How to Substitute into Formulae: KS3 Maths Step-by-Step Guide

Substitution means replacing a letter in a formula or expression with a number and then calculating the result. It is one of the most-tested algebra skills at KS3 and the foundation of every formula you will ever use in GCSE maths and science.

What substitution means

When you write a formula such as A = lw, the letters l and w stand for unknown lengths. If you are told that l = 8 and w = 5, you substitute — swap the letters for the numbers — and calculate: A = 8 × 5 = 40.

That is substitution in a single sentence: replace the letter, then calculate.

Step-by-step method

Write out the formula — copy it exactly.

Replace each letter with the given number — put the number in brackets to avoid sign errors.

Apply BIDMAS — calculate in the correct order (Brackets first, then Indices, then ÷/×, then +/−).

Write the answer with its units — area in cm², speed in m/s, etc.

Worked examples — positive values

Worked example 1 — one variable

Find the value of 3n + 5 when n = 4.

Step 1: Write the expression: 3n + 5

Step 2: Replace n with (4): 3(4) + 5

Step 3: Calculate — multiplication first: 12 + 5 = 17

Answer: 17

Worked example 2 — two variables

The formula for the perimeter of a rectangle is P = 2(l + w).

Find P when l = 7 cm and w = 3 cm.

Step 1: P = 2(l + w)

Step 2: Replace: P = 2(7 + 3)

Step 3: Brackets first: 7 + 3 = 10, so P = 2 × 10 = 20

Answer: P = 20 cm

Worked example 3 — formula with a squared term

The area of a circle is A = πr².

Find A when r = 5 cm (use π = 3.14).

Step 1: A = πr²

Step 2: Replace: A = 3.14 × (5)²

Step 3: Index first: 5² = 25, so A = 3.14 × 25 = 78.5

Answer: A = 78.5 cm²

Worked examples — negative values

Substituting negative numbers is where most KS3 pupils make errors. Using brackets every time prevents sign mistakes.

Worked example 4 — substituting a negative number

Find the value of 4x − 3 when x = −2.

Step 2: Replace x with (−2): 4(−2) − 3

Step 3: 4 × (−2) = −8, so −8 − 3 = −11

Answer: −11

Worked example 5 — a squared negative

Find the value of n² + 2n + 1 when n = −3.

Replace: (−3)² + 2(−3) + 1

Index first: (−3)² = 9

Then multiplication: 2 × (−3) = −6

Then add: 9 + (−6) + 1 = 9 − 6 + 1 = 4

Answer: 4

Key point: (−3)² = (−3) × (−3) = 9. A negative number squared gives a positive result. The brackets are essential: −3² without brackets would be read as −(3²) = −9, which is different.

Worked example 6 — two negative substitutions

The formula for speed is v = u + at.

Find v when u = 20, a = −3, and t = 4.

Replace: v = (20) + (−3)(4)

Multiplication first: (−3)(4) = −12

Then addition: v = 20 + (−12) = 20 − 12 = 8

Answer: v = 8 m/s

Worked examples — fractional values

Worked example 7 — substituting a fraction

Find the value of 6m − 2 when m = 1/2.

Replace: 6(1/2) − 2

Multiplication: 6 × 1/2 = 3

Subtraction: 3 − 2 = 1

Answer: 1

Substitution in common KS3 formulae

These formulae appear across KS3 maths and science. Practise substituting into each one.

Formula | Variables | Used for
P = 2(l + w) | l = length, w = width | Perimeter of a rectangle
A = lw | l = length, w = width | Area of a rectangle
A = ½bh | b = base, h = height | Area of a triangle
v = u + at | u = initial speed, a = acceleration, t = time | Physics (speed)
y = mx + c | m = gradient, c = y-intercept | Equation of a straight line
F = ma | m = mass, a = acceleration | Physics (force)

How substitution fits the KS3 national curriculum

The Department for Education's KS3 maths programme of study states that pupils should "substitute numerical values into formulae and expressions, including scientific formulae." The statutory national curriculum for Key Stage 3 and 4 positions substitution as a gateway skill: it connects number work (applying BIDMAS) to algebraic thinking, and is tested in every KS3 assessment and throughout GCSE. Students who are fluent with substitution find it far easier to rearrange formulae and solve equations later in the course.

Common mistakes

Mistake 1 — Forgetting BIDMAS. In 3n² + 1 with n = 2: calculate 3 × (2²) + 1 = 3 × 4 + 1 = 13, NOT (3 × 2)² + 1 = 6² + 1 = 37. The index applies to n, not to 3n.

Mistake 2 — Squaring only the digit, not the sign. (−4)² = 16. Writing −4² gives −16 because the square applies only to 4. Always bracket negative values.

Mistake 3 — Multiplying instead of substituting. 3n when n = 5 gives 3 × 5 = 15, not 35. The number replaces the letter; they do not sit next to each other as digits.

Mistake 4 — Ignoring units. An area formula gives an answer in cm² or m². Always state the unit — marks are often lost by leaving it out.

Practice substitution grid

Expression | n = 3 | n = −2 | n = 0
2n + 1 | 7 | −3 | 1
n² − n | 6 | 6 | 0
3n² + 2 | 29 | 14 | 2
10 − 4n | −2 | 18 | 10

Work through each cell and check your answers against those in the table above.

Frequently asked questions

What is the difference between substitution and solving an equation?

When you substitute, you are given the value of the variable and you calculate the result of the expression or formula. When you solve an equation, you are given the result and work backwards to find the variable. For example: substitution — "find 3x + 1 when x = 4" gives 13. Solving — "find x if 3x + 1 = 13" gives x = 4. The two processes are inverses of each other.

Does it matter which letter is used — x, n, or something else?

No. The letter is just a placeholder. If the formula uses h, replace h with the given number; if it uses a, replace a. Always match the letter in the formula to the letter given in the question. Never mix up two different letters in the same formula.

How do I handle a formula with several different letters?

Replace each letter one at a time, writing the number in brackets. Work through BIDMAS afterwards. For F = ma with m = 6 and a = −5: write F = (6)(−5) = −30. Tackling one letter at a time reduces errors.

Why do I put the substituted number in brackets?

Brackets prevent two types of error: first, they stop negative signs being dropped (e.g. 4 × (−3) is clear, whereas 4 × −3 can be misread); second, they make it obvious that the number has replaced the letter, so you do not accidentally treat the coefficient and the value as digits of a larger number.

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