Solving systems of equations in two variables - Math Planet

A system of a linear equation comprises two or more equations and one seeks a common solution to the equations. In a system of linear equations, ... Algebra2/ Howtosolvesystemoflinearequations/ Solvingsystemsofequationsintwovariables Asystemofalinearequationcomprisestwoormoreequationsandoneseeksacommonsolutiontotheequations.Inasystemoflinearequations,eachequationcorrespondswithastraightlinecorrespondsandoneseeksoutthepointwherethetwolinesintersect. Example Solvethefollowingsystemoflinearequations: $$\left\{\begin{matrix}y=2x+4\\y=3x+2\\\end{matrix}\right.$$ Sinceweareseekingoutthepointofintersection,wemaygraphtheequations: Weseeherethatthelinesintersecteachotheratthepointx=2,y=8.Thisisoursolutionandwemayrefertoitasagraphicsolutiontothetask. Buthowdoesonereachasolutionifthelinesneverintersect?Onecannot,thesystemofequationshavenosolution. Onemayalsoarriveatthecorrectanswerwiththehelpoftheeliminationmethod(alsocalledtheadditionmethodorthelinearcombinationmethod)orthesubstitutionmethod. Whenusingthesubstitutionmethodweusethefactthatiftwoexpressionsyandxareofequalvaluex=y,thenxmayreplaceyorviceversainanotherexpressionwithoutchangingthevalueoftheexpression. Example Solvethesystemsofequationsusingthesubstitutionmethod $$\left\{\begin{matrix}y=2x+4\\y=3x+2\\\end{matrix}\right.$$ Wesubstitutetheyinthetopequationwiththeexpressionforthesecondequation: $$\begin{array}{lcl}2x+4&=&3x+2\\4-2&=&3x-2x\\2&=&x\\\end{array}$$ Todeterminethey-value,wemayproceedbyinsertingourx-valueinanyoftheequations.Weselectthefirstequation: $$y=2x+4$$ Wepluginx=2andget $$y=2\cdot2+4=8$$ Wehavethusarrivedatpreciselythesameanswerasinthegraphicsolution. Theeliminationmethodrequiresustoaddorsubtracttheequationsinordertoeliminateeitherxory,oftenonemaynotproceedwiththeadditiondirectlywithoutfirstmultiplyingeitherthefirstorsecondequationbysomevalue. Example $$2x-2y=8$$ $$x+y=1$$ Wenowwishtoaddthetwoequationsbutitwillnotresultineitherxorybeingeliminated.Thereforewemustmultiplythesecondequationby2onbothsidesandget: $$2x-2y=8$$ $$2x+2y=2$$ Nowweattempttoaddoursystemofequations.Wecommencewiththex-termsontheleft,andthey-termsthereafterandfinallywiththenumbersontherightside: $$(2x+2x)+(-2y+2y)=8+2$$ They-termshavenowbeeneliminatedandwenowhaveanequationwithonlyonevariable: $$4x=10$$ $$x=\frac{10}{4}=2.5$$ Thereafter,inordertodeterminethey-valueweinsertx=2.5inoneoftheequations.Weselectthefirst: $$\begin{array}{lcl}2\cdot2.5-2y&=&8\\5-8&=&2y\\-3&=&2y\\\frac{-3}{2}&=&y\\y&=&-1.5\\\end{array}$$ Videolesson Solvethesystemofequations: $$\left\{\begin{matrix}2x-4y=0\\-4x+4y=-4\end{matrix}\right.$$ NextChapter: HOWTOSOLVESYSTEMOFLINEAREQUATIONS–Solvingsystemsofequationsinthreevariables Search MathPlayground Allcourses MathPlayground Pre-Algebra Allcourses Pre-Algebra Algebra1 Allcourses Algebra1 Algebra2 Allcourses Algebra2 Equationsandinequalities Algebra2 Equationsandinequalities Overview Solveequationsandsimplifyexpressions Lineplotsandstem-and-leafplots Absolutevalue Solveinequalities Howtographfunctionsandlinearequations Algebra2 Howtographfunctionsandlinearequations Overview Functionsandlinearequations Graphfunctionsandrelations Graphinequalities Howtosolvesystemoflinearequations Algebra2 Howtosolvesystemoflinearequations Overview Solvingsystemsofequationsintwovariables Solvingsystemsofequationsinthreevariables AboutMathplanet Cookiesettings Mattecentrum Matteboken Formelsamlingen Pluggakuten Matrices Algebra2 Matrices Overview Basicinformationaboutmatrices Howtooperatewithmatrices Determinants Usingmatriceswhensolvingsystemofequations Polynomialsandradicalexpressions Algebra2 Polynomialsandradicalexpressions Overview Simplifyexpressions Polynomials Factoringpolynomials Solvingradicalequations Complexnumbers Quadraticfunctionsandinequalities Algebra2 Quadraticfunctionsandinequalities Overview Howtographquadraticfunctions Howtosolvequadraticequations TheQuadraticformula Standarddeviationandnormaldistribution ConicSections Algebra2 ConicSections Overview Distancebetweentwopointsandthemidpoint Equationsofconicsections Polynomialfunctions Algebra2 Polynomialfunctions Overview Basicknowledgeofpolynomialfunctions Remainderandfactortheorems Rootsandzeros Descartes'ruleofsign Compositionoffunctions Rationalexpressions Algebra2 Rationalexpressions Overview Variation Operateonrationalexpressions Exponentialandlogarithmicfunctions Algebra2 Exponentialandlogarithmicfunctions Overview Exponentialfunctions Logarithmandlogarithmfunctions Logarithmproperty Sequencesandseries Algebra2 Sequencesandseries Overview Arithmeticsequencesandseries Geometricsequencesandseries Binomialtheorem Discretemathematicsandprobability Algebra2 Discretemathematicsandprobability Overview Countingprinciple Permutationsandcombinations Probabilities Trigonometry Algebra2 Trigonometry Overview Trigonometricfunctions Angles Sines Cosines Circularfunctions Inversefunctions Trigonometricidentities Geometry Allcourses Geometry SAT Allcourses SAT ACT Allcourses ACT