Day 11 Linear Equations, Inequalities & Systems
Transcript of Day 11 Linear Equations, Inequalities & Systems
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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.
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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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CardSort:SortingSystemsGotostudent.desmos.comEnterClassCode:27267G
Nowtrythisone!EnterClassCode:RU647Z
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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.
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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!
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Howcanwetell,withoutgraphing,thatSystem3hasnosolutions?
• Noticethatinoneequation3𝑥 + 4𝑦equals8,butintheotherequationitequals-4.Becauseitisimpossibleforthesameexpressiontoequal8and-4,theremustnotbeapairof𝑥-and𝑦-valuesthataresimultaneouslytrueforbothequations.Thistellsusthatthesystemhasnosolutions.
• Rearrangingeachequationintoslope-interceptformgives𝑦 = 2− !!𝑥and𝑦 = −1−
!!𝑥.Thetwographshavethesameslopebutthe𝑦-valuesoftheirverticalinterceptsaredifferent.Thistellsusthatthelinesareparallelandwillnevercross.