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Solution:, (a) Inthegivengraph,thelinesrepresentingthetwoequationsareparalleltoeachotherand, never intersect. Thus, the lines in this graph represent an inconsistent pair of, linearequationsin twovariablesastheydonothaveanysolution., (b) Inthegivengraph,thelinesrepresentingthetwoequationsintersectatapoint.Thus,the, lines in this graph represent a consistent pair of linear equations in two variables, astheyhaveauniquesolution., (c) In the given graph, the lines are coincident i.e., the same line represents the, twoequations.Thus,thelinesinthisgraphrepresentadependentpairoflinearequationsintw, ovariablesastheyhaveinfinite numberofsolutions., ExpressingGivenSituationsMathematically, Wecomeacrossmanysituationsinreallifewhenitiseasytofindthesolution, ifweexpressthem, mathematically., Letusseesuchasituation., The coach of the school cricket team buys 5 bats and 20 leather balls for Rs 3500., Aftersome time, some more boys joined the team for practice, so he buys another 4 bats, and 15ballsforRs2750.Supposethatthepriceofbatandballdoesnotchangeinthetimeperiod., Can we express this situation mathematically to find out the individual prices of a ball and, abat?, Letthepriceof abatbeRs xandthatofaballbeRsy., Itisgiventhat5batsand20ballscostRs 3500.
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As seen in the graph, the two lines intersect at the point (1, 0). Thus, the given pair of, linearequationsis consistentwith its solution as x =1andy =0., (b)x + y =, 52x+2y=10, Atablecanbedrawnforthecorrespondingvaluesofxandyfortheequation x +y=5as, x, , y=5−x, , 0, , 5−0=5, , 4, , 5−4=1, , Similarly, a table can be drawn for the corresponding values of x and y for the equation 2x, +2y=10as, x, , 3, 0, , Thesepointscannowbeplottedandjoinedtoobtainthegraphsofthelinesas
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As seen in the graph, the two equations are represented by the same line. This means, thatthe given pair of linear equations is dependent. Any point lying on this line is a, solution tothepairofequations., SubstitutionMethodOfSolvingPairsOfLinearEquations, , Inaclass,thenumberofboysis7morethantwicethenumberofgirls., Can you find the number of boys and girls in the, class?Let the number of boys be x and the number of girls, be y.Now,accordingtothegiven condition,, x–2y=7, Wecannotfindtheuniquevaluesof, xandybysolvingthisequationbecausetherearemultiplevaluesof xandyforwhichthis, equationholds true., We can write the above equation as x = 2y + 7. Thus, by taking different values of y, we, willobtaindifferent valuesof x., However,ifwearegivenonemorecondition,thenthevaluesof xandycanbeevaluated., To solve a linear equation in two variables, two linear equations in the same two, variablesarerequired., Letusconsideronemorecondition. Supposethetotalnumberofstudentsintheclassis52.
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Letusaddboththeequationsasfollows:, , ⇒12(x+y)=84, ⇒ x +y=7, , ...(iii), , Now,letussubtractequation(ii)fromequation(i)asfollows:, , ⇒2(x –y)=2, ⇒ x –y=1, , ...(iv), , Onaddingequations(iii)and(iv),weobtain, , ⇒ x =4, On putting the value of x in equation (i), we, obtain7×4+5y=43, ⇒28+5y=43, ⇒5y=15, ⇒ y=3, Thus,x=4andy=3., Example 5: Five units of a solution are obtained by mixing two liquids A and B. If, theliquidsAandBareusedintheratioas2:3thenthetotalcostcomesouttobeRs50.
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Thus,