Formulas where Subject Appears Twice (Edexcel GCSE Maths)

Revision Note

Written by: Mark Curtis

Reviewed by: Dan Finlay

If the subject appears twice, you will need to factorise at some point
- E.g. When making $x$ the subject of $x + x y = 3 - 2 y$
- Factorise out $x$ on the left to get $x (1 + y) = 3 - 2 y$
  - Notice that the subject now only appears once!
- Then divide both sides by $(1 + y)$ to get $x = \frac{3 - 2 y}{1 + y}$
If the subject appears twice, and any of these are inside a set of brackets, you will need to expand these brackets first
- E.g. When making $x$ the subject of $c (x + 2) - x = f$
- Expand the bracket first to $c x + 2 c - x = f$
- Keep the $x$ terms on one side $c x - x = f - 2 c$
- $x$ can then be made the subject using factorising as above
If the subject appears on two sides of a formula, you will need to bring those terms to the same side before you can factorise
- E.g. When making $x$ the subject of $3 x = y - p x$
- Add $p x$ to both sides first to form $3 x + p x = y$
- $x$ can then be made the subject using factorising as above

Rearrange the formula $p = \frac{2 - a x}{x - b}$ to make $x$ the subject.

Get rid of the fraction by multiplying both sides by the expression on the denominator

$p (x - b) = 2 - a x$

Expand the brackets on the left hand side to 'release' the $x$

$p x - p b = 2 - a x$

Bring the terms containing $x$ to one side of the equals sign and any other terms to the other side

$\begin{array}{rcl} p x - p b & = & 2 - a x \end{array}$

$\begin{array}{rcl} (+ a x) (+ a x) \end{array}$

$\begin{array}{rcl} p x - p b + a x & = & 2 \end{array}$

$\begin{array}{rcl} (+ p b) (+ p b) \end{array}$

$\begin{array}{rcl} p x + a x & = & 2 + p b \end{array}$

Factorise the left-hand side to bring $x$ outside of the brackets, so that it appears only once

$x (p + a) = 2 + p b$

Make $x$ the subject by dividing by the whole expression $(p + a)$

$x = \frac{2 + p b}{p + a}$

Last updated: 15 November 2024

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