Linear Equations in Two Variables (Definition and Solutions)

An equation is said to be linear equation in two variables if it is written in the form of ax + by + c=0, where a, b & c are real numbers and ... MathsMathArticleLinearEquationsInTwoVariables LinearEquationsInTwoVariables Linearequationsintwovariables,explainthegeometryoflinesorthegraphoftwolines,plottedtosolvethegivenequations.Aswealreadyknow,thelinearequationrepresentsastraightline.Theplottingofthesegraphswillhelpustosolvetheequations,whichconsistofunknownvariables.Previouslywehavelearnedtosolvelinearequationsinonevariable,herewewillfindthesolutionsfortheequationshavingtwovariables. Tableofcontents: Definition Solution Example UniqueSolution NoSolution SystemofLinearEquationinTwoVariables ProblemsandSolutions WordProblems FAQs Definition Anequationissaidtobelinearequationintwovariablesifitiswrittenintheformofax+by+c=0,wherea,b&carerealnumbersandthecoefficientsofxandy,i.eaandbrespectively,arenotequaltozero. Forexample,10x+4y=3and-x+5y=2arelinearequationsintwovariables. Thesolutionforsuchanequationisapairofvalues,oneforxandoneforywhichfurthermakesthetwosidesofanequationequal. Also,read:  LinearEquationsInTwoVariablesClass9 SolvingLinearEquations ApplicationsOfLinearEquations SolutionofLinearEquationsinTwoVariables Thesolutionoflinearequationsintwovariables,ax+by=c,isaparticularpointinthegraph,suchthatwhenx-coordinateismultipliedbyaandy-coordinateismultipliedbyb,thenthesumofthesetwovalueswillbeequaltoc.  Basically,forlinearequationintwovariables,thereareinfinitelymanysolutions. Example InordertofindthesolutionofLinearequationin2variables,twoequationsshouldbeknowntous. ConsiderforExample: 5x+3y=30 Theaboveequationhastwovariablesnamelyxandy. Graphicallythisequationcanberepresentedbysubstitutingthevariablestozero. Thevalueofxwheny=0is 5x+3(0)=30 ⇒ x = 6 andthevalueofywhenx=0is, 5(0)+3y=30 ⇒y=10 Itisnowunderstoodthattosolvelinearequationintwovariables,thetwoequationshavetobeknownandthenthesubstitutionmethodcanbefollowed.Let’sunderstandthiswithafewexamplequestions. UniqueSolution Forthegivenlinearequationsintwovariables,thesolutionwillbeuniqueforboththeequations,ifandonlyiftheyintersectatasinglepoint.  Theconditiontogettheuniquesolutionforthegivenlinearequationsis,theslopeofthelineformedbythetwoequations,respectively,shouldnotbeequal. Consider,m1andm2aretwoslopesofequationsoftwolinesintwovariables.So,iftheequationshaveauniquesolution,then: m1≠m2 NoSolution Ifthetwolinearequationshaveequalslopevalue,thentheequationswillhavenosolutions. m1 =m2 Thisisbecausethelinesareparalleltoeachotheranddonotintersect. SystemofLinearEquationsinTwoVariables Insteadoffindingthesolutionforasinglelinearequationintwovariables,wecantaketwosetsoflinearequations,bothhavingtwovariablesinthemandfindthesolutions.So,basicallythesystemoflinearequationsisdefinedwhenthereismorethanonelinearequation.  Forexample,a+b=15anda-b=5,arethesystemoflinearequationsintwovariables.Because,thepointa=10andb=5isthesolutionforbothequations,suchas: a+b=10+5=15 a-b=10-5=5 Hence,provedpoint(10,5)issolutionforbotha+b=15anda-b=5. ProblemsandSolutions Question:Findthevalueofvariableswhichsatisfiesthefollowingequation: 2x+5y=20and3x+6y=12. Solution: Usingthemethodofsubstitutiontosolvethepairoflinearequation,wehave: 2x+5y=20…………………….(i) 3x+6y=12……………………..(ii) Multiplyingequation(i)by3and(ii)by2,wehave: 6x+15y=60…………………….(iii) 6x+12y=24……………………..