## Oct 20, 2009

### Substitution Method: Solve Linear Equations

Substitution method is an algebraic method of solving the system of linear equations. Using this method, we can find the exact solution for the equations.
Let us explain this method using few examples.
Example 1:
Solve the system of linear equations using substitution method.
y = 2x
x + 2y = 5
Solution:
Step 1:
Given equations are y = 2x and x + 2y = 5.
The value of y is directly given in the first equation. So, substitute the value of y = 2x in the second equation x + 2y = 5 and simplify.
So, x + 2y = 5 becomes x + 2(2x) = 5.
x + 4x = 5
5x = 5
Divide by 5 on both the sides to isolate x.
x = 1
Step 2:
Substitute the value x = 1 to find the value of y.
Replace the value of x by 1 in the equation y = 2x.
y = 2 (1)
y = 2
Step 3:
So, the solution of the given system of equations is (1, 2).
Example 2:
Using substitution method, solve the given linear equations.
xy = 3
3x + 2y = 9
Solution:
Step 1:
Given equations are x - y = 3 and 3x + 2y = 9.
We need to find the value of variables by substituting the value of one variable in the other equation.
Let us find the value of x from the first equation.
To find the value of x, solve for x.
Add y on both the sides of the equation x - y = 3 to find the value of x and then simplify.
x - y + y = 3 + y
x = y + 3
Step 2:
Replace the value of x by y + 3 in the second equation 3x + 2y = 9 and simplify it.
3(y + 3) + 2y = 9
3y + 9 + 2y = 9
Combine the like terms.
(3y + 2y) + 9 = 9
5y + 9 = 9
Subtract 9 from both the sides.
5y + 9 - 9 = 9 - 9
5y = 0
Divide by 5 on both the sides.
y = 0
Step 3:
Now, substitute the value of y = 0 in any of the equation whichever is easy to solve.
So, substitute the value of y in the first equation x - y = 3.
x - 0 = 3
x = 3
Step 4:
So, the solution is (3, 0).
Example 3:
Use substitution method to solve the given system of linear equations.
2x + y = 4
4x + 2y = 8
Solution:
Step 1:

Given equations are 2x + y = 4 and 4x + 2y = 8.
We need to find the value of variables by substituting the value of one variable in the other equation.
Let us find the value of y from the first equation 2x + y = 4.
Subtract 2x from both the sides,
2x - 2x + y = 4 - 2x
y = 4 - 2x
Step 2:
Now, substitute the value of y = 4 - 2x in the second equation 4x + 2y = 8.
4x + 2(4 - 2x) = 8
4x + 8 - 4x = 8
8 = 8
Step 3:
We end up with a statement 8 = 8. This is a true statement which means that the given system of equations has infinitely many solutions.
Important note:
While solving the system of equations, we may arrive at some statements like 5 = 5, -5 = -5, and 3 = 4.
If we arrive at a true statement (E.g. 4 = 4), then the system has infinitely many solutions and the graph of both the equations are the same.
If we arrive at a false statement (E.g. 4 = 3), then the system has no solution and the lines are parallel.
Practice questions:
Use substitution method to solve the given system of equations.
a) -2x + 9y = -9; 9x – 2y = 2

b) x + 2y = -3; -2x + 4y = 10