4.1: Solve Systems of Linear Equations with Two Variables

A linear equation in two variables, such as 2x+y=7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line ... Skiptomaincontent LearningObjectivesDetermineWhetheranOrderedPairisaSolutionofaSystemofEquationsSYSTEMOFLINEAREQUATIONSSolutionSOFASYSTEMOFEQUATIONSExample\(\PageIndex{1}\)Example\(\PageIndex{2}\)Example\(\PageIndex{3}\)SolveaSystemofLinearEquationsbyGraphingExample\(\PageIndex{4}\):HowtoSolveaSystemofEquationsbyGraphingExample\(\PageIndex{5}\)Example\(\PageIndex{6}\)SOLVEASYSTEMOFLINEAREQUATIONSBYGRAPHING.Example\(\PageIndex{7}\)Example\(\PageIndex{8}\)Example\(\PageIndex{9}\)Example\(\PageIndex{10}\)Example\(\PageIndex{11}\)Example\(\PageIndex{12}\)Example\(\PageIndex{13}\)Example\(\PageIndex{14}\)Example\(\PageIndex{15}\)COINCIDENTLINESCONSISTENTANDINCONSISTENTSYSTEMSExample\(\PageIndex{16}\)Example\(\PageIndex{17}\)Example\(\PageIndex{18}\)SolveaSystemofEquationsbySubstitutionExample\(\PageIndex{19}\):HowtoSolveaSystemofEquationsbySubstitutionExample\(\PageIndex{20}\)Example\(\PageIndex{21}\)SOLVEASYSTEMOFEQUATIONSBYSUBSTITUTION.Example\(\PageIndex{22}\)Example\(\PageIndex{23}\)Example\(\PageIndex{24}\)SolveaSystemofEquationsbyEliminationExercise\(\PageIndex{25}\):HowtoSolveaSystemofEquationsbyEliminationExercise\(\PageIndex{26}\)Exercise\(\PageIndex{27}\)SOLVEASYSTEMOFEQUATIONSBYELIMINATION.Exercise\(\PageIndex{28}\)Exercise\(\PageIndex{29}\)Exercise\(\PageIndex{30}\)Exercise\(\PageIndex{31}\)Exercise\(\PageIndex{32}\)Exercise\(\PageIndex{33}\)Exercise\(\PageIndex{34}\)Exercise\(\PageIndex{35}\)Exercise\(\PageIndex{36}\)ChoosetheMostConvenientMethodtoSolveaSystemofLinearEquationsExample\(\PageIndex{37}\)Example\(\PageIndex{38}\)Example\(\PageIndex{39}\)KeyConceptsGlossary LearningObjectives Bytheendofthissection,youwillbeableto: Determinewhetheranorderedpairisasolutionofasystemofequations Solveasystemoflinearequationsbygraphing Solveasystemofequationsbysubstitution Solveasystemofequationsbyelimination Choosethemostconvenientmethodtosolveasystemoflinearequations Beforeyougetstarted,takethisreadinessquiz. Fortheequation\(y=\frac{2}{3}x−4\), ⓐIs\((6,0)\)asolution?ⓑIs\((−3,−2)\)asolution? Ifyoumissedthisproblem,review[link]. Findtheslopeandy-interceptoftheline\(3x−y=12\). Ifyoumissedthisproblem,review[link]. Findthex-andy-interceptsoftheline\(2x−3y=12\). Ifyoumissedthisproblem,review[link]. DetermineWhetheranOrderedPairisaSolutionofaSystemofEquations InSolvingLinearEquations,welearnedhowtosolvelinearequationswithonevariable.Nowwewillworkwithtwoormorelinearequationsgroupedtogether,whichisknownasasystemoflinearequations. SYSTEMOFLINEAREQUATIONS Whentwoormorelinearequationsaregroupedtogether,theyformasystemoflinearequations. Inthissection,wewillfocusourworkonsystemsoftwolinearequationsintwounknowns.Wewillsolvelargersystemsofequationslaterinthischapter. Anexampleofasystemoftwolinearequationsisshownbelow.Weuseabracetoshowthetwoequationsaregroupedtogethertoformasystemofequations. \[\left\{\begin{aligned}2x+y&=7\\x−2y&=6\end{aligned}\right.