Rearranging Formulae –

Rearranging formulae is one of the most frequently tested algebra skills in GCSE Maths. Whether you are working with science equations or pure algebra, you need to be confident changing the subject of a formula using inverse operations.

What Is Rearranging Formulae?

Rearranging a formula means rewriting it so that a different variable is on its own on one side of the equals sign. That variable becomes the new subject of the formula. For example, if you start with v = u + at and want to find t, you rearrange to make t the subject: t = (v - u) / a.

The key idea is to use inverse (opposite) operations to undo what has been done to the variable you want to isolate. If something has been added, you subtract it from both sides. If something has been multiplied, you divide both sides. You work step by step until only the target variable remains on one side.

On the Higher tier, you may also need to deal with squares, square roots, and cases where the new subject appears more than once in the formula, requiring factorisation.

Key Formulas

Step-by-Step Method

1. Identify the variable you want to make the subject.
2. If there is a fraction, multiply both sides by the denominator to clear it.
3. Use inverse operations to move all other terms away from the target variable — deal with addition/subtraction first, then multiplication/division.
4. If the target variable is inside a square or square root, apply the corresponding inverse (square root or squaring) as the final step.
5. Write the rearranged formula with the new subject on the left-hand side.

Worked Example 1 — Foundation Level

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

Working:

Step 1 — Subtract u from both sides: v - u = at.

Step 2 — Divide both sides by a: (v - u) / a = t.

Step 3 — Write with the subject on the left: t = (v - u) / a.

Answer: t = (v - u) / a

Worked Example 2 — Higher Level

Question: Make r the subject of the formula A = πr².

Working:

Step 1 — Divide both sides by π: A / π = r².

Step 2 — Take the positive square root of both sides (r is a length so must be positive): r = √(A / π).

Answer: r = √(A / π)

Worked Example 3 — Exam Style

Question: Make x the subject of the formula y = (3x + 5) / (x - 2). (4 marks)

Working:

Step 1 — Multiply both sides by (x - 2): y(x - 2) = 3x + 5.

Step 2 — Expand the left side: yx - 2y = 3x + 5.

Step 3 — Collect x terms on one side: yx - 3x = 5 + 2y.

Step 4 — Factorise: x(y - 3) = 5 + 2y.

Step 5 — Divide both sides by (y - 3): x = (5 + 2y) / (y - 3).

Answer: x = (5 + 2y) / (y - 3)

Common Mistakes

• Applying inverse operations in the wrong order. Always undo addition/subtraction before multiplication/division, working in reverse BIDMAS order.
• Forgetting ± when square-rooting. In pure algebra, x² = 9 gives x = ±3. In context (e.g. length), you may only need the positive root.
• Not multiplying every term when clearing a fraction. If y = (3x + 5) / 2, multiplying both sides by 2 gives 2y = 3x + 5, not 2y = 3x + 5/2.

Exam Tips

• Show each step clearly — examiners award method marks for each correct inverse operation.
• When the required subject appears twice, collect those terms on one side, factorise, then divide.
• Check your answer by substituting numbers into both the original and rearranged formula.

Practice Questions

Q1 (Foundation): Make b the subject of P = 2a + 2b.

Q2 (Foundation): Make h the subject of V = lwh.

Q3 (Higher): Make u the subject of v² = u² + 2as.

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Related Topics

• Solving Linear Equations
• Substitution into Expressions
• Rearranging Complex Formulae

Summary

• Rearranging means making a chosen variable the subject by using inverse operations.
• Work in reverse BIDMAS order: undo addition/subtraction first, then multiplication/division.
• Clear fractions early by multiplying both sides by the denominator.
• For squares, take the square root; for square roots, square both sides.
• When the subject appears twice, collect those terms and factorise before dividing.