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A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered ...
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CollegeAlgebra
Module13:SystemsofEquationsandInequalities
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SystemsofLinearEquations:TwoVariables
LearningOutcomes
Solvesystemsofequationsbygraphing, substitution,and addition.
Identifyinconsistentsystemsofequationscontainingtwovariables.
Expressthesolutionofasystemofdependentequationscontainingtwovariablesusingstandardnotations.
Askateboardmanufacturerintroducesanewlineofboards.Themanufacturertracksitscosts,whichistheamountitspendstoproducetheboards,anditsrevenue,whichistheamountitearnsthroughsalesofitsboards.Howcanthecompanydetermineifitismakingaprofitwithitsnewline?Howmanyskateboardsmustbeproducedandsoldbeforeaprofitispossible?Inthissectionwewillconsiderlinearequationswithtwovariablestoanswertheseandsimilarquestions.
(credit:ThomasSørenes)
IntroductiontoSolutionsofSystems
Inordertoinvestigatesituationssuchasthatoftheskateboardmanufacturer,weneedtorecognizethatwearedealingwithmorethanonevariableandlikelymorethanoneequation.Asystemoflinearequationsconsistsoftwoormorelinearequationsmadeupoftwoormorevariablessuchthatallequationsinthesystemareconsideredsimultaneously.Tofindtheuniquesolutiontoasystemoflinearequations,wemustfindanumericalvalueforeachvariableinthesystemthatwillsatisfyallequationsinthesystematthesametime.Somelinearsystemsmaynothaveasolutionandothersmayhaveaninfinitenumberofsolutions.Inorderforalinearsystemtohaveauniquesolution,theremustbeatleastasmanyequationsastherearevariables.Evenso,thisdoesnotguaranteeauniquesolution.
Inthissection,wewilllookatsystemsoflinearequationsintwovariables,whichconsistoftwoequationsthatcontaintwodifferentvariables.Forexample,considerthefollowingsystemoflinearequationsintwovariables.
[latex]\begin{align}2x+y&=15\\[1mm]3x-y&=5\end{align}[/latex]
Thesolutiontoasystemoflinearequationsintwovariablesisanyorderedpairthatsatisfieseachequationindependently.Inthisexample,theorderedpair[latex](4,7)[/latex]isthesolutiontothesystemoflinearequations.Wecanverifythesolutionbysubstitutingthevaluesintoeachequationtoseeiftheorderedpairsatisfiesbothequations.Shortlywewillinvestigatemethodsoffindingsuchasolutionifitexists.
[latex]\begin{align}2\left(4\right)+\left(7\right)&=15&&\text{True}\\[1mm]3\left(4\right)-\left(7\right)&=5&&\text{True}\end{align}[/latex]
Inadditiontoconsideringthenumberofequationsandvariables,wecancategorizesystemsoflinearequationsbythenumberofsolutions.Aconsistentsystemofequationshasatleastonesolution.Aconsistentsystemisconsideredtobeanindependentsystemifithasasinglesolution,suchastheexamplewejustexplored.Thetwolineshavedifferentslopesandintersectatonepointintheplane.Aconsistentsystemisconsideredtobeadependentsystemiftheequationshavethesameslopeandthesamey-intercepts.Inotherwords,thelinescoincidesotheequationsrepresentthesameline.Everypointonthelinerepresentsacoordinatepairthatsatisfiesthesystem.Thus,thereareaninfinitenumberofsolutions.
Anothertypeofsystemoflinearequationsisaninconsistentsystem,whichisoneinwhichtheequationsrepresenttwoparallellines.Thelineshavethesameslopeanddifferenty-intercepts.Therearenopointscommontobothlines;hence,thereisnosolutiontothesystem.
AGeneralNote:TypesofLinearSystems
Therearethreetypesofsystemsoflinearequationsintwovariables,andthreetypesofsolutions.
Anindependentsystemhasexactlyonesolutionpair[latex]\left(x,y\right)[/latex].Thepointwherethetwolinesintersectistheonlysolution.
Aninconsistentsystemhasnosolution.Noticethatthetwolinesareparallelandwillneverintersect.
Adependentsystemhasinfinitelymanysolutions.Thelinesarecoincident.Theyarethesameline,soeverycoordinatepaironthelineisasolutiontobothequations.
Belowisacomparisonof graphicalrepresentationsofeachtypeofsystem.
HowTo:Givenasystemoflinearequationsandanorderedpair,determinewhethertheorderedpairisasolution.
Substitutetheorderedpairintoeachequationinthesystem.
