GCSE MATHS • HIGHER TIER

Substitution Method Made Simple

Solve simultaneous equations by replacing one variable with an equivalent expression, solving, and substituting back.

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Clear method

Exam practice

Worked proofs

Substitution Method in Four Moves

1

Rearrange

Rearrange

2

Substitute

Replace that variable in the second equation with the equivalent expression.

3

Solve

Solve the resulting one-variable equation.

4

Substitute Back

Use the first solution to find the second variable, then check both values.

What You’ll Learn

Understand simultaneous equations.

Understand how substitution reduces two variables to one.

Know when substitution is a useful method.

Rearrange equations when necessary.

Use brackets correctly when substituting expressions.

Solve for the first variable and substitute back.

Check solutions in both original equations.

Key Rules

Choose the easiest variable to isolate.

Make one variable the subject first if needed.

Substitute the full expression into the other equation.

Always use brackets around substituted expressions.

Worked Proof: Substitution Method

Step-by-Step Method

  • 1. Choose one equation and make a variable the subject.
  • 2. Substitute this expression into the second equation.
  • 3. Solve the resulting equation.
  • 4. Find the first variable.
  • 5. Give the answer in exact or decimal form.

More Worked Examples

1

Question :
y = x + 2
2x + y = 11

Step 1 : Substitute y = x + 2 into the second equation.

2x + (x + 2) = 11
Step 2 : Simplify.
3x + 2 = 11
3x = 9
x = 3
Step 3 : Find y.
y = 3 + 2 = 5
Answer : x = 3, y = 5

2

Question :
y = 2x + 1
x + y = 10

Substitute

x + (2x + 1) = 10
3x + 1 = 10
3x = 9
x = 3
Find y.
y = 2(3) + 1 = 7
Answer : x = 3, y = 7

3

Question :
x = y + 4
x + y = 12

Substitute x.

(y + 4) + y = 12
2y + 4 = 12
2y = 8
y = 4
Find x.
x = 8
Answer : x = 8, y = 4.

4

Question :
y = 3x - 2
x + y = 14

Substitute.

x + (3x - 2) = 14
4x - 2 = 14
4x = 16
x = 4
y = 10
Answer : x = 4, y = 10.
  • Common GCSE Mistakes
  • Forgetting brackets when substituting an expression.
  • Substituting into the wrong equation.
  • Making algebraic simplification errors.
  • Forgetting to substitute back for the second variable.
  • Failing to check the final solution.
  • Exam Tips
  • Choose the equation with the easiest variable to isolate.
  • Use brackets around substituted expressions.
  • Write every algebraic step clearly.
  • Rearrange carefully before substituting when needed.
  • Substitute back into an original equation for the second variable.
  • Check both values in both original equations.

Skills Used in Substitution Method

Simultaneous Equations

Substitution

Rearranging

Linear Equations

Algebraic Manipulation

Checking Solutions

Graphs

Algebra

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Frequently Asked Questions

They are two equations containing the same variables, with a solution that satisfies both equations at the same time.

It replaces one variable with an equivalent expression so the second equation contains only one variable.

It is especially useful when one variable is already isolated or can easily be made the subject.

They help preserve the full substituted expression and reduce sign and simplification errors.

Rearrange one equation first, then apply substitution.

Substitute both values into the original equations and confirm that both equations are satisfied.