Simultaneous equations

Two unknowns need two equations. Find the values that make both true at once, using a graph or by eliminating a letter.

Practise this lesson free (7 questions)

What they are

x + y = 10 has many solutions: (1, 9), (6, 4), (7, 3) ...
x − y = 4 also has many: (5, 1), (7, 3), (9, 5) ...
Only x = 7, y = 3 makes both true. Those are the solutions of the simultaneous equations.

Solving with graphs

Draw both lines. The point where they cross is the solution, because it lies on both lines.
y = 2x and y = x + 3 cross at (3, 6), so x = 3 and y = 6.
Reading from a graph can give only an approximate answer if the lines cross between grid lines.
Parallel lines never cross, so there is no solution.

Solving by elimination

If one letter has the same coefficient in both equations, subtract to remove it.
3x + y = 17
x + y = 7
Subtract: 2x = 10, so x = 5. Substitute: 5 + y = 7, so y = 2.
Always check in both original equations. (You will learn more methods at GCSE.)
Worked example

Solve 2x + 3y = 18 and 2x + y = 10.

  1. Both have 2x, so subtract: 2y = 8, y = 4
  2. Substitute into 2x + y = 10: 2x + 4 = 10, so x = 3
  3. Check: 2(3) + 3(4) = 6 + 12 = 18

Answer: x = 3, y = 4

Key idea

Simultaneous equations share one solution: the point where their graphs cross. With matching coefficients, subtract to eliminate a letter, then substitute back and check.

Key words
Substitute
Replace each letter in an expression or formula with a number, then work it out.
Parallel
Lines that are always the same distance apart and never meet.
Parallel lines
Alternate angles are equal. Corresponding angles are equal. Co-interior angles add to 180°.
Coefficient
The number in front of a letter in a term. In 5x, the coefficient of x is 5.

All KS3 Mathematics key terms and definitions →

Common mistakes
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Extra practice

2 more questions

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  1. Solve x + y = 15 and x − y = 3. What is x?

    Show the answer

    Answer: 9

    Add the equations: 2x = 18, so x = 9 (and y = 6).

  2. Solve 4x + y = 23 and x + y = 8. What is y?

    Show the answer

    Answer: 3

    Subtract: 3x = 15, so x = 5. Then 5 + y = 8 gives y = 3.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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