Simultaneous equations (systems of equations) step by step
Solve two equations with two unknowns using elimination and substitution. Worked examples for GCSE simultaneous equations and Algebra 1 systems.
GCSEHigh SchoolCurriculum fit: GCSE Maths Algebra (Foundation: linear; Higher: linear and non-linear). US: Algebra I systems of equations.
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What is it?
Simultaneous equations are two equations with two unknowns, such as x and y, that must both be true at the same time. The solution is the pair of values that satisfies both.
On a graph, each equation is a straight line, and the solution is the point where the lines cross.
Key formulas and ideas
- Elimination: add or subtract the equations to remove one unknown.
- Substitution: rearrange one equation to make a variable the subject, and substitute into the other.
- Always check the solution in both original equations.
- Parallel lines have no solution; identical lines have infinitely many.
Worked examples
Example 1: Elimination
Question: Solve x + y = 10 and x − y = 2.
- Add the equations: 2x = 12, so x = 6.
- Substitute into x + y = 10: 6 + y = 10, so y = 4.
Answer: x = 6, y = 4
Example 2: Substitution
Question: Solve y = 2x and x + y = 9.
- Replace y with 2x: x + 2x = 9.
- 3x = 9, so x = 3.
- y = 2 × 3 = 6.
Answer: x = 3, y = 6
Example 3: Rearranging first
Question: Solve 3x + 2y = 16 and x − y = 2.
- Rearrange the second equation: x = y + 2.
- Substitute: 3(y + 2) + 2y = 16, so 5y + 6 = 16.
- y = 2, then x = 4.
Answer: x = 4, y = 2
Common mistakes
- Adding when you should subtract (or the reverse) in elimination.
- Finding one variable and forgetting the other.
- Forgetting to substitute back into an original equation.
- Making sign errors with negatives.
Practice questions
Frequently asked questions
Which method should I use?
Use elimination when the coefficients line up easily, and substitution when one equation already has a variable on its own.
Which method should I use?
Use elimination when the coefficients match or can be matched easily. Use substitution when one variable is already isolated.
Related guides
Solving linear equations step by step (with examples)
Learn the balance method for solving linear equations, including brackets and x on both sides, with worked examples for GCSE and Algebra 1.
KS3GCSEMiddle SchoolHigh SchoolStraight line graphs: gradient and y = mx + c
Plot straight lines, work out the gradient and read the equation from a graph. Covers y = mx + c (slope-intercept form), with worked examples.
GCSEHigh SchoolFind the equation of a line from two points
Work out the gradient (slope), find the y-intercept and write the equation of a line, plus parallel and perpendicular lines and midpoints.
GCSEA-LevelHigh SchoolSources and review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.