\nonumber\] Alinearequationintwovariables,suchas\(2x+y=7\),hasaninfinitenumberofsolutions.Itsgraphisaline.Remember,everypointonthelineisasolutiontotheequationandeverysolutiontotheequationisapointontheline. Tosolveasystemoftwolinearequations,wewanttofindthevaluesofthevariablesthataresolutionstobothequations.Inotherwords,wearelookingfortheorderedpairs\((x,y)\)thatmakebothequationstrue.Thesearecalledthesolutionsofasystemofequations. SolutionSOFASYSTEMOFEQUATIONS Thesolutionsofasystemofequationsarethevaluesofthevariablesthatmakealltheequationstrue.Asolutionofasystemoftwolinearequationsisrepresentedbyanorderedpair\((x,y)\). Todetermineifanorderedpairisasolutiontoasystemoftwoequations,wesubstitutethevaluesofthevariablesintoeachequation.Iftheorderedpairmakesbothequationstrue,itisasolutiontothesystem. Example\(\PageIndex{1}\) Determinewhethertheorderedpairisasolutiontothesystem\(\left\{\begin{array}{l}x−y=−1\\2x−y=−5\end{array}\right.\). ⓐ\((−2,−1)\)ⓑ\((−4,−3)\) Answer ⓐ ⓑ Example\(\PageIndex{2}\) Determinewhethertheorderedpairisasolutiontothesystem\(\left\{\begin{array}3x+y=0\\x+2y=−5\end{array}\right.\). ⓐ\((1,−3)\)ⓑ\((0,0)\) Answer ⓐyesⓑno Example\(\PageIndex{3}\) Determinewhethertheorderedpairisasolutiontothesystem\(\left\{\begin{array}x−3y=−8\\−3x−y=4\end{array}\right.\). ⓐ\((2,−2)\)ⓑ\((−2,2)\) Answer ⓐnoⓑyes SolveaSystemofLinearEquationsbyGraphing Inthissection,wewillusethreemethodstosolveasystemoflinearequations.Thefirstmethodwe’lluseisgraphing. Thegraphofalinearequationisaline.Eachpointonthelineisasolutiontotheequation.Forasystemoftwoequations,wewillgraphtwolines.Thenwecanseeallthepointsthataresolutionstoeachequation.And,byfindingwhatthelineshaveincommon,we’llfindthesolutiontothesystem. Mostlinearequationsinonevariablehaveonesolution,butwesawthatsomeequations,calledcontradictions,havenosolutionsandforotherequations,calledidentities,allnumbersaresolutions. Similarly,whenwesolveasystemoftwolinearequationsrepresentedbyagraphoftwolinesinthesameplane,therearethreepossiblecases,asshown. Figure\(\PageIndex{1}\) Eachtimewedemonstrateanewmethod,wewilluseitonthesamesystemoflinearequations.Attheendofthesectionyou’lldecidewhichmethodwasthemostconvenientwaytosolvethissystem. Example\(\PageIndex{4}\):HowtoSolveaSystemofEquationsbyGraphing Solvethesystembygraphing\(\left\{\begin{array}{l}2x+y=7\\x−2y=6\end{array}\right.\). Answer Example\(\PageIndex{5}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}x−3y=−3\\x+y=5\end{array}\right.\). Answer \((3,2)\) Example\(\PageIndex{6}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}−x+y=1\\3x+2y=12\end{array}\right.\) Answer \((2,3)\) Thestepstousetosolveasystemoflinearequationsbygraphingareshownhere. SOLVEASYSTEMOFLINEAREQUATIONSBYGRAPHING. Graphthefirstequation. Graphthesecondequationonthesamerectangularcoordinatesystem. Determinewhetherthelinesintersect,areparallel,orarethesameline. Identifythesolutiontothesystem. Ifthelinesintersect,identifythepointofintersection.Thisisthesolutiontothesystem. Ifthelinesareparallel,thesystemhasnosolution. Ifthelinesarethesame,thesystemhasaninfinitenumberofsolutions. Checkthesolutioninbothequations. Inthenextexample,we’llfirstre-writetheequationsintoslope–interceptformasthiswillmakeiteasyforustoquicklygraphthelines. Example\(\PageIndex{7}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}3x+y=−1\\2x+y=0\end{array}\right.\) Answer We’llsolvebothoftheseequationsfor\(y\)sothatwecaneasilygraphthemusingtheirslopesand\(y\)-intercepts.   Solvethefirstequationfory. Findtheslopeandy-intercept. Solvethesecondequationfory. Findtheslopeandy-intercept. Graphthelines. Determinethepointofintersection. Thelinesintersectat\((−1,2)\). Checkthesolutioninbothequations.     Thesolutionis\((−1,2)\). Example\(\PageIndex{8}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}−x+y=1\\2x+y=10\end{array}\right.