Free GCSE Maths lesson: Algebra
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Lesson 35 · GCSE / Key Stage 4 · Maths · Algebra

Simultaneous Equations

Learn how to solve two equations at the same time, check the pair of values, and recognise the graph intersection.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Algebra

GCSE specification fit

A core algebra topic for solving two conditions together.

Simultaneous equations are used when two statements are both true at the same time. GCSE questions may give the equations directly, hide them in a worded problem, or show them as two straight lines on a graph.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandAlgebra
Tier guidanceFoundation and Higher

What you will learn

  • What a simultaneous solution means.
  • How to solve by elimination when coefficients match.
  • How to multiply an equation before eliminating.
  • How to use substitution when one variable is already isolated or easy to rearrange.
  • How to check a solution in both original equations.
  • How a graph intersection represents the solution.
  • How to handle worded, graph and fractional-answer questions calmly.

Why this matters

Simultaneous equations let you solve problems with two unknowns, such as prices, ages, tickets, totals or straight-line intersections.

The best habit is to keep the equations lined up and write one small reason for each step. That makes sign errors easier to spot.

Prior knowledge

You should already be comfortable with:

  • solving linear equations,
  • expanding brackets and collecting like terms,
  • substituting values into expressions,
  • adding and subtracting negative numbers,
  • reading coordinates and straight-line graphs.

Clear explanation

The answer is a pair of values

A simultaneous equation solution gives values for both unknowns. For example, x = 4 and y = 1 is only correct if it works in both equations.

Equation 1: x + y = 5 → 4 + 1 = 5 Equation 2: x − y = 3 → 4 − 1 = 3 So x = 4 and y = 1 works in both equations.

Elimination removes one unknown

If the same number of x terms or y terms appears in both equations, add or subtract the equations so one unknown disappears.

2x + y = 11 x + y = 7 Subtract the second equation from the first: x = 4 Substitute into x + y = 7: 4 + y = 7, so y = 3

Sometimes you multiply first

If the coefficients do not match, multiply one equation so a letter has the same coefficient in both equations.

x + 2y = 13 3x + 2y = 21 Subtract: 2x = 8, so x = 4 Then 4 + 2y = 13, so 2y = 9 and y = 4.5

Graphs solve simultaneous equations too

If two straight lines are drawn on the same axes, the solution is their intersection point. At that point, both equations have the same x and y values.

Worked examples

Example 1: Eliminate y

Solve 3x + y = 14 and x + y = 6.

Subtract the second equation from the first: 2x = 8 x = 4 4 + y = 6, so y = 2
Answer: x = 4 and y = 2.

Example 2: Add to eliminate

Solve 2x + 3y = 17 and 2x − 3y = 7.

Add the equations: 4x = 24 x = 6 12 + 3y = 17, so 3y = 5 and y = 5/3
Answer: x = 6 and y = 5/3.

Example 3: Substitute

Solve y = 2x + 1 and x + y = 10.

x + 2x + 1 = 10 3x = 9, so x = 3 y = 2 × 3 + 1 = 7
Answer: x = 3 and y = 7.

Example 4: Worded problem

Two adult tickets and one child ticket cost £19. One adult ticket and one child ticket cost £11. Find each price.

Let a be an adult ticket and c be a child ticket. 2a + c = 19 and a + c = 11 Subtract: a = 8, then c = 3
Answer: Adult ticket £8, child ticket £3.

Quick checks

Choose an answer, then check your thinking.

1. What does a simultaneous equation solution have to do?

2. Solve x + y = 9 and x − y = 3. What is x?

3. On a graph, where is the solution to two straight-line equations?

Practice questions

Question 1

Check whether x = 5 and y = 2 solves x + y = 7 and x − y = 3.

Reveal answer and marking guidance

Answer: Yes, it solves both equations.

Marking: Substitute into both equations: 5 + 2 = 7 and 5 − 2 = 3.

Question 2

Solve x + y = 12 and x − y = 4.

Reveal answer and marking guidance

Answer: x = 8 and y = 4.

Marking: Add the equations to get 2x = 16, so x = 8. Then 8 + y = 12, so y = 4.

Question 3

Solve 2x + y = 13 and x + y = 8.

Reveal answer and marking guidance

Answer: x = 5 and y = 3.

Marking: Subtract the second equation from the first to get x = 5, then substitute into x + y = 8.

Question 4

Solve 3x + 2y = 20 and x + 2y = 10.

Reveal answer and marking guidance

Answer: x = 5 and y = 2.5.

