Substituting Values into Formulae and Expressions Revision Notes | GCSE AQA Higher Maths

1. Why Substitution Matters

Every formula you will ever use in maths or science — speed, area, kinetic energy, the quadratic formula — only becomes useful when you plug numbers in. That process is called substitution, and it is one of the most frequently tested skills across all GCSE maths papers.

Think of a formula as a machine. The variables are the input slots, and the result is the output. Substitution is the act of feeding numbers into those slots and turning the handle. Get the substitution right, and the machine gives you the correct answer every time.

This skill appears on both questions without a calculator and calculator-allowed exam questions , so you need to be comfortable working with and without a calculator. The method is identical either way — only the arithmetic changes.

2. The Core Method: Substitute, Then Calculate

The method has two clear stages. First, write out the expression or formula and replace every letter with its given value — use brackets around negative numbers to keep the signs safe. Second, evaluate using BIDMAS: sort out any brackets first, then indices (powers and roots), then multiplication and division, and finally addition and subtraction.

Keeping these two stages separate — substitute first, calculate second — prevents the most common errors. Students who try to do both at once often drop a negative sign or skip a squaring step.

3. Substituting into Expressions

Start with expressions — these have no equals sign and no rearranging to do. Your only job is to replace the letters and evaluate. Here is a straightforward example to anchor the method before things get more involved.

Worked example: Evaluating an Expression

Find the value of $3x^2 - 2y + 5$ when $x = 4$ and $y = -3$.

1. $3x^2 - 2y + 5$ — Copy the expression exactly before touching any numbers.
2. $3(4)^2 - 2(-3) + 5$ — Replace $x$ with $4$ and $y$ with $-3$. Wrap $-3$ in brackets to protect the sign. [M1]
3. $(4)^2 = 4 \times 4 = 16$ — Indices come before multiplication in BIDMAS. Expand the square longhand.
4. $3 \times 16 = 48$ and $2 \times (-3) = -6$ — Deal with the coefficients now that the index is resolved.
5. $48 - (-6) + 5 = 48 + 6 + 5 = 59$ — Subtracting a negative is the same as adding. Final answer. [A1]

Answer: $59$

4. Substituting into Formulae

Formulae work in exactly the same way as expressions — you still substitute and evaluate — but now the formula has an equals sign and usually a physical meaning. The higher-tier specification requires you to handle scientific formulae too, so the values you substitute will sometimes carry units.

A reliable habit is to write the formula in its original form first, then substitute on the next line, then evaluate. This three-line layout earns method marks even if your arithmetic slips at the end.

Worked example: Substituting into a Scientific Formula

A ball of mass $2.5$ kg moves at a speed of $6$ m/s. Calculate its kinetic energy using $KE = \frac{1}{2}mv^2$.

1. $KE = \frac{1}{2}mv^2$ — Always start by stating the formula. This earns the first method mark. [M1]
2. $KE = \frac{1}{2} \times 2.5 \times (6)^2$ — Replace $m$ with $2.5$ and $v$ with $6$. Bracket the $6$ before squaring. [M1]
3. $(6)^2 = 6 \times 6 = 36$ — Square the speed first — indices before multiplication in BIDMAS.
4. $KE = \frac{1}{2} \times 2.5 \times 36 = \frac{1}{2} \times 90 = 45$ — $2.5 \times 36 = 90$, then $\frac{1}{2} \times 90 = 45$. [A1]
5. $KE = 45 \text{ J}$ — Kinetic energy is measured in joules. Missing units costs a mark in context questions.

Answer: $KE = 45 \text{ J}$

5. Handling Negative and Fractional Values

Negative values and fractions are where most marks are dropped. The rules do not change — you still substitute and apply BIDMAS — but the arithmetic demands more care. Two situations catch students out repeatedly: squaring a negative number, and substituting a fraction into a term with a coefficient.

When a negative number is squared, the result is always positive. This is because a negative multiplied by a negative gives a positive. The bracket is what makes this visible on the page: $(-3)^2 = (-3) \times (-3) = 9$, not $-9$.

Worked example: Substituting a Negative Value

Find the value of $5 - 2x^2$ when $x = -3$.

1. $5 - 2x^2$ — Copy the expression before substituting.
2. $5 - 2(-3)^2$ — Wrap $-3$ in brackets. The square applies to $-3$, not just $3$. [M1]
3. $(-3)^2 = (-3) \times (-3) = 9$ — Negative times negative is positive. This step is where errors most often occur.
4. $2 \times 9 = 18$ — Now apply the coefficient of $2$.
5. $5 - 18 = -13$ — The final answer is negative here — that is perfectly valid. [A1]

Answer: $-13$

6. Scientific Formulae in Context

The higher-tier specification explicitly requires you to substitute into scientific formulae. These appear in maths papers as well as science papers, so you need to be comfortable with them in both settings. The formula is always given to you — your job is accurate substitution and evaluation.

Two scientific formulae that appear regularly in GCSE Higher maths contexts are the area of a circle and the formula for speed. Both follow the same two-stage method: write the formula, substitute, then evaluate in BIDMAS order.

Worked example: Substituting into the Area of a Circle Formula

A circular pond has radius $r = 7$ m. Calculate its area using $A = \pi r^2$. Give your answer to 1 decimal place.

1. $A = \pi r^2$ — State the formula first. [M1]
2. $A = \pi \times (7)^2$ — Replace $r$ with $7$. Bracket the $7$ before squaring.
3. $(7)^2 = 7 \times 7 = 49$ — Square first, then multiply by $\pi$.
4. $A = \pi \times 49 = 49\pi \approx 153.93804...$ — On a calculator-allowed exam question, use the $\pi$ button for full accuracy. Keep the unrounded value before rounding.
5. $A \approx 153.9 \text{ m}^2$ — Round to 1 decimal place as instructed. Include the unit $\text{m}^2$. [A1]

Answer: $A \approx 153.9 \text{ m}^2$

7. Deliberate Practice

Now that the method is clear, work through these problems independently. For each one, write the formula or expression first, substitute on the next line, then evaluate step by step. Check your answers below only after attempting each question.

8. Exam Technique and Common Mistakes

Substitution questions are reliable mark-earners — the method is always the same. The marks you lose here are almost always avoidable. The three traps below account for the vast majority of errors seen in mark schemes.

Substitution underpins almost every calculation topic in GCSE maths and science. Master the two-stage habit — substitute first, evaluate second using BIDMAS — and you have a reliable method that works for every expression and formula you will encounter at this level.