\). Answer \((3,4)\) Example\(\PageIndex{9}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}2x+y=6\\x+y=1\end{array}\right.\). Answer \((5,−4)\) Inallthesystemsoflinearequationssofar,thelinesintersectedandthesolutionwasonepoint.Inthenexttwoexamples,we’lllookatasystemofequationsthathasnosolutionandatasystemofequationsthathasaninfinitenumberofsolutions. Example\(\PageIndex{10}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}y=\tfrac{1}{2}x-3\\x-2y=4\end{array}\right.\). Answer   Tographthefirstequation,wewilluseits slopeandy-intercept. Tographthesecondequation,wewilluse theintercepts.   Graphthelines. Determinethepointsofintersection. Thelinesareparallel. Sincenopointisonbothlines,thereisno orderedpairthatmakesbothequations true.Thereisnosolutiontothissystem. Example\(\PageIndex{11}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}y=-\tfrac{1}{4}x+2\\x+4y=4\end{array}\right.\). Answer nosolution Example\(\PageIndex{12}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}y=3x-1\\6x-2y=6\end{array}\right.\). Answer nosolution Sometimestheequationsinasystemrepresentthesameline.Sinceeverypointonthelinemakesbothequationstrue,thereareinfinitelymanyorderedpairsthatmakebothequationstrue.Thereareinfinitelymanysolutionstothesystem. Example\(\PageIndex{13}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}y=2x-3\\-6x+3y=9\end{array}\right.\). Answer   Findtheslopeandy-interceptofthefirstequation. Findtheinterceptsofthesecondequation.   Graphthelines.   Thelinesarethesame! Sinceeverypointonthelinemakesboth equationstrue,thereareinfinitelymany orderedpairsthatmakebothequationstrue. Thereareinfinitelymanysolutionstothissystem. Ifyouwritethesecondequationinslope-interceptform,youmayrecognizethattheequationshavethesameslopeandsamey-intercept. Example\(\PageIndex{14}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}y=-3x-6\\6x+2y=-12\end{array}\right.\). Answer infinitelymanysolutions Example\(\PageIndex{15}\) Solvethesystembygraphing:\(\left\{\begin{array}{l}y=\tfrac{1}{2}x-4\\2x-4y=16\end{array}\right.\). Answer infinitelymanysolutions Whenwegraphedthesecondlineinthelastexample,wedrewitrightoverthefirstline.Wesaythetwolinesarecoincident.Coincidentlineshavethesameslopeandsamey-intercept. COINCIDENTLINES Coincidentlineshavethesameslopeandsamey-intercept. ThesystemsofequationsinExampleandExampleeachhadtwointersectinglines.Eachsystemhadonesolution. InExample,theequationsgavecoincidentlines,andsothesystemhadinfinitelymanysolutions. Thesystemsinthosethreeexampleshadatleastonesolution.Asystemofequationsthathasatleastonesolutioniscalledaconsistentsystem. Asystemwithparallellines,likeExample,hasnosolution.Wecallasystemofequationslikethisinconsistent.Ithasnosolution. CONSISTENTANDINCONSISTENTSYSTEMS Aconsistentsystemofequationsisasystemofequationswithatleastonesolution. Aninconsistentsystemofequationsisasystemofequationswithnosolution. Wealsocategorizetheequationsinasystemofequationsbycallingtheequationsindependentordependent.Iftwoequationsareindependent,theyeachhavetheirownsetofsolutions.Intersectinglinesandparallellinesareindependent. Iftwoequationsaredependent,allthesolutionsofoneequationarealsosolutionsoftheotherequation.Whenwegraphtwodependentequations,wegetcoincidentlines. Let’ssumthisupbylookingatthegraphsofthethreetypesofsystems.SeebelowandTable. Lines Intersecting Parallel Coincident Numberofsolutions 1point Nosolution Infinitelymany Consistent/inconsistent Consistent Inconsistent Consistent Dependent/independent Independent Independent Dependent Example\(\PageIndex{16}\) Withoutgraphing,determinethenumberofsolutionsandthenclassifythesystemofequations. ⓐ\(\left\{\begin{array}{l}y=3x−1\\6x−2y=12\end{array}\right.