Elimination, substitution, and the graphical method — every type covered with full worked examples.
Simultaneous equations appear on virtually every GCSE Maths paper. They're one of those topics where students either have a reliable method they trust — and fly through the question — or they don't, and panic. This guide gives you a reliable method for every type of simultaneous equations question you'll encounter.
There are three methods: elimination, substitution, and graphical. You won't need all three in every question — part of the skill is recognising which to use. We'll cover that decision process too.
Elimination is the most commonly taught method and works best when both equations are in the form ax + by = c. The idea is to manipulate the equations so that one variable has the same coefficient in both — then add or subtract the equations to eliminate that variable.
Write equation (1) and equation (2). This makes your working clear and helps you track what you're doing. Examiners reward clear working even when the final answer is wrong.
Look at both equations and decide which variable to eliminate. If one variable already has the same coefficient — great. If not, multiply one or both equations by appropriate numbers to make the coefficients match.
If the matched coefficients have the same sign — subtract. If they have opposite signs — add. This eliminates one variable, leaving a simple one-step equation.
Solve for the remaining variable, then substitute back into one of the original equations to find the other. Always verify by substituting both values into the equation you didn't use — this catches arithmetic errors and takes only seconds.
The y coefficients are already matched (both have 2y, opposite signs). Add the equations:
3x + 5x + 2y − 2y = 16 + 8 → 8x = 24 → x = 3
Substitute x = 3 into equation (1): 3(3) + 2y = 16 → 9 + 2y = 16 → 2y = 7 → y = 3.5
Verify using equation (2): 5(3) − 2(3.5) = 15 − 7 = 8 ✓
Often the coefficients won't match and you'll need to multiply. Example:
To eliminate y: multiply equation (2) by 3 to get 9x + 3y = 33. Now subtract equation (1) from this: 9x − 2x + 3y − 3y = 33 − 13 → 7x = 20 → x = 20/7.
Sometimes both equations need multiplying. If equations are (1): 2x + 3y = 7 and (2): 3x + 2y = 8, multiply (1) by 3 and (2) by 2 to get 6x in both. Then subtract.
Same signs → subtract. Different signs → add. Getting this backwards is one of the most common errors in elimination questions. Write it on your working paper before you start if you need to.
Substitution is best used when one equation has a variable already isolated, or when one equation is non-linear (like a quadratic). It's also the method you must use for simultaneous equations involving a quadratic and a linear equation — which is a Higher tier skill but appears regularly.
Equation (1) has y isolated. Substitute 3x − 1 in place of y in equation (2):
2x + (3x − 1) = 9 → 5x − 1 = 9 → 5x = 10 → x = 2
Then y = 3(2) − 1 = 5. Solution: x = 2, y = 5.
This type appears at Higher tier and is worth understanding clearly. You'll be given one linear equation and one quadratic equation. The method is always substitution — rearrange the linear equation to make one variable the subject, then substitute into the quadratic.
From (2): y = x + 1. Substitute into (1):
x² + (x + 1)² = 25
x² + x² + 2x + 1 = 25
2x² + 2x − 24 = 0
x² + x − 12 = 0
(x + 4)(x − 3) = 0
x = −4 or x = 3
Two x-values means two solution pairs. When x = 3: y = 3 + 1 = 4. When x = −4: y = −4 + 1 = −3.
Always give both solution pairs. Leaving out one of them is a very common way to lose the final mark.
When solving quadratic simultaneous equations, there are almost always two solution pairs. Write both clearly at the end: "When x = 3, y = 4. When x = −4, y = −3." Examiners want to see both stated explicitly, not just the x-values.
The graphical method involves drawing both equations as lines on a graph and reading off the coordinates where they intersect. This method is less precise than algebraic methods (especially if the intersection isn't at whole-number coordinates) but it's sometimes specifically asked for in exam questions.
To draw a linear equation as a graph, find at least three coordinate pairs by substituting x-values. For x = 0, x = 2, x = 4 — find the corresponding y-values and plot. Draw the line through those points. Repeat for the second equation. The intersection is your solution.
The key skill here is accurate plotting. Use a ruler. The OCR GCSE specification explicitly tests the ability to interpret graphical solutions, and you can find their full details on the OCR GCSE Mathematics page.
This is the question students ask most often, and the honest answer is: it depends on the question, and with practice you'll just know.
If you're ever unsure, elimination is a safe default for linear pairs. It's systematic and the working is easy for an examiner to follow, which means you pick up method marks even if you make an arithmetic error.
PaperPlus generates unlimited GCSE simultaneous equations questions — linear pairs, quadratic types, and graphical — with full mark schemes. Free for all exam boards.
Start Practising Free →Forgetting to find both variables. After eliminating and solving for x, students sometimes write down x = 3 and stop. You must find y too. Both values together form the solution.
Not checking the solution. Substituting your answers back into both original equations takes thirty seconds and catches sign errors, arithmetic slips, and coefficient mistakes. Do it every time.
With word problems — not defining variables. When simultaneous equations are presented as a word problem (e.g. "two numbers add to 14 and their difference is 4"), start by writing "let x = ..." and "let y = ..." before setting up the equations. Marks are often awarded for correctly forming the equations, separate from solving them.
If you want to see the specific content assessed across all exam boards, the AQA specification and the Edexcel specification both lay out exactly which simultaneous equations skills are tested at Foundation and Higher.