Formulas where Subject Appears Once (Edexcel GCSE Maths)

Revision Note

Simple Rearranging

What are formulas?

  • A formula is a rule, definition or relationship between different quantities, written in shorthand using letters (variables)

    • They include an equals sign

  • Some examples 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

How do I rearrange formulas?  

  • The letter (variable) that is on its own on one side is called the subject

    • y  is the subject of y = mx + c

  • To make a different letter the subject, we need to rearrange the formula

    • This is also called changing the subject

  • The method is as follows:

    • First, remove any fractions

      • Multiply both sides by the lowest common denominator

    • Then use inverse (opposite) operations to get the variable on its own

      • This is similar to solving equations

  • For example, make x the subject of fraction numerator 5 x plus 6 over denominator 2 end fraction equals y

    • First remove fractions

      • Multiply both sides by 2
        5 x plus 6 equals 2 y

    • Then get x on its own

      • Subtract 6 from both sides
        5 x equals 2 y minus 6

      • Divide both sides by 5
        x equals fraction numerator 2 y minus 6 over denominator 5 end fraction

    • There may be more than one correct way to write an answer

      • The following are acceptable alternative forms 
        x equals fraction numerator 2 y over denominator 5 end fraction minus 6 over 5
        x equals fraction numerator 2 open parentheses y minus 3 close parentheses over denominator 5 end fraction
        x equals 0.4 open parentheses y minus 3 close parentheses
        x equals 0.4 y minus 1.2

Should I expand brackets?

  • Expand brackets if it releases the variable you want from inside the brackets

    • If not, you can leave them in

  • To make x the subject of 3 left parenthesis 1 plus x right parenthesis equals y

    • x is inside the brackets, so expand

      • 3 plus 3 x equals y

    • Rearrange

      • table row cell 3 x end cell equals cell y minus 3 end cell end table
          table row x equals cell fraction numerator y minus 3 over denominator 3 end fraction end cell end table

  • To make x the subject of open parentheses 1 plus k close parentheses x equals y

    • x is not inside the brackets, so you do not need to expand

    • Instead, divide both sides by the bracket open parentheses 1 plus k close parentheses

      • x equals fraction numerator y over denominator 1 plus k end fraction

What if I get fractions in fractions?

  • Some rearrangements can lead to fractions in fractions 

    • x equals fraction numerator space 3 over t space over denominator 2 end fraction

  • Either rewrite with a divide sign, divided by, then use the method of dividing two fractions

    • x equals 3 over t divided by 2
      x equals 3 over t divided by 2 over 1
x equals 3 over t cross times 1 half
x equals fraction numerator 3 over denominator 2 t end fraction

  • Or multiply top and bottom by the the lowest common denominator of the two fractions and cancel

    • x equals fraction numerator space space begin display style 5 over y end style space space over denominator begin display style t over 8 end style end fraction becomes x equals fraction numerator space space 5 over y cross times 8 y space space over denominator t over 8 cross times 8 y end fraction equals fraction numerator 40 over denominator t y end fraction

What if I end up dividing by a negative?

  • Remember that fraction numerator a over denominator negative b end fraction (minus below) is the same as fraction numerator negative a over denominator b end fraction (minus above) and the same as negative a over b (minus outside)

    • Though be careful, as fraction numerator negative a over denominator negative b end fraction is a over b

  • negative 2 x equals y minus 3 becomes x equals fraction numerator y minus 3 over denominator negative 2 end fraction (minus below)

    • This is the same as x equals fraction numerator negative open parentheses y minus 3 close parentheses over denominator 2 end fraction (minus above) or negative space fraction numerator y minus 3 over denominator 2 end fraction (minus outside)

      • brackets are required for minus above

      • brackets are assumed for minus outside

    • You can also expand the brackets
      fraction numerator negative open parentheses y minus 3 close parentheses over denominator 2 end fraction equals fraction numerator negative y plus 3 over denominator 2 end fraction equals fraction numerator 3 minus y over denominator 2 end fraction

Examiner Tips and Tricks

  • Mark schemes will accept different forms of the same answer, as long as they are correct and fully simplified

Worked Example

Make x the subject of the following.

(a) 4 m plus 5 x equals 3

Get 5x  on its own by subtracting 4m  from both sides

5 x equals 3 minus 4 m

Get x  on its own by dividing both sides by 5

bold italic x bold equals fraction numerator bold 3 bold minus bold 4 bold italic m over denominator bold 5 end fraction

(b) 3 t equals 2 over x

Remove fractions by multiplying both sides by the denominator, x

3 t x equals 2

Get x  on its own by dividing both sides by 3t

bold italic x bold equals fraction numerator bold 2 over denominator bold 3 bold italic t end fraction

(c) A equals fraction numerator 9 open parentheses 1 minus 4 x close parentheses over denominator 2 g end fraction

Remove fractions by multiplying both sides by the denominator, 2g

2 g A equals 9 open parentheses 1 minus 4 x close parentheses

is inside the brackets
Expand the brackets to release the x  term

2 g A equals 9 minus 36 x

One way to get x  on its own is by subtracting 9 then dividing by -36
Or you can first add 36 to both sides, to create positive 36x  on the left

2 g A plus 36 x equals 9

Now get x  on its own by subtracting 2gA then dividing by 36

table row cell 36 x end cell equals cell 9 minus 2 g A end cell row x equals cell fraction numerator 9 minus 2 g A over denominator 36 end fraction end cell end table

bold italic x bold equals fraction numerator bold 9 bold minus bold 2 bold g bold A over denominator bold 36 end fraction

Other accepted forms of the answer are fraction numerator 2 g A minus 9 over denominator negative 36 end fraction space comma space space space fraction numerator negative open parentheses 2 g A minus 9 close parentheses over denominator 36 end fraction space comma space space space minus fraction numerator 2 g A minus 9 over denominator 36 end fraction space comma space space space 1 fourth minus fraction numerator g A over denominator 18 end fraction


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Mark Curtis

Author: Mark Curtis

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.