AIR OF L IN INEAR TWO E V QUATIONS ARIABLES 3

1 IntroductionYou must have come across situations like the one given below :Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheeland play Hoopla (a game in which you throw a ring on the items kept in a stall, and ifthe ring covers any object completely, you get it). The number of times she playedHoopla is half the number of rides she had on the Giant Wheel. If each ride costsRs 3, and a game of Hoopla costs Rs 4, how would you find out the number of ridesshe had and how many times she played Hoopla, provided she spent Rs be you will try it by considering different cases. If she has one ride, is itpossible? Is it possible to have two rides? And so on. Or you may use the knowledgeof Class IX, to represent such situations as linear equations in two OF LINEAR EQUATIONSIN TWO VARIABLESPAIR OF LINEAR EQUATIONS IN TWO VARIABLES39 Let us try this the number of rides that Akhila had by x, and the number of times sheplayed Hoopla by y.

2 Now the situation can be represented by the two equations:y =12x(1)3x + 4y = 20(2)Can we find the solutions of this pair of equations? There are several ways offinding these, which we will study in this Pair of Linear Equations in Two VariablesRecall, from Class IX, that the following are examples of linear equations in twovariables:2x + 3y =5x 2y 3 = 0andx 0y = 2, ,x = 2 You also know that an equation which can be put in the form ax + by + c = 0,where a, b and c are real numbers, and a and b are not both zero, is called a linearequation in two variables x and y. (We often denote the condition a and b are not bothzero by a2 + b2 0). You have also studied that a solution of such an equation is apair of values, one for x and the other for y, which makes the two sides of theequation example, let us substitute x = 1 and y = 1 in the left hand side (LHS) of theequation 2x + 3y = 5.

3 ThenLHS = 2(1) + 3(1) = 2 + 3 = 5,which is equal to the right hand side (RHS) of the , x = 1 and y = 1 is a solution of the equation 2x + 3y = let us substitute x = 1 and y = 7 in the equation 2x + 3y = 5. Then,LHS = 2(1) + 3(7) = 2 + 21 = 23which is not equal to the , x = 1 and y = 7 is not a solution of the , what does this mean? It means that the point (1, 1) lies on the linerepresenting the equation 2x + 3y = 5, and the point (1, 7) does not lie on it. So, everysolution of the equation is a point on the line representing fact, this is true for any linear equation, that is, each solution (x, y) of alinear equation in two variables, ax + by + c = 0, corresponds to a point on theline representing the equation, and vice , consider Equations (1) and (2) given above.

4 These equations, takentogether, represent the information we have about Akhila at the two linear equations are in the same two variables x and y. Equationslike these are called a pair of linear equations in two us see what such pairs look like general form for a pair of linear equations in two variables x and y isa1x + b1y + c1 =0anda2x + b2y + c2 =0,where a1, b1, c1, a2, b2, c2 are all real numbers and a12 + b12 0, a22 + b22 examples of pair of linear equations in two variables are:2x + 3y 7 = 0 and 9x 2y + 8 = 05x = yand 7x + 2y + 3 = 0x + y = 7 and 17 = yDo you know, what do they look like geometrically?Recall, that you have studied in Class IX that the geometrical ( , graphical)representation of a linear equation in two variables is a straight line.

5 Can you nowsuggest what a pair of linear equations in two variables will look like, geometrically?There will be two straight lines, both to be considered have also studied in Class IX that given two lines in a plane, only one of thefollowing three possibilities can happen:(i) The two lines will intersect at one point.(ii)The two lines will not intersect, , they are parallel.(iii)The two lines will be show all these possibilities in Fig. :In Fig. (a), they Fig. (b), they are Fig. (c), they are OF LINEAR EQUATIONS IN TWO VARIABLES41 Fig. ways of representing a pair of linear equations go hand-in-hand thealgebraic and the geometric ways. Let us consider some 1 : Let us take the example given in Section Akhila goes to a fair withRs 20 and wants to have rides on the Giant Wheel and play Hoopla.

