How to solve simultaneous equations at GCSE
Simultaneous equations are two equations that share the same unknowns, and you solve them together to find the values that satisfy both at the same time. Once you learn the two main methods – elimination and substitution – you can handle any pair the exam throws at you.
This guide walks through both methods step by step, with worked examples for linear pairs (foundation and higher) and a quadratic pair (higher tier only). It also covers when to use each method and the mistakes that cost students the most marks.
Just
2
methods to learn – elimination and substitution – and between them they cover every simultaneous equations question at GCSE
What are simultaneous equations?
Simultaneous equations are a pair of equations with two unknowns, usually x and y. The goal is to find the one pair of values that makes both equations true at the same time.
For example, x + y = 10 and x – y = 4. The solution is x = 7 and y = 3, because those values work in both equations. If you only had one equation, there would be infinitely many solutions. The second equation narrows it down to exactly one pair.
At foundation tier, both equations will be linear (no squared terms). At higher tier, one equation might be quadratic – for example, x² + y² = 25 paired with y = x + 1. The method changes slightly for quadratic pairs, but the underlying logic is the same: Use one equation to remove one unknown, then solve what remains.
The elimination method step by step
Elimination works by adding or subtracting the two equations so that one variable cancels out. It is the go-to method when both equations are linear and already have matching or near-matching coefficients.
Here is the method in five steps.
Step 1: Write both equations clearly, one above the other, and label them equation 1 and equation 2.
Step 2: Look at the coefficients of x and y. If the coefficients of one variable already match (ignoring sign), go straight to step 3. If not, multiply one or both equations so that the coefficients of one variable become equal.
Step 3: Add or subtract the equations to eliminate that variable. If the matching coefficients have the same sign, subtract. If they have opposite signs, add.
Step 4: Solve the resulting equation for the remaining variable.
Step 5: Substitute your answer back into either original equation to find the other variable. Then check both values in the equation you did not use.
Elimination method worked example
Solve 2x + 3y = 12 (equation 1) and 5x + 3y = 21 (equation 2).
The coefficients of y are already the same (both 3), so subtract equation 1 from equation 2.
(5x + 3y) – (2x + 3y) = 21 – 12, which gives 3x = 9, so x = 3.
Substitute x = 3 into equation 1: 2(3) + 3y = 12, so 6 + 3y = 12, meaning 3y = 6 and y = 2.
Check in equation 2: 5(3) + 3(2) = 15 + 6 = 21. Correct.
The solution is x = 3, y = 2.
Always check your answer in the equation you did not use for substitution. If it does not work, you have made an arithmetic error somewhere – go back and look for it.
The substitution method step by step
Substitution works by rearranging one equation to express one variable in terms of the other, then replacing that variable in the second equation. It is the best choice when one equation already has a variable on its own – for example, y = 3x + 1 – and it is essential for quadratic pairs at higher tier.
Step 1: Pick the equation that is easiest to rearrange. Ideally, one variable already has a coefficient of 1.
Step 2: Rearrange that equation to make one variable the subject (for example, y = ...).
Step 3: Substitute the expression into the other equation. Replace every occurrence of that variable with the expression you found.
Step 4: Solve the resulting equation. For a linear pair this gives a single value. For a quadratic pair you will need to expand, rearrange to a quadratic, and solve (by factorising or using the quadratic formula).
Step 5: Substitute back to find the other variable. If there are two solutions (quadratic case), find the matching y-value for each x-value.
Substitution worked example – linear pair
Solve y = 2x – 1 (equation 1) and 3x + 2y = 16 (equation 2).
Equation 1 already gives y in terms of x, so substitute y = 2x – 1 into equation 2.
3x + 2(2x – 1) = 16. Expand: 3x + 4x – 2 = 16. Simplify: 7x – 2 = 16, so 7x = 18 and x = 18/7.
That fraction is fine – leave it exact unless the question says otherwise. Substitute back: Y = 2(18/7) – 1 = 36/7 – 7/7 = 29/7.
Check in equation 2: 3(18/7) + 2(29/7) = 54/7 + 58/7 = 112/7 = 16. Correct.
The solution is x = 18/7, y = 29/7.
