Simultaneous equations are a key part of algebra for GCSE maths. To do well with simultaneous equations you need to be confident solving difficult linear equations. Check out our linear equations guide and worksheet if you need to revise this before going any further. Once you’re confident with linear equations read on and find out how to solve simultaneous equations.
Up to this point you have learnt about solving one equation with one unknown (represented by a letter). To solve the equation you find the value of the unknown. Simultaneous equations allow you to find two unknowns, as long as you have two equations and both have the same two unknowns. The two equations are solved simultaneously (at the same time) to find the values of both unknowns.
You need to learn two methods to solve simultaneous equations for GCSE maths: the elimination method and the substitution method. I have set out guides to both methods below, including some examples. I’ve written in key parts of working that you need to show the examiner. You will use the elimination method most often, so let’s start there.
Simultaneous Equations – Elimination Method
With this method you will need to combine the two equations by either adding or subtracting one from the other. In doing so you need to eliminate one of the unknowns. You then solve the remaining equation to find one unknown and substitute your answer back into one of the original equations to find the second unknown. Take a look at the worked example below.
Worked Example 1
Firstly, label the equations number one and two to help with your thinking and to help the examiner see your working.
Next, combine the equations and eliminate the x (we eliminate x rather than y here because the coefficients of x are the same) by subtracting equation one from equation two. Make sure you subtract everything, including the x, y and numbers after the equals sign.
We now know the value of y. Substitute y = 1 into one of the original equations. I’ve used equation one here:
Then solve this final equation to find the value of x.
That’s it. We’ve solved the simultaneous equations to find y = 1 and x = 2.
Worked Example 2
Label the equations one and two:
This time we will eliminate the y because the coefficients of y are already the same. Notice that equation one includes 2y and equation two includes -2y. When the terms you want to eliminate have different signs, you should add the two equations together. This will ensure that you eliminate the y.
Next, substitute x = 3 into equation one.
Worked Example 3
Notice that this pair of simultaneous equations do not have an unknown with the same coefficient (as the earlier examples did). When this is the case, you will need to do an extra step at the beginning. After labelling the equations, you need to find the lowest common multiple of the coefficients for x or for y. Multiply up to this lowest common multiple to ensure the coefficients of one of the unknowns are the same. Then follow the same process as with the earlier examples to combine the equations and solve.
Substitute y = 2 into equation one:
Simultaneous Equations – Substitution Method
The second method you need to know to solve some simultaneous equations for GCSE maths is the substitution method. As with the elimination method: we will combine the two equations, solve for one unknown and then use that answer to find the second unknown. However, with this method we combine the equations initially by substituting part of one equation into the other (hence the name: substitution method).
Worked Example 4
Start by labelling the equations one and two. Then substitute equation two into equation one. In equation two y is the subject. This allows us to substitute (x – 8) in for y.
Finally, substitute x = 6 into equation two and solve for y.
That is everything you need to know about simultaneous equations for GCSE maths. Make sure you practice both the elimination and substitution methods once you have read through the information above a few times. It is only by doing lots of questions yourself that you will properly learn and understand the topic. Follow the links below to try some past paper questions as well. Let us know if you have any questions as you continue revising. You can contact us directly through our website to ask any questions or book an expert maths tutor for a personalised lesson.