Page 1 38 MATHEMA TICS 3 3.1 Introduction You 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 Wheel and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs ` 3, and a game of Hoopla costs ` 4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ` 20. May be you will try it by considering different cases. If she has one ride, is it possible? Is it possible to have two rides? And so on. Or you may use the knowledge of Class IX, to represent such situations as linear equations in two variables. PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 202021 Page 2 38 MATHEMA TICS 3 3.1 Introduction You 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 Wheel and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs ` 3, and a game of Hoopla costs ` 4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ` 20. May be you will try it by considering different cases. If she has one ride, is it possible? Is it possible to have two rides? And so on. Or you may use the knowledge of Class IX, to represent such situations as linear equations in two variables. PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 202021 PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 39 Let us try this approach. Denote the number of rides that Akhila had by x, and the number of times she played Hoopla by y. Now the situation can be represented by the two equations: y = 1 2 x (1) 3x + 4y = 20 (2) Can we find the solutions of this pair of equations? There are several ways of finding these, which we will study in this chapter. 3.2 Pair of Linear Equations in T wo V ariables Recall, from Class IX, that the following are examples of linear equations in two variables: 2x + 3y = 5 x – 2y – 3 = 0 and x – 0y = 2, i.e., 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 linear equation in two variables x and y. (We often denote the condition a and b are not both zero by a 2 + b 2 ? 0). You have also studied that a solution of such an equation is a pair of values, one for x and the other for y, which makes the two sides of the equation equal. For example, let us substitute x = 1 and y = 1 in the left hand side (LHS) of the equation 2x + 3y = 5. Then LHS = 2(1) + 3(1) = 2 + 3 = 5, which is equal to the right hand side (RHS) of the equation. Therefore, x = 1 and y = 1 is a solution of the equation 2x + 3y = 5. Now let us substitute x = 1 and y = 7 in the equation 2x + 3y = 5. Then, LHS = 2(1) + 3(7) = 2 + 21 = 23 which is not equal to the RHS. Therefore, x = 1 and y = 7 is not a solution of the equation. Geometrically, what does this mean? It means that the point (1, 1) lies on the line representing the equation 2x + 3y = 5, and the point (1, 7) does not lie on it. So, every solution of the equation is a point on the line representing it. 202021 Page 3 38 MATHEMA TICS 3 3.1 Introduction You 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 Wheel and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs ` 3, and a game of Hoopla costs ` 4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ` 20. May be you will try it by considering different cases. If she has one ride, is it possible? Is it possible to have two rides? And so on. Or you may use the knowledge of Class IX, to represent such situations as linear equations in two variables. PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 202021 PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 39 Let us try this approach. Denote the number of rides that Akhila had by x, and the number of times she played Hoopla by y. Now the situation can be represented by the two equations: y = 1 2 x (1) 3x + 4y = 20 (2) Can we find the solutions of this pair of equations? There are several ways of finding these, which we will study in this chapter. 3.2 Pair of Linear Equations in T wo V ariables Recall, from Class IX, that the following are examples of linear equations in two variables: 2x + 3y = 5 x – 2y – 3 = 0 and x – 0y = 2, i.e., 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 linear equation in two variables x and y. (We often denote the condition a and b are not both zero by a 2 + b 2 ? 