11.1: Systems of Linear Equations - Two Variables

A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system ... Skiptomaincontent LearningObjectivesIntroductiontoSystemsofEquationsTYPESOFLINEARSYSTEMSGivenasystemoflinearequationsandanorderedpair,determinewhethertheorderedpairisasolutionExample\(\PageIndex{1}\):DeterminingWhetheranOrderedPairIsaSolutiontoaSystemofEquationsExercise\(\PageIndex{1}\)SolvingSystemsofEquationsbyGraphingExample\(\PageIndex{2}\):SolvingaSystemofEquationsinTwoVariablesbyGraphingExercise\(\PageIndex{2}\)Q&ASolvingSystemsofEquationsbySubstitutionHowto:Givenasystemoftwoequationsintwovariables,solveusingthesubstitutionmethod.Example\(\PageIndex{3}\):SolvingaSystemofEquationsinTwoVariablesbySubstitutionExercise\(\PageIndex{3}\)Q&ASolvingSystemsofEquationsinTwoVariablesbytheAdditionMethodHowto:Givenasystemofequations,solveusingtheadditionmethod.Example\(\PageIndex{4}\):SolvingaSystembytheAdditionMethodExample\(\PageIndex{5}\):UsingtheAdditionMethodWhenMultiplicationofOneEquationIsRequiredExercise\(\PageIndex{4}\)Example\(\PageIndex{6}\):UsingtheAdditionMethodWhenMultiplicationofBothEquationsIsRequiredExample\(\PageIndex{7}\):UsingtheAdditionMethodinSystemsofEquationsContainingFractionsExercise\(\PageIndex{5}\)IdentifyingInconsistentSystemsofEquationsContainingTwoVariablesExample\(\PageIndex{8}\):SolvinganInconsistentSystemofEquationsExercise\(\PageIndex{6}\)ExpressingtheSolutionofaSystemofDependentEquationsContainingTwoVariablesExample\(\PageIndex{9}\):FindingaSolutiontoaDependentSystemofLinearEquationsExercise\(\PageIndex{7}\)UsingSystemsofEquationstoInvestigateProfitsExample\(\PageIndex{10}\):FindingtheBreak-EvenPointandtheProfitFunctionUsingSubstitutionExample\(\PageIndex{11}\):WritingandSolvingaSystemofEquationsinTwoVariablesExercise\(\PageIndex{8}\)MediaKeyConcepts LearningObjectives Solvesystemsofequationsbygraphing. Solvesystemsofequationsbysubstitution. Solvesystemsofequationsbyaddition. Identifyinconsistentsystemsofequationscontainingtwovariables. Expressthesolutionofasystemofdependentequationscontainingtwovariables. Askateboardmanufacturerintroducesanewlineofboards.Themanufacturertracksitscosts,whichistheamountitspendstoproducetheboards,anditsrevenue,whichistheamountitearnsthroughsalesofitsboards.Howcanthecompanydetermineifitismakingaprofitwithitsnewline?Howmanyskateboardsmustbeproducedandsoldbeforeaprofitispossible?Inthissection,wewillconsiderlinearequationswithtwovariablestoanswertheseandsimilarquestions. Figure\(\PageIndex{1}\) IntroductiontoSystemsofEquations Inordertoinvestigatesituationssuchasthatoftheskateboardmanufacturer,weneedtorecognizethatwearedealingwithmorethanonevariableandlikelymorethanoneequation.Asystemoflinearequationsconsistsoftwoormorelinearequationsmadeupoftwoormorevariablessuchthatallequationsinthesystemareconsideredsimultaneously.Tofindtheuniquesolutiontoasystemoflinearequations,wemustfindanumericalvalueforeachvariableinthesystemthatwillsatisfyallequationsinthesystematthesametime.Somelinearsystemsmaynothaveasolutionandothersmayhaveaninfinitenumberofsolutions.Inorderforalinearsystemtohaveauniquesolution,theremustbeatleastasmanyequationsastherearevariables.Evenso,thisdoesnotguaranteeauniquesolution. Inthissection,wewilllookatsystemsoflinearequationsintwovariables,whichconsistoftwoequationsthatcontaintwodifferentvariables.Forexample,considerthefollowingsystemoflinearequationsintwovariables. \[\begin{align*}2x+y&=15\\3x–y&=5\end{align*}\] Thesolutiontoasystemoflinearequationsintwovariablesisanyorderedpairthatsatisfieseachequationindependently.Inthisexample,theorderedpair\((4,7)\)isthesolutiontothesystemoflinearequations.Wecanverifythesolutionbysubstitutingthevaluesintoeachequationtoseeiftheorderedpairsatisfiesbothequations.Shortlywewillinvestigatemethodsoffindingsuchasolutionifitexists. \[\begin{align*}2(4)+(7)&=15\text{True}\\3(4)−(7)&=5\text{True}\end{align*}\] Inadditiontoconsideringthenumberofequationsandvariables,wecancategorizesystemsoflinearequationsbythenumberofsolutions.Aconsistentsystemofequationshasatleastonesolution.Aconsistentsystemisconsideredtobeanindependentsystemifithasasinglesolution,suchastheexamplewejustexplored.Thetwolineshavedifferentslopesandintersectatonepointintheplane.Aconsistentsystemisconsideredtobeadependentsystemiftheequationshavethesameslopeandthesamey-intercepts.Inotherwords,thelinescoincidesotheequationsrepresentthesameline.Everypointonthelinerepresentsacoordinatepairthatsatisfiesthesystem.Thus,thereareaninfinitenumberofsolutions. Anothertypeofsystemoflinearequationsisaninconsistentsystem,whichisoneinwhichtheequationsrepresenttwoparallellines.Thelineshavethesameslopeanddifferenty-intercepts.Therearenopointscommontobothlines;hence,thereisnosolutiontothesystem. TYPESOFLINEARSYSTEMS Therearethreetypesofsystemsoflinearequationsintwovariables,andthreetypesofsolutions. Anindependentsystemhasexactlyonesolutionpair\((x,y)\).Thepointwherethetwolinesintersectistheonlysolution. Aninconsistentsystemhasnosolution.Noticethatthetwolinesareparallelandwillneverintersect. Adependentsystemhasinfinitelymanysolutions.Thelinesarecoincident.Theyarethesameline,soeverycoordinatepaironthelineisasolutiontobothequations. Figure\(\PageIndex{2}\)comparesgraphicalrepresentationsofeachtypeofsystem. Figure\(\PageIndex{2}\) Givenasystemoflinearequationsandanorderedpair,determinewhethertheorderedpairisasolution Substitutetheorderedpairintoeachequationinthesystem. Determinewhethertruestatementsresultfromthesubstitutioninbothequations;ifso,theorderedpairisasolution. Example\(\PageIndex{1}\):DeterminingWhetheranOrderedPairIsaSolutiontoaSystemofEquations Determinewhethertheorderedpair\((5,1)\)isasolutiontothegivensystemofequations. \[\begin{align*}x+3y&=8\\2x−9&=y\end{align*}\] Solution Substitutetheorderedpair\((5,1)\)intobothequations. \[\begin{align*}(5)+3(1)&=8\\8&=8\text{True}\\2(5)−9&=(1)\\1&=1\text{True}\end{align*}\] Theorderedpair\((5,1)\)satisfiesbothequations,soitisthesolutiontothesystem. Analysis Wecanseethesolutionclearlybyplottingthegraphofeachequation.Sincethesolutionisanorderedpairthatsatisfiesbothequations,itisapointonbothofthelinesandthusthepointofintersectionofthetwolines.SeeFigure\(\PageIndex{3}\). Figure\(\PageIndex{3}\) Exercise\(\PageIndex{1}\) Determinewhethertheorderedpair\((8,5)\)isasolutiontothefollowingsystem. \[\begin{align*}5x−4y&=20\\2x+1&=3y\end{align*}\] Answer