# Approach to solve Simultaneous Equations : Elimination Method

**Approach 1 : Elimination Method **

In this method we try to eliminate one of the two variables in the linear equations. After elimination, an equation with only one variable is left. Simplifying the equation, we get the solution to one of the variables and subsequently solution to the other variable.

To eliminate a variable, the first and foremost step is to make the coefficient of the variable equal in both the equations and then the equations are either added or subtracted depending on the sign of the variable in both the equations.

There may be 2 cases possible in elimination method

i.) Coefficient of one of the variables is same in both the equations.

ii.) Coefficient of both the variables is different in both the equations.

**Case i. Coefficient of one of the variables is same in both the equations. **

There may be instances when the problem statement results in equations with variable having same coefficient. The two equations may be added or subtracted to eliminate one variable very easily resulting in a simple one variable equation which can be reduced to provide solution.

**E.g.: 3x + y = 23 **……………(a)

** 8x – y = 21** ……………(b)

Here we can see that variable x has different coefficient, but variable ‘y’ has same coefficient in different sign. To eliminate variable ‘y’ we just add two equations.

So, on adding the two equations, we get

11x = 44

x = 4

Now, substituting the value of ‘x’ in the first equation, we get

y = 23 – 3x = 11

The solution is (4,11)

Let's look at the graph of the two equations3x + y = 23 and 8x - y = 21on the same plot.

**Case ii. Coefficient of both variables is different in the two equations.**

There may be other instances when the problem statement results in equations with variable having different coefficient.

In this case, we either multiply or divide one/both equations with real numbers (except 0) which will result in equations with variable having same coefficient. After this we can follow as discussed in Case i as above.

**E.g.: x + 2y = 4 **…………..(a)** 3x – y = 5** …………..(b)

Multiplying 1st equation (a) by 3 we get 3x + 6y = 12 ………………(c)

Subtracting (b) from (c), we get 7y = 7

y = 1

x = 4 – 2y = 2

The solution is (2,1)

We can also multiply 2nd equation (b) by 2 to get 6x – 2y = 10 ……………(d)

Adding (a) and (d), we get 7x = 14

x = 2

y = 3x -5 = 3(2) – 5 = 1

The solution is (2,1)

Let's look at the graph of the two equationsx + 2y = 4and3x - y = 5on the same plot.

There are two other approach by which we can solve simultaneous equations.*Approach 2 : Substitution MethodApproach 3 : Graphical Method*

We will discuss these approaches in the next post.