0). You have also studied that a solution of such an equation is a pair of values, one for x and the other for y, which makes the two sides of the equation equal. For example, let us substitute x = 1 and y = 1 in the left hand side (LHS) of the equation 2x + 3y = 5. Then LHS = 2(1) + 3(1) = 2 + 3 = 5, which is equal to the right hand side (RHS) of the equation. Therefore, x = 1 and y = 1 is a solution of the equation 2x + 3y = 5. Now let us substitute x = 1 and y = 7 in the equation 2x + 3y = 5. Then, LHS = 2(1) + 3(7) = 2 + 21 = 23 which is not equal to the RHS. Therefore, x = 1 and y = 7 is not a solution of the equation. Geometrically, what does this mean? It means that the point (1, 1) lies on the line representing the equation 2x + 3y = 5, and the point (1, 7) does not lie on it. So, every solution of the equation is a point on the line representing it. 202021 40 MATHEMA TICS In fact, this is true for any linear equation, that is, each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa. Now, consider Equations (1) and (2) given above. These equations, taken together, represent the information we have about Akhila at the fair. These two linear equations are in the same two variables x and y. Equations like these are called a pair of linear equations in two variables. Let us see what such pairs look like algebraically. The general form for a pair of linear equations in two variables x and y is a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0, where a 1 , b 1 , c 1 , a 2 , b 2 , c 2 are all real numbers and a 1 2 + b 1 2 ? 0, a 2 2 + b 2 2 ? 0. Some examples of pair of linear equations in two variables are: 2x + 3y – 7 = 0 and 9x – 2y + 8 = 0 5x = y and –7x + 2y + 3 = 0 x + y = 7 and 17 = y Do you know, what do they look like geometrically? Recall, that you have studied in Class IX that the geometrical (i.e., graphical) representation of a linear equation in two variables is a straight line. Can you now suggest what a pair of linear equations in two variables will look like, geometrically? There will be two straight lines, both to be considered together. You have also studied in Class IX that given two lines in a plane, only one of the following three possibilities can happen: (i) The two lines will intersect at one point. (ii) The two lines will not intersect, i.e., they are parallel. (iii) The two lines will be coincident. We show all these possibilities in Fig. 3.1: In Fig. 3.1 (a), they intersect. In Fig. 3.1 (b), they are parallel. In Fig. 3.1 (c), they are coincident. 202021 Page 4 38 MATHEMA TICS 3 3.1 Introduction You 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 Wheel and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs ` 3, and a game of Hoopla costs ` 4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ` 20. May be you will try it by considering different cases. If she has one ride, is it possible? Is it possible to have two rides? And so on. Or you may use the knowledge of Class IX, to represent such situations as linear equations in two variables. PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 202021 PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 39 Let us try this approach. Denote the number of rides that Akhila had by x, and the number of times she played Hoopla by y. Now the situation can be represented by the two equations: y = 1 2 x (1) 3x + 4y = 20 (2) Can we find the solutions of this pair of equations? There are several ways of finding these, which we will study in this chapter. 3.2 Pair of Linear Equations in T wo V ariables Recall, from Class IX, that the following are examples of linear equations in two variables: 2x + 3y = 5 x – 2y – 3 = 0 and x – 0y = 2, i.e., 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 linear equation in two variables x and y. (We often denote the condition a and b are not both zero by a 2 + b 2 ? 0). You have also studied that a solution of such an equation is a pair of values, one for x and the other for y, which makes the two sides of the equation equal. For example, let us substitute x = 1 and y = 1 in the left hand side (LHS) of the equation 2x + 3y = 5. Then LHS = 2(1) + 3(1) = 2 + 3 = 5, which is equal to the right hand side (RHS) of the equation. Therefore, x = 1 and y = 1 is a solution of the equation 2x + 3y = 5. Now let us substitute x = 1 and y = 7 in the equation 2x + 3y = 5. Then, LHS = 2(1) + 3(7) = 2 + 21 = 23 which is not equal to the RHS. Therefore, x = 1 and y = 7 is not a solution of the equation. Geometrically, what does this mean? It means that the point (1, 1) lies on the line representing the equation 2x + 3y = 5, and the point (1, 7) does not lie on it. So, every solution of the equation is a point on the line representing it. 202021 40 MATHEMA TICS In fact, this is true for any linear equation, that is, each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa. Now, consider Equations (1) and (2) given above. These equations, taken together, represent the information we have about Akhila at the fair. These two linear equations are in the same two variables x and y. Equations like these are called a pair of linear equations in two variables. Let us see what such pairs look like algebraically. The general form for a pair of linear equations in two variables x and y is a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0, where a 1 , b 1 , c 1 , a 2 , b 2 , c 2 are all real numbers and a 1 2 + b 1 2 ? 