Changing the subject

Objective. Rearrange formulae involving powers, roots and fractions.

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Learn: the key idea

Changing the subject of a formula means rearranging it so a different letter stands alone, even when the formula involves powers, roots or fractions. The approach is to undo operations in reverse order using inverse operations, treating the formula like an equation: add or subtract terms, then multiply or divide, then deal with powers or roots last. For example, to make x the subject of y = √(x − 3) + 1, first subtract 1, then square both sides, then add 3. Always isolate the term containing the required letter before applying a root or power.

Worked example

Make x the subject of the formula y = √(2x + 5), rearranging the formula correctly.

1 $Square both sides to remove the root: y² = 2x + 5.$
2 $Subtract 5 from both sides: y² - 5 = 2x.$
3 $Divide both sides by 2: x = (y² - 5)/2.$
4 $Check by substituting back: \sqrt(2 \times (y² - 5)/2 + 5) = \sqrt(y²) = y.$

Answer: x = (y² − 5)/2

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Common mistake. Pupils square or root both sides too early, before isolating the term containing the target letter, leading to extra unwanted terms being squared. Always isolate the relevant term fully before applying the inverse power or root.

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