Rearranging Formulae | Mathematics for GCSE/IGCSE - Class 10 PDF Download

Simple Rearranging

What are formulae?

• A formula (plural, formulae) represents a mathematical relationship involving variables, constants, and an equals sign.
• In your IGCSE studies, you will encounter various formulae, such as those for areas and volumes of shapes, equations of lines and curves, and the correlation between speed, distance, and time.
• Examples of formulae you should recognize include:
  • The equation of a straight line
    
  • The area of a trapezium
    
  • Pythagoras' theorem
    
• You will also be tasked with rearranging unfamiliar formulae.

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

• 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 h 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 (a + b)
      
    • You can now rewrite this with the subject (h) 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 x the subject of y = ax⁵
    • Divide both sides by a 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
•  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
    • Raise both sides to the power of 3 first
      m³ = 2ab
    • Divide both sides by 2b
      

Are there any common formulae to be aware of?

• The formula for the equation of a straight line is often used
  y = mx + c
• 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

Subject Appears Twice

How to Rearrange Formulas where the Subject Appears Twice?

• If the subject appears twice, you will need to factorize the equation. Factorizing involves placing an expression into brackets, with the subject positioned outside the brackets.
  • Factorizing entails enclosing an expression in brackets, with the subject situated outside the brackets.
• In scenarios where the subject is enclosed within a set of brackets, you must expand these brackets before proceeding with rearrangements.
• If the subject is present on both sides of a formula, you must consolidate those terms onto the same side before factorizing.