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Algebra · Foundation & Higher

Rearranging formulae (changing the subject)

Changing the subject of a formula means rearranging it so a different variable appears on its own on one side. Use inverse operations — whatever is done on one side must be done on the other.

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Key facts to remember

  • 1Do the opposite operation: + and − are opposites; × and ÷ are opposites; square and square root are opposites.
  • 2Always do the same operation to both sides.
  • 3When the new subject appears twice, factorise to collect it on one side.
  • 4Square both sides to remove a square root; take the square root to remove a square (remember ±).
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Formulas

Rearrangement template
y = mx + c → x = (y − c) / m
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Worked examples

Example 1

Make r the subject of A = πr².

Working

  1. Divide both sides by π: A/π = r²
  2. Square root both sides: r = √(A/π)
Answerr = √(A/π)
Example 2

Make x the subject of y = (2x + 3) / (x − 1).

Working

  1. Multiply by (x − 1): y(x − 1) = 2x + 3
  2. Expand: yx − y = 2x + 3
  3. Collect x: yx − 2x = y + 3
  4. Factorise: x(y − 2) = y + 3
  5. Divide: x = (y + 3) / (y − 2)
Answerx = (y + 3) / (y − 2)
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Common mistakes

✗Only doing the operation on one side of the equation.
✗Forgetting to factorise when the new subject appears twice.
✗Taking a square root and forgetting the ± sign.
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Exam tips

✓Work in small steps and show each rearrangement.
✓If the variable appears twice, collect the terms on one side, then factorise.
✓Check by substituting numbers into the original and rearranged versions.

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