\)ⓑ\(\left\{\begin{array}{l}2x+y=−3\\x−5y=5\end{array}\right.\) Answer ⓐWewillcomparetheslopesandinterceptsofthetwolines. \(\begin{array}{lll}{}&{}&{\left\{\begin{array}{l}{y=3x-1}\\{6x−2y=12}\end{array}\right.}\\{}&{}&{y=3x-1}\\{\text{Thefirstequationisalreadyinslope-interceptform.}}&{}&{}\\{\text{Writethesecondequationinslope-interceptform.}}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{}&{}&{6x-2y=12}\\{}&{}&{-2y=-6x+12}\\{}&{}&{\frac{-2y}{-2}=\frac{-6x+12}{-2}}\\{}&{}&{y=3x-6}\\{}&{y=3x-1}&{y=3x-6}\\{}&{m=3}&{m=3}\\{}&{b=-1}&{b=-6}\\{\text{Findtheslopeandinterceptofeachline.}}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{}&{\text{Sincetheslopesarethesameandy-interceptsare}}&{}\\{}&{\text{different,thelinesareparallel.}}&{}\\\end{array}\) ⓑWewillcomparetheslopeandinterceptsofthetwolines. \(\begin{array}{lll}{}&{}&{}\\{}&{\left\{\begin{array}{l}2x+y=-3\\x-5y=5\\\end{array}\right.}&{}\\{\text{Writebothequationsinslope–interceptform.}}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{}&{2x+y=-3}&{x-5y=5}\\{}&{y=-2x-3}&{-5y=-x+5}\\{}&{}&{\frac{-5y}{-5}=\frac{-x+5}{-5}}\\{}&{}&{y=\frac{1}{5}-1}\\{}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{}&{}&{}\\{\text{Findtheslopeandinterceptofeachline.}}&{}&{}\\{}&{}&{}\\{}&{y=-2x-3}&{y=\frac{1}{5}-1}\\{}&{m=-2}&{m=\frac{1}{5}}\\{}&{b=-3}&{b=-1}\\{}&{}&{}\\{}&{\text{Sincetheslopesaredifferent,thelinesintersect.}}&{}\\\end{array}\) Asystemofequationswhosegraphsareintersecthas1solutionandisconsistentandindependent. Example\(\PageIndex{17}\) Withoutgraphing,determinethenumberofsolutionsandthenclassifythesystemofequations. ⓐ\(\left\{\begin{array}{l}y=−2x−4\\4x+2y=9\end{array}\right.\)ⓑ\(\left\{\begin{array}{l}3x+2y=2\\2x+y=1\end{array}\right.\) Answer ⓐnosolution,inconsistent,independentⓑonesolution,consistent,independent Example\(\PageIndex{18}\) Withoutgraphing,determinethenumberofsolutionsandthenclassifythesystemofequations. ⓐ\(\left\{\begin{array}{l}y=\frac{1}{3}x−5\\x−3y=6\end{array}\right.\)ⓑ\(\left\{\begin{array}{l}x+4y=12\\−x+y=3\end{array}\right.\) Answer ⓐnosolution,inconsistent,independentⓑonesolution,consistent,independent Solvingsystemsoflinearequationsbygraphingisagoodwaytovisualizethetypesofsolutionsthatmayresult.However,therearemanycaseswheresolvingasystembygraphingisinconvenientorimprecise.Ifthegraphsextendbeyondthesmallgridwithxandybothbetween\(−10\)and10,graphingthelinesmaybecumbersome.Andifthesolutionstothesystemarenotintegers,itcanbehardtoreadtheirvaluespreciselyfromagraph. SolveaSystemofEquationsbySubstitution Wewillnowsolvesystemsoflinearequationsbythesubstitutionmethod. Wewillusethesamesystemweusedfirstforgraphing. \[\left\{\begin{array}{l}2x+y=7\\x−2y=6\end{array}\right.\nonumber\] Wewillfirstsolveoneoftheequationsforeitherxory.Wecanchooseeitherequationandsolveforeithervariable—butwe’lltrytomakeachoicethatwillkeeptheworkeasy. Thenwesubstitutethatexpressionintotheotherequation.Theresultisanequationwithjustonevariable—andweknowhowtosolvethose! Afterwefindthevalueofonevariable,wewillsubstitutethatvalueintooneoftheoriginalequationsandsolvefortheothervariable.Finally,wecheckoursolutionandmakesureitmakesbothequationstrue. Example\(\PageIndex{19}\):HowtoSolveaSystemofEquationsbySubstitution Solvethesystembysubstitution:\(\left\{\begin{array}{l}2x+y=7\\x−2y=6\end{array}\right.