Determinewhethertruestatementsresultfromthesubstitutioninbothequations;ifso,theorderedpairisasolution.
Example:DeterminingWhetheranOrderedPairIsaSolutiontoaSystemofEquations
Determinewhethertheorderedpair[latex]\left(5,1\right)[/latex]isasolutiontothegivensystemofequations.
[latex]\begin{align}x+3y&=8\\2x-9&=y\end{align}[/latex]
ShowSolution
Substitutetheorderedpair[latex]\left(5,1\right)[/latex]intobothequations.
[latex]\begin{align}\left(5\right)+3\left(1\right)&=8\\[1mm]8&=8&&\text{True}\\[3mm]2\left(5\right)-9&=\left(1\right)\\[1mm]1&=1&&\text{True}\end{align}[/latex]
Theorderedpair[latex]\left(5,1\right)[/latex]satisfiesbothequations,soitisthesolutiontothesystem.
AnalysisoftheSolution
Wecanseethesolutionclearlybyplottingthegraphofeachequation.Sincethesolutionisanorderedpairthatsatisfiesbothequations,itisapointonbothofthelinesandthusthepointofintersectionofthetwolines.
TryIt
Determinewhethertheorderedpair[latex]\left(8,5\right)[/latex]isasolutiontothefollowingsystem.
[latex]\begin{align}5x-4y&=20\\2x+1&=3y\end{align}[/latex]
ShowSolution
Notasolution.
SolvingSystemsofEquationsbyGraphing
Therearemultiplemethodsofsolvingsystemsoflinearequations.Forasystemoflinearequationsintwovariables,wecandetermineboththetypeofsystemandthesolutionbygraphingthesystemofequationsonthesamesetofaxes.
Example:SolvingaSystemofEquationsinTwoVariablesbyGraphing
Solvethefollowingsystemofequationsbygraphing.Identifythetypeofsystem.
[latex]\begin{align}2x+y&=-8\\x-y&=-1\end{align}[/latex]
ShowSolution
Solvethefirstequationfor[latex]y[/latex].
[latex]\begin{align}2x+y&=-8\\y&=-2x-8\end{align}[/latex]
Solvethesecondequationfor[latex]y[/latex].
[latex]\begin{align}x-y&=-1\\y&=x+1\end{align}[/latex]
Graphbothequationsonthesamesetofaxes:
Thelinesappeartointersectatthepoint[latex]\left(-3,-2\right)[/latex].Wecanchecktomakesurethatthisisthesolutiontothesystembysubstitutingtheorderedpairintobothequations.
[latex]\begin{align}2\left(-3\right)+\left(-2\right)&=-8\\[1mm]-8=-8&&\text{True}\\[3mm]\left(-3\right)-\left(-2\right)&=-1\\[1mm]-1&=-1&&\text{True}\end{align}[/latex]
Thesolutiontothesystemistheorderedpair[latex]\left(-3,-2\right)[/latex],sothesystemisindependent.
TryIt
Solvethefollowingsystemofequationsbygraphing.
[latex]\begin{gathered}2x-5y=-25\\-4x+5y=35\end{gathered}[/latex]
ShowSolution
Thesolutiontothesystemistheorderedpair[latex]\left(-5,3\right)[/latex].
Q&A
Cangraphingbeusedifthesystemisinconsistentordependent?
Yes,inbothcaseswecanstillgraphthesystemtodeterminethetypeofsystemandsolution.Ifthetwolinesareparallel,thesystemhasnosolutionandisinconsistent.Ifthetwolinesareidentical,thesystemhasinfinitesolutionsandisadependentsystem.
Tryit
Plotthethreedifferentsystemswithanonlinegraphingtool.Categorize eachsolutionaseitherconsistentorinconsistent.Ifthesystemisconsistentdeterminewhetheritisdependentorindependent.Youmayfinditeasiertoploteachsystemindividually,thenclearoutyourentriesbeforeyouplotthenext.
1)
[latex]5x-3y=-19[/latex]
[latex]x=2y-1[/latex]
2)
[latex]4x+y=11[/latex]
[latex]-2y=-25+8x[/latex]
3)
[latex]y=-3x+6[/latex]
[latex]-\frac{1}{3}y+2=x[/latex]
ShowSolution
Onesolution–consistent,independent
Nosolutions,inconsistent,neitherdependentnorindependent
Manysolutions– consistent,dependent
SolvingSystemsofEquationsbySubstitution
Solvingalinearsystemintwovariablesbygraphingworkswellwhenthesolutionconsistsofintegervalues,butifoursolutioncontainsdecimalsorfractions,itisnotthemostprecisemethod.Wewillconsidertwomoremethodsofsolvingasystemoflinearequationsthataremoreprecisethangraphing.Onesuchmethodissolvingasystemofequationsbythesubstitutionmethod,inwhichwesolveoneoftheequationsforonevariableandthensubstitutetheresultintothesecondequationtosolveforthesecondvariable.Recallthatwecansolveforonlyonevariableatatime,whichisthereasonthesubstitutionmethodisbothvaluableandpractical.
