What Are Simultaneous Equations?
Simultaneous equations are two (or more) equations that share the same unknown variables. You need to find values that satisfy both equations at the same time, which is exactly why they are called "simultaneous". In GCSE Maths you will almost always work with two equations in two unknowns, usually and .
Here is the idea in plain terms. The equation has endless solutions on its own: works, so does . The equation also has endless solutions on its own. But only one pair satisfies both at once: . Finding that pair is what solving simultaneous equations means.
On the exam, these questions typically carry 3 to 5 marks, and most of those marks are method marks, so showing clear working matters as much as the final answer. If you want the full topic breakdown with practice questions, the simultaneous equations topic page covers what each board expects at each tier. This guide focuses on the step-by-step methods.
- Two equations, two unknowns (usually $x$ and $y$)
- One equation alone has many solutions; the pair together usually pins down exactly one
- Typically worth 3 to 5 marks, mostly method marks
- Appears on both Foundation (linear only) and Higher (including one linear, one quadratic)
Method 1: Elimination, Step by Step
Elimination is the workhorse method at GCSE. The plan: add or subtract the two equations so that one variable cancels out, leaving a single equation in one unknown that you can solve directly.
The rule for deciding whether to add or subtract: look at the signs of the matching coefficients. If the coefficients of one variable are the same sign (both , say), subtract one equation from the other. If they are opposite signs ( and ), add the equations. A memory aid many teachers use is "same signs subtract, different signs add".
Work through the example below slowly. Notice how each line of working is written out: label the equations (1) and (2), state the operation you are doing, and keep the equals signs lined up. Examiners award method marks for exactly this kind of visible reasoning, so a correct answer with no working can lose most of the marks on a "show your working" question.
Solve and
- 1
Label the equations: (1) and (2)
- 2
Both equations have , so subtract (1) from (2) to eliminate :
- 3
This gives , so
- 4
Substitute into (1): , so , giving and
- 5
Check in (2): ✓
Elimination When the Coefficients Don't Match
Most exam questions are not as tidy as the first example. If neither variable has matching coefficients, you must multiply one or both equations first to create a match. You can multiply an equation by any number you like, as long as you multiply every term, including the number on the right-hand side. Forgetting the right-hand side is one of the most common ways to lose marks on this topic.
Choose the variable that needs the least work. In the example below, the coefficients are and : multiplying the second equation by 3 gives , which matches with opposite signs, so the equations can simply be added. That is quicker than matching the coefficients, which would need both equations multiplied.
When you have to multiply both equations, aim for the lowest common multiple of the two coefficients. For and , multiply to get in each (times 3 and times 2), not .
Solve and
- 1
Label: (1) and (2)
- 2
Multiply (2) by 3 so the coefficients match: — call this (3)
- 3
The terms are and (different signs), so add (1) and (3):
- 4
Divide by 11:
- 5
Substitute into (2): , so , giving
- 6
Check in (1): ✓
Method 2: Substitution, Step by Step
Substitution works best when one equation is already rearranged for a single variable, such as . Instead of cancelling a variable, you replace it: wherever appears in the other equation, write in its place. That turns two equations in two unknowns into one equation in one unknown.
The crucial habit is to substitute with brackets. Writing happens to work here, but the moment the substituted expression is multiplied by something, missing brackets cause sign and coefficient errors. Train yourself to write every time.
Substitution is not just an alternative to elimination. On Higher papers it becomes essential, because the linear-and-quadratic questions in the next section can only sensibly be done by substitution.
Solve and
- 1
The first equation already gives in terms of , so substitute it into the second:
- 2
Collect like terms:
- 3
Subtract 1: , so
- 4
Substitute back into :
- 5
Check in the second equation: ✓
Higher Tier: One Linear, One Quadratic
Higher papers extend the topic to a pair where one equation is linear and one is quadratic, most commonly a circle such as or a curve like . These are usually worth 5 or 6 marks and sit in the second half of the paper.
The method is always substitution: rearrange the linear equation for one variable, substitute into the quadratic, and solve the resulting quadratic equation. Expect two pairs of solutions, because a straight line generally crosses a curve or circle at two points. If you are rusty on solving the quadratic you end up with, work through how to solve quadratic equations first, because that skill carries most of the marks here.
One presentation warning: pair your answers correctly. Writing " or , or " is ambiguous and can cost the final mark, because it suggests four possible pairings when only two are valid. Write the solutions as coordinate pairs, each matched to the that produced it.