\) Answer Example\(\PageIndex{20}\) Solvethesystembysubstitution:\(\left\{\begin{array}{l}−2x+y=−11\\x+3y=9\end{array}\right.\) Answer \((6,1)\) Example\(\PageIndex{21}\) Solvethesystembysubstitution:\(\left\{\begin{array}{l}2x+y=−1\\4x+3y=3\end{array}\right.\) Answer \((−3,5)\) SOLVEASYSTEMOFEQUATIONSBYSUBSTITUTION. Solveoneoftheequationsforeithervariable. SubstitutetheexpressionfromStep1intotheotherequation. Solvetheresultingequation. SubstitutethesolutioninStep3intoeitheroftheoriginalequationstofindtheothervariable. Writethesolutionasanorderedpair. Checkthattheorderedpairisasolutiontobothoriginalequations. Beverycarefulwiththesignsinthenextexample. Example\(\PageIndex{22}\) Solvethesystembysubstitution:\(\left\{\begin{array}{l}4x+2y=4\\6x−y=8\end{array}\right.\) Answer Weneedtosolveoneequationforonevariable.Wewillsolvethefirstequationfory.   Solvethefirstequationfory. Substitute\(−2x+2\)foryinthesecondequation. Replacetheywith\(−2x+2\). Solvetheequationforx. Substitute\(x=54\)into\(4x+2y=4\)tofindy.   Theorderedpairis\((54,−12)\). Checktheorderedpairinbothequations.     Thesolutionis\((54,−12)\). Example\(\PageIndex{23}\) Solvethesystembysubstitution:\(\left\{\begin{array}{l}x−4y=−4\\−3x+4y=0\end{array}\right.\) Answer \((2,32)\) Example\(\PageIndex{24}\) Solvethesystembysubstitution:\(\left\{\begin{array}{l}4x−y=0\\2x−3y=5\end{array}\right.\) Answer \((−12,−2)\) SolveaSystemofEquationsbyElimination Wehavesolvedsystemsoflinearequationsbygraphingandbysubstitution.Graphingworkswellwhenthevariablecoefficientsaresmallandthesolutionhasintegervalues.Substitutionworkswellwhenwecaneasilysolveoneequationforoneofthevariablesandnothavetoomanyfractionsintheresultingexpression. ThethirdmethodofsolvingsystemsoflinearequationsiscalledtheEliminationMethod.Whenwesolvedasystembysubstitution,westartedwithtwoequationsandtwovariablesandreducedittooneequationwithonevariable.Thisiswhatwe’lldowiththeeliminationmethod,too,butwe’llhaveadifferentwaytogetthere. TheEliminationMethodisbasedontheAdditionPropertyofEquality.TheAdditionPropertyofEqualitysaysthatwhenyouaddthesamequantitytobothsidesofanequation,youstillhaveequality.WewillextendtheAdditionPropertyofEqualitytosaythatwhenyouaddequalquantitiestobothsidesofanequation,theresultsareequal. Foranyexpressionsa,b,c,andd. \[\begin{array}{ll}{\text{if}}&{a=b}\\{\text{and}}&{c=d}\\{\text{then}}&{a+c=b+d.}\\\nonumber\end{array}\] Tosolveasystemofequationsbyelimination,westartwithbothequationsinstandardform.Thenwedecidewhichvariablewillbeeasiesttoeliminate.Howdowedecide?Wewanttohavethecoefficientsofonevariablebeopposites,sothatwecanaddtheequationstogetherandeliminatethatvariable. Noticehowthatworkswhenweaddthesetwoequationstogether: \[\left\{\begin{array}{l}3x+y=5\\\underline{2x−y=0}\end{array}\right.\nonumber\] \[5x=5\nonumber\] They’saddtozeroandwehaveoneequationwithonevariable. Let’stryanotherone: \[\left\{\begin{array}x+4y=2\\2x+5y=−2\end{array}\right.