MathematicsExam code: 0580
Rearranging Formulae
Algebra and Sequences
Gathering and factorising
- Where the wanted letter appears more than once, it must end up appearing once before it can be isolated
- Expand any brackets containing it first
- Collect every term containing that letter on one side, and everything else on the other
- Factorise to bring the letter outside a bracket, which leaves it appearing once
- Divide by the bracket to finish
Worked example
The subject appearing on both sides
Make x the subject of k(x − 3) = 2x + 1.
Solution:
- x appears twice, so it cannot simply be isolated by undoing operations: it has to be collected first
- Expand the bracket so nothing is hidden inside it: kx − 3k = 2x + 1
- Gather every term containing x on one side and everything else on the other
- Subtract 2x from both sides and add 3k to both sides: kx − 2x = 1 + 3k
- The left-hand side now has x in both terms, so factorise it out: this is the step the marks concentrate on
- x(k − 2) = 1 + 3k
- x is now multiplied by a single bracket, so divide both sides by it
- The rearranged formula is x = (1 + 3k) ÷ (k − 2)
- Check with k = 4: the original reads 4(x − 3) = 2x + 1, giving 4x − 12 = 2x + 1 and x = 6.5, and the formula gives (1 + 12) ÷ 2 = 6.5
- The answer keeps k in it, which is expected: a formula rearranged for one letter is still a formula
- The same method handles a formula such as at = 5 − bt, which becomes t(a + b) = 5 and then t = 5 ÷ (a + b)
- A squared subject appearing twice factorises in exactly the same way, and the root is taken at the very end
- ax² = b − cx² gathers to x²(a + c) = b, so x² = b ÷ (a + c) and x = ±√(b ÷ (a + c))
- The factorising step is what the question is testing, so it must be shown
Exam tip
Two easy-to-skip lines carry marks
Examiners mark a rearrangement move by move, and a wrong final answer still keeps the earlier marks: up to 3 of 4, or 2 of 3. Two lines that are easy to do in your head carry a mark each, so write them down: the expanded line after clearing a fraction, and the square on its own, such as x² = b ÷ (a + c) above, before taking the root.
Worked example
The subject above and below a fraction line
Make x the subject of y = (2x + 1) ÷ (x + 3).
Solution:
- x is on the top and on the bottom, so it appears twice: clear the fraction first, then gather as above
- Multiply both sides by the whole denominator, kept in a bracket: y(x + 3) = 2x + 1
- Expand the bracket: xy + 3y = 2x + 1
- Gather the x terms on the left and everything else on the right: xy − 2x = 1 − 3y
- Factorise: x(y − 2) = 1 − 3y
- Divide both sides by the bracket
- The rearranged formula is x = (1 − 3y) ÷ (y − 2)
- Check with x = 2: the original gives y = 5 ÷ 5 = 1, and the formula gives (1 − 3) ÷ (1 − 2) = 2
- Multiplying the top and the bottom by −1 gives (3y − 1) ÷ (2 − y), the same formula written another way