Substitution GCSE Questions and Worked Answers

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Start with the meaning

What you need to know about substitution

One notebook costs £3. Four notebooks cost 3 × 4 = £12. Instead of rewriting that rule for every order, let n stand for the number of notebooks. Then the cost is 3 × n, usually written 3n. For an order of four, replace n by 4. That replacement is substitution.

See the idea first

Keep the rule; replace its input

A letter is a placeholder for a value. The same letter has the same value everywhere in one calculation. Writing letters next to each other means multiplication: ab means a × b.

Notebook cost at £3 each
n: notebooks	Replace n in 3n	Cost (£)
2	3 × 2	6
4	3 × 4	12
7	3 × 7	21

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Evaluate an expression

What the problem asks: Find 5x + 2 when x = 3.

How to solve it: Replace x with 3 to get 5 × 3 + 2 = 17. Multiplication happens before addition.

Use a negative input

What the problem asks: Find x² when x = −4.

How to solve it: The whole input is negative four, so write (−4)² = (−4) × (−4) = 16. Brackets preserve what is being squared.

Use a formula

What the problem asks: Use A = bh ÷ 2 to find triangle area for b = 8 cm and h = 5 cm.

How to solve it: A represents area, b base and h perpendicular height. Replace both inputs: 8 × 5 ÷ 2 = 20 cm².

A reliable routine

Evaluate a given expression or formula

Use this when the values of the letters are supplied and you need the resulting value, rather than solving for an unknown. The expression stays equivalent because each letter is replaced by the number it stands for.

Write the expression with every letter replaced.
Put brackets around negative or fractional inputs, particularly under powers.
Calculate brackets and powers, then multiplication/division, then addition/subtraction.
Include units if the formula describes a measurement.

Check: 2x² means 2 × x × x. It does not mean (2x)².

Fully worked

Substitution GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

One input

2 marks

Question

Find $4x-7$ when $x=5$ .

4(5)-7

=20-7

=13

The 4 multiplies x; it is not added to it.

Example 2

Two inputs

2 marks

Question

Find $3a+2b$ when $a=4$ and $b=-3$ .

3(4)+2(-3)

=12+(-6)

=6

Keep the sign attached to b.

Example 3

A bracket

2 marks

Question

Find $5(p-2)$ when $p=-1$ .

5((-1)-2)

=5(-3)

=-15

Calculate inside the bracket first.

Example 4

Powers and coefficients

3 marks

Question

Find $2x^2-3x$ when $x=-4$ .

2(-4)^2-3(-4)

=2(16)+12

=32+12

=44

Squaring −4 gives 16. Subtracting −12 adds 12.

Example 5

A fraction formula

3 marks

Question

Find $v=\dfrac{a+b}{c}$ when $a=7$ , $b=5$ and $c=4$ .

v=\frac{7+5}{4}

=\frac{12}{4}

=3

The fraction bar groups the entire numerator.

Example 6

A scientific formula

3 marks

Question

Use $s=ut+\frac12at^2$ to find displacement s in metres when $u=3$ m/s, $a=2$ m/s² and $t=4$ s.

s=3(4)+\frac12(2)(4)^2

=12+\frac12(2)(16)

=12+16

=28\text{ m}

Tip: square t, not the entire product at.

10 original questions · total 19 marks

Substitution GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 24 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

1

Multiplication

1 mark

Find $7n$ when $n=6$ .

Show worked answer

7(6)=42

Adjacent number and letter mean multiplication.

2

Subtract

2 marks

Find $6x-5$ when $x=3$ .

Show worked answer

6(3)-5=18-5=13

Multiply before subtracting.

3

Negative input

2 marks

Find $4a+9$ when $a=-2$ .

Show worked answer

4(-2)+9

=-8+9

=1

4

Two letters

2 marks

Find $ab$ when $a=-3$ , $b=5$ .

Show worked answer

(-3)(5)=-15

This is a product, not a two-digit number.

5

Square

2 marks

Find $y^2+2$ when $y=-5$ .

Show worked answer

(-5)^2+2

=25+2

=27

6

Cube

2 marks

Find $x^3$ when $x=-2$ .

Show worked answer

(-2)^3=(-2)(-2)(-2)

=4(-2)

=-8

An odd number of negative factors leaves a negative product.

7

Decimal input

2 marks

Find $8p+1$ when $p=0.25$ .

Show worked answer

8(0.25)+1

=2+1

=3

8

Fraction input

2 marks

Find $3t^2$ when $t=\frac23$ .

Show worked answer

3\left(\frac23\right)^2

=3\cdot\frac49

=\frac43

Square numerator and denominator.

9

Formula with units

2 marks

A rectangle has perimeter $P=2l+2w$ . Find P when l = 9 cm and w = 4 cm.

Show worked answer

P=2(9)+2(4)

=18+8

=26\text{ cm}

10

An invalid input

2 marks

Can you evaluate $\frac{12}{x-2}$ at x = 2? Explain.

Show worked answer

The denominator becomes $2-2=0$ . Division by zero is undefined, so the expression has no value for this input.

Examiner-style feedback

Common substitution mistakes

Joining digits

If a = 5, 2a is 10, not 25.

Losing a negative sign

Replace x = −3 by (−3), especially when x is squared or subtracted.

Ignoring a denominator

A fraction bar groups everything above and below it. Check for division by zero.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Replace all letters consistently.
Preserve signs with brackets.
Follow the order of operations.
Check units and allowed inputs.

Quick answers

Substitution FAQ

Is substitution the same as solving?

No. Substitution uses given values. Solving finds a value that makes an equation true.

Must I expand brackets first?

Usually not when all inputs are known. Substituting first often gives simpler arithmetic.

Build connected skills

What to revise next

Order of operations

BIDMAS

Calculate the replaced expression correctly.

Revise bidmas

Turn values into a curve

Quadratic graphs

Build a table by substitution.

Revise quadratic graphs

Content standards

Curriculum and rights review

A1–A2 numerical substitution across tiers, including unfamiliar supplied formulae. This is not the simultaneous-equation substitution method. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content