Notasolution. SolvingSystemsofEquationsbyGraphing Therearemultiplemethodsofsolvingsystemsoflinearequations.Forasystemoflinearequationsintwovariables,wecandetermineboththetypeofsystemandthesolutionbygraphingthesystemofequationsonthesamesetofaxes. Example\(\PageIndex{2}\):SolvingaSystemofEquationsinTwoVariablesbyGraphing Solvethefollowingsystemofequationsbygraphing.Identifythetypeofsystem. \[\begin{align*}2x+y&=−8\\x−y&=−1\end{align*}\] Solution Solvethefirstequationfor\(y\). \[\begin{align*}2x+y&=−8\\y&=−2x−8\end{align*}\] Solvethesecondequationfor\(y\). \[\begin{align*}x−y&=−1\\y&=x+1\end{align*}\] GraphbothequationsonthesamesetofaxesasinFigure\(\PageIndex{4}\). Figure\(\PageIndex{4}\) Thelinesappeartointersectatthepoint\((−3,−2)\).Wecanchecktomakesurethatthisisthesolutiontothesystembysubstitutingtheorderedpairintobothequations. \[\begin{align*}2(−3)+(−2)&=−8\\−8&=−8\text{True}\\(−3)−(−2)&=−1\\−1&=−1\text{True}\end{align*}\] Thesolutiontothesystemistheorderedpair\((−3,−2)\),sothesystemisindependent. Exercise\(\PageIndex{2}\) Solvethefollowingsystemofequationsbygraphing. \[\begin{align*}2x−5y&=−25\\−4x+5y&=35\end{align*}\] Answer Thesolutiontothesystemistheorderedpair\((−5,3)\). Figure\(\PageIndex{5}\) Q&A Cangraphingbeusedifthesystemisinconsistentordependent? Yes,inbothcaseswecanstillgraphthesystemtodeterminethetypeofsystemandsolution.Ifthetwolinesareparallel,thesystemhasnosolutionandisinconsistent.Ifthetwolinesareidentical,thesystemhasinfinitesolutionsandisadependentsystem. SolvingSystemsofEquationsbySubstitution Solvingalinearsystemintwovariablesbygraphingworkswellwhenthesolutionconsistsofintegervalues,butifoursolutioncontainsdecimalsorfractions,itisnotthemostprecisemethod.Wewillconsidertwomoremethodsofsolvingasystemoflinearequationsthataremoreprecisethangraphing.Onesuchmethodissolvingasystemofequationsbythesubstitutionmethod,inwhichwesolveoneoftheequationsforonevariableandthensubstitutetheresultintothesecondequationtosolveforthesecondvariable.Recallthatwecansolveforonlyonevariableatatime,whichisthereasonthesubstitutionmethodisbothvaluableandpractical. Howto:Givenasystemoftwoequationsintwovariables,solveusingthesubstitutionmethod. Solveoneofthetwoequationsforoneofthevariablesintermsoftheother. Substitutetheexpressionforthisvariableintothesecondequation,thensolvefortheremainingvariable. Substitutethatsolutionintoeitheroftheoriginalequationstofindthevalueofthefirstvariable.Ifpossible,writethesolutionasanorderedpair. Checkthesolutioninbothequations. Example\(\PageIndex{3}\):SolvingaSystemofEquationsinTwoVariablesbySubstitution Solvethefollowingsystemofequationsbysubstitution. \[\begin{align*}−x+y&=−5\\2x−5y&=1\end{align*}\] Solution