Rearranging Formulae (AQA GCSE Further Maths) : Revision Note

Simple Rearranging

What are formulae?

• A formula (plural, formulae) is a mathematical relationship consisting of variables, constants and an equals sign

• You will come across many formulae in your course, including

  • the formulae for areas and volumes of shapes

  • equations of lines and curves

  • the relationship between speed, distance and time

• Some examples of formulae you should be familiar with are

  • The equation of a straight line

    •   or    

  • The area of a trapezium

    • 

  • Pythagoras' theorem

    • 

• You will also be expected to rearrange formulae that you are not familiar with

How do I rearrange formulae where the subject appears only once?  

• Rearranging formulae can also be called changing the subject

  • The subject is the variable that you want to find out, or get on its own on one side of the formula

• The method for changing the subject is the same as the method used for solving linear equations

  • STEP 1 
    Remove any fractions or brackets

    • Remove fractions by multiplying both sides by anything on the denominator

    • Expand any brackets only if it helps to release the variable, if not it may be easier to leave the bracket there

  • STEP 2
    Carry out inverse operations to isolate the variable you are trying to make the subject

    • This works in the same way as with linear equations, however you will create expressions rather than carry out calculations

  • For example, to rearrange  so that  is the subject (note that the letters  and  represent different variables)

    • Multiply by 2

• Expanding the bracket will not help here as we would end up with the subject appearing twice, so instead divide by the whole expression

• You can now rewrite this with the subject () on the left hand side

How do I rearrange formulae that include powers or roots?  

• If the formula contains a power of n, use the nth root to reverse this operation 
  

  • For example to make  the subject of 

    • Divide both sides by first 

• Then take the 5th root of both sides

• If n is even then there will be two answers: a positive and a negative

  • For example if then

• If the formula contains an nth root, reverse this operation by raising both sides to the power of n
  

  • For example to make  the subject of  

    • Raise both sides to the power of 3 first

• Divide both sides by 

Rearranging When Subject Appears Twice

How do I rearrange formulae where the subject appears twice?  

• If the subject appears twice, you will need to factorise at some point

  • Factorising means putting an expression into brackets, with the subject on the outside of the brackets

• If the subject appears inside a set of brackets, you will need to expand these brackets before you can begin rearranging

• 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