Equations with Unknowns on Both Sides (Cambridge (CIE) IGCSE International Maths)

Revision Note

Written by: Mark Curtis

Reviewed by: Dan Finlay

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 $4 x - 7 = 11 + x$
Let's remove the x term on the right-hand side
x has been added, so subtract x from both sides

$\begin{array}{rcl} 4 x - 7 & = & 11 + x \end{array}$

$\begin{array}{rcl} (- x) (- x) \end{array}$

$\begin{array}{rcl} 3 x - 7 & = & 11 \end{array}$

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

$\begin{array}{rcl} 3 x - 7 & = & 11 \\ 3 x & = & 18 \\ x & = & 6 \end{array}$

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

$4 - 5 x = 6 x - 29$

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

$\begin{array}{rcl} 4 - 5 x + 5 x & = & 6 x - 29 + 5 x \\ 4 & = & 11 x - 29 \end{array}$

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)

$4 + 29 = 11 x$

Work out 4 + 29

$33 = 11 x$

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

$\frac{33}{11} = x$

Work out 33 ÷ 11

$3 = x$

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 = ...

$x = 3$

Last updated: 4 June 2024

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