\nonumber\] Thistimewedon’tseeavariablethatcanbeimmediatelyeliminatedifweaddtheequations. Butifwemultiplythefirstequationby\(−2\),wewillmakethecoefficientsofxopposites.Wemustmultiplyeverytermonbothsidesoftheequationby\(−2\). Thenrewritethesystemofequations. Nowweseethatthecoefficientsofthextermsareopposites,soxwillbeeliminatedwhenweaddthesetwoequations. Oncewegetanequationwithjustonevariable,wesolveit.Thenwesubstitutethatvalueintooneoftheoriginalequationstosolvefortheremainingvariable.And,asalways,wecheckouranswertomakesureitisasolutiontobothoftheoriginalequations. Nowwe’llseehowtouseeliminationtosolvethesamesystemofequationswesolvedbygraphingandbysubstitution. Exercise\(\PageIndex{25}\):HowtoSolveaSystemofEquationsbyElimination Solvethesystembyelimination:\(\left\{\begin{array}{l}2x+y=7\\x−2y=6\end{array}\right.\) Answer Exercise\(\PageIndex{26}\) Solvethesystembyelimination:\(\left\{\begin{array}{l}3x+y=5\\2x−3y=7\end{array}\right.\) Answer \((2,−1)\) Exercise\(\PageIndex{27}\) Solvethesystembyelimination:\(\left\{\begin{array}{l}4x+y=−5\\−2x−2y=−2\end{array}\right.\) Answer \((−2,3)\) Thestepsarelistedhereforeasyreference. SOLVEASYSTEMOFEQUATIONSBYELIMINATION. Writebothequationsinstandardform.Ifanycoefficientsarefractions,clearthem. Makethecoefficientsofonevariableopposites. Decidewhichvariableyouwilleliminate. Multiplyoneorbothequationssothatthecoefficientsofthatvariableareopposites. AddtheequationsresultingfromStep2toeliminateonevariable. Solvefortheremainingvariable. SubstitutethesolutionfromStep4intooneoftheoriginalequations.Thensolvefortheothervariable. Writethesolutionasanorderedpair. Checkthattheorderedpairisasolutiontobothoriginalequations. Nowwe’lldoanexamplewhereweneedtomultiplybothequationsbyconstantsinordertomakethecoefficientsofonevariableopposites. Exercise\(\PageIndex{28}\) Solvethesystembyelimination:\(\left\{\begin{array}{l}4x−3y=9\\7x+2y=−6\end{array}\right.\) Answer Inthisexample,wecannotmultiplyjustoneequationbyanyconstanttogetoppositecoefficients.Sowewillstrategicallymultiplybothequationsbydifferentconstantstogettheopposites.   Bothequationsareinstandardform. Togetoppositecoefficientsofy,wewill multiplythefirstequationby2andthe secondequationby3. Simplify. Addthetwoequationstoeliminatey. Solveforx. Substitutex=0x=0intooneoftheoriginalequations. Solvefory. Writethesolutionasanorderedpair. Theorderedpairis\((0,−3)\). Checkthattheorderedpairisasolutionto bothoriginalequations.     Thesolutionis\((0,−3)\). Exercise\(\PageIndex{29}\) Solvethesystembyelimination:\(\left\{\begin{array}{l}3x−4y=−9\\5x+3y=14\end{array}\right.\) Answer \((1,3)\) Exercise\(\PageIndex{30}\) Solveeachsystembyelimination:\(\left\{\begin{array}{l}7x+8y=4\\3x−5y=27\end{array}\right.\) Answer \((4,−3)\) Whenthesystemofequationscontainsfractions,wewillfirstclearthefractionsbymultiplyingeachequationbytheLCDofallthefractionsintheequation. Exercise\(\PageIndex{31}\) Solvethesystembyelimination:\(\left\{\begin{array}{l}x+\tfrac{1}{2}y=6\\\tfrac{3}{2}x+\tfrac{2}{3}y=\tfrac{17}{2}\end{array}\right.\) Answer Inthisexample,bothequationshavefractions.OurfirststepwillbetomultiplyeachequationbytheLCDofallthefractionsintheequationtoclearthefractions.   Toclearthefractions,multiplyeach equationbyitsLCD. Simplify. Nowwearereadytoeliminateone ofthevariables.Noticethatbothequationsarein standardform.   Wecaneliminateybymultiplyingthetopequationby\(−4\). Simplifyandadd. Substitute\(x=3\)intooneoftheoriginalequations. Solvefory. Writethesolutionasanorderedpair. Theorderedpairis\((3,6)\). Checkthattheorderedpairisasolutionto bothoriginalequations.     Thesolutionis\((3,6)\). Exercise\(\PageIndex{32}\) Solveeachsystembyelimination:\(\left\{\begin{array}{l}\tfrac{1}{3}x−\tfrac{1}{2}y=1\\\tfrac{3}{4}x−y=\tfrac{5}{2}\end{array}\right.