(iv) Subtractingequation(iv)from(iii) 3y=36 ⇒ y = 12 Substitutingthevalueofyinanyoftheequation(i)or(ii),wehave 2x+5(12)=20 ⇒x=−20 Therefore,x=-20andy=12isthepointwherethegivenequationsintersect. Now,itisimportanttoknowthesituationalexampleswhicharealsoknownaswordproblemsfromlinearequationsin2variables. Check: LinearEquationsCalculator WordProblems Question1: Aboatrunningdownstreamcoversadistanceof20kmin2hourswhileforcoveringthesamedistanceupstream,ittakes5hours.Whatisthespeedoftheboatinstillwater? Solution: Thesetypesofquestionsarethereal-timeexamplesoflinearequationsintwovariables. Inwater,thedirectionalongthestreamiscalleddownstream.And,thedirectionagainstthestreamiscalledupstream. Letusconsiderthespeedofaboatisukm/handthespeedofthestreamisvkm/h,then: SpeedDownstream=(u+v)km/h SpeedUpstream=(u–v)km/h Weknowthat,Speed=Distance/Time So,thespeedofboatwhenrunningdownstream= (20⁄2) km/h = 10 km/h Thespeedofboatwhenrunningupstream=(20⁄5) km/h =4km/h Fromabove,u+v=10…….(1) u–v=4……….(2) Addingequation1and2,weget:2u=1 u=7km/h Also,v=3km/h Therefore,thespeedoftheboatinstillwater=u=7km/h Question2:Aboatrunningupstreamtakes6hours30minutestocoveracertaindistance,whileittakes3hourstocoverthesamedistancerunningdownstream.Whatistheratiobetweenthespeedoftheboatandspeedofthewatercurrent,respectively? Solution:Ifthespeeddownstreamis a km/hr andthespeedupstreamis b km/hr,then Speedinstillwater=(a+b)/2km/h Rateofstream=½(a−b)kmph LettheBoat’srateupstreambe x kmphandthatdownstreambe y kmph. Then,distancecoveredupstreamin6hrs30min=Distancecovereddownstreamin3hrs. ⇒x×6.5hrs= y×3hrs ⇒13/2x = 3y ⇒y =13x/6 \(\begin{array}{l}The\required\ratio\is\\frac{y+x}{2}~:~\frac{y–x}{2}\end{array}\) \(\begin{array}{l}\Rightarrow~\frac{\frac{13x}{6}~+~x}{2}~:~\frac{\frac{13x}{6}~-~x}{2}\end{array}\) \(\begin{array}{l}\Rightarrow~\frac{\frac{19x}{6}}{2}~:~\frac{\frac{7x}{6}}{2}\end{array}\) =19:7 FrequentlyAskedQuestions–FAQsHowtosolvelinearequationintwovariables?Forasystemoflinearequationsintwovariables,wecanfindthesolutionsbytheeliminationmethod.Howmanysolutionsarethereforlinearequationsintwovariables?Forlinearequationsintwovariables,thereareinfinitelymanysolutions.Whatisthetwo-variableequation?Alinearequationintwovariablesisanequationwhichhastwodifferentsolutions.Whatarethecoefficientsoftheequation3x-6y=-13?Thecoefficientofxis3andthecoefficientofyis-6.Whatistheconstantoftheequation3x-6y=-13?Theconstantoftheequation3x-6y=-13is-13. Quizonlinearequationintwovariables Q5 PutyourunderstandingofthisconcepttotestbyansweringafewMCQs.Click‘StartQuiz’tobegin! Selectthecorrectanswerandclickonthe“Finish”buttonCheckyourscoreandanswersattheendofthequiz StartQuiz Congrats! VisitBYJU’SforallMathsrelatedqueriesandstudymaterials Yourresultisasbelow 0outof0arewrong 0outof0arecorrect 0outof0areUnattempted ViewQuizAnswersandAnalysis MATHSRelatedLinks WhatIsMathematics LinearEquationsInOneVariable DivisionAlgorithmForPolynomials DifferentialEquation ExponentialDistribution SurfaceAreaOfRectangle EquationOfASphere RightAngledTriangle AreaOfEquilateralTriangle GraphingOfLinearEquations 3Comments SohailAhmad September7,2020at8:58pm Excellentservice Reply shahzadhyder December30,2020at5:45pm excellentjob Reply MANZOORAHMADBHAT April7,2022at6:53pm Helpingthestudentswell Reply LeaveaCommentCancelreplyYourMobilenumberandEmailidwillnotbepublished.Requiredfieldsaremarked* * SendOTP DidnotreceiveOTP? 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