Simultaneous equations | AQA GCSE Maths Notes

Visual model

The solution is where two lines meet

solutionsame x and same y

Gold-standard guide

20 mins

What you will learn

Find values that satisfy two equations at the same time.

Use a clear step-by-step method for simultaneous equations.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier algebra skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it

Step 1

Choose the elimination method and scale one equation

Multiply the second equation by 2: 2x − 2y = 4

Step 2

Add the equations to eliminate y

(3x + 2y) + (2x − 2y) = 16 + 4 → 5x = 20 → x = 4

Step 3

Substitute x = 4 into one original equation to find y

4 − y = 2 → y = 2

Watch out

Students add when they should subtract (or vice versa)

Elimination

Make one variable coefficient match, then add or subtract equations.

Check

Substitute both values into both original equations.

Worked example

Solve 3x + 2y = 16 and x − y = 2 simultaneously.

Choose the elimination method and scale one equation: Multiply the second equation by 2: 2x − 2y = 4.Now the y-coefficients are 2 and −2.

Add the equations to eliminate y: (3x + 2y) + (2x − 2y) = 16 + 4 → 5x = 20 → x = 4.

Substitute x = 4 into one original equation to find y: 4 − y = 2 → y = 2.

Check in the other original equation: 3(4) + 2(2) = 12 + 4 = 16. ✓

Final answer

x = 4, y = 2

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Solve 3x + 2y = 16 and x − y = 2 simultaneously.

4 marks4 minssimultaneous-equations-worked

Show solution

Worked solution

1.Choose the elimination method and scale one equation: Multiply the second equation by 2: 2x − 2y = 4.Now the y-coefficients are 2 and −2.
2.Add the equations to eliminate y: (3x + 2y) + (2x − 2y) = 16 + 4 → 5x = 20 → x = 4.
3.Substitute x = 4 into one original equation to find y: 4 − y = 2 → y = 2.
4.Check in the other original equation: 3(4) + 2(2) = 12 + 4 = 16. ✓

Final answer

x = 4, y = 2

Mark points

M1: choose the elimination method and scale one equation
M1: add the equations to eliminate y
M1: substitute x = 4 into one original equation to find y
M1: check in the other original equation
A1: x = 4, y = 2

Watch out

Students add when they should subtract (or vice versa). If the coefficients are the same sign, subtract the equations.If they are opposite signs, add. Writing the operation explicitly above the equation avoids this slip.

Diagnosticrecall

Solve 2x + y = 11 and x + y = 7.

1 mark2 minssimultaneous-equations-q1

Show solution

Worked solution

1.Spot the skill: Elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.
2.Use the choose the elimination method and scale one equation stage first, then add the equations to eliminate y.
3.Keep the final answer visible: x = 4, y = 3.

Final answer

x = 4, y = 3

Mark points

M1: use the correct elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.substitution: rearrange one equation for a variable and substitute into the other.check both values in both original equations.
A1: x = 4, y = 3

Watch out

Easyprocedure

Solve 5x − 2y = 1 and 3x + 2y = 15.

2 marks3 minssimultaneous-equations-q2

Show solution

Worked solution

1.Spot the skill: Elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.
2.Use the add the equations to eliminate y stage first, then substitute x = 4 into one original equation to find y.
3.Keep the final answer visible: x = 2, y = $\frac{9}{2}$ .

Final answer

x = 2, y = $\frac{9}{2}$

Mark points

M1: use the correct elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.substitution: rearrange one equation for a variable and substitute into the other.check both values in both original equations.
A1: x = 2, y = $\frac{9}{2}$

Watch out

Mediumreasoning

Solve x + 3y = 10 and 2x − y = 1 by substitution.

3 marks4 minssimultaneous-equations-q3

Show solution

Worked solution

1.Spot the skill: Elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.
2.Use the substitute x = 4 into one original equation to find y stage first, then check in the other original equation.
3.Keep the final answer visible: x = 1, y = 3.

Final answer

x = 1, y = 3

Mark points

M1: use the correct elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.substitution: rearrange one equation for a variable and substitute into the other.check both values in both original equations.
A1: x = 1, y = 3

Watch out

Hardproblem solving

Solve 4x + 3y = 18 and 2x − y = 2.

3 marks5 minssimultaneous-equations-q4

Show solution

Worked solution

1.Spot the skill: Elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.
2.Use the check in the other original equation stage first, then choose the elimination method and scale one equation.
3.Keep the final answer visible: x = 3, y = 2.

Final answer

x = 3, y = 2

Mark points

M1: use the correct elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.substitution: rearrange one equation for a variable and substitute into the other.check both values in both original equations.
A1: x = 3, y = 2

Watch out

Exam-stylemulti-step

Two numbers sum to 40. Their difference is 8. Find them.

4 marks6 minssimultaneous-equations-q5

Show solution

Worked solution

1.Spot the skill: Elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.
2.Use the choose the elimination method and scale one equation stage first, then add the equations to eliminate y.
3.Keep the final answer visible: 24 and 16.

Final answer

24 and 16

Mark points

M1: use the correct elimination: multiply to make one variable's coefficient equal in both equations, then add or subtract to remove it.substitution: rearrange one equation for a variable and substitute into the other.check both values in both original equations.
A1: 24 and 16

Watch out

Grade 9 stretchproblem solving

Solve 2x + y = 13 and x - y = 2.

4 marks7 minslinear-simultaneous-g9

Show solution

Worked solution

1.Add the equations to eliminate y.
2.Find x.
3.Substitute to find y.

Final answer

x = 5, y = 3

Mark points

M1: obtain 3x = 15
A1: x = 5
A1: y = 3

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Simultaneous equations - 2 marksSolve 2x + y = 11 and x + y = 7.Mark answer

Answer

x = 4, y = 3

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer

Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer

Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer

Answer

3x(2x + 3)

Mastery check

I can explain the method for simultaneous equations.
I can show clear working without skipping key steps.
I can avoid this mistake: Students add when they should subtract (or vice versa).If the coefficients are the same sign, subtract the equations. If they are opposite signs, add.Writing the operation explicitly above the equation avoids this slip.

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