0, a 2 2 + b 2 2 ? 0. Some examples of pair of linear equations in two variables are: 2x + 3y – 7 = 0 and 9x – 2y + 8 = 0 5x = y and –7x + 2y + 3 = 0 x + y = 7 and 17 = y Do you know, what do they look like geometrically? Recall, that you have studied in Class IX that the geometrical (i.e., graphical) representation of a linear equation in two variables is a straight line. Can you now suggest what a pair of linear equations in two variables will look like, geometrically? There will be two straight lines, both to be considered together. You have also studied in Class IX that given two lines in a plane, only one of the following three possibilities can happen: (i) The two lines will intersect at one point. (ii) The two lines will not intersect, i.e., they are parallel. (iii) The two lines will be coincident. We show all these possibilities in Fig. 3.1: In Fig. 3.1 (a), they intersect. In Fig. 3.1 (b), they are parallel. In Fig. 3.1 (c), they are coincident. 202021 PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 41 Fig. 3.1 Both ways of representing a pair of linear equations go handinhand—the algebraic and the geometric ways. Let us consider some examples. Example 1 : Let us take the example given in Section 3.1. Akhila goes to a fair with ` 20 and wants to have rides on the Giant Wheel and play Hoopla. Represent this situation algebraically and graphically (geometrically). Solution : The pair of equations formed is : y = 1 2 x i.e., x – 2y = 0 (1) 3x + 4y = 20 (2) Let us represent these equations graphically. For this, we need at least two solutions for each equation. We give these solutions in Table 3.1. Table 3.1 x 0 2 x 0 20 3 4 y = 2 x 0 1 y = 20 3 4 x  5 0 2 (i) (ii) Recall from Class IX that there are infinitely many solutions of each linear equation. So each of you can choose any two values, which may not be the ones we have chosen. Can you guess why we have chosen x = 0 in the first equation and in the second equation? When one of the variables is zero, the equation reduces to a linear 202021 Page 5 38 MATHEMA TICS 3 3.1 Introduction You 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 Wheel and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs ` 3, and a game of Hoopla costs ` 4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ` 20. May be you will try it by considering different cases. If she has one ride, is it possible? Is it possible to have two rides? And so on. Or you may use the knowledge of Class IX, to represent such situations as linear equations in two variables. PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 202021 PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 39 Let us try this approach. Denote the number of rides that Akhila had by x, and the number of times she played Hoopla by y. Now the situation can be represented by the two equations: y = 1 2 x (1) 3x + 4y = 20 (2) Can we find the solutions of this pair of equations? There are several ways of finding these, which we will study in this chapter. 3.2 Pair of Linear Equations in T wo V ariables Recall, from Class IX, that the following are examples of linear equations in two variables: 2x + 3y = 5 x – 2y – 3 = 0 and x – 0y = 2, i.e., 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 linear equation in two variables x and y. (We often denote the condition a and b are not both zero by a 2 + b 2 ? 