# What is a linear equation?

The pair of the linear equation is in the following form-

The most commonly used algebraic methods of solving a pair of linear equations in two variables are –

a) Substitution method

Let’s discuss these methods one by one:

## Substitution Method

Here are the steps which we follow while solving a pair linear equations:

Step I – Obtain the two equations. Let the equations be

........................(i)

........................(ii)

Step II – Choose either of the two equations, say (i), and find the value of one variable, say y, in terms of the other, i.e. x.

Step III – Substitute the value of y, obtained in step II, in other equation i.e. (ii) to get an equation in x.

Step IV – Solve the equation obtained in step III to get the value of x.

Step V – Substitute the value of x obtained in step IV in the expression for y in terms of x obtained in step II to get the value of y.

Step VI – The values of x and y obtained in steps IV and V respectively constitute the solution of the given system of two linear equations.

Example: Solve the following system of equations by using the method of substitution:

3x – 5y = -1

x – y = -1

Let, 3x – 5y = -1                                                                               … (i)

and x – y = -1                                                                                  … (ii)

From (ii), we get y = x + 1

Substituting y = x + 1 in (i), we get

3x – 5(x + 1) = -1

-2x – 5 = -1

-2x = 4

x = -2

Putting x = -2 in y = x + 1 we get y = -1

Hence, the solution of the given system of equations is x = -2 and y = -1.

Try these questions now:

1. Solve the following system of equations by using the method of substitution:

x + 2y = -1
2x – 3y = 12

(Answer: x = 3 and y = -2)

2. Solve the following system of equations by using the method of substitution:

2x + 3y = 9
3x + 4y = 5

(Answer: x = -21 and y = 17)

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