Day 11 Linear Equations, Inequalities & Systems
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NAME DATE SECTION Day 11 Linear Equations, Inequalities & Systems Systems of Linear Equations and Their Solutions A Curious System Andre is trying to solve this system of equations: + = 3 4 = 12 − 4 Looking at the first equation, he thought, "The solution to the system is a pair of numbers that add up to 3. I wonder which two numbers they are." 1. Choose any two numbers that add up to 3. Let the first one be the -value and the second one be the -value. 2. The pair of values you chose is a solution to the first equation. Check if it is also a solution to the second equation. 3. How many solutions does the system have? Use what you know about equations or about solving systems to show that you are right.
Transcript of Day 11 Linear Equations, Inequalities & Systems
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.
CardSort:SortingSystemsGotostudent.desmos.comEnterClassCode:27267G
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.
