SIMULTANEOUS LINEAR EQUATION - SUBSTITUTION METHOD
Substitution Method –
There are some steps of substitution method are given below –
Step.1) Using one of the equations write ‘y’ in terms of ‘x’ or ‘x’ in terms of ‘y’ and the constant.
Step.2) Substitute the expression for ‘y’ or ‘x’ in the second equation.
Step.3) Solve the resulting linear equations in ‘x’ or ‘y’.
Step.4) Substitute the value of ‘x’ or ‘y’ in either of the equations
Step.5) Solve the resulting linear equations in ‘y’ or ‘x’
Step.6) Verify the correctness of the solution by substituting the values of ‘x’ & ‘y’ in the given equations.
There are some examples are given below for your better understanding –
Example.1) Solve the equations 5a + b = 10 and 14a + 3b = 18
Ans.) that has observed that the given equations are 5a + b = 10 ………………………………. (1)
14a + 3b = 18 …………………………………….. (2)
From the equation (1), 5a + b = 10 or b = 10 – 5a
Now, we would like to substitute the expression 10 – 5a for b in equation (2)
So, 14a + 3b = 18
or, 14a + 3 (10 – 5a) = 18
or, 14a + 30 – 15a = 18
or, 14a – 15a = - 30 + 18
or, - a = - 12
or, a = 12
substituting the value of a = 12 in equation (1) we get
so, b = 10 – 5a
or, b = 10 – 5 (12) = 10 – 60 = - 50
so, the required value is a = 12 & b = - 50 (Ans.)
Example.2) Solve the equations x + 8y = 20 and 13x + 14y = - 10
Ans.) that has observed that the given equations are x + 8y = 20 ………………………………. (1)
13x + 14y = - 10 …………………………………….. (2)
From the equation (1), x + 8y = 20 or x = 20 – 8y
Now, we would like to substitute the expression 20 – 8y for b in equation (2)
So, 13x + 14y = - 10
or, 13 (20 – 8y) + 14y = - 10
or, 260 – 104y + 14y = - 10
or, 260 + 10 = 104y – 14y
or, 90y = 270
or, y = 270/90 = 3
substituting the value of y = 3 in equation (1) we get
so, x + 8y = 20
or, x + (8 X 3) = 20
or, x + 24 = 20
or, x = - 24 + 20
or, x = - 4
so, the required value is x = - 4 & y = 3 (Ans.)
Example.3) Solve the equations 5x + y = 20 and 10y + 5x = 10
Ans.) That has observed that the given equations are 5x + y = 20 ………………………………. (1)
10y + 5x = 10 …………………………………….. (2)
From the equation (1), 5x + y = 20 or y = 20 – 5x
Now, we would like to substitute the expression 20 – 5x for y in equation (2)
So, 10y + 5x = 10
or, 10 (20 – 5x) + 5x = 10
or, 200 – 50x + 5x = 10
or, 200 - 10 = 50x – 5x
or, 45x = 190
or, x = 190/45 = 4.2
substituting the value of x = 4.2 in equation (1) we get
so, 5x + y = 20
or, (5 X 4.2) + y = 20
or, 21 + y = 20
or, x = - 21 + 20
or, x = - 1
so, the required value is x = -1 and y = 4.2 (Ans.)