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Changing the Subject

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on changing the subject helps you revise rearranging formulae to change the subject. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Use inverse operations to isolate the new subject, like solving an equation.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Changing the subject of a formula means rearranging it so a different letter stands alone on one side. The formula still describes the same relationship, just written from a different starting point.

The method is identical to solving an equation. You undo the operations affecting the letter you want, doing the same to both sides each time, working in reverse order.

The difference is that you are working with letters rather than numbers, so you cannot simplify at the end. The answer is an expression, not a value, and it should have the new subject alone on the left-hand side.

Revision notes

Working in reverse order

Identify what has been done to the letter you want, then undo those operations backwards.

To make \(x\) the subject of \(y = 3x + 2\): subtract \(2\) from both sides to get \(y - 2 = 3x\), then divide by \(3\), giving \(x = \frac{y-2}{3}\).

Dealing with fractions and roots

Multiply to clear a fraction, and square to remove a square root.

To make \(x\) the subject of \(y = \sqrt{x + 1}\): square both sides to get \(y^2 = x + 1\), then subtract \(1\), giving \(x = y^2 - 1\).

When the subject appears twice

Collect all terms containing the new subject on one side, then factorise it out.

From \(ax = bx + c\): subtract \(bx\) to get \(ax - bx = c\), factorise to \(x(a - b) = c\), then divide, giving \(x = \frac{c}{a-b}\).

Key points

  • Rearranging a formula uses the same rules as solving an equation.
  • Do the same to both sides.
  • Undo operations in reverse order.
  • Square to remove a square root.
  • Factorise when the subject appears twice.
  • The answer is an expression, not a number.

Worked examples

Example 1

Make \(x\) the subject of \(y = 5x - 4\).

Working

\[y + 4 = 5x\]add 4 to both sides
\[x = \frac{y + 4}{5}\]divide both sides by 5

Example 2

Make \(r\) the subject of \(A = \pi r^2\).

Working

\[\frac{A}{\pi} = r^2\]divide both sides by pi
\[r = \sqrt{\frac{A}{\pi}}\]take the square root of both sides

Example 3

Make \(x\) the subject of \(y = \frac{x + 3}{2}\).

Working

\[2y = x + 3\]multiply both sides by 2
\[x = 2y - 3\]subtract 3 from both sides

Common mistakes

  • Undoing operations in the wrong order.

    In y = 3x + 2, remove the +2 before dividing by 3.

  • Only applying an operation to part of a side.

    Multiplying by 2 must apply to the whole of the other side, not one term.

  • Forgetting to factorise when the subject appears twice.

    ax − bx must become x(a − b) before dividing.

  • Losing the square root or leaving the answer squared.

    If r² is isolated, the final step is to take the root.

Exam tips

  • Underline the letter you want as the subject before starting.
  • Write the operation you are doing beside each line.
  • Take extra care to apply each step to the whole side.
  • Check by substituting numbers into both the original and rearranged forms.

Key terms

Subject
The letter on its own, usually on the left of a formula.
Rearrange
To rewrite a formula with a different subject.
Inverse operation
The operation that undoes another.
Factorise
To take a common factor outside a bracket.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.