First,wewillsolvethefirstequationfor\(y\). \[\begin{align*}−x+y&=−5\\y&=x−5\end{align*}\] Nowwecansubstitutetheexpression\(x−5\)for\(y\)inthesecondequation. \[\begin{align*}2x−5y&=1\\2x−5(x−5)&=1\\2x−5x+25&=1\\−3x&=−24\\x&=8\end{align*}\] Now,wesubstitute\(x=8\)intothefirstequationandsolvefor\(y\). \[\begin{align*}−(8)+y&=−5\\y&=3\end{align*}\] Oursolutionis\((8,3)\). Checkthesolutionbysubstituting\((8,3)\)intobothequations. \[\begin{align*}−x+y&=−5\\−(8)+(3)&=−5\text{True}\\2x−5y&=1\\2(8)−5(3)&=1\text{True}\end{align*}\] Exercise\(\PageIndex{3}\) Solvethefollowingsystemofequationsbysubstitution. \[\begin{align*}x&=y+3\\4&=3x−2y\end{align*}\] Answer \((−2,−5)\) Q&A Canthesubstitutionmethodbeusedtosolveanylinearsystemintwovariables? Yes,butthemethodworksbestifoneoftheequationscontainsacoefficientof\(1\)or\(–1\)sothatwedonothavetodealwithfractions. SolvingSystemsofEquationsinTwoVariablesbytheAdditionMethod Athirdmethodofsolvingsystemsoflinearequationsistheadditionmethod.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\(\PageIndex{4}\):SolvingaSystembytheAdditionMethod Solvethegivensystemofequationsbyaddition. \[\begin{align*}x+2y&=−1\\−x+y&=3\end{align*}\] Solution Bothequationsarealreadysetequaltoaconstant.Noticethatthecoefficientof\(x\)inthesecondequation,\(–1\),istheoppositeofthecoefficientof\(x\)inthefirstequation,\(1\).Wecanaddthetwoequationstoeliminate\(x\)withoutneedingtomultiplybyaconstant. \[\begin{align*}x+2y&=-1\\\underline{-x+y}&=\underline{3}\\3y&=2\\\end{align*}\] Nowthatwehaveeliminated\(x\),wecansolvetheresultingequationfor\(y\). \[\begin{align*}3y&=2\\y&=\dfrac{2}{3}\end{align*}\] Then,wesubstitutethisvaluefor\(y\)intooneoftheoriginalequationsandsolvefor\(x\). \[\begin{align*}−x+y&=3\\−x+\dfrac{2}{3}&=3\\−x&=3−\dfrac{2}{3}\\−x&=\dfrac{7}{3}\\x&=−\dfrac{7}{3}\end{align*}\] Thesolutiontothissystemis\(\left(−\dfrac{7}{3},\dfrac{2}{3}\right)\). Checkthesolutioninthefirstequation. \[\begin{align*}x+2y&=−1\\\left(−\dfrac{7}{3}\right)+2\left(\dfrac{2}{3}\right)&=\\−\dfrac{7}{3}+\dfrac{4}{3}&=−\dfrac{3}{3}\\−1&=−1\;\;\;\;\;\;\;\;\text{True}\end{align*}\] Analysis Wegainanimportantperspectiveonsystemsofequationsbylookingatthegraphicalrepresentation.SeeFigure\(\PageIndex{6}\)tofindthattheequationsintersectatthesolution.Wedonotneedtoaskwhethertheremaybeasecondsolutionbecauseobservingthegraphconfirmsthatthesystemhasexactlyonesolution. Figure\(\PageIndex{6}\) Example\(\PageIndex{5}\):UsingtheAdditionMethodWhenMultiplicationofOneEquationIsRequired Solvethegivensystemofequationsbytheadditionmethod. \[\begin{align*}3x+5y&=−11\\x−2y&=11\end{align*}\] Solution Addingtheseequationsaspresentedwillnoteliminateavariable.However,weseethatthefirstequationhas\(3x\)initandthesecondequationhas\(x\).Soifwemultiplythesecondequationby\(−3\),thex-termswilladdtozero. \[\begin{align*}x−2y&=11\\−3(x−2y)&=−3(11)\;\;\;\;\;\;\;\;\text{Multiplybothsidesby}−3.\\−3x+6y&=−33\;\;\;\;\;\;\;\;\;\text{Usethedistributiveproperty.