Marking: Subtract the second equation from the first to get 2x = 10, so x = 5. Then 5 + 2y = 10, so y = 2.5.

Question 5

Solve y = 3x − 2 and x + y = 10.

Reveal answer and marking guidance

Answer: x = 3 and y = 7.

Marking: Substitute y = 3x − 2 into x + y = 10 to get 4x − 2 = 10, so x = 3 and y = 7.

Question 6

Two notebooks and one pen cost £5. One notebook and one pen cost £3.20. Find the cost of one notebook and one pen.

Reveal answer and marking guidance

Answer: Notebook £1.80 and pen £1.40.

Marking: Let n be a notebook and p be a pen. 2n + p = 5 and n + p = 3.20. Subtract to get n = 1.80, then p = 1.40.

Question 7

Solve 2x + 3y = 12 and 5x + 2y = 19.

Reveal answer and marking guidance

Answer: x = 3 and y = 2.

Marking: Multiply the first equation by 2 and the second by 3 to make 4x + 6y = 24 and 15x + 6y = 57. Subtract to get 11x = 33, so x = 3, then substitute to get y = 2.

Question 8

The straight lines y = 2x + 1 and y = −x + 7 intersect. Find the intersection point.

Reveal answer and marking guidance

Answer: (2, 5).

Marking: Set 2x + 1 = −x + 7, so 3x = 6 and x = 2. Substitute into either equation to get y = 5, then give the answer as a coordinate pair.

Question 9

At a school concert, 3 adult tickets and 2 child tickets cost £31. Two adult tickets and 5 child tickets cost £39. Find the cost of one adult ticket and one child ticket.

Reveal answer and marking guidance

Answer: Adult ticket £7, child ticket £5.

Marking: Let a be an adult ticket and c be a child ticket. 3a + 2c = 31 and 2a + 5c = 39. Multiply by 5 and 2 to get 15a + 10c = 155 and 4a + 10c = 78, then subtract to get 11a = 77, so a = 7. Substitute to get c = 5.

Question 10

A cinema sells adult tickets and student tickets. Three adult tickets and four student tickets cost £54. Two adult tickets and five student tickets cost £50. Find the price of one adult ticket and one student ticket.

Reveal answer and marking guidance

Answer: Adult ticket £10 and student ticket £6.

Marking: Let a be an adult ticket and s be a student ticket. 3a + 4s = 54 and 2a + 5s = 50. Multiply by 2 and 3 to get 6a + 8s = 108 and 6a + 15s = 150, then subtract to get 7s = 42, so s = 6. Substitute to get a = 10.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For simultaneous equations, marks usually come from writing the two equations clearly, choosing a valid elimination or substitution step, preserving negative signs, substituting back to find the second unknown, and checking both values in the original equations. In worded problems, define the variables and include units such as pounds where needed.

Common mistakes

  • Only checking one equation: the answer must work in both equations.
  • Eliminating the wrong way: if signs are the same, subtract; if signs are opposite, add.
  • Losing negative signs: write each line carefully when subtracting a whole equation.
  • Forgetting to find the second unknown: solving for x is only half the answer.
  • Not defining variables: in worded problems, state what x and y represent.

Extension challenge

Solve 4x + 3y = 29 and 2x − y = −3.

Reveal answer

Answer: x = 2 and y = 7.

Multiply the second equation by 3 to get 6x − 3y = −9, then add it to 4x + 3y = 29. This gives 10x = 20, so x = 2. Substitute into 2x − y = −3 to get y = 7.

Exam-board guidance

Simultaneous equations are assessed by every GCSE Maths board. Questions may ask for algebraic solving, a graph intersection, or equations formed from a real-life context.

AQA GCSE Maths

Show the elimination step clearly, state the ordered pair when a graph is used, and check your x and y values in both original equations.

OCR GCSE Maths

If the coefficients do not match, multiply one or both whole equations first, then add or subtract so one letter disappears cleanly.

Pearson Edexcel GCSE Maths

Worded problems often need you to define x and y first, build two equations, solve them and answer in the units from the question.

Eduqas GCSE Maths

Keep equations lined up so addition or subtraction eliminates exactly one variable, and check the final pair in both equations.

WJEC Wales

Expect some questions to hide the two equations inside a real-life context such as tickets, prices or quantities, so define variables before solving.

CCEA GCSE Maths

Write enough working to show how one unknown was removed, then substitute into an original equation to finish the pair.

Next lesson

Next, move into Quadratic Expressions and Equations.

Previous lesson: Coordinates and Straight-Line Graphs Next lesson: Quadratic Expressions and Equations Back to GCSE Maths lessons