Rearranging Formulae (Cambridge O Level Maths)

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
    • 
  • 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
    • 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 

Are there any common formulae to be aware of?

• The formula for the equation of a straight line is often used
  • 
• Formulae for accelerating objects are often used
  
  • 
  • 
  • 
  • 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 substitute numbers into them

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