}\end{align*}\] Now,let’saddthem. \[\begin{align*}3x+5y&=-11\\\underline{-3x+6y}&=\underline{-33}\\11y&=-44\\y&=-4\end{align*}\] Forthelaststep,wesubstitute\(y=−4\)intooneoftheoriginalequationsandsolvefor\(x\). \[\begin{align*}3x+5y&=−11\\3x+5(−4)&=−11\\3x−20&=−11\\3x&=9\\x&=3\end{align*}\] Oursolutionistheorderedpair\((3,−4)\).SeeFigure\(\PageIndex{7}\).Checkthesolutionintheoriginalsecondequation. \[\begin{align*}x−2y&=11\\(3)−2(−4)&=3+8\\&=11\;\;\;\;\;\;\;\;\;\;\text{True}\end{align*}\] Figure\(\PageIndex{7}\) Exercise\(\PageIndex{4}\) Solvethesystemofequationsbyaddition. \[\begin{align*}2x−7y&=2\\3x+y&=−20\end{align*}\] Answer \((−6,−2)\) Example\(\PageIndex{6}\):UsingtheAdditionMethodWhenMultiplicationofBothEquationsIsRequired Solvethegivensystemofequationsintwovariablesbyaddition. \[\begin{align*}2x+3y&=−16\\5x−10y&=30\end{align*}\] Solution Oneequationhas\(2x\)andtheotherhas\(5x\).Theleastcommonmultipleis\(10x\)sowewillhavetomultiplybothequationsbyaconstantinordertoeliminateonevariable.Let’seliminate\(x\)bymultiplyingthefirstequationby\(−5\)andthesecondequationby\(2\). \[\begin{align*}−5(2x+3y)&=−5(−16)\\−10x−15y&=80\\2(5x−10y)&=2(30)\\10x−20y&=60\end{align*}\] Then,weaddthetwoequationstogether. \[\begin{align*}-10x-15y&=80\\\underline{10x-20y}&=\underline{60}\\-35y&=140\\y&=-4\end{align*}\] Substitute\(y=−4\)intotheoriginalfirstequation. \[\begin{align*}2x+3(−4)&=−16\\2x−12&=−16\\2x&=−4\\x&=−2\end{align*}\] Thesolutionis\((−2,−4)\).Checkitintheotherequation. \[\begin{align*}5x−10y&=30\\5(−2)−10(−4)&=30\\−10+40&=30\\30&=30\end{align*}\] SeeFigure\(\PageIndex{8}\). Figure\(\PageIndex{8}\) Example\(\PageIndex{7}\):UsingtheAdditionMethodinSystemsofEquationsContainingFractions Solvethegivensystemofequationsintwovariablesbyaddition. \[\begin{align*}\dfrac{x}{3}+\dfrac{y}{6}&=3\\\dfrac{x}{2}−\dfrac{y}{4}&=1\end{align*}\] Solution Firstcleareachequationoffractionsbymultiplyingbothsidesoftheequationbytheleastcommondenominator. \[\begin{align*}6\left(\dfrac{x}{3}+\dfrac{y}{6}\right)&=6(3)\\2x+y&=18\\4\left(\dfrac{x}{2}−\dfrac{y}{4}\right)&=4(1)\\2x−y&=4\end{align*}\] Nowmultiplythesecondequationby\(−1\)sothatwecaneliminatethex-variable. \[\begin{align*}−1(2x−y)&=−1(4)\\−2x+y&=−4\end{align*}\] Addthetwoequationstoeliminatethe\(x\)-variableandsolvetheresultingequation. \[\begin{align*}2x+y&=18\\−2x+y&=−4\\2y&=14\\y&=7\end{align*}\] Substitute\(y=7\)intothefirstequation. \[\begin{align*}2x+(7)&=18\\2x&=11\\x&=\dfrac{11}{2}\\&=7.5\end{align*}\] Thesolutionis\(\left(\dfrac{11}{2},7\right)\).Checkitintheotherequation. \[\begin{align*}\dfrac{x}{2}−\dfrac{y}{4}&=1\\\dfrac{\dfrac{11}{2}}{2}−\dfrac{7}{4}&=1\\\dfrac{11}{4}−\dfrac{7}{4}&=1\\\dfrac{4}{4}&=1\end{align*}\] Exercise\(\PageIndex{5}\) Solvethesystemofequationsbyaddition. \[\begin{align*}2x+3y&=8\\3x+5y&=10\end{align*}\] Answer \((10,−4)\) IdentifyingInconsistentSystemsofEquationsContainingTwoVariables Nowthatwehaveseveralmethodsforsolvingsystemsofequations,wecanusethemethodstoidentifyinconsistentsystems.Recallthataninconsistentsystemconsistsofparallellinesthathavethesameslopebutdifferenty-intercepts.Theywillneverintersect.Whensearchingforasolutiontoaninconsistentsystem,wewillcomeupwithafalsestatement,suchas\(12=0\). Example\(\PageIndex{8}\):SolvinganInconsistentSystemofEquations Solvethefollowingsystemofequations. \[\begin{align*}x&=9−2y\\x+2y&=13\end{align*}\] Solution