♦ Introduction

♦ Pair of Linear Equations in Two Variables

♦ Pair of Linear Equations in Two Variables

● 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 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?

● 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)

● There are several ways to find the solutions of this pair of equations, which we will study in this chapter.

● 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?

● 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)

● There are several ways to find the solutions of this pair of equations, which we will study in this chapter.

● As the following are examples of linear equations in two variables:

`2x + 3y = 5`

`x – 2y – 3 = 0`

and `x – 0.y = 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`).

`=>` 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, 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

`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`

● 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.

`=>` Both ways of representing a pair of linear equations go hand-in-hand—the algebraic and the geometric ways. Let us consider some examples.

`2x + 3y = 5`

`x – 2y – 3 = 0`

and `x – 0.y = 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`).

`=>` 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, 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

`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`

● 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.

`=>` Both ways of representing a pair of linear equations go hand-in-hand—the algebraic and the geometric ways. Let us consider some examples.

Q 3149167913

Let us take the example given in Section 3.1. Akhila goes to a fair with

Rs 20 and wants to have rides on the Giant Wheel and play Hoopla. Represent this

situation algebraically and graphically (geometrically).

Class 10 Chapter 3 Example 1

Rs 20 and wants to have rides on the Giant Wheel and play Hoopla. Represent this

situation algebraically and graphically (geometrically).

Class 10 Chapter 3 Example 1

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.

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

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.

Q 3159167914

Romila went to a stationery shop and purchased 2 pencils and 3 erasers

for 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. Represent this

situation algebraically and graphically.

Class 10 Chapter 3 Example 2

for 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. Represent this

situation algebraically and graphically.

Class 10 Chapter 3 Example 2

Let us denote the cost of 1 pencil by Rs. x and one eraser by Rs. 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.

These solutions are given below in Table 3.2.

We plot these points in a graph

paper and draw the lines. We find that

both the lines coincide (see Fig. 3.3).

This is so, because, both the

equations are equivalent, i.e., one can

be derived from the other .

Q 3179167916

Two rails are represented by the equations

`x + 2y – 4 = 0` and` 2x + 4y – 12 = 0`.

Represent this situation geometrically.

Class 10 Chapter 3 Example 3

`x + 2y – 4 = 0` and` 2x + 4y – 12 = 0`.

Represent this situation geometrically.

Class 10 Chapter 3 Example 3

Two solutions of each of

the equations :

`x + 2y – 4 = 0` (1)

`2x + 4y – 12 = 0` (2)

are given in Table 3.3

To represent the equations graphically, we plot the points R(0, 2) and S(4, 0), to

get the line RS and the points P(0, 3) and Q(6, 0) to get the line PQ.

We observe in Fig. 3.4, that the

lines do not intersect anywhere, i.e.,

they are parallel.

So, we have seen several

situations which can be represented

by a pair of linear equations. We

have seen their algebraic and

geometric representations. In the

next few sections, we will discuss

how these representations can be

used to look for solutions of the pair

of linear equations.