What is substitution?
Substitution means replacing a letter (variable) in an expression or formula with a given number. It is used throughout GCSE Maths — from evaluating expressions to using real-world formulae in physics, engineering and finance.
The key rules:
- Always write brackets around negative values before substituting
- Apply BIDMAS (powers before multiplication, multiplication before addition)
- Be especially careful with negative numbers raised to a power: (−3)² = 9, not −9
Basic substitution
Worked Example 1
If x = 4 and y = −3, find the value of 3x − 2y
1
Replace x with 4 and y with −3: 3(4) − 2(−3)
2
Multiply: 12 − (−6) = 12 + 6 = 18
Answer18
Substitution with powers
Worked Example 2
If a = 3 and b = −2, find 4a − 3b²
1
Substitute: 4(3) − 3(−2)²
2
Apply power first (BIDMAS): (−2)² = 4, so 12 − 3(4)
3
Complete: 12 − 12 = 0
Answer0
Substituting into formulae
Worked Example 3
The formula for the area of a trapezium is A = ½(a + b)h. Find A when a = 5, b = 9, h = 4
1
Substitute: A = ½(5 + 9) × 4
2
Bracket first: ½(14) × 4 = 7 × 4 = 28
AnswerA = 28
Worked Example 4 — Negative input
Find the value of x³ − 2x² + 3x − 1 when x = −2
1
Substitute x = −2: (−2)³ − 2(−2)² + 3(−2) − 1
2
Evaluate each term: −8 − 2(4) + (−6) − 1
3
Simplify: −8 − 8 − 6 − 1 = −23
Answer−23
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1If x = 3 and y = −2, find the value of 2x + yFoundation
Q2If a = 4 and b = 2, find a² − 3bFoundation
Q3Find the value of 3x² − 2x + 1 when x = −3Foundation
Q4If p = 2 and q = −1, find 4p − 3q²Foundation
Q5Using s = ut + ½at², find s when u = 0, a = 10, t = 5Foundation
Higher (Grade 5–7)
Q6Find the value of x³ − 2x² + 3x − 1 when x = −2Higher
Q7If a = 3 and b = −4, find (a + b)² − (a − b)Higher
Q8Find the value of 4x² − 1 when x = ½Higher
Q9If m = 2 and n = −3, find m²n − mn²Higher
Q10Using v = u + at, find v when u = 5, a = −3, t = 2Higher
Higher — Hard (Grade 8–9)
Q11If a = 9 and b = 4, find √a + √bGrade 8–9
Q12Find the value of x⁴ − 3x³ + x² − 2 when x = 2Grade 8–9
Q13If p = 3 and q = −2, find 2p² − pq + q²Grade 8–9
Q14If a = 4 and b = −1, find (2a − b) ÷ (a + b)Grade 8–9
Q15Substitute n = 1, 2, 3 into the expression n² + n and show that the result is always even. What algebraic argument proves this for all positive integers?Grade 8–9
Answers
Foundation (Q1–Q5)
Q14(2(3) + (−2) = 6 − 2)
Q210(4² − 3(2) = 16 − 6)
Q334(3(9) − 2(−3) + 1 = 27 + 6 + 1)
Q45(4(2) − 3(−1)² = 8 − 3)
Q5125(0 + ½(10)(25) = 125)
Higher (Q6–Q10)
Q6−23(−8 − 8 − 6 − 1)
Q7−6((3+(−4))² − (3−(−4)) = (−1)² − 7 = 1 − 7)
Q80(4(¼) − 1 = 1 − 1)
Q9−30((4)(−3) − (2)(9) = −12 − 18)
Q10−1(5 + (−3)(2) = 5 − 6)
Higher — Hard (Q11–Q15)
Q115(√9 + √4 = 3 + 2)
Q12−6(16 − 24 + 4 − 2)
Q1328(2(9) − (3)(−2) + 4 = 18 + 6 + 4)
Q143((8+1) ÷ (4−1) = 9 ÷ 3)
Q15n=1: 2; n=2: 6; n=3: 12 (all even). n² + n = n(n+1) — the product of consecutive integers, one of which is always even, so the product is always even.
Common mistakes
Common Mistake 1
Squaring a negative and getting a negative
If x = −3, then x² = (−3)² = 9, not −9. The negative sign is inside the brackets and gets squared too. Always write brackets when substituting negative values.
Common Mistake 2
Applying the wrong order of operations
For 3x² when x = 2: the correct answer is 3 × 4 = 12. Students sometimes compute (3 × 2)² = 36. Remember — indices come before multiplication in BIDMAS: evaluate 2² = 4 first, then multiply by 3.
Common Mistake 3
Subtracting a negative incorrectly
For 5 − (−3), students sometimes write 5 − 3 = 2. This is wrong. Subtracting a negative is the same as adding: 5 − (−3) = 5 + 3 = 8.
Common Mistake 4
Not substituting every occurrence of the variable
In 2x² − x + 3 with x = 4, students sometimes only substitute the first x: 2(16) − x + 3. Every x must be replaced: 2(16) − 4 + 3 = 32 − 4 + 3 = 31.
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