IGCSE Formulae: Substitution and Changing the Subject (H)
Formulae are the bridges between abstract algebra and the real world. Whether determining the speed of a car or the resistance of an electrical circuit, formulae allow us to calculate unknown values. In the Edexcel IGCSE 4MA1 Higher Tier exam, you must master two key skills: substituting values accurately (especially negatives) and rearranging complex formulae to change the subject.
This guide covers Topic 2.3 of the IGCSE Algebra Hub, taking you from basic substitution to Grade 9 rearranging problems where the subject appears twice or is trapped inside a root.
1. Substitution: The Foundation
Substitution involves replacing the variables (letters) in a formula with specific numbers to calculate a value.
The "Golden Rule" of Substitution
When substituting a number, especially a negative number, always place it in brackets. This ensures you follow the correct BIDMAS and negative numbers rules.
- Formula: y = x² - 4x
- Substitute x = -3:
- Correct: y = (-3)² - 4(-3) = 9 + 12 = 21.
- Incorrect: y = -3² - 4 - 3 = -9 - 7 = -16. (Calculator error: squaring only the 3, not the negative).
2. Rearranging Formulae (Changing the Subject)
To "make x the subject" means to manipulate the formula until you have x = ... on one side, with no x terms on the other side.
The Balance Method (Reverse BIDMAS)
To isolate a variable, you generally perform operations in the reverse order of BIDMAS (SAMDIB). Unwrap the layers surrounding the subject.
Complex Cases (Higher Tier)
For Grades 7-9, you will encounter three specific challenges:
- Roots and Powers: To remove a square root, square both sides. To remove a square, take the square root.
- Fractions: Always multiply to remove fractions as your first step.
- Subject Appears Twice: If the subject (e.g., x) appears on both sides or in the top and bottom of a fraction, you must collect terms and factorise.
Key Formulas & Operations
Memorize the inverse operations required to "unwrap" a variable.
| Operation | Inverse Operation |
|---|---|
| Addition (+) | Subtraction (-) |
| Multiplication (×) | Division (÷) |
| Square (x²) | Square Root (√x) |
| Nth Power (xⁿ) | Nth Root (ⁿ√x) |
| Reciprocal (1/x) | Reciprocal (1/y) (Flip both sides) |
Step-by-Step Worked Examples
Question: Use the formula v = u + at to find v when u = 15, a = -9.8, and t = 4.
Solution:
- Substitute with brackets: v = (15) + (-9.8)(4).
- Multiply first (BIDMAS): v = 15 + (-39.2).
- Add/Subtract: v = 15 - 39.2 = -24.2.
Answer: v = -24.2.
Question: Make r the subject of the formula: A = 4πr².
Methodology: Reverse BIDMAS. Remove the multiply by 4π first, then the power.
Solution:
- Divide by 4π: A4π = r².
- Square root both sides: A4π = r.
- (Since r represents a length, we assume the positive root).
- Simplifying the root (Optional but good practice): r = √(A4π) = √A2√π.
Answer: r = √(A4π).
Question: Make x the subject of the formula: y = 3x + 5x - 2.
Methodology: The subject x is on top and bottom. Use the "Collect → Factorise → Divide" algorithm.
Solution:
- Multiply by the denominator (x - 2):
y(x - 2) = 3x + 5 - Expand the brackets:
xy - 2y = 3x + 5 - Collect all x terms on one side, non-x terms on the other:
xy - 3x = 5 + 2y - Factorise x out of the LHS (This is the crucial step):
x(y - 3) = 5 + 2y - Divide by the bracket (y - 3):
x = 5 + 2yy - 3
Answer: x = 2y + 5y - 3.
Real-World Application (Global Context)
Rearranging formulae is a daily task for physicists and engineers.
Scenario: The Simple Pendulum
The time period (T) of a swinging pendulum is given by the formula:
T = 2π√(Lg)
Where L is the length of the string and g is gravity. If an engineer wants to build a clock that ticks exactly once every second (T=1), they cannot simply plug numbers in. They must rearrange the formula to find the required Length (L).
- Divide by 2π: T2π = √(Lg).
- Square both sides: (T2π)² = Lg.
- Multiply by g: L = g(T2π)².
This rearranged formula allows the engineer to calculate the exact length needed for any time period.
Video Masterclass: Visualizing the 'Subject Appears Twice' & Engineer's Puzzle
Exam Technique and Common Pitfalls
1. The "Square Root" Trap
When moving a term like +5 away from a square root, you must move the 5 before squaring.
- Given: √x + 5 = y
- Incorrect: x + 25 = y² (Squaring individual terms).
- Correct: √x = y - 5 → x = (y - 5)².
2. Factorising is Key
If you end up with x = 5 + 2y - xy, you have not finished. The subject (x) is still on the right-hand side. You must group the x terms and factorise (as shown in Example 3).
3. Checking Your Answer
You can check algebraic rearrangement by using simple numbers.
In Example 3, if x=3, then y = (9+5)/(3-2) = 14.
Test your answer: x = (2(14)+5)/(14-3) = 33/11 = 3. It works!
If you want to ensure you can handle these high-tariff questions, mastering high-tariff algebraic questions is the best way to secure the top grades.
Summary Checklist and Next Steps
Checklist:
- [ ] I can substitute positive and negative numbers into formulae (using brackets).
- [ ] I can rearrange simple linear formulae (Reverse BIDMAS).
- [ ] I can rearrange formulae involving squares and roots.
- [ ] (H) I can rearrange formulae where the subject appears on both sides (Collect & Factorise).
- [ ] (H) I can check my rearrangement using substitution.
Next Steps:
Now that you can manipulate variables, you are ready to solve for them. The next topic is Topic 2.4 Linear Equations, where we find the specific value of an unknown.