6 Represent thissituation algebraically and graphically (geometrically).Solution : The pair of equations formed is :y = ,x 2y = 0(1)3x + 4y = 20(2)Let us represent these equations graphically. For this, we need at least twosolutions for each equation. We give these solutions in Table = 2x01y = 2034x 50 2(i)(ii)Recall from Class IX that there are infinitely many solutions of each linearequation. So each of you can choose any two values, which may not be the ones wehave chosen. Can you guess why we have chosen x = 0 in the first equation and in thesecond equation? When one of the variables is zero, the equation reduces to a linear42 mathematics equation in one variable, which can be solved easily. For instance, putting x = 0 inEquation (2), we get 4y = 20, , y = 5.

7 Similarly, putting y = 0 in Equation (2), we get3x = 20, , x = 203. But as 203 isnot an integer, it will not be easy toplot exactly on the graph paper. So,we choose y = 2 which gives x = 4,an integral the points A(0, 0), B(2, 1)and P(0, 5), Q(4, 2), correspondingto the solutions in Table Nowdraw the lines AB and PQ,representing the equationsx 2y = 0 and 3x + 4y = 20, asshown in Fig. Fig. , observe that the two lines representing the two equations areintersecting at the point (4, 2). We shall discuss what this means in the next 2 : Romila went to a stationery shop and purchased 2 pencils and 3 erasersfor Rs 9. Her friend Sonali saw the new variety of pencils and erasers with Romila,and she also bought 4 pencils and 6 erasers of the same kind for Rs 18.

8 Represent thissituation algebraically and : Let us denote the cost of 1 pencil by Rs x and one eraser by Rs y. Then thealgebraic representation is given by the following equations:2x + 3y = 9(1)4x + 6y = 18(2)To obtain the equivalent geometric representation, we find two points on the linerepresenting each equation. That is, we find two solutions of each OF LINEAR EQUATIONS IN TWO VARIABLES43 Fig. solutions are given below in Table = 923x 30y = 18 46x 31(i)(ii)We plot these points in a graphpaper and draw the lines. We find thatboth the lines coincide (see Fig. ).This is so, because, both theequations are equivalent, , one canbe derived from the 3 : Two rails arerepresented by the equationsx + 2y 4 = 0 and 2x + 4y 12 = this : Two solutions of each ofthe equations :x + 2y 4 = 0(1)2x + 4y 12 = 0(2)are given in Table = 42x 20y = 12 24x 30(i)(ii)To represent the equations graphically, we plot the points R(0, 2) and S(4, 0), toget the line RS and the points P(0, 3) and Q(6, 0) to get the line observe in Fig.

9 , that thelines do not intersect anywhere, ,they are , we have seen severalsituations which can be representedby a pair of linear equations. Wehave seen their algebraic andgeometric representations. In thenext few sections, we will discusshow these representations can beused to look for solutions of the pairof linear tells his daughter, Seven years ago, I was seven times as old as you were , three years from now, I shall be three times as old as you will be. (Isn t thisinteresting?) Represent this situation algebraically and coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys anotherbat and 2 more balls of the same kind for Rs 1300. Represent this situation algebraicallyand cost of 2 kg of apples and 1kg of grapes on a day was found to be Rs 160.

10 After amonth, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situationalgebraically and Graphical Method of Solution of a Pair of Linear EquationsIn the previous section, you have seen how we can graphically represent a pair oflinear equations as two lines. You have also seen that the lines may intersect, or maybe parallel, or may coincide. Can we solve them in each case? And if so, how? Weshall try and answer these questions from the geometrical point of view in this us look at the earlier examples one by the situation of Example 1, find out how many rides on the Giant WheelAkhila had, and how many times she played Fig. , you noted that the equations representing the situation aregeometrically shown by two lines intersecting at the point (4, 2).