If you are asked to solve a pair of simultaneous equations what you need to do is find the values of x and y which satisfy both equations. There are two methods that you can use to solve simultaneous equations – substitution method or elimination method. In this article we shall be solving the simultaneous equations by elimination.

To solve simultaneous equations by elimination follow these simple steps:

**Step 1** Make the coefficients of x (or y) the same in both equations by multiplying the equations. Sometimes only 1 equation may need to be multiplied to make the coefficients the same. Or in easier examples the coefficients of x (or y) may already be the same in both equations so this step can be left out.

Note: A coefficient is the number before the letter.

**Step 2** Once the coefficients are the same take the two equations away.

**Step 3** After step 2 one of the variables will be eliminated, so solve the equation that is left to find the value of the first letter.

**Step 4** Find the value of the other variable (letter) by substituting your answer from step 3 back into the first equation.

**Step 5** Now you have found both values, substitute the values back into the second equation to check that they work.

Let’s take a look at an example:

**Example 1**

Solve this pair of simultaneous equations by using elimination:

3x + 5y = 27 (1)

9x + 2y = 42 (2)

Step 1:

Let’s make the x coefficients the same. Do this by multiplying the first equation by 3, and leaving the second equation unchanged.

9x + 15y = 81 (3)

9x + 2y = 42 (4)

(Notice all the numbers are multiplied by 3)

Step 2:

Take the two equations away.

13y = 39

Step 3:

Solve this equation:

13y = 39 (÷3)

y = 3

Step 4:

Now substitute y = 3 into the first equation to find the value of x.

3x + 5y = 27 (1)

3x + 5×3 = 27

3x + 15 = 27 (-15)

3x = 12 (÷3)

x = 4

Step 5:

Since we now know that x = 4 and y = 3, check that these values work in the second equation to confirm they are correct.

9x + 2y = 42 (2)

9 × 4 + 2 × 3 = 42

36 + 6 = 42

42 = 42

Since both sides are equal we know our two variables are correct.

For some more examples on solving by elimination click here.

For some harder examples click here.