Substitution into Formulae and Expressions — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers replacing letters with numbers and working out the result. By the end of this guide you should be able to substitute into expressions and formulae, handle powers and brackets correctly, and work confidently with negative values.

You should also be able to use standard formulae such as those for area, speed and the circumference of a circle, substitute into a formula and then solve for a letter that is not the subject, and round an answer sensibly.

The organising idea is that substitution is a two-stage job: replace, then evaluate — and the two must not be mixed. Write the expression out with the numbers in place of the letters before doing any arithmetic. That single habit removes almost every error in this topic, because the moment a number replaces a letter the hidden multiplication signs become visible, the brackets around negative values get written down, and the order of operations can be applied to something you can actually see. Students who substitute and calculate in one mental step are the ones who lose marks.

Key terms and definitions

Expression — a collection of terms with no equals sign, such as 3x + 2.

Formula — a rule connecting quantities, written with an equals sign, such as A = lw.

Variable — a letter standing for a number.

Substitute — replace each letter with its given value.

Term — a part of an expression separated by a plus or minus sign.

Coefficient — the number multiplying a letter. In 5x the coefficient is 5.

Subject — the letter on its own, usually on the left of a formula.

Core concepts

The hidden multiplication signs

Algebra leaves multiplication signs out, and they must be put back when substituting.

3a means 3 × a, so with a = 4 it is 12, not 34.

ab means a × b, so with a = 3 and b = 5 it is 15, not 35.

5(x + 2) means 5 × (x + 2), so the bracket is worked out first and then multiplied.

Writing the multiplication sign in as you substitute costs nothing and prevents the commonest error of all, which is writing the digits side by side.

Order of operations

Once the numbers are in place, apply BIDMAS as normal: brackets, then indices, then division and multiplication, then addition and subtraction.

For 2a² with a = 3: the index applies to the a alone, so a² = 9 first, then 2 × 9 = 18. Working left to right and squaring the 6 would give 36, which is the answer to (2a)² instead.

This distinction matters throughout. 3x² squares only the x; (3x)² squares the whole thing.

For 5 + 2b with b = 4: the multiplication comes first, giving 5 + 8 = 13, not 7 × 4.

Powers

A power tells you how many times to multiply the value by itself. With a = 5, a² = 5 × 5 = 25 and a³ = 125.

The most frequent slip is multiplying by the index instead: a² is not 5 × 2.

Fractional and negative indices follow the usual rules, so with a = 4, a^(1/2) is the square root, giving 2, and a^(−1) is the reciprocal, giving 0.25.

Brackets

Work out the inside of the bracket completely before doing anything with it.

For 3(a + b) with a = 2 and b = 4: the bracket gives 6, then 3 × 6 = 18.

For (a + b)² with the same values: the bracket gives 6, then 6² = 36. Squaring the terms separately would give 4 + 16 = 20, which is a different and wrong answer.

Negative numbers

This is where most marks are lost, and brackets are the defence.

Write a negative value inside brackets as you substitute. With x = −3, write x² as (−3)², which is clearly 9, whereas writing −3² invites the reading −(3²) = −9.

The two are genuinely different: (−3)² = 9 but −3² = −9. Which one a question means is decided by whether the letter itself is negative.

Odd powers keep the sign: with x = −2, x³ = (−2)³ = −8.

Subtracting a negative adds: with a = 5 and b = −3, a − b = 5 − (−3) = 8.

Multiplying two negatives gives a positive: with x = −4 and y = −2, xy = 8.

Substituting into standard formulae

Many questions use formulae you are expected to know or are given.

Area of a rectangle: A = lw. Area of a triangle: A = ½bh. Circumference: C = πd. Speed: speed = distance ÷ time.

The method never changes — replace the letters, then evaluate — but two extra things matter.

First, check the units match before substituting. A formula mixing centimetres and metres will give a meaningless answer, so convert first.

Second, give the units in the answer, including squares for areas and compound units such as m/s for speeds.

Finding a letter that is not the subject

Sometimes the value you need is not the one the formula gives directly. Substitute everything you know, then solve the resulting equation.

For A = lw with A = 48 and l = 6: substituting gives 48 = 6w, so w = 8.

This works for any formula. Substituting first and solving afterwards is usually easier than rearranging the formula and then substituting, because you are solving with numbers rather than letters.

Substituting into fractions

Where the expression is a fraction, the bar acts as a bracket around the whole of the top and the whole of the bottom.