Wecanapproachthisproblemintwoways.Becauseoneequationisalreadysolvedfor\(x\),themostobviousstepistousesubstitution. \[\begin{align*}x+2y&=13\\(9−2y)+2y&=13\\9+0y&=13\\9&=13\end{align*}\] Clearly,thisstatementisacontradictionbecause\(9≠13\).Therefore,thesystemhasnosolution. Thesecondapproachwouldbetofirstmanipulatetheequationssothattheyarebothinslope-interceptform.Wemanipulatethefirstequationasfollows. \[\begin{align*}x&=9−2y\\2y&=−x+9\\y&=−\dfrac{1}{2}x+\dfrac{9}{2}\end{align*}\] Wethenconvertthesecondequationexpressedtoslope-interceptform. \[\begin{align*}x+2y&=13\\2y&=−x+13\\y&=−\dfrac{1}{2}x+\dfrac{13}{2}\end{align*}\] Comparingtheequations,weseethattheyhavethesameslopebutdifferent\(y\)-intercepts.Therefore,thelinesareparallelanddonotintersect. \[\begin{align*}y&=−\dfrac{1}{2}x+\dfrac{9}{2}\\y&=−\dfrac{1}{2}x+\dfrac{13}{2}\end{align*}\] Analysis Writingtheequationsinslope-interceptformconfirmsthatthesystemisinconsistentbecausealllineswillintersecteventuallyunlesstheyareparallel.Parallellineswillneverintersect;thus,thetwolineshavenopointsincommon.ThegraphsoftheequationsinthisexampleareshowninFigure\(\PageIndex{9}\). Figure\(\PageIndex{9}\) Exercise\(\PageIndex{6}\) Solvethefollowingsystemofequationsintwovariables. \[\begin{align*}2y−2x&=2\\2y−2x&=6\end{align*}\] Answer Nosolution.Itisaninconsistentsystem. ExpressingtheSolutionofaSystemofDependentEquationsContainingTwoVariables Recallthatadependentsystemofequationsintwovariablesisasysteminwhichthetwoequationsrepresentthesameline.Dependentsystemshaveaninfinitenumberofsolutionsbecauseallofthepointsononelinearealsoontheotherline.Afterusingsubstitutionoraddition,theresultingequationwillbeanidentity,suchas\(0=0\). Example\(\PageIndex{9}\):FindingaSolutiontoaDependentSystemofLinearEquations Findasolutiontothesystemofequationsusingtheadditionmethod. \[\begin{align*}x+3y&=2\\3x+9y&=6\end{align*}\] Solution Withtheadditionmethod,wewanttoeliminateoneofthevariablesbyaddingtheequations.Inthiscase,let’sfocusoneliminating\(x\).Ifwemultiplybothsidesofthefirstequationby\(−3\),thenwewillbeabletoeliminatethex-variable. \[\begin{align*}x+3y&=2\\(−3)(x+3y)&=(−3)(2)\\−3x−9y&=−6\end{align*}\] Nowaddtheequations. \[\begin{align*}-3x-9y&=-6\\\underline{+\space3x+9y}&=\underline{6}\\0&=0\\\end{align*}\] Wecanseethattherewillbeaninfinitenumberofsolutionsthatsatisfybothequations. Analysis Ifwerewrotebothequationsintheslope-interceptform,wemightknowwhatthesolutionwouldlooklikebeforeadding.Let’slookatwhathappenswhenweconvertthesystemtoslope-interceptform. \[\begin{align*}x+3y&=2\\3y&=−x+2\\y&=−\dfrac{1}{3}x+\dfrac{2}{3}\\3x+9y&=6\\9y&=−3x+6\\y&=−\dfrac{3}{9}x+\dfrac{6}{9}\\y&=−\dfrac{1}{3}x+\dfrac{2}{3}\end{align*}\] SeeFigure\(\PageIndex{10}\).Noticetheresultsarethesame.Thegeneralsolutiontothesystemis\(\left(x,−\dfrac{1}{3}x+\dfrac{2}{3}\right)\). Figure\(\PageIndex{10}\) Exercise\(\PageIndex{7}\) Solvethefollowingsystemofequationsintwovariables. \[\begin{align*}y−2x&=5\\−3y+6x&=−15\end{align*}\] Answer Thesystemisdependentsothereareinfinitesolutionsoftheform\((x,2x+5)\). UsingSystemsofEquationstoInvestigateProfits