\) Answer \((6,2)\) Exercise\(\PageIndex{33}\) Solveeachsystembyelimination:\(\left\{\begin{array}{l}x+\tfrac{3}{5}y=−\tfrac{1}{5}\\−\tfrac{1}{2}x−\tfrac{2}{3}y=\tfrac{5}{6}\end{array}\right.\) Answer \((1,−2)\) Whenwesolvedthesystembygraphing,wesawthatnotallsystemsoflinearequationshaveasingleorderedpairasasolution.Whenthetwoequationswerereallythesameline,therewereinfinitelymanysolutions.Wecalledthataconsistentsystem.Whenthetwoequationsdescribedparallellines,therewasnosolution.Wecalledthataninconsistentsystem. Thesameistrueusingsubstitutionorelimination.Iftheequationattheendofsubstitutionoreliminationisatruestatement,wehaveaconsistentbutdependentsystemandthesystemofequationshasinfinitelymanysolutions.Iftheequationattheendofsubstitutionoreliminationisafalsestatement,wehaveaninconsistentsystemandthesystemofequationshasnosolution. Exercise\(\PageIndex{34}\) Solvethesystembyelimination:\(\left\{\begin{array}{l}3x+4y=12\\y=3−\tfrac{3}{4}x\end{array}\right.\) Answer \(\begin{array}{ll}{}&{\left\{\begin{array}{l}3x+4y=12\\y=3−\frac{3}{4}x\end{array}\right.}\\{}&{}\\{\text{Writethesecondequationinstandardform.}}&{\left\{\begin{array}{l}3x+4y=12\\\frac{3}{4}x+y=3\end{array}\right.}\\{}&{}\\{\text{Clearthefractionsbymultiplyingthe}\\\text{secondequationby4.}}&{\left\{\begin{array}{l}3x+4y=12\\4(\frac{3}{4}x+y)=4(3)\end{array}\right.}\\{}&{}\\{\text{Simplify.}}&{\left\{\begin{array}{l}3x+4y=12\\3x+4y=12\end{array}\right.}\\{}&{}\\{\text{Toeliminateavariable,wemultiplythe}\\\text{secondequationby−1.Simplifyandadd.}}&{\begin{array}{l}{\left\{\begin{array}{l}3x+4y=12\\\underline{-3x-4y=-12}\end{array}\right.}\\{\hspace{16mm}0=0}\end{array}}\\\end{array}\) Thisisatruestatement.Theequationsareconsistentbutdependent.Theirgraphswouldbethesameline.Thesystemhasinfinitelymanysolutions. Afterweclearedthefractionsinthesecondequation,didyounoticethatthetwoequationswerethesame?Thatmeanswehavecoincidentlines. Exercise\(\PageIndex{35}\) Solvethesystembyelimination:\(\left\{\begin{array}{l}5x−3y=15\\5y=−5+\tfrac{5}{3}x\end{array}\right.\) Answer infinitelymanysolutions Exercise\(\PageIndex{36}\) Solvethesystembyelimination:\(\left\{\begin{array}{l}x+2y=6\\y=−\tfrac{1}{2}x+3\end{array}\right.\) Answer infinitelymanysolutions ChoosetheMostConvenientMethodtoSolveaSystemofLinearEquations Whenyousolveasystemoflinearequationsininanapplication,youwillnotbetoldwhichmethodtouse.Youwillneedtomakethatdecisionyourself.Soyou’llwanttochoosethemethodthatiseasiesttodoandminimizesyourchanceofmakingmistakes. \[\textbf{ChoosetheMostConvenientMethodtoSolveaSystemofLinearEquations}\\\begin{array}{lll}{\underline{\textbf{Graphing}}}&{\underline{\textbf{Substitution}}}&{\underline{\textbf{Elimination}}}\\{\text{Usewhenyouneeda}}&{\text{Usewhenoneequationis}}&{\text{Usewhentheequationsa}}\\{\text{pictureofthesituation.}}&{\text{alreadysolvedorcanbe}}&{\text{reinstandardform.}}\\{\text{}}&{\text{easilysolvedforone}}&{\text{}}\\{\text{}}&{\text{variable.}}&{\text{}}\\\end{array}\nonumber\] Example\(\PageIndex{37}\) Foreachsystemoflinearequations,decidewhetheritwouldbemoreconvenienttosolveitbysubstitutionorelimination.Explainyouranswer. ⓐ\(\left\{\begin{array}{l}3x+8y=40\\7x−4y=−32\end{array}\right.\)ⓑ\(\left\{\begin{array}{l}5x+6y=12\\y=\tfrac{2}{3}x−1\end{array}\right.\) Answer ⓐ \[\left\{\begin{array}{l}3x+8y=40\\7x−4y=−32\end{array}\right.