HowTo:Givenasystemoftwoequationsintwovariables,solveusingthesubstitutionmethod.
Solveoneofthetwoequationsforoneofthevariablesintermsoftheother.
Substitutetheexpressionforthisvariableintothesecondequation,thensolvefortheremainingvariable.
Substitutethatsolutionintoeitheroftheoriginalequationstofindthevalueofthefirstvariable.Ifpossible,writethesolutionasanorderedpair.
Checkthesolutioninbothequations.
Example:SolvingaSystemofEquationsinTwoVariablesbySubstitution
Solvethefollowingsystemofequationsbysubstitution.
[latex]\begin{align}-x+y&=-5\\2x-5y&=1\end{align}[/latex]
ShowSolution
First,wewillsolvethefirstequationfor[latex]y[/latex].
[latex]\begin{align}-x+y&=-5\\y&=x-5\end{align}[/latex]
Nowwecansubstitutetheexpression[latex]x-5[/latex]for[latex]y[/latex]inthesecondequation.
[latex]\begin{align}2x-5y&=1\\2x-5\left(x-5\right)&=1\\2x-5x+25&=1\\-3x&=-24\\x&=8\end{align}[/latex]
Now,wesubstitute[latex]x=8[/latex]intothefirstequationandsolvefor[latex]y[/latex].
[latex]\begin{align}-\left(8\right)+y&=-5\\y&=3\end{align}[/latex]
Oursolutionis[latex]\left(8,3\right)[/latex].
Checkthesolutionbysubstituting[latex]\left(8,3\right)[/latex]intobothequations.
[latex]\begin{align}-x+y&=-5\\-\left(8\right)+\left(3\right)&=-5&&\text{True}\\[3mm]2x-5y&=1\\2\left(8\right)-5\left(3\right)&=1&&\text{True}\end{align}[/latex]
TryIt
Solvethefollowingsystemofequationsbysubstitution.
[latex]\begin{align}x&=y+3\\4&=3x-2y\end{align}[/latex]
ShowSolution
[latex]\left(-2,-5\right)[/latex]
Q&A
Canthesubstitutionmethodbeusedtosolveanylinearsystemintwovariables?
Yes,butthemethodworksbestifoneoftheequationscontainsacoefficientof1or–1sothatwedonothavetodealwithfractions.
Thefollowingvideois~10minuteslongandprovidesamini-lessononusingthesubstitutionmethodtosolveasystemoflinearequations. Wepresentthreedifferentexamples,andalsouseagraphingtooltohelpsummarizethesolutionforeachexample.
SolvingSystemsofEquationsinTwoVariablesbytheAdditionMethod
Athirdmethodofsolvingsystemsoflinearequationsistheadditionmethod, thismethodisalsocalledthe eliminationmethod.Inthismethod,weaddtwotermswiththesamevariable,butoppositecoefficients,sothatthesumiszero.Ofcourse,notallsystemsaresetupwiththetwotermsofonevariablehavingoppositecoefficients.Oftenwemustadjustoneorbothoftheequationsbymultiplicationsothatonevariablewillbeeliminatedbyaddition.
HowTo:Givenasystemofequations,solveusingtheadditionmethod.
Writebothequationswithx–andy-variablesontheleftsideoftheequalsignandconstantsontheright.
Writeoneequationabovetheother,liningupcorrespondingvariables.Ifoneofthevariablesinthetopequationhastheoppositecoefficientofthesamevariableinthebottomequation,addtheequationstogether,eliminatingonevariable.Ifnot,usemultiplicationbyanonzeronumbersothatoneofthevariablesinthetopequationhastheoppositecoefficientofthesamevariableinthebottomequation,thenaddtheequationstoeliminatethevariable.
Solvetheresultingequationfortheremainingvariable.
Substitutethatvalueintooneoftheoriginalequationsandsolveforthesecondvariable.
Checkthesolutionbysubstitutingthevaluesintotheotherequation.
Example:SolvingaSystembytheAdditionMethod
Solvethegivensystemofequationsbyaddition.
