirSolutions

ACuriousSystem

Andreistryingtosolvethissystemofequations: 𝑥 + 𝑦 = 34𝑥 = 12− 4𝑦

Lookingatthefirstequation,hethought,"Thesolutiontothesystemisapairofnumbersthataddupto3.Iwonderwhichtwonumberstheyare."

1. Chooseanytwonumbersthataddupto3.Letthefirstonebethe𝑥-valueandthesecondonebethe𝑦-value.

2. Thepairofvaluesyouchoseisasolutiontothefirstequation.Checkifitisalsoa

solutiontothesecondequation.3. Howmanysolutionsdoesthesystemhave?Usewhatyouknowaboutequationsor

aboutsolvingsystemstoshowthatyouareright.

What'stheDeal?

SalineRecCenterisofferingspecialpricesonitspoolpassesandgymmembershipsforthesummer.Onthefirstdayoftheoffering,afamilypaid$96for4poolpassesand2gymmemberships.Laterthatday,anindividualboughtapoolpassforherself,apoolpassforafriend,and1gymmembership.Shepaid$72.

1. Writeasystemofequationsthatrepresentstherelationshipsbetweenpoolpasses,gymmemberships,andthecosts.Besuretostatewhateachvariablerepresents.

2. Findthepriceofapoolpassandthepriceofagymmembershipbysolvingthesystem

algebraically.Explainorshowyourreasoning.

3. UseDesmostographtheequationsinthesystem.Sketchthegraphsbelow.Make1-2observationsaboutyourgraphs.

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Nowtrythisone!EnterClassCode:RU647Z

Day11Summary

Wehaveseenmanyexamplesofasystemwhereonepairofvaluessatisfiesbothequations.Notallsystems,however,haveonesolution.Somesystemshavemanysolutions,andothershavenosolutions.

Let'slookatthreesystemsofequationsandtheirgraphs.

System1: 3𝑥 + 4𝑦 = 83𝑥 − 4𝑦 = 8

ThegraphsoftheequationsinSystem1intersectatonepoint.Thecoordinatesofthepointaretheonepairofvaluesthataresimultaneouslytrueforbothequations.Whenwesolvetheequations,wegetexactlyonesolution.

System2: 3𝑥 + 4𝑦 = 86𝑥 + 8𝑦 = 16

ThegraphsoftheequationsinSystem2appeartobethesameline.Thissuggeststhateverypointonthelineisasolutiontobothequations,orthatthesystemhasinfinitelymanysolutions.

System3: 3𝑥 + 4𝑦 = 83𝑥 + 4𝑦 = −4

ThegraphsoftheequationsinSystem3appeartobeparallel.Ifthelinesneverintersect,thenthereisnocommonpointthatisasolutiontobothequationsandthesystemhasnosolutions.

Howcanwetell,withoutgraphing,thatSystem2indeedhasmanysolutions?

• Noticethat3𝑥 + 4𝑦 = 8and6𝑥 + 8𝑦 = 16areequivalentequations.Multiplyingthefirstequationby2givesthesecondequation.Multiplyingthesecondequationby!

!

givesthefirstequation.Thismeansthatanysolutiontothefirstequationisasolutiontothesecond.

• Rearranging3𝑥 + 4𝑦 = 8intoslope-interceptformgives𝑦 = !!!!!,or𝑦 = 2− !

!𝑥.

Rearranging6𝑥 + 8𝑦 = 16gives𝑦 = !"!!!!,whichisalso𝑦 = 2− !

!𝑥.Bothlineshave

thesameslopeandthesame𝑦-valuefortheverticalintercept!

Howcanwetell,withoutgraphing,thatSystem3hasnosolutions?

• Noticethatinoneequation3𝑥 + 4𝑦equals8,butintheotherequationitequals-4.Becauseitisimpossibleforthesameexpressiontoequal8and-4,theremustnotbeapairof𝑥-and𝑦-valuesthataresimultaneouslytrueforbothequations.Thistellsusthatthesystemhasnosolutions.

• Rearrangingeachequationintoslope-interceptformgives𝑦 = 2− !!𝑥and𝑦 = −1−

!!𝑥.Thetwographshavethesameslopebutthe𝑦-valuesoftheirverticalinterceptsaredifferent.Thistellsusthatthelinesareparallelandwillnevercross.