Simultaneous Equations

Topic overview

What you need to know

Simultaneous Equations is a key part of GCSE Mathematics. Build fluency with the method, then apply it to unfamiliar and multi-step problems. Two or more equations with the same unknowns that are solved together to find values satisfying all equations. Add or subtract the equations to eliminate one variable, then solve for the other. May need to multiply first. Rearrange one equation to express a variable in terms of the other, then substitute into the second equation.

Exam tip

Write every stage of your method. Method marks can still be earned when the final answer is wrong.

Revision notes

Core ideas

Secure these ideas first. Say each definition in your own words, then connect it to the topic overview.

Simultaneous Equations: Two or more equations with the same unknowns that are solved together to find values satisfying all equations.
Elimination Method: Add or subtract the equations to eliminate one variable, then solve for the other. May need to multiply first.
Substitution Method: Rearrange one equation to express a variable in terms of the other, then substitute into the second equation.
Linear Simultaneous Equations: Both equations are linear (degree 1). The solution is the point where two straight lines intersect.

Revision notes

Apply it in the exam

The exam will rarely ask for an isolated definition. Practise selecting the right idea and using it as part of a complete explanation or method.

Non-linear Simultaneous Equations: One equation is quadratic and one is linear. Solved by substituting the linear into the quadratic.
Graphical Solution: Plot both equations on a graph; the solution is the coordinates of the point(s) of intersection.
No Solution: Parallel lines have the same gradient but different y-intercepts, so they never meet — no solution exists.
Forming Simultaneous Equations: Translate a word problem into two equations with two unknowns, then solve. E.g. 'Total of 20 coins worth £3.50'.

Core knowledge

Key facts for Simultaneous Equations

8 essentials

Two or more equations with the same unknowns that are solved together to find values satisfying all equations.

Elimination Method

Add or subtract the equations to eliminate one variable, then solve for the other. May need to multiply first.

Substitution Method

Rearrange one equation to express a variable in terms of the other, then substitute into the second equation.

Linear Simultaneous Equations

Both equations are linear (degree 1). The solution is the point where two straight lines intersect.

Non-linear Simultaneous Equations

One equation is quadratic and one is linear. Solved by substituting the linear into the quadratic.

Graphical Solution

Plot both equations on a graph; the solution is the coordinates of the point(s) of intersection.

No Solution

Parallel lines have the same gradient but different y-intercepts, so they never meet — no solution exists.

Forming Simultaneous Equations

Translate a word problem into two equations with two unknowns, then solve. E.g. 'Total of 20 coins worth £3.50'.

Active recall

Close the notes and answer these

1.Without looking, explain simultaneous equations and give one example or consequence.
2.Without looking, explain elimination method and give one example or consequence.
3.Without looking, explain substitution method and give one example or consequence.
4.Without looking, explain linear simultaneous equations and give one example or consequence.

Content reviewed 23 July 2026 against the current linked specification.

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