Substitution Method

Observe the steps how to solve the system of linear equations by using the substitution method.

(i) Find the value of one variable in terms of the other from one of the given equations.

(ii) Substitute the value of this variable in the other equation.

(iii) Solve the equation and get the value of one of the variables.

(iv) Substitute the value of this variable in any of the equation to get the value of other variable.

Follow the instructions along with the method of solution of the two simultaneous equations given below to find the value of x and y.

7x – 3y = 31 --------- (i)

9x – 5y = 41 --------- (ii)

Step I:

From equation (i) 7x – 3y = 31, express y in terms of x

From equation (i) 7x – 3y = 31, we get;

– 3y = 31 – 7x

or, 3y = 7x – 31

or, 3y/3 = (7x – 31)/3

Therefore, y = (7x – 31)/3 --------- (iii)

Step II:

Substitute the value of y obtained from equation (iii) (7x – 31)/3 in equation (ii) 9x – 5y = 41

Putting the value of y obtained from equation (iii) in equation (ii) we get;

9x – 5 × (7x – 31)/3 = 41 --------- (iv)

Step III:

Now, solve equation (iv) 9x – 5 × (7x – 31)/3 = 41

Simplifying equation (iv) 9x – 5 × (7x – 31)/3 = 41 we get;

(27x – 35x + 155)/3 = 41

or, 27x – 35x + 155 = 41 × 3

or, 27x – 35x + 155 = 123

or, –8x + 155 = 123

or, –8x + 155 – 155 = 123 – 155

or, –8x = –32

or, 8x/8 = 32/8

Therefore, x = 4

Step IV:

Putting the value of x in equation (iii)

y = (7x – 31)/3, find the value of y

Putting x = 4 in equation (iii), we get;

y = (7 × 4 – 31)/3

or, y = (28 – 31)/3

or, y = –3/3

Therefore, y = –1

Step V:

Write down the required solution of the two simultaneous linear equations by using the substitution method

Therefore, x= 4 and y = –1

In this case, the general method obtained for solving simultaneous equations as follows:

1. To express y in terms of x from any one of the equations.

2. To substitute this value of y in the other equation.

3. One value of x will be obtained, by solving the equation in x thus obtained.

4. Substituting this value of x in any of the equations, we will get the corresponding value of y.

5. Solution of the two given simultaneous equations will be given by this pair of values of x and y.

6. Similarly expressing x in terms of y from an equation and substituting in the other, we can find the value of y. Putting this value of y in any one of the equations, we can find the value of x and thus we can solve the two linear simultaneous equations.

As in this method of solution, we express one unknown quantity in terms of the other and substitute in an equation; o we call this method as ‘Method of Substitution’.

Keep these instructions in your mind and notice how the following simultaneous equations can be solved.

Worked-out examples on two variables linear equations by using the substitution method:

2/x + 3/y = 2 --------- (i)

5/x + 10/y = 5⁵/₆ --------- (ii)

From equation (i), we get:

3/y = 2 – 2/x

or, 3/y = (2x – 2)/x

or, y/3 = x/(2x – 2)

or, y = 3x/(2x – 2) --------- (iii)

Substituting 3x/(2x – 2) in place of y in equation (ii),

or, 5/x + 10 ÷ 3x/(2x – 2) = 35/6

or, 5/x + 10(2x – 2)/3x = 35/6

or, 1/x + 2(2x – 2)/3x = 7/6

or, (3 + 4x – 4)/3x = 7/6

or, (4x – 1)/3x = 7/6

or, (4x – 1)/x = 7/2