[latex]\begin{align}x+2y&=-1\\-x+y&=3\end{align}[/latex]
ShowSolution
Bothequationsarealreadysetequaltoaconstant.Noticethatthecoefficientof[latex]x[/latex]inthesecondequation,–1,istheoppositeofthecoefficientof[latex]x[/latex]inthefirstequation,1.Wecanaddthetwoequationstoeliminate[latex]x[/latex]withoutneedingtomultiplybyaconstant.
[latex]\begin{align}x+2y&=-1\\-x+y&=3\\\hline3y&=2\end{align}[/latex]
Nowthatwehaveeliminated[latex]x[/latex],wecansolvetheresultingequationfor[latex]y[/latex].
[latex]\begin{align}3y&=2\\y&=\dfrac{2}{3}\end{align}[/latex]
Then,wesubstitutethisvaluefor[latex]y[/latex]intooneoftheoriginalequationsandsolvefor[latex]x[/latex].
[latex]\begin{align}-x+y&=3\\-x+\frac{2}{3}&=3\\-x&=3-\frac{2}{3}\\-x&=\frac{7}{3}\\x&=-\frac{7}{3}\end{align}[/latex]
Thesolutiontothissystemis[latex]\left(-\frac{7}{3},\frac{2}{3}\right)[/latex].
Checkthesolutioninthefirstequation.
[latex]\begin{align}x+2y&=-1\\\left(-\frac{7}{3}\right)+2\left(\frac{2}{3}\right)&=\\-\frac{7}{3}+\frac{4}{3}&=\\\-\frac{3}{3}&=\\-1&=-1&&\text{True}\end{align}[/latex]
AnalysisoftheSolution
Wegainanimportantperspectiveonsystemsofequationsbylookingatthegraphicalrepresentation.Seethegraphbelow tofindthattheequationsintersectatthesolution.Wedonotneedtoaskwhethertheremaybeasecondsolutionbecauseobservingthegraphconfirmsthatthesystemhasexactlyonesolution.
TryIT
Example:UsingtheAdditionMethodWhenMultiplicationofOneEquationIsRequired
Solvethegivensystemofequationsbytheadditionmethod.
[latex]\begin{align}3x+5y&=-11\\x-2y&=11\end{align}[/latex]
ShowSolution
Addingtheseequationsaspresentedwillnoteliminateavariable.However,weseethatthefirstequationhas[latex]3x[/latex]initandthesecondequationhas[latex]x[/latex].Soifwemultiplythesecondequationby[latex]-3,\text{}[/latex]thex-termswilladdtozero.
[latex]\begin{align}x-2y&=11\\-3\left(x-2y\right)&=-3\left(11\right)&&\text{Multiplybothsidesby}-3\\-3x+6y&=-33&&\text{Usethedistributiveproperty}.\end{align}[/latex]
Now,let’saddthem.
[latex]\begin{align}3x+5y&=−11\\−3x+6y&=−33\\\hline11y&=−44\\y&=−4\end{align}[/latex]
Forthelaststep,wesubstitute[latex]y=-4[/latex]intooneoftheoriginalequationsandsolvefor[latex]x[/latex].
[latex]\begin{align}3x+5y&=-11\\3x+5\left(-4\right)&=-11\\3x-20&=-11\\3x&=9\\x&=3\end{align}[/latex]
Oursolutionistheorderedpair[latex]\left(3,-4\right)[/latex].Checkthesolutionintheoriginalsecondequation.
[latex]\begin{align}x-2y&=11\\\left(3\right)-2\left(-4\right)&=3+8\\&=11&&\text{True}\end{align}[/latex]
TryIt
Solvethesystemofequationsbyaddition.
[latex]\begin{align}2x-7y&=2\\3x+y&=-20\end{align}[/latex]
ShowSolution
[latex]\left(-6,-2\right)[/latex]
Example:UsingtheAdditionMethodWhenMultiplicationofBothEquationsIsRequired
Solvethegivensystemofequationsintwovariablesbyaddition.
[latex]\begin{align}2x+3y&=-16\\5x-10y&=30\end{align}[/latex]
ShowSolution
Oneequationhas[latex]2x[/latex]andtheotherhas[latex]5x[/latex].Theleastcommonmultipleis[latex]10x[/latex]sowewillhavetomultiplybothequationsbyaconstantinordertoeliminateonevariable.Let’seliminate[latex]x[/latex]bymultiplyingthefirstequationby[latex]-5[/latex]andthesecondequationby[latex]2[/latex].
