Substitution
GCSE Mathematics🔄 What is Substitution?
Substitution means replacing variables (letters) with given numerical values to evaluate an expression. It's like following a recipe - when you know the actual ingredients (numbers), you replace the placeholders (variables) to get the final result.
📝 How Substitution Works
🔢 Simple Substitution
Replace each variable with its given value and calculate.
Example 1
Expression: 3x + 5
Given: x = 7
Solution: 3(7) + 5 = 21 + 5 = 26
Example 2
Expression: 4y - 3
Given: y = 6
Solution: 4(6) - 3 = 24 - 3 = 21
Example 3
Expression: 2a + 7
Given: a = 4
Solution: 2(4) + 7 = 8 + 7 = 15
📦 Substitution with Brackets
Remember BIDMAS/BODMAS - calculate brackets first!
Example 1
Expression: 2(x + 3)
Given: x = 5
Solution: 2(5 + 3) = 2(8) = 16
Example 2
Expression: 3(2y - 1)
Given: y = 4
Solution: 3(2×4 - 1) = 3(8 - 1) = 3(7) = 21
Example 3
Expression: 5(2a + 3b)
Given: a = 2, b = 3
Solution: 5(2×2 + 3×3) = 5(4 + 9) = 5(13) = 65
⬆️ Substitution with Powers
Calculate powers before multiplication and addition.
Example 1
Expression: x² + 4
Given: x = 3
Solution: 3² + 4 = 9 + 4 = 13
Example 2
Expression: 2y² - 5
Given: y = 4
Solution: 2(16) - 5 = 32 - 5 = 27
Example 3
Expression: a³ + 2a
Given: a = 3
Solution: 27 + 6 = 33
🔀 Multiple Variables
Substitute all variables with their given values.
Example 1
Expression: 2x + 3y - z
Given: x = 4, y = 5, z = 2
Solution: 2(4) + 3(5) - 2 = 8 + 15 - 2 = 21
Example 2
Expression: 3a² - 2b + c
Given: a = 3, b = 4, c = 7
Solution: 3(9) - 2(4) + 7 = 27 - 8 + 7 = 26
🧮 Interactive Substitution Calculator
Enter values for variables and see the result instantly!
✏️ Practice Substitution
If x = 4 and y = 3, find the value of 2x + 5y
Given values:
x = 4, y = 3
Expression:
2x + 5y
Your Progress: 0/0 correct
🌍 Real-World Applications
💰 Salary Calculation
Formula: Weekly wage = 15h + 100
Where: h = hours worked
Example: If h = 35, wage = 15(35) + 100 = £625
📏 Area of Rectangle
Formula: A = l × w
Where: l = length, w = width
Example: l = 8cm, w = 5cm, A = 40cm²
🌡️ Temperature Conversion
Formula: F = (C × 9/5) + 32
Example: C = 20°C, F = (20 × 9/5) + 32 = 68°F
🏃 Distance Formula
Formula: d = rt
Where: r = rate, t = time
Example: r = 60 mph, t = 2.5h, d = 150 miles
⚠️ Common Mistakes to Avoid
❌ Wrong Order
Expression: 2 + 3x when x = 4
Wrong: 2 + 3 = 5, then 5 × 4 = 20
Correct: 2 + 3(4) = 2 + 12 = 14
✅ BIDMAS/BODMAS Rule
Always follow the order:
Brackets → Indices → Division → Multiplication → Addition → Subtraction
❌ Sign Errors
Expression: -3x when x = -2
Wrong: -3 × -2 = -6
Correct: -3 × -2 = +6
💡 Remember
Negative × Negative = Positive
Negative × Positive = Negative
🌟 Challenge Questions
If a = 3, b = -2, and c = 4, evaluate: 2a² - 3b + c
If x = 5, evaluate: 3(x + 2)² - 4x
If p = -3, q = 4, evaluate: p³ + 2pq - q²