Substitution worked example – quadratic pair (higher tier)
Solve y = x + 3 (equation 1) and x² + y² = 29 (equation 2).
Substitute y = x + 3 into equation 2: X² + (x + 3)² = 29.
Expand (x + 3)²: X² + x² + 6x + 9 = 29. Simplify: 2x² + 6x + 9 = 29, so 2x² + 6x – 20 = 0. Divide through by 2: X² + 3x – 10 = 0.
Factorise: (x + 5)(x – 2) = 0, so x = –5 or x = 2.
Substitute back into equation 1. When x = –5, y = –5 + 3 = –2. When x = 2, y = 2 + 3 = 5.
Check both pairs in equation 2. For (–5, –2): 25 + 4 = 29. Correct. For (2, 5): 4 + 25 = 29. Correct.
The solutions are x = –5, y = –2 and x = 2, y = 5. Notice there are two pairs of answers – this is normal for simultaneous equations involving a quadratic.
When to use which method
| Situation | Best method | Why |
|---|---|---|
| Both equations are linear with similar coefficients | Elimination | One quick multiply (or none) and an add/subtract gets rid of a variable |
| One equation already has a variable as the subject (e.g. y = ...) | Substitution | You can plug the expression straight in – no rearranging needed |
| One equation is quadratic (e.g. x² + y² = ...) | Substitution | Elimination cannot remove a squared term; substitution is the only option |
| Coefficients are awkward and neither variable is easily isolated | Either | Pick whichever feels more comfortable – both will work |
In practice, most foundation-tier questions suit elimination because both equations are linear and the coefficients are designed to cancel neatly. At higher tier, if one equation is quadratic you must use substitution – there is no way around it.
If you are unsure, look at the equations for a few seconds before you start writing. If you can see matching coefficients, go with elimination. If one equation is already rearranged into y = ... or x = ..., go with substitution. Picking the right method will not change the answer, but it will save you time and reduce the chance of errors.
Common mistakes to avoid
The single most common error is a sign mistake when subtracting equations. Students regularly forget to subtract every term, especially when there is a negative in the second equation. Writing out the subtraction term by term – rather than trying to do it in your head – prevents most of these errors.
Another frequent mistake is substituting back into the wrong equation. After finding one variable, substitute into the simpler equation. Then check in the other equation to verify. If you substitute and check in the same equation, you will not catch an error.
With quadratic pairs, students often forget to expand the bracket fully. When you substitute y = x + 3 into y², you get (x + 3)², which expands to x² + 6x + 9 – not x² + 9. Missing the middle term (6x in this case) throws off everything that follows.
Finally, remember that quadratic simultaneous equations usually have two solutions. If you stop after finding one x-value, you will lose half the marks. Always check whether your quadratic gives two roots.
When subtracting equations in the elimination method, write out each term separately: (5x + 3y) – (2x + 3y) = 5x – 2x + 3y – 3y. This makes sign errors much less likely.
Simultaneous equations exam checklist
Run through this list every time you tackle a simultaneous equations question.
- Read both equations and decide whether they are both linear or one is quadratic
- Choose your method: Elimination (matching coefficients) or substitution (variable already isolated or quadratic involved)
- If using elimination, multiply to match coefficients, then add or subtract
- If using substitution, rearrange one equation, then substitute into the other
- Solve for the first variable
- Substitute back to find the second variable
- Check both values in the equation you did not use for substitution
- For quadratic pairs, check whether you have two pairs of solutions
- Write your final answer clearly, stating both x and y values
Putting it all together
Simultaneous equations come down to a single idea: Use one equation to get rid of one unknown, solve for what is left, then work backwards to find the other unknown. Whether you eliminate or substitute, the logic is identical.
At foundation tier, you will see straightforward linear pairs designed to practise the method. At higher tier, expect at least one question pairing a linear equation with a quadratic – typically a circle equation like x² + y² = r² or a quadratic curve like y = x² + bx + c. The substitution method handles both cases.
The best way to build confidence is deliberate practice. Work through a set of linear pairs using elimination, then a set using substitution, and finally tackle some quadratic pairs. Once you have seen the pattern a few times, the method becomes automatic.
