Rearranging Formulae
Rearranging formulae is a vital algebra skill that allows you to change the “subject” of a formula to find any unknown value. It’s like having a recipe for 4 people and adapting it to cook for 8 – you’re changing the formula to suit your needs!
The Core Method: Balance and Inverses
Think of a formula as a balanced scale. Whatever you do to one side, you must do to the other. To isolate a variable and make it the new subject, we “undo” the operations around it using inverse (opposite) operations.
Key Inverse Operations
- Addition (+) ↔ Subtraction (−)
- Multiplication (×) ↔ Division (÷)
- Squaring ($x^2$) ↔ Square Root ($\sqrt{x}$)
- Cubing ($x^3$) ↔ Cube Root ($\sqrt[3]{x}$)
Worked Examples
Example 1: One-Step (Add/Subtract)
Make $V$ the subject of $P = V + C$.
To undo adding $C$, subtract $C$ from both sides.
$P – C = V$
$V = P – C$
Example 2: One-Step (Multiply/Divide)
Make $S$ the subject of $D = ST$.
To undo multiplying by $T$, divide both sides by $T$.
$S = \frac{D}{T}$
Example 3: Two-Step Rearrangement
Make $x$ the subject of $y = 4x – 7$.
Undo operations in reverse BIDMAS order.
- Undo the subtraction — add 7 to both sides: $y + 7 = 4x$.
- Undo the multiplication — divide by 4: $\frac{y + 7}{4} = x$.
Answer: $x = \frac{y + 7}{4}$
Example 4: Subject on Both Sides
Make $x$ the subject of $ax + b = cx$.
- Collect $x$ terms on one side: $ax – cx + b = 0$.
- Move other terms to the other side: $ax – cx = -b$.
- Factorise out the subject: $x(a – c) = -b$.
- Divide to isolate $x$: $x = \frac{-b}{a – c}$.
Answer: $x = \frac{-b}{a – c}$ (equivalently, $x = \frac{b}{c – a}$)
Tutor Insights
🤔 Common Misunderstandings
- Order of operations: Forgetting to “undo” in the reverse order of BIDMAS — deal with addition/subtraction before multiplication/division.
- Signs: Forgetting to change the sign when moving a term across the equals sign.
- Division errors: When dividing a whole side, forgetting to divide every term. For $2x = y + z$, the answer is $x = \frac{y + z}{2}$, not $x = \frac{y}{2} + z$.
📝 Common Exam Mistakes
- Incorrect inverse operations: Squaring instead of square rooting, or adding instead of subtracting.
- Not factorising when necessary: Forgetting to factorise out the subject when it appears in multiple terms — a common higher-tier mistake.
- Messy working: Disorganised steps can easily lead to errors. Show one step per line.
Practice Questions
Try these questions to test your skills!
- Make $k$ the subject of: $m = k – 5$.
- Make $a$ the subject of: $P = 3a + b$.
- Make $r$ the subject of: $C = 2\pi r$.
- Make $h$ the subject of: $V = \frac{1}{3}\pi r^2 h$.
- Make $k$ the subject of: $7k – 2m = 3k + 5m$.
Show Answers
- Working: Add 5 to both sides.
Answer: $k = m + 5$ - Working: Subtract $b$, then divide by 3.
Answer: $a = \frac{P – b}{3}$ - Working: Divide both sides by $2\pi$.
Answer: $r = \frac{C}{2\pi}$ - Working: Multiply both sides by 3, then divide by $\pi r^2$.
Answer: $h = \frac{3V}{\pi r^2}$ - Working: $7k – 3k = 5m + 2m \implies 4k = 7m$.
Answer: $k = \frac{7m}{4}$
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