[latex]\begin{align}-5\left(2x+3y\right)&=-5\left(-16\right)\\-10x-15y&=80\\[3mm]2\left(5x-10y\right)&=2\left(30\right)\\10x-20y&=60\end{align}[/latex]
Then,weaddthetwoequationstogether.
[latex]\begin{align}−10x−15y&=80\\10x−20y&=60\\\hline−35y&=140\\y&=−4\end{align}[/latex]
Substitute[latex]y=-4[/latex]intotheoriginalfirstequation.
[latex]\begin{align}2x+3\left(-4\right)&=-16\\2x-12&=-16\\2x&=-4\\x&=-2\end{align}[/latex]
Thesolutionis[latex]\left(-2,-4\right)[/latex].Checkitintheotherequation.
[latex]\begin{align}5x-10y&=30\\5\left(-2\right)-10\left(-4\right)&=30\\-10+40&=30\\30&=30\end{align}[/latex]
Example:UsingtheAdditionMethodinSystemsofEquationsContainingFractions
Solvethegivensystemofequationsintwovariablesbyaddition.
[latex]\begin{align}\frac{x}{3}+\frac{y}{6}&=3\\[1mm]\frac{x}{2}-\frac{y}{4}&=1\end{align}[/latex]
ShowSolution
Firstcleareachequationoffractionsbymultiplyingbothsidesoftheequationbytheleastcommondenominator.
[latex]\begin{align}6\left(\frac{x}{3}+\frac{y}{6}\right)&=6\left(3\right)\\[1mm]2x+y&=18\\[3mm]4\left(\frac{x}{2}-\frac{y}{4}\right)&=4\left(1\right)\\[1mm]2x-y&=4\end{align}[/latex]
Nowmultiplythesecondequationby[latex]-1[/latex]sothatwecaneliminate x.
[latex]\begin{align}-1\left(2x-y\right)&=-1\left(4\right)\\[1mm]-2x+y&=-4\end{align}[/latex]
Addthetwoequationstoeliminate x andsolvetheresultingequationfory.
[latex]\begin{align}2x+y&=18\\−2x+y&=−4\\\hline2y&=14\\y&=7\end{align}[/latex]
Substitute[latex]y=7[/latex]intothefirstequation.
[latex]\begin{align}2x+\left(7\right)&=18\\2x&=11\\x&=\frac{11}{2}\\&=7.5\end{align}[/latex]
Thesolutionis[latex]\left(\frac{11}{2},7\right)[/latex].Checkitintheotherequation.
[latex]\begin{align}\frac{x}{2}-\frac{y}{4}&=1\\[1mm]\frac{\frac{11}{2}}{2}-\frac{7}{4}&=1\\[1mm]\frac{11}{4}-\frac{7}{4}&=1\\[1mm]\frac{4}{4}&=1\end{align}[/latex]
TryIt
Solvethesystemofequationsbyaddition.
[latex]\begin{align}2x+3y&=8\\3x+5y&=10\end{align}[/latex]
ShowSolution
[latex]\left(10,-4\right)[/latex]
inthefollowingvideowepresentmoreexamplesofhowtousetheaddition(elimination)methodtosolveasystemoftwolinearequations.
ClassifySolutionstoSystems
Nowthatwehaveseveralmethodsforsolvingsystemsofequations,wecanusethemethodstoidentifyinconsistentsystems.Recallthataninconsistentsystemconsistsofparallellinesthathavethesameslopebutdifferent[latex]y[/latex]-intercepts.Theywillneverintersect.Whensearchingforasolutiontoaninconsistentsystem,wewillcomeupwithafalsestatement,suchas[latex]12=0[/latex].
Example:SolvinganInconsistentSystemofEquations
Solvethefollowingsystemofequations.
[latex]\begin{gathered}&x=9-2y\\&x+2y=13\end{gathered}[/latex]
ShowSolution
Wecanapproachthisproblemintwoways.Becauseoneequationisalreadysolvedfor[latex]x[/latex],themostobviousstepistousesubstitution.
[latex]\begin{align}x+2y&=13\\\left(9-2y\right)+2y&=13\\9+0y&=13\\9&=13\end{align}[/latex]
Clearly,thisstatementisacontradictionbecause[latex]9\ne13[/latex].Therefore,thesystemhasnosolution.
Thesecondapproachwouldbetofirstmanipulatetheequationssothattheyarebothinslope-interceptform.Wemanipulatethefirstequationasfollows.
[latex]\begin{gathered}x=9-2y\\2y=-x+9\\y=-\frac{1}{2}x+\frac{9}{2}\end{gathered}[/latex]
Wethenconvertthesecondequationexpressedtoslope-interceptform.
