Algebra
Grade 5-7Rearranging Formulae
Rearranging a formula means changing which letter is the subject, using the same inverse-operation steps as solving an equation, but working with letters throughout instead of numbers. This lesson covers rearranging a simple linear formula, rearranging formulae with a square or square root, rearranging formulae where the subject appears on both sides, and rearranging formulae where the subject is in a fraction.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Rearranging a simple linear formula
Undo the operations applied to the subject in reverse order, applying the same inverse operation to both sides of the formula.
To make x the subject of y = 3x + 2, subtract 2 from both sides to get y − 2 = 3x, then divide both sides by 3: x = (y − 2) ÷ 3.
Rearranging with a square or square root
Undo a square by taking the square root of both sides, and undo a square root by squaring both sides, applying this to the whole side, not just part of it.
To make x the subject of y = x² + 3, subtract 3 from both sides to get x² = y − 3, then take the square root of both sides: x = √(y − 3).
Rearranging with the subject on both sides
Collect every term containing the subject onto one side of the formula, factorise the subject out of these terms, then divide by the remaining factor.
To make x the subject of ax + 3 = bx + 7, rearrange to get ax − bx = 4, factorise to get x(a − b) = 4, then divide by (a − b): x = 4 ÷ (a − b).
Rearranging with the subject in a fraction
Multiply both sides by the denominator to clear the fraction, then continue rearranging using the usual inverse operations.
To make x the subject of y = 3 ÷ (x + 2), multiply both sides by (x + 2) to get y(x + 2) = 3, divide by y to get x + 2 = 3 ÷ y, then subtract 2: x = (3 ÷ y) − 2.
Worked Examples
Three exam-style questions, fully solved.
Make x the subject of y = 3x + 2.
Easy- 1.Subtract 2 from both sides: y − 2 = 3x
- 2.Divide both sides by 3: x = (y − 2) ÷ 3
Answer: x = (y − 2) ÷ 3
Make x the subject of 3x + c = dx + 8.
Medium- 1.Collect the x terms on one side: 3x − dx = 8 − c
- 2.Factorise x out: x(3 − d) = 8 − c
- 3.Divide by (3 − d): x = (8 − c) ÷ (3 − d)
Answer: x = (8 − c) ÷ (3 − d)
Make x the subject of y = 3 ÷ (x + 2).
Hard- 1.Multiply both sides by (x + 2): y(x + 2) = 3
- 2.Divide both sides by y: x + 2 = 3 ÷ y
- 3.Subtract 2 from both sides: x = (3 ÷ y) − 2
Answer: x = (3 ÷ y) − 2
Avoid These
The most common mistakes students make.
Not applying the same operation to every term on both sides of the formula, especially when a side has more than one term.
Undoing the operations in the wrong order, such as dividing before subtracting instead of subtracting first, when isolating the subject.
When the subject appears on both sides, forgetting to collect all the subject terms onto one side and factorise before dividing.
Taking the square root of only part of an expression, such as writing √(y − 3) as √y − 3 instead of the square root of the whole right-hand side.
When the subject is inside a fraction, forgetting to multiply by the denominator first, or making an error isolating the subject after clearing the fraction.
FAQ
Questions parents and students ask.
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