## Oct 25, 2009

### Elimination Method: Linear Equations (elimination method using addition and subtraction)

Elimination method is one of the best methods of solving the linear equations. In this method, we eliminate one variable from the equations and then find the value of the variable. Then substitute the value of the variable in either original equation to find the value of the eliminated variable.
Solving the system of linear equations by elimination method can be done in three ways. They are:-
b) Elimination using subtraction.
c) Elimination using multiplication.
In this post we are going to discuss elimination using addition and subtraction. Elimination method using multiplication will be discussed in next post.

In this method, we eliminate one like variable by adding the two equations and solve the resulting equation for the other variable and substitute the value of this variable in either of the given equations to find the value of the eliminated variable.
Elimination method of addition is easy to use for the equations with at least one like variable whose co-efficient are additive inverse of each other.
Let us explain this method with an example.
Example:
Solve the system of equations by elimination method.
4x - 2y = 14
6x + 2y = 16
Solution:
Step 1:
The given system of equations is
4x – 2y = 14
6x + 2y = 16
Observe that the coefficients of y terms are -2 and 2 which are additive inverses. So, by adding the above two equations, we can eliminate the y variable and find the value of the variable x.

Step 2:
Write the two equations one below the other as shown and add.
4x - 2y = 14
(+) 6x + 2y = 16
----------------------
10x      = 30
Now, the variable y is eliminated.
Divide by 10 on both the sides to find the value of x.
So, x = 3.
Step 3:
Now, substitute the value of x = 3 in either of the given equations to find the value of y. Let us substitute in the first equation 4x - 2y = 14 and simplify.
4(3) - 2y = 14
12 - 2y = 14
Subtract 12 from both the sides and simplify.
12 - 2y - 12 = 14 - 12
-2y = 2
Divide both the sides by -2 to isolate y.
y = -1
Step 4:
So, the solution of the given system of linear equations is (3, -1).

Elimination Method Using Subtraction:
In this method, we eliminate one variable (with the same coefficient) by subtracting the two equations to find the value of other variable and then substitute the value of this variable in either of the given equations to find the value of the eliminated variable.
Let us explain this method with an example.
Example:
Solve the system of equations by elimination method.
5m + 6n = 7
5m + 2n = -1
Solution:
Step 1:
Observe that the co-efficient of m terms are the same.
Subtract the given equations to eliminate the m terms and find the value of the variable n.
Step 2:
Write the two equations one below the other as shown and subtract.
5m + 6n = 7
(-) 5m + 2n = -1                     [-(5m + 2n = -1) = -5m - 2n = 1]
---------------------
4n = 8
Now, the variable m is eliminated.
Divide both the sides by 4 to find the value of n.
n = 2
Step 3:
Now, substitute the value of n = 2 in either of the equations to find the value of m.
Let us substitute in the second equation 5m + 2n = -1 and simplify.
5m + 2(2) = -1
5m + 4 = -1
Subtract 4 from both sides of the equation and simplify.
5m + 4 - 4 = -1 - 4
5m = -5
Divide by 5 on both the sides to isolate m.
m = -1
Step 4:
So, (-1, 2) is a solution for the given system of equations.

Practice Questions:
Solve the following linear equations using Elimination method:
a) 2x – 7y = 12; -3x + 7y = -11

b) 6x + 5y = -5; -6x – 7y = -5