[latex]\begin{gathered}x+2y=13\\2y=-x+13\\y=-\frac{1}{2}x+\frac{13}{2}\end{gathered}[/latex]
Comparingtheequations,weseethattheyhavethesameslopebutdifferenty-intercepts.Therefore,thelinesareparallelanddonotintersect.
[latex]\begin{gathered}y=-\frac{1}{2}x+\frac{9}{2}\\y=-\frac{1}{2}x+\frac{13}{2}\end{gathered}[/latex]
AnalysisoftheSolution
Writingtheequationsinslope-interceptformconfirmsthatthesystemisinconsistentbecausealllineswillintersecteventuallyunlesstheyareparallel.Parallellineswillneverintersect;thus,thetwolineshavenopointsincommon.Thegraphsoftheequationsinthisexampleareshownbelow.
TryIt
Solvethefollowingsystemofequationsintwovariables.
[latex]\begin{gathered}2y-2x=2\\2y-2x=6\end{gathered}[/latex]
ShowSolution
Nosolution.Itisaninconsistentsystem.
ExpressingtheSolutionofaSystemofDependentEquationsContainingTwoVariables
Recallthatadependentsystemofequationsintwovariablesisasysteminwhichthetwoequationsrepresentthesameline.Dependentsystemshaveaninfinitenumberofsolutionsbecauseallofthepointsononelinearealsoontheotherline.Afterusingsubstitutionoraddition,theresultingequationwillbeanidentity,suchas[latex]0=0[/latex].
Example:FindingaSolutiontoaDependentSystemofLinearEquations
Findasolutiontothesystemofequationsusingtheadditionmethod.
[latex]\begin{gathered}x+3y=2\\3x+9y=6\end{gathered}[/latex]
ShowSolution
Withtheadditionmethod,wewanttoeliminateoneofthevariablesbyaddingtheequations.Inthiscase,let’sfocusoneliminating[latex]x[/latex].Ifwemultiplybothsidesofthefirstequationby[latex]-3[/latex],thenwewillbeabletoeliminatethe[latex]x[/latex]-variable.
[latex]\begin{align}x+3y&=2\\\left(-3\right)\left(x+3y\right)&=\left(-3\right)\left(2\right)\\-3x-9y&=-6\end{align}[/latex]
Nowaddtheequations.
[latex]\begin{align}−3x−9y&=−6\\+3x+9y&=6\\\hline0&=0\end{align}[/latex]
Wecanseethattherewillbeaninfinitenumberofsolutionsthatsatisfybothequations.
AnalysisoftheSolution
Ifwerewrotebothequationsintheslope-interceptform,wemightknowwhatthesolutionwouldlooklikebeforeadding.Let’slookatwhathappenswhenweconvertthesystemtoslope-interceptform.
[latex]\begin{align}\begin{gathered}x+3y=2\\3y=-x+2\\y=-\frac{1}{3}x+\frac{2}{3}\end{gathered}\hspace{2cm}\begin{gathered}3x+9y=6\\9y=-3x+6\\y=-\frac{3}{9}x+\frac{6}{9}\\y=-\frac{1}{3}x+\frac{2}{3}\end{gathered}\end{align}[/latex]
Lookatthegraphbelow.Noticetheresultsarethesame.Thegeneralsolutiontothesystemis[latex]\left(x,-\frac{1}{3}x+\frac{2}{3}\right)[/latex].
Writingthegeneralsolution
Inthepreviousexample,wepresentedananalysisofthesolutiontothefollowingsystemofequations:
[latex]\begin{gathered}x+3y=2\\3x+9y=6\end{gathered}[/latex]
Afteralittlealgebra,wefoundthatthesetwoequationswereexactlythesame.Wethenwrotethegeneralsolutionas [latex]\left(x,-\frac{1}{3}x+\frac{2}{3}\right)[/latex].Whywouldwewritethesolutionthisway?Insomeways,thisrepresentationtellsusalot. Ittellsusthatxcanbeanything,xisx. Italsotellsusthatyisgoingtodependonx,justlikewhenwewriteafunctionrule. Inthiscase,dependingonwhatyouputinforx,ywillbedefinedintermsofxas[latex]-\frac{1}{3}x+\frac{2}{3}[/latex].
Inotherwords,thereareinfinitelymany(x,y)pairsthatwillsatisfythissystemofequations,andtheyallfallontheline [latex]f(x)-\frac{1}{3}x+\frac{2}{3}[/latex].
TryIt
Solvethefollowingsystemofequationsintwovariables.
