How to Solve: Changing the Subject of a Formula

How to Solve: Changing the Subject of a Formula

GCSE & A-Level Maths

Introduction

"If you can rearrange a formula, you can unlock 5–10 marks in your GCSE or A-Level exam—whether it’s physics equations, interest rates, or even medical dosage calculations. One question, done right, could be the difference between a grade 4 and a 5, or a B and an A."

What You Need To Know First

Before you start, make sure you’re confident with: 1. Solving linear equations (e.g., 2x + 3 = 7 → x = 2). 2. Basic algebra rules (e.g., multiplying/dividing both sides, expanding brackets). 3. Fractions and indices (e.g., a/b = c → a = bc, x² = 9 → x = ±3).

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Identify the subject you need

• Look at the question: "Make [variable] the subject."
• Circle the variable you’re solving for.

Step 2: Write down the original formula

• Copy the formula exactly as given.

Step 3: Undo operations in reverse order

• Use inverse operations to move terms away from the subject.
• Rule: Whatever you do to one side, do to the other.

Step 4: Isolate the subject

• Keep moving terms until the subject is alone on one side.

Step 5: Simplify (if needed)

• Factorise, expand, or cancel terms to tidy up.

Step 6: Check your answer

• Plug in numbers to test if it works.

Worked Example (Using Steps)

Question: Make u the subject of v = u + at.

Step 1: Subject = u. Step 2: Original formula: v = u + at. Step 3: Subtract at from both sides: v – at = u. Step 4: u is now isolated. Step 5: No simplification needed. Step 6: Check: If u = 5, a = 2, t = 3, then v = 5 + 6 = 11. Rearranged: u = 11 – 6 = 5 ✔️

Final answer: u = v – at

Worked Examples

Example 1 – Basic

Question: Make r the subject of A = πr².

Working: 1. Original: A = πr² 2. Divide both sides by π: A/π = r² 3. Square root both sides: r = ±√(A/π)

What we did and why: - We divided by π to isolate r², then took the square root to solve for r. - Note: Square roots give two answers (±), but in real-world problems (e.g., radius), we usually take the positive value.

Example 2 – Medium

Question: Make x the subject of y = (3x + 2)/5.

Working: 1. Original: y = (3x + 2)/5 2. Multiply both sides by 5: 5y = 3x + 2 3. Subtract 2: 5y – 2 = 3x 4. Divide by 3: x = (5y – 2)/3

What we did and why: - We first multiplied by the denominator to eliminate the fraction. - Then, we moved the constant (+2) before dividing by the coefficient of x (3).

Example 3 – Exam-Style

Question (A-Level): The formula for kinetic energy is E = ½mv². Rearrange to make v the subject.

Working: 1. Original: E = ½mv² 2. Multiply both sides by 2: 2E = mv² 3. Divide by m: 2E/m = v² 4. Square root both sides: v = ±√(2E/m)

What we did and why: - We multiplied by 2 first to remove the fraction. - Then, we isolated v² before taking the square root. - Exam tip: If the question doesn’t specify, include ± in your answer.

Common Mistakes

Exam Traps

1-Minute Recap

"Right, listen up—this is your 60-second cheat sheet for rearranging formulas. First, circle the subject you need. Then, undo operations in reverse order: if it’s +5, subtract 5; if it’s ×3, divide by 3. Fractions? Multiply by the denominator first. Squares? Take the square root last—and don’t forget the ±! Always check your answer by plugging numbers back in. Examiners love hiding fractions or squares, so slow down and do one step at a time. You’ve got this—now go smash those 5 marks!