\nonumber\] Sincebothequationsareinstandardform,usingeliminationwillbemostconvenient. ⓑ \[\left\{\begin{array}{l}5x+6y=12\\y=\tfrac{2}{3}x−1\end{array}\right.\nonumber\] Sinceoneequationisalreadysolvedfory,usingsubstitutionwillbemostconvenient. Example\(\PageIndex{38}\) Foreachsystemoflinearequationsdecidewhetheritwouldbemoreconvenienttosolveitbysubstitutionorelimination.Explainyouranswer. ⓐ\(\left\{\begin{array}{l}4x−5y=−32\\3x+2y=−1\end{array}\right.\)ⓑ\(\left\{\begin{array}{l}x=2y−1\\3x−5y=−7\end{array}\right.\) Answer ⓐSincebothequationsareinstandardform,usingeliminationwillbemostconvenient.ⓑSinceoneequationisalreadysolvedforx,usingsubstitutionwillbemostconvenient. Example\(\PageIndex{39}\) Foreachsystemoflinearequationsdecidewhetheritwouldbemoreconvenienttosolveitbysubstitutionorelimination.Explainyouranswer. ⓐ\(\left\{\begin{array}{l}y=2x−1\\3x−4y=−6\end{array}\right.\)ⓑ\(\left\{\begin{array}{l}6x−2y=12\\3x+7y=−13\end{array}\right.\) Answer ⓐSinceoneequationisalreadysolvedfory,usingsubstitutionwillbemostconvenient.ⓑSincebothequationsareinstandardform,usingeliminationwillbemostconvenient. KeyConcepts Howtosolveasystemoflinearequationsbygraphing. Graphthefirstequation. Graphthesecondequationonthesamerectangularcoordinatesystem. Determinewhetherthelinesintersect,areparallel,orarethesameline. Identifythesolutiontothesystem. Ifthelinesintersect,identifythepointofintersection.Thisisthesolutiontothesystem. Ifthelinesareparallel,thesystemhasnosolution. Ifthelinesarethesame,thesystemhasaninfinitenumberofsolutions. Checkthesolutioninbothequations. Howtosolveasystemofequationsbysubstitution. Solveoneoftheequationsforeithervariable. SubstitutetheexpressionfromStep1intotheotherequation. Solvetheresultingequation. SubstitutethesolutioninStep3intoeitheroftheoriginalequationstofindtheothervariable. Writethesolutionasanorderedpair. Checkthattheorderedpairisasolutiontobothoriginalequations. Howtosolveasystemofequationsbyelimination. Writebothequationsinstandardform.Ifanycoefficientsarefractions,clearthem. Makethecoefficientsofonevariableopposites. Decidewhichvariableyouwilleliminate. Multiplyoneorbothequationssothatthecoefficientsofthatvariableareopposites. AddtheequationsresultingfromStep2toeliminateonevariable. Solvefortheremainingvariable. SubstitutethesolutionfromStep4intooneoftheoriginalequations.Thensolvefortheothervariable. Writethesolutionasanorderedpair. Checkthattheorderedpairisasolutiontobothoriginalequations.\[\textbf{ChoosetheMostConvenientMethodtoSolveaSystemofLinearEquations}\\\begin{array}{lll}{\underline{\textbf{Graphing}}}&{\underline{\textbf{Substitution}}}&{\underline{\textbf{Elimination}}}\\{\text{}}&{\text{Usewhenoneequationis}}&{\text{}}\\{\text{Usewhenyouneeda}}&{\text{alreadysolvedorcanbe}}&{\text{Usewhentheequationsa}}\\{\text{pictureofthesituation.}}&{\text{easilysolvedforone}}&{\text{reinstandardform.}}\\{\text{}}&{\text{variable.}}&{\text{}}\\\end{array}\nonumber\] Glossary coincidentlines Coincidentlineshavethesameslopeandsamey-intercept. consistentandinconsistentsystems Consistentsystemofequationsisasystemofequationswithatleastonesolution;inconsistentsystemofequationsisasystemofequationswithnosolution. solutionsofasystemofequations Solutionsofasystemofequationsarethevaluesofthevariablesthatmakealltheequationstrue;solutionisrepresentedbyanorderedpair(x,y).(x,y). systemoflinearequations Whentwoormorelinearequationsaregroupedtogether,theyformasystemoflinearequations.