A7AlgebraFoundation & Higher

Simultaneous Equations

Simultaneous equations are two equations with two unknowns that are true at the same time. Solving them finds the pair of values that works for both — the point where two lines cross.

45 min Video by Maths Genie AQA GCSE Maths

Solving Simultaneous EquationsWatch the walkthrough, then read the notes below.

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What you'll learn

Solve simultaneous equations by elimination
Solve by substitution
Interpret the solution as an intersection point
Check both equations

Elimination

Make the coefficients of one letter match, then add or subtract the equations to eliminate it. Solve for the remaining letter, then substitute back to find the other.

1Add the equations:

2 x = 14

, so

x = 7

2Substitute into the first:

7 + y = 10

, so

y = 3

Answer

x = 7, y = 3

Tip — Same signs → subtract; different signs → add (to eliminate a letter).

Substitution

Rearrange one equation to make a letter the subject, then substitute that expression into the other equation. This is the go-to method when one equation already has a single letter on its own.

Remember these

match a coefficient, then add/subtract

Elimination method.

solution = point where the lines cross

Graphical meaning.

Watch out for these

Adding when you should subtract (or vice versa).

Add to eliminate when signs differ; subtract when they match.

Finding one letter and forgetting the other.

Substitute back to find the second unknown, and check both equations.

Key takeaways

Elimination: match a coefficient, add or subtract.
Substitution: make a letter the subject, then sub in.
The solution is where the two lines intersect.

Test yourself

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