Solving a pair of simultaneous equations using the substitution method.

In this article I will show you how to solve a pair of simultaneous equations by substitution. Make sure you can solve linear equations before you go any further. Remember, when you are solving simultaneous equations you are after the values and x and y that satisfy both equations.

When you are asked to solve simultaneous equations by substitution normally y (or another letter) will be the subject of one of the equations. Therefore, all you need to do is substitute this equation into the other equation which will give you one equation involving x. If you simplify and solve this resulting equation you will be able to find the value of x. Once x is known, you can find the value of y by substituting x into either equation 1 or equation 2.

Let’s take a look at a couple of questions on solving simultaneous equations by substitution.

Question 1

Solve this pair of simultaneous equations using the substitution method.

4x – 3y = 9 (equation 1)

y = 10 - 3x (equation 2)

Since y is the subject of equation 2 you can substitute this equation into equation 1.

This gives:

4x – 3(10 – 3x) = 9

Now multiply out the brackets and simplify. Take care with the negative sign before your bracket.

4x -30 + 9x = 9

13x – 30 = 9

Now solve this linear equation to find the value of x.

13x = 39

x = 3

Now since x is found substitute this into equation 1 to find the value of y.

4x – 3y = 9

12 – 3y = 9

-3y = -3

y = 1

Let’s take a look at one more question on solving simultaneous equations by substitution.

Question 2

Solve this pair of simultaneous equations using the substitution method.

5x + 2y = 23 (equation 1)

y = 16 - 4x (equation 2)

Since y is the subject of equation 2 you can substitute this equation into equation 1.

This gives:

5x + 2(16-4x) = 23

Now multiply out the brackets and simplify. Take care with the negative sign before your bracket.

5x + 32 -8x = 23

-3x + 32 = 23

Now solve this linear equation to find the value of x.

-3x = -9

x = 3

Now since x is found substitute this into equation 2 to find the value of y.

y = 16 - 4x

y = 16 – 4 × 3

y = 4

So the solution are x = 3 and y = 4

For some extra help try these links out:

More help on solving by substitution.

Trickier questions on solving simultaneous equations by substitution.