Solve and
- 1
Substitute into the quadratic:
- 2
Expand the bracket:
- 3
Simplify: , and divide every term by 2:
- 4
Factorise: , so or
- 5
Find each matching from : when , ; when ,
- 6
Check both pairs in : ✓ and ✓
What the Answer Means on a Graph
Every simultaneous equations question has a picture behind it. Each linear equation is a straight line, and the solution is the point where the two lines cross. When you solved and and got , you found that the two lines intersect at the point . That is also why the linear-quadratic case has two answers: a straight line typically cuts a circle or parabola at two points.
This picture explains the awkward cases too. Two parallel lines never meet, so a pair like and has no solution; the algebra collapses to something impossible like . And if the two equations are secretly the same line, such as and , every point on the line works, so there are infinitely many solutions.
Some Foundation questions ask you to solve simultaneous equations by drawing: plot both lines and read off the crossing point. Accurate plotting matters, and so does reading the scales carefully. If graph work is a weak spot, the straight-line graphs topic page is the place to practise it.
- The solution is the point where the two graphs intersect
- Parallel lines: no solution; the same line twice: infinitely many
- A line crossing a circle or parabola gives the two solution pairs seen at Higher tier
- Graphical questions ask you to plot both lines and read off the crossing point
How the Marks Are Awarded
Understanding how mark schemes reward this topic changes how you should write your answers. A typical linear pair carries one or two method marks for a correct elimination or substitution strategy, an accuracy mark for the first variable, and a final mark for the second variable. The linear-quadratic version adds method marks for the substitution and for solving the quadratic.
Two consequences follow. First, an arithmetic slip does not wreck the question: if your method is sound and clearly shown, you usually keep the method marks and lose only accuracy marks. Second, finding only one variable throws marks away. The question asks for and ; stopping at leaves the last mark, sometimes two, on the table.
Always spend the last ten seconds substituting your pair back into the equation you have not yet used. It is the fastest self-check in algebra: if both sides balance, your answer is right, and if they don't, you have caught an error the examiner would otherwise catch for you. Mock exams on Exam Ladder are graded against real exam-board grade boundaries, so practising under those conditions shows you exactly what these part-marks do to a final grade.
- Method marks reward a visible, correct strategy even if the arithmetic slips
- State both values clearly: "$x = 4$, $y = 2$", not just one of them
- Check your pair in the equation you didn't use to substitute
- Label equations (1) and (2) and say what you are doing to them
Common Mistakes to Avoid
Almost every dropped mark on this topic comes from a short list of avoidable errors, and sign slips top it. Subtracting equations with negative terms is the classic trap: is , not . If subtracting makes you nervous, multiply one equation by first and then add instead; adding is far harder to get wrong.
The second family of errors comes from multiplying an equation but missing a term, most often the right-hand side. If you turn into , everything after that line is wrong, and you may only keep one method mark. Say the multiplication out loud as you do it: "times 3, times 3, times 3".
Finally, read the question to the end. Some questions dress the equations in words: two teas and three coffees cost a given amount, and so on. Define your letters, build the two equations, then solve as normal. For instance, if 3 teas and 2 coffees cost £6.60 while 5 teas and 2 coffees cost £9.40, subtracting gives , so a tea is £1.40 and a coffee is £1.20. The algebra is identical; only the packaging changes. More of these recurring slip-ups, across all of algebra, are collected in the most common GCSE maths mistakes.
- Watch signs when subtracting: $3y - (-2y) = 5y$
- Multiply every term, including the right-hand side
- Find BOTH unknowns and present them clearly
- In worded questions, define your variables before writing the equations
Frequently asked questions
How do I know whether to use elimination or substitution?
Use elimination when both equations are in the form ax + by = c, especially if the coefficients of one variable already match or can be matched with one multiplication. Use substitution when one equation is already rearranged, like y = 2x + 3, and always for the Higher-tier linear and quadratic pairs.
Can simultaneous equations have no solution?
Yes. If the equations represent parallel lines (same gradient, different intercept) there is no solution, and the algebra collapses to a false statement like 0 = 4. If both equations describe the same line, there are infinitely many solutions.
Are simultaneous equations on Foundation GCSE?
Yes. Linear simultaneous equations appear on Foundation papers, solved by elimination, substitution or graphically. The linear-and-quadratic version is Higher tier only.
How many marks are simultaneous equations questions worth?
A typical linear pair is worth 3 to 4 marks, and the Higher-tier linear and quadratic version is usually 5 or 6. Most are method marks, so clearly shown working protects you even if you make an arithmetic slip.
Do I need to check my answers in the exam?
It takes seconds and is worth doing every time: substitute both values into the original equation you did not use during your working. If both sides balance, your solution is correct.