Rearranging Formulae (AQA GCSE Maths)

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Amber

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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 IGCSE 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
      • y space equals space m x space plus space c
    • The area of a trapezium
      • Area space equals space fraction numerator open parentheses a space plus space b close parentheses h over denominator 2 end fraction
    • Pythagoras' theorem
      • a to the power of 2 space end exponent plus space b to the power of 2 space end exponent equals space c squared
  • 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 A space equals space fraction numerator open parentheses a space plus space b close parentheses h over denominator 2 end fraction so that h is the subject
      • Multiply by 2

2 A space equals space open parentheses a space plus space b close parentheses h

      • Expanding the bracket will not help here as we would end up with the subject appearing twice, so instead divide by the whole expression open parentheses a space plus space b close parentheses

fraction numerator 2 A over denominator a space plus space b end fraction space equals space h

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

h space equals space fraction numerator 2 A space over denominator a space plus space b end fraction

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 x the subject of y space equals space a x to the power of 5
      • Divide both sides by a first 

y over a space equals space x to the power of 5

      • Then take the 5th root of both sides

fifth root of y over a end root space equals space x

  • If n is even then there will be two answers: a positive and a negative
    • For example if y equals x squared then x equals plus-or-minus square root of y
  • If the formula contains an nth root, reverse this operation by raising both sides to the power of n
    • For example to make a the subject of  m space equals space cube root of 2 a b end root
      • Raise both sides to the power of 3 first

m to the power of 3 space end exponent equals space 2 a b

      • Divide both sides by 2 b

a space equals fraction numerator space m cubed over denominator 2 b end fraction

Are there any common formulae to be aware of?

  • Formulae for accelerating objects are often used
    • v equals u plus a t
    • v squared equals u squared plus 2 a s
    • s equals u t plus 1 half a t squared
  • The letters mean the following:
    • t stands for the amount of time something accelerates for (in seconds)
    • u stands for its initial speed (in m/s) - the speed at the beginning
    • v stands for its final speed (in m/s) - the speed after t seconds
    • a stands for its acceleration (in m/s2) during in that time
    • s stands for the distance covered in t seconds
  • You do not need to memorise these formulae, but you should know how to rearrange them
    • You should also understand the difference between initial speed (u) and final speed (v)

Examiner Tip

  • If you are unsure about the order in which you would carry out the inverse operations, try substituting numbers in and reverse the order that you would carry out the substitution

Worked example

Make x the subject of y space equals space square root of a x to the power of 2 space end exponent minus space b end root.

Use inverse operations to isolate x.

Square both sides.

table row cell space space space space space space space space space space space space y squared space space end cell equals cell open parentheses square root of a x squared space minus space b space space end root close parentheses squared end cell row cell space space y squared space space end cell equals cell space a x squared space minus space b end cell end table

Add b to both sides.

table row cell space space space space space space space space space space space space y squared space space end cell equals cell space space a x squared space minus space b space space end cell end table

table row blank blank cell open parentheses plus b close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses plus b close parentheses end cell end table

table row cell space space space y squared space plus space b space end cell equals cell space space a x squared space end cell end table

Divide both sides by a.

table row cell y squared space plus space b space end cell equals cell space a x squared end cell end table

table row cell open parentheses divided by a close parentheses space space space space space space space space space space space space space space space space space space space end cell cell space space end cell cell space space space space space space space space space space space space space space space space space open parentheses divided by a close parentheses end cell end table

table row cell fraction numerator y squared space plus space b over denominator a end fraction space end cell equals cell space x squared end cell end table

Square root both sides.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign right center left columnspacing 0px 0px 0px end attributes row cell square root of fraction numerator y squared space plus space b over denominator a end fraction space end root end cell equals cell space square root of x squared end root end cell end table end cell row blank blank cell table attributes columnalign right center left columnspacing 0px 0px 0px end attributes row cell square root of fraction numerator y squared space plus space b over denominator a end fraction space end root end cell equals cell space x end cell end table end cell end table

The equation is fully correct as it is and will gain full marks, however the two sides can be swapped if preferred. Remember that when you square root you get two answers (a positive and a negative).

bold italic x bold equals bold plus-or-minus square root of fraction numerator bold italic y to the power of bold 2 bold space bold plus bold space bold italic b over denominator bold italic a end fraction end root

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

Worked example

Rearrange the formula p equals fraction numerator 2 space minus space a x over denominator x space minus space b space end fraction to make x the subject.

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

p open parentheses x space minus space b close parentheses space equals space 2 space minus space a x

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

p x space minus space p b space equals space 2 space minus space a x

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

table row cell p x space minus space p b space end cell equals cell space 2 space minus space a x end cell end table

table row blank blank cell open parentheses plus a x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses plus a x close parentheses space end cell end table

table row cell p x space minus space p b space plus space a x space end cell equals cell space 2 space end cell end table

table row blank blank cell open parentheses plus p b close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses plus p b close parentheses space end cell end table

table row cell p x space plus space a x space end cell equals cell space 2 space plus space p b end cell end table

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

x open parentheses p space plus space a close parentheses space equals space 2 space plus space p b

Isolate x by dividing by the whole expression open parentheses p space plus space a close parentheses.

bold italic x bold space bold equals bold space fraction numerator bold 2 bold space bold plus bold space bold italic p bold italic b over denominator bold italic p bold space bold plus bold space bold italic a end fraction

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.