What you'll learn
Substitution means replacing letters in a formula or expression with given numbers and working out the result. In this guide you will learn how to substitute values correctly, how to handle powers, brackets and negative numbers, how to use the order of operations, and how to substitute into common formulae. This is a fundamental skill used across all of GCSE Maths and science.
Key terms and definitions
Substitution — replacing a variable with a given value.
Formula — a rule connecting variables, e.g. area = length × width.
Expression — a combination of numbers and letters with no equals sign.
Order of operations — the sequence (BIDMAS) for carrying out calculations.
Variable — a letter standing for a number.
Core concepts
Substituting values
To substitute, replace each letter with its given value and calculate. For example, if a = 3 and b = 4, then a + b = 3 + 4 = 7, and ab = 3 × 4 = 12. Write the calculation out clearly before evaluating to avoid mistakes.
Using the order of operations
Follow BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) when evaluating. For example, 2a² with a = 3 means 2 × 3² = 2 × 9 = 18 (do the index first, then multiply) — not (2 × 3)² = 36.
Powers and brackets
Be careful with powers: a² means a × a, so with a = 5, a² = 25. With brackets, work out the inside first. For 3(a + b) with a = 2, b = 4: 3 × (2 + 4) = 3 × 6 = 18.
Negative numbers
Take care substituting negative values, especially with powers. For a = −2: a² = (−2)² = 4 (negative squared is positive), but a³ = (−2)³ = −8. Use brackets around negative values when substituting to keep signs correct.
Substituting into formulae
Many problems use standard formulae — area, perimeter, speed, etc. Substitute the given values, then evaluate using BIDMAS. For example, speed = distance ÷ time: with distance 100 and time 4, speed = 100 ÷ 4 = 25.
Worked examples
Example 1: Simple substitution
Find 4x + 5 when x = 3.
4 × 3 + 5 = 12 + 5 = 17.
Example 2: With a power
Find 3y² when y = 4.
3 × 4² = 3 × 16 = 48 (square first, then multiply).
Example 3: Negative value
Find x² − 2x when x = −3.
(−3)² − 2(−3) = 9 + 6 = 15.
Common mistakes and how to avoid them
Ignoring the order of operations. Do indices and brackets before multiplying or adding.
Squaring after multiplying. 2a² means 2 × a², not (2a)².
Sign errors with negatives. Use brackets: (−2)² = 4, (−2)³ = −8.
Forgetting brackets in the original. 3(a + b) needs the bracket evaluated first.
Rushing the calculation. Write out each step before evaluating.
Exam technique for Substitution
Replace each letter with its value, using brackets for negatives.
Apply BIDMAS — indices and brackets before multiplication and addition.
Handle powers carefully, squaring or cubing the value itself.
Write out the steps rather than doing it all at once.
Check signs at the end, especially with negative inputs.
Quick revision summary
Substitution replaces letters with given numbers, then evaluates. Always follow the order of operations (BIDMAS): brackets and indices before division/multiplication, then addition/subtraction. Be careful that 2a² means 2 × a² (square first), not (2a)². With powers, a² = a × a. With brackets, evaluate the inside first (3(a + b)). Take special care with negative numbers and powers: (−2)² = 4 but (−2)³ = −8 — use brackets around negative values when you substitute. For formulae (area, speed, etc.), put in the values and evaluate with BIDMAS. The usual mistakes are ignoring the order of operations, squaring the coefficient too, sign slips with negatives, and skipping brackets. Replace each letter carefully (brackets for negatives), apply BIDMAS, treat powers correctly, write out the steps, and check signs — and substitution becomes quick and reliable across maths and science.