Simultaneous equations
Solving pairs of equations by elimination and substitution.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 2.3.1Solve simultaneous linear equations by elimination, substitution and graphically.
6 minute read
Simultaneous equations
Two unknowns need two equations. The aim is always to get rid of one unknown so an ordinary equation remains.
Elimination
- Multiply one or both equations so that one unknown has the same coefficient in both.
- Add if the signs are opposite; subtract if the signs are the same.
- Solve for the remaining unknown.
- Substitute back into either original equation to find the other.
- Check in the equation you did not use for the substitution.
A memory aid for step 2: Same Signs Subtract.
Substitution
Better when one equation already gives a subject, such as y = 3x + 1. Replace y in the other equation and solve.
Graphically
Two straight lines meet at exactly one point, and its coordinates are the solution. Parallel lines never meet, which is why some pairs have no solution.
Word problems
Define your variables explicitly first — "let x be the cost of one pen" — then form one equation per piece of information. Most lost marks here come from unstated or muddled definitions rather than the algebra.
Think of it like this
Elimination is cancelling out a common ingredient so only one variable is left to taste. Substitution is swapping in a known recipe for one ingredient — the same destination reached by a different route.
Worked examples
Method, step by step
Solve 3x + 2y = 16 and x − 2y = 0.
- 1The y coefficients are +2 and −2 — opposite signs, so add the equations.
- 2(3x + 2y) + (x − 2y) = 16 + 0 gives 4x = 16, so x = 4.
- 3Substitute x = 4 into the second equation: 4 − 2y = 0, so 2y = 4 and y = 2.
- 4Check in the first: 3(4) + 2(2) = 12 + 4 = 16 ✓
x = 4, y = 2
Common misconceptions
- Adding when the signs are the same, which doubles a term instead of cancelling it.
- Finding one unknown and stopping. Both values are required, and both are usually marked.
- Forgetting to multiply **every** term when scaling an equation, including the constant on the right.
In the exam
- Number your equations (1) and (2) and label the ones you create, e.g. (3) = (1) × 2. Examiners follow the method more easily and method marks are safer.
- Check the final pair in the equation you did not use to substitute — that genuinely verifies both values.