Find the pair of values that fits two equations at once, by elimination or substitution, including from a worded set-up. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Two equations, two unknowns. Elimination: make one coefficient match (multiply through), add or subtract to kill that letter, solve for the other, substitute back. Substitution: rearrange one equation to y=… and replace y in the other. Addition when the matching coefficients have opposite signs; subtraction when they have the same sign.
Write the pair as x=…,y=…. A failed check in both original equations means the add/subtract choice was wrong or an arithmetic slip in the multiply-through. Worded 'two numbers…' questions are simultaneous equations in disguise.
Elimination: make one letter’s coefficients equal, add or subtract. Substitution: rearrange the simpler equation and plug in. Label the equations, and substitute back to find the second letter. Two solutions need two pairs.
Objectives
Solve a pair for both x and y.
Give only x, or only y.
Add two left-hand sides.
Key points
Solve two linear equations by adding or subtracting to eliminate one letter.
Multiply one or both equations first so a pair of coefficients match.
Solve by substitution when one equation is already written as y=….
Form two equations from a context such as prices of tickets, drinks or pens.
Worked examples
Example 1
Solve 2x+y=7 and x+y=4.
2 marks
Subtract the second equation from the first to eliminate y: (2x - x) = (7 - 4), so x = 3
Substitute x = 3 into x + y = 4 to find y: 3 + y = 4, so y = 1
State the full pair: x=3, y=1
Answer: x=3, y=1
Common slips
Slip: Aaron added the equations Subtracting eliminates the identical +y terms.
Slip: Amber stopped at x = 3 Both variables must be evaluated.
Slip: Aaron wrote x=1, y=3 Keep x and y values in their correct positions.
Example 2
Solve 3x+2y=16 and x+2y=8, and give only the value of x.
2 marks
Subtract the second equation from the first: (3x - x) = (16 - 8), so 2x = 8
Divide by 2 to solve for x: x = 4
State the requested value: x is 4
Answer: 4
Common slips
Slip: Boris subtracted x from 2y Subtract corresponding terms on each side.
Slip: Boris solved for y instead The question asks specifically for x.
Slip: Becky gave both x and y Provide only the single value requested.
Example 3
Solve 3x + y = 13 and x + y = 5.
3 marks
Subtract to remove y: 2x = 8, so x = 4
Substitute x = 4 into x + y = 5: y = 1
The solution is x = 4, y = 1
Answer: x=4, y=1
Common slips
Slip: Added the expressions Subtract the equations when the y terms match.
Slip: Stopped after finding x The pair needs both letters.
Slip: Wrote 6x+8y Simultaneous equations give a pair of values.
Questions and answers
GCSE Maths simultaneous equations questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Solve 3x+2y=16 and x+2y=8.
Show answer
Answer: x=4, y=2
Subtract equations: 2x = 8, so x = 4.
Substitute: 4 + 2y = 8 gives y = 2.
M1 — Subtract equations: 2x = 8, so x = 4.
A1 — Substitute: 4 + 2y = 8 gives y = 2.
Question 2 Easy2 marks
Solve 4x+y=9 and 2x+y=5.
Show answer
Answer: x=2, y=1
Subtract equations: 2x = 4, so x = 2.
Substitute: 2(2) + y = 5 gives y = 1.
M1 — Subtract equations: 2x = 4, so x = 2.
A1 — Substitute: 2(2) + y = 5 gives y = 1.
Question 3 Medium2 marks
Solve 5x+2y=19 and x+2y=7, and give only the value of x.
Show answer
Answer: 3
Subtract equations: 4x = 12.
Divide by 4: x = 3.
M1 — Subtract equations: 4x = 12.
A1 — Divide by 4: x = 3.
Question 4 Medium2 marks
Solve 4x+y=9 and 2x+y=5, and give only the value of x.
Show answer
Answer: 2
Subtract equations: 2x = 4.
Divide by 2: x = 2.
M1 — Subtract equations: 2x = 4.
A1 — Divide by 2: x = 2.
Question 5 Hard2 marks
Solve 2x + y = 11 and x + y = 4.
Show answer
Answer: x=7, y=-3
Subtract: x = 7.
Then y = 4 − 7 = −3.
M1 — Subtract: x = 7.
A1 — Then y = 4 − 7 = −3.
Question 6 Hard2 marks
Solve 3x + y = 14 and x + y = 6.
Show answer
Answer: x=4, y=2
Subtract: 2x = 8, so x = 4.
Then y = 6 − 4 = 2.
M1 — Subtract: 2x = 8, so x = 4.
A1 — Then y = 6 − 4 = 2.
Question 7 Exam-style3 marks
Solve 5x+2y=19 and x+2y=7.
Show answer
Answer: x=3, y=2
Subtract equations: 4x = 12, so x = 3.
Substitute x = 3 into x + 2y = 7 to get 2y = 4 and y = 2.
M1 — Subtracts the equations to eliminate y: 4x = 12
A1 — x = 3
A1 — y = 2
Common mistakes
Most lost marks on this topic are one slip.
Adding equations when coefficients have the same sign
Subtract to eliminate matching signs; add for opposite signs
Giving only one variable when both are requested
Always check if the question asks for x, y, or both
Multiplying instead of adding when combining expressions
Collect like terms by adding their coefficients
Spot the slip
Solve 2x+y=7 and x+y=4. Ada subtracts and then says y is 3. What is the pair?
x=3, y=1
x=3, y=3
x=4, y=7
Show the slip
x=3, y=1 Subtracting gives x = 3, then y = 4 − 3 = 1.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Label the equations (1) and (2) and say what you are doing ('(1) ×5') — the marker can follow and award method marks.
In a worded question, define your letters in a sentence first; it is often the first mark.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.