Maths Library

Simultaneous equations: Year 13 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 13 (age 17 to 18, 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 non-linear simultaneous equations arising from the intersections of lines, curves, and circles.
  • Determine intersections and parallel or perpendicular lines within coordinate geometry and vector frameworks.

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 13 maths

Qs
6
Marks
30
NameDateScore / 30
1.
Solve:
x + y = 10
3x − y = 14
[5]
2.
Solve:
x + y = 8
2x − y = 4
[5]
3.
Solve:
x + y = 9
3x − y = 11
[5]
4.
Solve:
x + y = 5
4x − y = 0
[5]
5.
Solve:
x + y = 4
4x − y = 1
[5]
6.
Solve:
x + y = 6
2x − y = 0
[5]

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