A2: Substitution
Substitute numerical values into formulae and expressions
📋 Key Concepts
Definition: Substitution means replacing variables (letters) with their numerical values, then calculating the result.
Method:
- Replace each letter with its given value
- Calculate following BIDMAS/BODMAS rules
- Give your final answer
📝 Basic Substitution
Example 1
Find the value of 3x + 5 when x = 4
Solution:
3x + 5 = 3(4) + 5
= 12 + 5
= 17
Example 2
Find the value of 2a - 3b when a = 7 and b = 2
Solution:
2a - 3b = 2(7) - 3(2)
= 14 - 6
= 8
📝 Substitution with Powers
Remember: Powers are calculated before multiplication. x² means x × x, then multiply by any coefficient.
Example 3
Find the value of 2x² when x = 3
Solution:
2x² = 2 × 3²
= 2 × 9
= 18
Note: This is NOT (2×3)² = 36
Example 4
Find the value of x² + 3x - 2 when x = 5
Solution:
x² + 3x - 2 = 5² + 3(5) - 2
= 25 + 15 - 2
= 40 - 2
= 38
📝 Substitution with Brackets
Example 5
Find the value of 4(x - 2) when x = 7
Solution:
4(x - 2) = 4(7 - 2)
= 4 × 5
= 20
Example 6
Find the value of (a + b)² when a = 3 and b = 1
Solution:
(a + b)² = (3 + 1)²
= 4²
= 16
📝 Substitution in Formulae
Formula: A formula shows how quantities are related. Substitute values to find unknown quantities.
Example 7
The formula for the area of a circle is A = πr². Find A when r = 5 (use π = 3.14)
Solution:
A = πr² = 3.14 × 5²
= 3.14 × 25
= 78.5
Example 8
The formula for speed is s = d⁄t. Find s when d = 120 and t = 3
Solution:
s = 120⁄3 = 40
📝 Negative Numbers in Substitution
Warning: Be careful with negative signs, especially when substituting negative values.
Example 9
Find the value of 3x + 7 when x = -4
Solution:
3x + 7 = 3(-4) + 7
= -12 + 7
= -5
Example 10
Find the value of x² - 2x when x = -3
Solution:
x² - 2x = (-3)² - 2(-3)
= 9 + 6
= 15
🧮 Worked Examples — Exam Style
Substitution is guaranteed marks if you follow two rules: put EVERY substituted number in brackets, and square the whole bracket when squaring negatives.
Evaluate 3x² − 2x when x = −3
Substitute in brackets: 3(−3)² − 2(−3)
Square first: (−3)² = 9, so 3 × 9 = 27
−2(−3) = +6 → 27 + 6 = 33
M1 for 3 × 9 or correct handling of −2(−3) = +6, A1 for 33. Writing 3 −3² = 3 − 9 (no brackets) gives −30 — the classic error.
Use the formula v = u + at to find v when u = 5, a = −2, t = 8.
v = 5 + (−2)(8) = 5 + (−16) = 5 − 16 = −3
A negative answer can be correct — here it means the object is moving backwards.
M1 for correct substitution, A1 for −3. Formula questions tell you the equation — read it off the page, never work from memory.
Exam wording to expect: "Find the value of … when …", or a named formula (area, speed, v = u + at) to substitute into.
❓ Practice Questions
Q1: Find the value of 5x - 2 when x = 3.
Q2: Find the value of 2a + 3b when a = 4 and b = 5.
Q3: Find the value of 3x² when x = 4.
Q4: Find the value of (m + 2)(n - 1) when m = 5 and n = 4.
Q5: Use the formula A = 1⁄2bh. Find A when b = 10 and h = 7.
Q6: Find the value of x² + 2x - 3 when x = -2.
Q7: Evaluate 3x² − 2x when x = −3
Q8: Use v = u + at to find v when u = 5, a = −2, t = 8
✅ Answers
- 13
- 23
- 48
- 21
- 35
- -3
🧮 Worked Examples — Harder Exam Style
The cost £C of hiring a van for d days is C = 45 + 20d. Find the cost of 3 days, and the number of days affordable with £165.
Cost: C = 45 + 20 × 3 = 45 + 60 = £105
Reverse: 165 = 45 + 20d → 120 = 20d → d = 6 days
M1 for 20 × 3, A1 for 105; M1 for 165 − 45 = 120, A1 for 6. Formula questions often ask forward AND reverse — the reverse is just equation solving (A17).
