Using Formulae: Substitute and Solve

A10Formulae

Rules you can calculate with.

A formula connects quantities with letters — substitute to use it.

Taxi fare: C = 3 + 2m (£3 flag fee plus £2 per mile).

C = 3 + 2m

5-mile trip: C = 3 + 2 × 5 = 3 + 10 = £13

Formulas you already know: P = 2 × (l + w) · area = l × w · V = l × w × h.

Learn the 3-Step Method

1Read what each letter means.

C = cost, m = miles. The formula is a sentence in symbols.

C, m defined

2Substitute the values you know.

m = 5: C = 3 + 2 × 5.

3 + 2 × 5

3Calculate with the order of operations.

Multiply first: 3 + 10 = 13. So C = £13.

£13

💡 Key IdeaFormulas can run backwards too: paid £17? Then 2m = 14, so m = 7 miles — a two-step equation in disguise.

Worked Example

Using P = 2 × (l + w): find the perimeter of a 9 cm by 4 cm rectangle — then find w when P = 30 and l = 10.

Forwards

P = 2 × (9 + 4)
= 2 × 13 = 26

26 cm

Backwards

30 ÷ 2 = 15
15 − 10 = 5

w = 5 cm

Watch Out! Common Mistakes

Ignoring the brackets.

2 × (9 + 4) = 18 + 4 = 22 ✕

Brackets first! 9 + 4 = 13, then × 2 = 26.

Adding before multiplying.

C = 3 + 2 × 5 = 5 × 5 = 25 ✕

Order of operations! 2 × 5 first: 3 + 10 = £13.

Muddling which letter is which.

Label first! Write ‘m = 5’ before substituting. Swapping cost and miles produces nonsense answers.

⬇ Cheat Sheet

FORMULAE

Read the letter definitions first.
Substitute known values.
Brackets, then × and ÷, then + and −.
Backwards questions become equations — undo step by step.

Examples: C = 3 + 2m, m = 5 → £13 · P = 2×(l+w), 9 and 4 → 26 cm.

📖 Continue Your Journey

✓ A1 What Is Algebra?✓ A2 Function Machines ✓ A3 Writing Expressions ✓ A4 Substitution ✓ A5 Simplifying Expressions ✓ A6 One-Step Equations ✓ A7 Two-Step Equations ✓ A8 Sequences ✓ A9 The nth Term A10 Formulae A11 Bar Models for Unknowns A12 Algebra Problem Solving

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Next Lesson: A11 Bar Models for UnknownsDraw the problem as bars.

📄 Free printable: Download the Formulae cheat sheet (PDF) — keep it by your child's desk.

Quick summary

A formula states a rule in symbols; read the letter definitions first.
Substitute the known values, then follow the order of operations.
C = 3 + 2m with m = 5 gives £13.
Backwards use turns the formula into an equation — undo step by step.

Frequently asked questions

What is a formula in maths?

A rule written with letters that connects quantities — like C = 3 + 2m for a taxi's cost, or P = 2 × (l + w) for a rectangle's perimeter.

How do you use a formula?

Substitute the values you know for their letters, then calculate following brackets and the order of operations.

Can you work a formula backwards?

Yes — if C = 17 in C = 3 + 2m, subtract 3 (2m = 14) and divide by 2 (m = 7). It's a two-step equation.

A11 Bar Models for Unknowns