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
❓ 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.
✅ Answers
- 13
- 23
- 48
- 18
- 35
- -3
🎯 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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