Simultaneous equations are two equations with two unknowns that must both be true at the same time. You solve them by elimination (adding or subtracting equations to remove one unknown) or substitution (replacing one unknown with an expression). Both methods lead to the same answer.

What are simultaneous equations?

Two equations share two unknowns — usually x and y. A single equation has infinitely many solutions; a pair of simultaneous equations narrows that down to exactly one point where both are satisfied at once. That point is the intersection of their two straight-line graphs.

Term Meaning
Simultaneous Both equations are true at the same time
Elimination Add/subtract equations to remove one unknown
Substitution Replace one unknown using one equation, then solve the other

How does the elimination method work?

Elimination removes one variable by adding or subtracting the two equations. The secret is to make the coefficient (the number in front) of one letter the same in both equations first.

Worked example — elimination:

Solve:
2x + 3y = 12
2x + y = 8

  1. The coefficients of x are already both 2, so subtract the second equation from the first.
  2. (2x + 3y) − (2x + y) = 12 − 8
  3. 2y = 4, so y = 2.
  4. Substitute y = 2 into the second equation: 2x + 2 = 8, so 2x = 6, giving x = 3.
  5. Check in the first: 2(3) + 3(2) = 6 + 6 = 12. ✓

How does the substitution method work?

Substitution works well when one equation already has a single letter isolated (or is easy to rearrange).

Worked example — substitution:

Solve:
y = 2x − 1
3x + y = 14

  1. The first equation gives y in terms of x. Substitute into the second.
  2. 3x + (2x − 1) = 14
  3. 5x − 1 = 14, so 5x = 15, giving x = 3.
  4. Back-substitute: y = 2(3) − 1 = 5, so y = 5.
  5. Check in the second: 3(3) + 5 = 9 + 5 = 14. ✓

When should you use elimination versus substitution?

Choose whichever makes the algebra simpler:

  • Elimination is usually faster when both equations are in the form ax + by = c and the coefficients of one letter match (or become equal after multiplying).
  • Substitution is usually faster when one equation already says y = … or x = ….

At GCSE you will see both, so practise each until you can spot which is quicker at a glance.

What if the coefficients do not match?

Multiply one (or both) equations by a suitable number to make the coefficients of one unknown equal, then eliminate.

Solve:
3x + 2y = 13
x + 4y = 11

  1. Multiply the second equation by 3: 3x + 12y = 33.
  2. Subtract the first: (3x + 12y) − (3x + 2y) = 33 − 13, giving 10y = 20, so y = 2.
  3. Substitute back: x + 4(2) = 11, so x = 3, giving x = 3.
  4. Check: 3(3) + 2(2) = 9 + 4 = 13. ✓

How do you verify your answer?

Always substitute both values into the equation you did not use to find the second variable. If the left-hand side equals the right-hand side, both unknowns are correct. This step takes ten seconds and catches most arithmetic slips.

Why do simultaneous equations appear on GCSE papers?

Simultaneous equations test whether you can combine two pieces of information to pin down two unknowns — a fundamental skill in algebra, science, and economics. They appear on both foundation and higher GCSE tiers and carry several marks per question, making them well worth mastering early.

Frequently asked questions

What does it mean to solve simultaneous equations?

It means finding values of x and y (or any two unknowns) that satisfy both equations at the same time. There is usually exactly one solution, which corresponds to the point where the two lines cross on a graph.

Which method is better — elimination or substitution?

Neither is universally better. Elimination is often quickest when coefficients match; substitution is quicker when one equation isolates a variable. At GCSE, practise both so you can choose the faster option during an exam.

What if after solving I get a contradiction like 0 = 5?

A contradiction (for example, 0 = 5) means the two equations represent parallel lines that never cross — there is no solution. If you get an identity like 0 = 0, the lines are the same and there are infinitely many solutions. Both are edge-case GCSE Higher questions.

How do I check my simultaneous-equation answer?

Substitute both values back into each original equation and confirm both sides are equal. Checking in only one equation is not enough — verify in both.

For Socratic simultaneous-equations practice at KS3 and GCSE, see aitutors.me.