Equations with Unknowns on Both Sides (Cambridge International IGCSE Mathematics)

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 $4x-7=11+x$

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

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

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

$\begin{gathered} ( - x ) ( - x ) \end{gathered}$

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

• 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

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

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

Worked Example Solve the equation

$4-5x=6x-29$

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)

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

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

$4+29=11x$

Work out 4 + 29

$33=11x$

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 = ...

$x=3$