Usingwhatwehavelearnedaboutsystemsofequations,wecanreturntotheskateboardmanufacturingproblematthebeginningofthesection.Theskateboardmanufacturer’srevenuefunctionisthefunctionusedtocalculatetheamountofmoneythatcomesintothebusiness.Itcanberepresentedbytheequation\(R=xp\),where\(x\)=quantityand\(p\)=price.TherevenuefunctionisshowninorangeinFigure\(\PageIndex{11}\). Thecostfunctionisthefunctionusedtocalculatethecostsofdoingbusiness.Itincludesfixedcosts,suchasrentandsalaries,andvariablecosts,suchasutilities.ThecostfunctionisshowninblueinFigure\(\PageIndex{11}\).The\(x\)-axisrepresentsquantityinhundredsofunits.The\(y\)-axisrepresentseithercostorrevenueinhundredsofdollars. Figure\(\PageIndex{11}\) Thepointatwhichthetwolinesintersectiscalledthebreak-evenpoint.Wecanseefromthegraphthatif\(700\)unitsareproduced,thecostis\($3,300\)andtherevenueisalso\($3,300\).Inotherwords,thecompanybreakseveniftheyproduceandsell\(700\)units.Theyneithermakemoneynorlosemoney. Theshadedregiontotherightofthebreak-evenpointrepresentsquantitiesforwhichthecompanymakesaprofit.Theshadedregiontotheleftrepresentsquantitiesforwhichthecompanysuffersaloss.Theprofitfunctionistherevenuefunctionminusthecostfunction,writtenas\(P(x)=R(x)−C(x)\).Clearly,knowingthequantityforwhichthecostequalstherevenueisofgreatimportancetobusinesses. Example\(\PageIndex{10}\):FindingtheBreak-EvenPointandtheProfitFunctionUsingSubstitution Giventhecostfunction\(C(x)=0.85x+35,000\)andtherevenuefunction\(R(x)=1.55x\),findthebreak-evenpointandtheprofitfunction. Solution Writethesystemofequationsusing\(y\)toreplacefunctionnotation. \[\begin{align*}y&=0.85x+35,000\\y&=1.55x\end{align*}\] Substitutetheexpression\(0.85x+35,000\)fromthefirstequationintothesecondequationandsolvefor\(x\). \[\begin{align*}0.85x+35,000&=1.55x\\35,000&=0.7x\\50,000&=x\end{align*}\] Then,wesubstitute\(x=50,000\)intoeitherthecostfunctionortherevenuefunction. \(1.55(50,000)=77,500\) Thebreak-evenpointis\((50,000,77,500)\). Theprofitfunctionisfoundusingtheformula\(P(x)=R(x)−C(x)\). \[\begin{align*}P(x)&=1.55x−(0.85x+35,000)\\&=0.7x−35,000\end{align*}\] Theprofitfunctionis\(P(x)=0.7x−35,000\). Analysis Thecosttoproduce\(50,000\)unitsis\($77,500\),andtherevenuefromthesalesof\(50,000\)unitsisalso\($77,500\).Tomakeaprofit,thebusinessmustproduceandsellmorethan\(50,000\)units.SeeFigure\(\PageIndex{12}\). Figure\(\PageIndex{12}\) WeseefromthegraphinFigure\(\PageIndex{13}\)thattheprofitfunctionhasanegativevalueuntil\(x=50,000\),whenthegraphcrossesthe\(x\)-axis.Then,thegraphemergesintopositive\(y\)-valuesandcontinuesonthispathastheprofitfunctionisastraightline.Thisillustratesthatthebreak-evenpointforbusinessesoccurswhentheprofitfunctionis\(0\).Theareatotheleftofthebreak-evenpointrepresentsoperatingataloss. Figure\(\PageIndex{13}\) Example\(\PageIndex{11}\):WritingandSolvingaSystemofEquationsinTwoVariables Thecostofatickettothecircusis\($25.00\)forchildrenand\($50.00\)foradults.Onacertainday,attendanceatthecircusis\(2,000\)andthetotalgaterevenueis\($70,000\).Howmanychildrenandhowmanyadultsboughttickets? Solution