For (a + b)/(a − b) with a = 7 and b = 3: the top gives 10 and the bottom gives 4, so the value is 2.5.

Evaluate the top completely and the bottom completely, then divide. Dividing partway through is the usual error, and on a calculator it means bracketing the numerator and denominator before pressing divide.

Substituting one expression into another

Some questions give two related formulae and ask you to combine them.

If y = 3x + 1 and z = 2y, then substituting gives z = 2(3x + 1), which expands to 6x + 2.

The bracket is essential. Writing z = 2 × 3x + 1 would multiply only the first term and give the wrong rule entirely.

The same care applies when a value is itself an expression: substituting x = 2t into 5x gives 10t, not 5 × 2 + t.

Checking an answer is sensible

A quick check catches most slips before they cost marks.

Ask whether the size is plausible. A rectangle with sides of 6 cm and 8 cm cannot have an area of 14 cm², which is what adding rather than multiplying would give.

Ask whether the sign is plausible. A length, an area or a speed cannot be negative, so a negative answer to such a question signals an error — usually a mishandled negative value.

Substituting a different, easy value can also confirm a rearranged formula behaves as expected.

Rounding and accuracy

Keep full accuracy through the working and round only the final answer, following the instruction in the question.

Where a formula contains π, use the calculator's π key rather than 3.14, unless told otherwise. Questions asking for an answer "in terms of π" want the symbol left in, not a decimal.

Worked examples

Example 1: Powers and order of operations

Work out 4x² − 3x when x = 5.

Substitute first, writing the multiplication signs in: 4 × 5² − 3 × 5.

Now apply the order of operations. The index goes first: 5² = 25.

Then the multiplications: 4 × 25 = 100 and 3 × 5 = 15.

Finally the subtraction: 100 − 15 = 85.

Squaring 4 × 5 instead would give 400, which is the answer to (4x)² and a different expression entirely.

Example 2: A negative value

Work out 2x² + 3x − 1 when x = −4.

Substitute with brackets around the negative value: 2 × (−4)² + 3 × (−4) − 1.

The index first: (−4)² = 16, positive because the power is even.

Then the multiplications: 2 × 16 = 32 and 3 × (−4) = −12.

Combine: 32 − 12 − 1 = 19.

Without the brackets, the first term would have been read as 2 × −16 = −32, giving −45 — an error worth three marks and entirely avoidable.

Example 3: Solving for a letter that is not the subject

The area of a triangle is A = ½bh. A triangle has an area of 36 cm² and a base of 9 cm. Find its perpendicular height.

Substitute everything known: 36 = ½ × 9 × h.

Simplify the right-hand side: 36 = 4.5h.

Solve: h = 36 ÷ 4.5 = 8 cm.

Note the units: a length, so centimetres rather than square centimetres.

Common mistakes and how to avoid them

Writing digits side by side. 3a with a = 4 is 12, not 34. Put the multiplication sign in.

Multiplying by the index. a² with a = 5 is 25, not 10.

Squaring the coefficient too. 3x² squares only the x; (3x)² squares both.

Losing the sign of a negative value. Substitute negatives inside brackets every time.

Evaluating left to right. Apply BIDMAS once the numbers are in place.

Mixing units. Convert everything to the same unit before substituting.

Dropping the units from the answer. Areas need squares, speeds need compound units.

Exam technique for "Substitution"

Write the substituted expression as a separate line before calculating anything. That line frequently carries a method mark on its own and makes every later step checkable.

Put brackets around every negative value as you substitute, without exception.

Include multiplication signs when you substitute, then remove them naturally as you evaluate.

Work down the page one operation at a time rather than combining steps.

State the units with the final answer, and check they are the right kind — length, area or a compound unit.

When a value you need is not the subject, substitute everything else first and solve the equation that results, rather than rearranging the formula first.

Quick revision summary

Substitute first, then evaluate — write the expression out with the numbers in place before doing any arithmetic.

Algebra hides its multiplication signs: 3a means 3 × a, and ab means a × b.

Apply BIDMAS to the substituted expression. An index attaches only to the letter beside it, so 3x² squares only the x, while (3x)² squares both.

A power means repeated multiplication, so a² with a = 5 is 25, not 10.

Bracket every negative value as you substitute: (−3)² = 9 but −3² = −9. Odd powers keep the sign.

For a formula, check the units match before substituting and state them in the answer, with squares for areas.

To find a letter that is not the subject, substitute everything you know and solve the resulting equation.

Keep full accuracy throughout and round only at the end.