[latex]\begin{gathered}y-2x=5\\-3y+6x=-15\end{gathered}[/latex]
ShowSolution
Thesystemisdependentsothereareinfinitelymanysolutionsoftheform[latex]\left(x,2x+5\right)[/latex].
UsingSystemsofEquationstoInvestigateProfits
Usingwhatwehavelearnedaboutsystemsofequations,wecanreturntotheskateboardmanufacturingproblematthebeginningofthesection.Theskateboardmanufacturer’srevenuefunctionisthefunctionusedtocalculatetheamountofmoneythatcomesintothebusiness.Itcanberepresentedbytheequation[latex]R=xp[/latex],where[latex]x=[/latex]quantityand[latex]p=[/latex]price.Therevenuefunctionisshowninorangeinthegraphbelow.
Thecostfunctionisthefunctionusedtocalculatethecostsofdoingbusiness.Itincludesfixedcosts,suchasrentandsalaries,andvariablecosts,suchasutilities.Thecostfunctionisshowninblueinthegraphbelow.Thex-axisrepresentsquantityinhundredsofunits.They-axisrepresentseithercostorrevenueinhundredsofdollars.
Thepointatwhichthetwolinesintersectiscalledthebreak-evenpoint.Wecanseefromthegraphthatif700unitsareproduced,thecostis$3,300andtherevenueisalso$3,300.Inotherwords,thecompanybreakseveniftheyproduceandsell700units.Theyneithermakemoneynorlosemoney.
Theshadedregiontotherightofthebreak-evenpointrepresentsquantitiesforwhichthecompanymakesaprofit.Theshadedregiontotheleftrepresentsquantitiesforwhichthecompanysuffersaloss.Theprofitfunctionistherevenuefunctionminusthecostfunction,writtenas[latex]P\left(x\right)=R\left(x\right)-C\left(x\right)[/latex].Clearly,knowingthequantityforwhichthecostequalstherevenueisofgreatimportancetobusinesses.
Example:FindingtheBreak-EvenPointandtheProfitFunctionUsingSubstitution
Giventhecostfunction[latex]C\left(x\right)=0.85x+35{,}000[/latex]andtherevenuefunction[latex]R\left(x\right)=1.55x[/latex],findthebreak-evenpointandtheprofitfunction.
ShowSolution
Writethesystemofequationsusing[latex]y[/latex]toreplacefunctionnotation.
[latex]\begin{align}y&=0.85x+35{,}000\\y&=1.55x\end{align}[/latex]
Substitutetheexpression[latex]0.85x+35{,}000[/latex]fromthefirstequationintothesecondequationandsolvefor[latex]x[/latex].
[latex]\begin{gathered}0.85x+35{,}000=1.55x\\35{,}000=0.7x\\50{,}000=x\end{gathered}[/latex]
Then,wesubstitute[latex]x=50{,}000[/latex]intoeitherthecostfunctionortherevenuefunction.
[latex]1.55\left(50{,}000\right)=77{,}500[/latex]
Thebreak-evenpointis[latex]\left(50{,}000,77{,}500\right)[/latex].
Theprofitfunctionisfoundusingtheformula[latex]P\left(x\right)=R\left(x\right)-C\left(x\right)[/latex].
[latex]\begin{align}P\left(x\right)&=1.55x-\left(0.85x+35{,}000\right)\\&=0.7x-35{,}000\end{align}[/latex]
Theprofitfunctionis[latex]P\left(x\right)=0.7x-35{,}000[/latex].
AnalysisoftheSolution
Thecosttoproduce50,000unitsis$77,500,andtherevenuefromthesalesof50,000unitsisalso$77,500.Tomakeaprofit,thebusinessmustproduceandsellmorethan50,000units.
Weseefromthegraphbelowthattheprofitfunctionhasanegativevalueuntil[latex]x=50{,}000[/latex],whenthegraphcrossesthex-axis.Then,thegraphemergesintopositivey-valuesandcontinuesonthispathastheprofitfunctionisastraightline.Thisillustratesthatthebreak-evenpointforbusinessesoccurswhentheprofitfunctionis0.Theareatotheleftofthebreak-evenpointrepresentsoperatingataloss.
WritingaSystemofLinearEquationsGivenaSituation
Itisraretobegivenequationsthatneatlymodelbehaviorsthatyouencounterinbusiness,rather,youwillprobablybefacedwithasituationforwhichyouknowkeyinformationasintheexampleabove.Below,wesummarizethreekeyfactorsthatwillhelpguideyouintranslatingasituationintoasystem.
HowTo:Givenasituationthatrepresentsasystemoflinearequations,writethesystemofequationsandidentifythesolution.