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 5x − 2 = 5(3) − 2 = 15 − 2 = 13
Q2. 2a + 3b = 2(4) + 3(5) = 8 + 15 = 23
Q3. 3x² = 3(4²) = 3 × 16 = 48. Square FIRST, then multiply by 3 — not (3x)².
Q4. (m + 2)(n − 1) = (5 + 2)(4 − 1) = 7 × 3 = 21
Q5. A = ½bh = ½ × 10 × 7 = ½ × 70 = 35
Q6. x² + 2x − 3 with x = −2: (−2)² + 2(−2) − 3 = 4 − 4 − 3 = −3
Q7. 3(−3)² − 2(−3) = 27 + 6 = 33
Q8. v = 5 + (−2)(8) = 5 − 16 = −3
📌 Key Facts to Memorise
- Substitute with BRACKETS: 3x² at x = −3 means 3(−3)², never 3 −3²
- Order of operations: powers before multiplication, multiplication before addition
- (−3)² = +9 but −3² = −9 — the bracket changes everything
- Formula questions give you the formula — copy it, do not rely on memory
- Two-step formulas (v = u + at): substitute all values, then compute in BIDMAS order
🗺️ In the Exam
Substitution: 2–3 marks on nearly every Paper 1, plus the front end of every formula-sheet question on Papers 2/3. Negative-value substitution is the discriminating version.
Links to: index laws (N7), formulas and sequences (A23), compound measures (R11).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always substitute carefully, putting values in brackets
- Remember: 3x² is different from (3x)²
- Use BIDMAS - calculate powers before multiplication
- Be careful with negative numbers - (-3)² = 9, not -9
- Check your answer by substituting back if time allows
🧠 Problem-Solving Strategies
Problem-Solving
When substituting, always put values in brackets to avoid sign errors. Use BIDMAS carefully — powers come before multiplication. Check whether you need (2x)² or 2x² as they give different results.
Multi-Step Problem
The formula for kinetic energy is E = ½mv². Find the energy when m = 8 and v = 5, then find the new energy if v doubles.
Solution:
Step 1: E = ½ × 8 × 5² = ½ × 8 × 25 = 4 × 25 = 100 J
Step 2: If v doubles, v = 10: E = ½ × 8 × 10² = ½ × 8 × 100 = 400 J
The energy is 4 times larger (because v is squared, doubling v gives 2² = 4 times the energy).
⚠️ Common Errors
Watch Out!
1. Wrong: 2x² when x = 3 gives (2×3)² = 36 Correct: 2 × 3² = 2 × 9 = 18
2. Wrong: (-3)² = -9 Correct: (-3)² = (-3)×(-3) = +9
3. Wrong: Substituting x = -2 into 3x + 1 gives 3-2+1 = 2 Correct: 3(-2) + 1 = -6 + 1 = -5 (use brackets!)
✍️ 6-Mark Exam Question
Extended Answer
6 marks: The volume of a cone is V = 1⁄3πr²h. (a) Find V when r = 5 and h = 12, giving your answer in terms of π. (b) Find V when r = 5, h = 12 and π = 3.14. (c) The radius doubles and the height stays the same. What happens to the volume?
(a) V = 1⁄3 × π × 5² × 12 = 1⁄3 × π × 25 × 12 = 300⁄3π = 100π cm³
(b) V = 100 × 3.14 = 314 cm³
(c) New V = 1⁄3π(2r)²h = 1⁄3π × 4r² × h = 4 times original volume.
Mark scheme: (a) 2 marks for correct substitution and simplification. (b) 1 mark for numerical answer. (c) 3 marks — 1 for (2r)², 1 for showing 4r², 1 for stating 4 times.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
The surface area of a sphere is A = 4πr². Earth has radius approximately 6400 km.
(a) Calculate the surface area of Earth in terms of π.
(b) If 70% of the surface is water, write an expression for the water surface area.
(c) Explain why your answer to (a) is only approximate.
Answers: (a) A = 4π(6400)² = 4π × 40,960,000 = 163,840,000π km². (b) 0.7 × 163,840,000π = 114,688,000π km². (c) The radius 6400 km is approximate — Earth is not a perfect sphere.
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