0). You have also studied that a solution of such an equation is a pair of values, one for x and the other for y, which makes the two sides of the equation equal. For example, let us substitute x = 1 and y = 1 in the left hand side (LHS) of the equation 2x + 3y = 5. Then LHS = 2(1) + 3(1) = 2 + 3 = 5, which is equal to the right hand side (RHS) of the equation. Therefore, x = 1 and y = 1 is a solution of the equation 2x + 3y = 5. Now let us substitute x = 1 and y = 7 in the equation 2x + 3y = 5. Then, LHS = 2(1) + 3(7) = 2 + 21 = 23 which is not equal to the RHS. Therefore, x = 1 and y = 7 is not a solution of the equation. Geometrically, what does this mean? It means that the point (1, 1) lies on the line representing the equation 2x + 3y = 5, and the point (1, 7) does not lie on it. So, every solution of the equation is a point on the line representing it. 202021 40 MATHEMA TICS In fact, this is true for any linear equation, that is, each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa. Now, consider Equations (1) and (2) given above. These equations, taken together, represent the information we have about Akhila at the fair. These two linear equations are in the same two variables x and y. Equations like these are called a pair of linear equations in two variables. Let us see what such pairs look like algebraically. The general form for a pair of linear equations in two variables x and y is a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0, where a 1 , b 1 , c 1 , a 2 , b 2 , c 2 are all real numbers and a 1 2 + b 1 2 ? 0, a 2 2 + b 2 2 ? 0. Some examples of pair of linear equations in two variables are: 2x + 3y – 7 = 0 and 9x – 2y + 8 = 0 5x = y and –7x + 2y + 3 = 0 x + y = 7 and 17 = y Do you know, what do they look like geometrically? Recall, that you have studied in Class IX that the geometrical (i.e., graphical) representation of a linear equation in two variables is a straight line. Can you now suggest what a pair of linear equations in two variables will look like, geometrically? There will be two straight lines, both to be considered together. You have also studied in Class IX that given two lines in a plane, only one of the following three possibilities can happen: (i) The two lines will intersect at one point. (ii) The two lines will not intersect, i.e., they are parallel. (iii) The two lines will be coincident. We show all these possibilities in Fig. 3.1: In Fig. 3.1 (a), they intersect. In Fig. 3.1 (b), they are parallel. In Fig. 3.1 (c), they are coincident. 202021 PAIR OF LINEAR EQUA TIONS IN TWO VARIABLES 41 Fig. 3.1 Both ways of representing a pair of linear equations go handinhand—the algebraic and the geometric ways. Let us consider some examples. Example 1 : Let us take the example given in Section 3.1. Akhila goes to a fair with ` 20 and wants to have rides on the Giant Wheel and play Hoopla. Represent this situation algebraically and graphically (geometrically). Solution : The pair of equations formed is : y = 1 2 x i.e., x – 2y = 0 (1) 3x + 4y = 20 (2) Let us represent these equations graphically. For this, we need at least two solutions for each equation. We give these solutions in Table 3.1. Table 3.1 x 0 2 x 0 20 3 4 y = 2 x 0 1 y = 20 3 4 x  5 0 2 (i) (ii) Recall from Class IX that there are infinitely many solutions of each linear equation. So each of you can choose any two values, which may not be the ones we have chosen. Can you guess why we have chosen x = 0 in the first equation and in the second equation? When one of the variables is zero, the equation reduces to a linear 202021 42 MATHEMA TICS equation in one variable, which can be solved easily. For instance, putting x = 0 in Equation (2), we get 4y = 20, i.e., y = 5. Similarly, putting y = 0 in Equation (2), we get 3x = 20, i.e., x = 20 3 . But as 20 3 is not an integer, it will not be easy to plot exactly on the graph paper. So, we choose y = 2 which gives x = 4, an integral value. Plot the points A(0, 0), B(2, 1) and P(0, 5), Q(4, 2), corresponding to the solutions in Table 3.1. Now draw the lines AB and PQ, representing the equations x – 2y = 0 and 3x + 4y = 20, as shown in Fig. 3.2. In Fig. 3.2, observe that the two lines representing the two equations are intersecting at the point (4, 2). We shall discuss what this means in the next section. Example 2 : Romila went to a stationery shop and purchased 2 pencils and 3 erasers for ` 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 ` 18. Represent this situation algebraically and graphically. Solution : Let us denote the cost of 1 pencil by ` x and one eraser by ` y. Then the algebraic 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 line representing each equation. That is, we find two solutions of each equation. Fig. 3.2 202021Read More
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