Let\(c\)=thenumberofchildrenand\(a\)=thenumberofadultsinattendance. Thetotalnumberofpeopleis\(2,000\).Wecanusethistowriteanequationforthenumberofpeopleatthecircusthatday. \(c+a=2,000\) Therevenuefromallchildrencanbefoundbymultiplying\($25.00\)bythenumberofchildren,\(25c\).Therevenuefromalladultscanbefoundbymultiplying\($50.00\)bythenumberofadults,\(50a\).Thetotalrevenueis\($70,000\).Wecanusethistowriteanequationfortherevenue. \(25c+50a=70,000\) Wenowhaveasystemoflinearequationsintwovariables. \(c+a=2,000\) \(25c+50a=70,000\) Inthefirstequation,thecoefficientofbothvariablesis\(1\).Wecanquicklysolvethefirstequationforeither\(c\)or\(a\).Wewillsolvefor\(a\). \[\begin{align*}c+a&=2,000\\a&=2,000−c\end{align*}\] Substitutetheexpression\(2,000−c\)inthesecondequationforaaandsolvefor\(c\). \[\begin{align*}25c+50(2,000−c)&=70,000\\25c+100,000−50c&=70,000\\−25c&=−30,000\\c&=1,200\end{align*}\] Substitute\(c=1,200\)intothefirstequationtosolvefor\(a\). \[\begin{align*}1,200+a&=2,000\\a&=800\end{align*}\] Wefindthat\(1,200\)childrenand\(800\)adultsboughtticketstothecircusthatday. Exercise\(\PageIndex{8}\) Mealticketsatthecircuscost\($4.00\)forchildrenand\($12.00\)foradults.If\(1,650\)mealticketswereboughtforatotalof,\($14,200\),howmanychildrenandhowmanyadultsboughtmealtickets? Answer \(700\)children,\(950\)adults Media Accesstheseonlineresourcesforadditionalinstructionandpracticewithsystemsoflinearequations. SolvingSystemsofEquationsUsingSubstitution SolvingSystemsofEquationsUsingElimination ApplicationsofSystemsofEquations KeyConcepts Asystemoflinearequationsconsistsoftwoormoreequationsmadeupoftwoormorevariablessuchthatallequationsinthesystemareconsideredsimultaneously. Thesolutiontoasystemoflinearequationsintwovariablesisanyorderedpairthatsatisfieseachequationindependently.SeeExample\(\PageIndex{1}\). Systemsofequationsareclassifiedasindependentwithonesolution,dependentwithaninfinitenumberofsolutions,orinconsistentwithnosolution. Onemethodofsolvingasystemoflinearequationsintwovariablesisbygraphing.Inthismethod,wegraphtheequationsonthesamesetofaxes.SeeExample\(\PageIndex{2}\). Anothermethodofsolvingasystemoflinearequationsisbysubstitution.Inthismethod,wesolveforonevariableinoneequationandsubstitutetheresultintothesecondequation.SeeExample\(\PageIndex{3}\). Athirdmethodofsolvingasystemoflinearequationsisbyaddition,inwhichwecaneliminateavariablebyaddingoppositecoefficientsofcorrespondingvariables.SeeExample\(\PageIndex{4}\). Itisoftennecessarytomultiplyoneorbothequationsbyaconstanttofacilitateeliminationofavariablewhenaddingthetwoequationstogether.SeeExample\(\PageIndex{5}\),Example\(\PageIndex{6}\),andExample\(\PageIndex{7}\). Eithermethodofsolvingasystemofequationsresultsinafalsestatementforinconsistentsystemsbecausetheyaremadeupofparallellinesthatneverintersect.SeeExample\(\PageIndex{8}\). Thesolutiontoasystemofdependentequationswillalwaysbetruebecausebothequationsdescribethesameline.SeeExample\(\PageIndex{9}\). Systemsofequationscanbeusedtosolvereal-worldproblemsthatinvolvemorethanonevariable,suchasthoserelatingtorevenue,cost,andprofit.SeeExample\(\PageIndex{10}\)andExample\(\PageIndex{11}\).