Factorising and Rearranging Formulae

What is Rearranging with Factorising?

Rearranging with factorising means rewriting an equation to isolate a variable (make it the "subject") when that variable appears in multiple terms. Factorising helps to group these terms together, making it easier to solve or rearrange.

Think of factorising like "tidying up" an equation—grouping common parts so the variable you need is neatly on one side.

 

General Steps for Rearranging with Factorising

1. Identify the Variable: Find the variable you need to make the subject. It might appear in more than one term.
2. Group Terms: Bring all terms with the chosen variable to one side of the equation.
3. Factorise: Take out the common factor (usually the variable) to simplify.
4. Solve for the Variable: Divide or rearrange further to isolate the variable.

 

Example

Rearrange $y = 3x + px$ to make $x$ the subject

Step 1: Identify the terms with $x$: $$y = 3x + px$$

Step 2: Factorise $x$ from the terms: $$y = x(p + 3)$$

Step 3: Divide by $(p + 3)$ to isolate $x$: $$x = \frac{y}{p + 3}$$

 

 

 

 

Example

Rearrange $z = \frac{2x}{x + 1}$ to make $x$ the subject

Multiply through by $(x + 1)$: $$z(x + 1) = 2x$$

Expand: $$zx + z = 2x$$

Reorganise: $$zx - 2x = -z$$

factorise: $$x(z - 2) = -z$$

Divide: $$x = \frac{-z}{z - 2}$$