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
Exam tip

Clear fractions before anything else, and do one operation per line to both sides — a term never crosses the equals sign just by changing its sign. Where the letter appears twice, show the gathering line and the factorised line separately, because that is where the marks concentrate. A rearranged formula may perfectly well end as a fraction.

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