Identifytheinputandoutputofeachlinearmodel.
Identifytheslopeandy-interceptofeachlinearmodel.
Findthesolutionbysettingthetwolinearfunctionsequaltoanotherandsolvingforx,orfindthepointofintersectiononagraph.
Nowlet’spracticeputtingthesekeyfactorstowork.Inthenextexample,wedeterminehowmanydifferenttypesofticketsaresoldgiveninformationaboutthetotalrevenueandamountofticketssoldtoanevent.
Example:WritingandSolvingaSystemofEquationsinTwoVariables
Thecostofatickettothecircusis$25.00forchildrenand$50.00foradults.Onacertainday,attendanceatthecircusis2,000andthetotalgaterevenueis$70,000.Howmanychildrenandhowmanyadultsboughttickets?
ShowSolution
Letc=thenumberofchildrenanda=thenumberofadultsinattendance.
Thetotalnumberofpeopleis2,000.Wecanusethistowriteanequationforthenumberofpeopleatthecircusthatday.
[latex]c+a=2{,}000[/latex]
Therevenuefromallchildrencanbefoundbymultiplying$25.00bythenumberofchildren,[latex]25c[/latex].Therevenuefromalladultscanbefoundbymultiplying$50.00bythenumberofadults,[latex]50a[/latex].Thetotalrevenueis$70,000.Wecanusethistowriteanequationfortherevenue.
[latex]25c+50a=70{,}000[/latex]
Wenowhaveasystemoflinearequationsintwovariables.
[latex]\begin{gathered}c+a=2,000\\25c+50a=70{,}000\end{gathered}[/latex]
Inthefirstequation,thecoefficientofbothvariablesis1.Wecanquicklysolvethefirstequationforeither[latex]c[/latex]or[latex]a[/latex].Wewillsolvefor[latex]a[/latex].
[latex]\begin{gathered}c+a=2{,}000\\a=2{,}000-c\end{gathered}[/latex]
Substitutetheexpression[latex]2{,}000-c[/latex]inthesecondequationfor[latex]a[/latex]andsolvefor[latex]c[/latex].
[latex]\begin{align}25c+50\left(2{,}000-c\right)&=70{,}000\\25c+100{,}000-50c&=70{,}000\\-25c&=-30{,}000\\c&=1{,}200\end{align}[/latex]
Substitute[latex]c=1{,}200[/latex]intothefirstequationtosolvefor[latex]a[/latex].
[latex]\begin{align}1{,}200+a&=2{,}000\\a&=800\end{align}[/latex]
Wefindthat1,200childrenand800adultsboughtticketstothecircusthatday.
TryIt
Mealticketsatthecircuscost$4.00forchildrenand$12.00foradults.If1,650mealticketswereboughtforatotalof$14,200,howmanychildrenandhowmanyadultsboughtmealtickets?
ShowSolution
700children,950adults
Sometimes,asystemofequationscaninformadecision. Inournextexample,wehelpanswerthequestion,“Whichtruckrentalcompanywillgivethebestvalue?”
Example:BuildingaSystemofLinearModelstoChooseaTruckRentalCompany
Jamalischoosingbetweentwotruck-rentalcompanies.Thefirst,KeeponTrucking,Inc.,chargesanup-frontfeeof$20,then59centsamile.Thesecond,MoveItYourWay,chargesanup-frontfeeof$16,then63centsamile.[1]WhenwillKeeponTrucking,Inc.bethebetterchoiceforJamal?
ShowSolution
Thetwoimportantquantitiesinthisproblemarethecostandthenumberofmilesdriven.Becausewehavetwocompaniestoconsider,wewilldefinetwofunctions.
Input
d,distancedriveninmiles
Outputs
K(d):cost,indollars,forrentingfromKeeponTruckingM(d)cost,indollars,forrentingfromMoveItYourWay
InitialValue
Up-frontfee:K(0)=20andM(0)=16
RateofChange
K(d)=$0.59/mileandP(d)=$0.63/mile
Alinearfunctionisoftheform[latex]f\left(x\right)=mx+b[/latex].Usingtheratesofchangeandinitialcharges,wecanwritetheequations
[latex]\begin{align}K\left(d\right)=0.59d+20\\M\left(d\right)=0.63d+16\end{align}[/latex]
Usingtheseequations,wecandeterminewhenKeeponTrucking,Inc.,willbethebetterchoice.Becauseallwehavetomakethatdecisionfromisthecosts,wearelookingforwhenMoveItYourWay,willcostless,orwhen[latex]K\left(d\right)
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