O LevelMathsCambridge (CIE)Revision NotesAlgebra & SequencesLinear Equations & InequalitiesEquations with Unknowns on Both Sides

Equations with Unknowns on Both Sides (Cambridge (CIE) O Level Maths): Revision Note

Exam code: 4024

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Unknown on both sides

How do I solve linear equations with x terms on both sides?

  • If a linear equation contains the unknown variable on both sides, collect the x terms together on one side 

    • To do this, choose a side you wish to remove the x  term from

      •  this should be the side with the lowest value x  term

    • Apply the opposite of that term to both sides

  • For example, solve  4 x minus 7 equals 11 plus x

  • Let's remove the x  term on the right-hand side

  • x  has been added, so subtract x  from both sides 

table row cell 4 x minus 7 end cell equals cell 11 plus x end cell end table

table row blank blank cell open parentheses negative 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 open parentheses negative x close parentheses end cell end table

table row cell 3 x minus 7 end cell equals 11 end table

  • There are no longer any x  terms on the right

  • The 4x  has become 3x  on the left

  • This now has the same form as equations with one variable

    • Solve it by adding 7 then dividing by 3         

table row cell 3 x minus 7 end cell equals 11 row cell 3 x end cell equals 18 row x equals 6 end table

Examiner Tips and Tricks

  • In the exam, substitute your answer back into the original equation to check you got it right!

Worked Example

Solve the equation

4 minus 5 x equals 6 x minus 29 

Answer:

5x  is smaller than 6x  so it's easier to remove this term first
Add 5x  to both sides and simplify (collect like terms)

table row cell 4 minus 5 x plus 5 x end cell equals cell 6 x minus 29 plus 5 x end cell row 4 equals cell 11 x minus 29 end cell end table

This could also have been written as 4 = -29 + 11x
Leave the x's on the right-hand side (avoid 'reflecting' the equation)
Get 11x  on its own (by adding 29 to both sides)

4 plus 29 equals 11 x

Work out 4 + 29

33 equals 11 x 

Get x  on its own (by dividing both sides by 11) 

33 over 11 equals x

Work out 33 ÷ 11

3 equals x 

The answer currently has x  on the right-hand side
At this point, you can reflect the equation to present your final answer as x = ... 

bold italic x bold equals bold 3

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    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

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.