what do you mean by pair in linear equation in two variables by giving suitable examples explain graphical and Arabic to find out the solution of the variable
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3.1 Introduction
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 Rs 3, and a game of Hoopla costs Rs 4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent Rs 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.
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 (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 Two Variables
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 a2 + b2 ≠ 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.
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
ax1 + b1y + c1 = 0
and a2x + b2y + c2 = 0,
where a1, b1, c1, a2, b2, c2 are all real numbers and a12 + b12 ≠ 0, a22 + b22 ≠ 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:
The two lines will intersect at one point.
The two lines will not intersect, i.e., they are parallel.
The two lines will be coincident.
We show all these possibilities in Fig. 3
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 Rs 3, and a game of Hoopla costs Rs 4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent Rs 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.
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 (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 Two Variables
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 a2 + b2 ≠ 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.
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
ax1 + b1y + c1 = 0
and a2x + b2y + c2 = 0,
where a1, b1, c1, a2, b2, c2 are all real numbers and a12 + b12 ≠ 0, a22 + b22 ≠ 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:
The two lines will intersect at one point.
The two lines will not intersect, i.e., they are parallel.
The two lines will be coincident.
We show all these possibilities in Fig. 3
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