Maths Library

Simultaneous equations: Year 12 maths

This topic covers solving pairs of linear equations simultaneously using elimination and substitution methods. The worksheet provides systematic practise from matching coefficients to solving algebraic word problems.

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What this covers

Simultaneous equations is taught in Year 12 (age 16 to 17, KS5) as a topic to introduce. It sits in the Algebra strand.

Worked example

Solve simultaneously: 2x + y = 11 and 3x - y = 9

  1. Add the two equations to eliminate y: (2x + 3x) + (y - y) = 11 + 9
  2. Solve for x: 5x = 20, so x = 4
  3. Substitute x = 4 into the first equation: 2(4) + y = 11 => 8 + y = 11, so y = 3

Answer: x = 4, y = 3

Difficulty scale

The sheet generator offers five difficulty settings for this topic.

  1. Coefficients of one variable already match with opposite signs (direct addition)
  2. Coefficients of one variable match with the same sign (direct subtraction)
  3. Multiply one equation to match a coefficient before eliminating
  4. Multiply both equations to eliminate a variable
  5. Solving simultaneous equations involving fractions, decimals, or contextual word problems

Common mistakes

  • Adding equations when signs are identical, failing to eliminate the target variable
  • Solving correctly for the first variable but forgetting to substitute back to find the second

Curriculum links

Indicative matches to National Curriculum for England. Check wording and codes against the official document.

  • Solve simultaneous equations involving one linear and one non-linear quadratic equation.

Sample sheet and answers

Six questions at the level's usual difficulty. Open the editor to change, add or regenerate any of them.

Simultaneous equations

Year 12 maths

Qs
6
Marks
24
NameDateScore / 24
1.
Solve:
x + y = 10
3x − y = 14
[4]
2.
Solve:
x + y = 8
2x − y = 4
[4]
3.
Solve:
x + y = 9
3x − y = 11
[4]
4.
Solve:
x + y = 5
4x − y = 0
[4]
5.
Solve:
x + y = 4
4x − y = 1
[4]
6.
Solve:
x + y = 6
2x − y = 0
[4]

Answer key

  1. x = 6, y = 4Add the equations to remove y: 4x = 24; x = 6. Then y = 10 − 6 = 4
  2. x = 4, y = 4Add the equations to remove y: 3x = 12; x = 4. Then y = 8 − 4 = 4
  3. x = 5, y = 4Add the equations to remove y: 4x = 20; x = 5. Then y = 9 − 5 = 4
  4. x = 1, y = 4Add the equations to remove y: 5x = 5; x = 1. Then y = 5 − 1 = 4
  5. x = 1, y = 3Add the equations to remove y: 5x = 5; x = 1. Then y = 4 − 1 = 3
  6. x = 2, y = 4Add the equations to remove y: 3x = 6; x = 2. Then y = 6 − 2 = 4