Equations with Brackets & Fractions (Edexcel IGCSE Maths)

Spec point — 2.4A

Equations with brackets & fractions

How do I solve linear equations with brackets?

• If a linear equation involves brackets, expand the brackets first

• For example, solve $2 ( x - 3 ) = 10$

  • Expand the brackets

$2x-6=10$

• This now has the same form as before

  • Solve by adding 6 then dividing by 2

$\begin{gathered} 2 x & = 16 \\ x = \frac{16}{2} \\ x = 8 \end{gathered}$

• Expanding brackets first will always work, but you can also divide first

  • Dividing both sides of $2 ( x - 3 ) = 10$ by 2 gives $( x - 3 ) = 5$

    • which solves to $x=8$

  • This method works but can lead to harder fractions

How do I solve linear equations with fractions?

• If a linear equation contains fractions, multiply both sides by the lowest common denominator

• For example, solve $\frac{x}{5}+4=\frac{9}{2}$

  • The lowest common denominator of 5 and 2 is 10

  • Multiply all terms on both sides by 10

    • Don't forget to do it to the 4

$\begin{gathered} 10 \times \frac{x}{5} + 10 \times 4 & = 10 \times \frac{9}{2} \end{gathered}$

• Simplify the fractions

  • Imagine the 10 is in the numerator and cancel

$\begin{gathered} 2 x + 40 & = 45 \end{gathered}$

• This now has the same form as before

  • Solve by subtracting 40 then dividing by 2

$\begin{gathered} 2 x & = 5 \\ x = \frac{5}{2} \end{gathered}$

• Unless the question asks, you can leave the answer like this

  • A decimal or mixed number would also be accepted

• Sometimes equations have the unknown on the denominator

• For example $\frac{4}{x-2}=3$

  • Multiply both sides of the equation by the denominator

$\frac{4}{x - 2} \times ( x - 2 ) = 3 ( x - 2 )$

• Simplify the fractions, and expand any brackets

$4 = 3 ( x - 2 )$

$4=3x-6$

• This now looks like a standard linear equation

  • Solve by adding 6 to both sides, then dividing by 3

$10=3x\,\frac{10}{3}=x$

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

Worked Example (a) Solve the equation

$5 ( 3 - 4 x ) + 1 = 26$

Answer:

First expand the brackets on the left-hand side

$\begin{gathered} 15 - 20 x + 1 & = 26 \end{gathered}$

Simplify the left-hand side (by adding 15 and 1)

$\begin{gathered} 16 - 20 x & = 26 \end{gathered}$

Remember 16 - 20x is the same as -20x + 16 Subtract 16 from both sides

$\begin{gathered} - 20 x & = 26 - 16 \\ - 20 x & = 10 \end{gathered}$

Divide both sides by -20 and simplify

$\begin{gathered} x = \frac{10}{- 20} \end{gathered}$

$x=-\frac{1}{2}$

(b) Solve the equation

$\frac{5x}{4}=\frac{1}{2}$

Answer:

The lowest common denominator of 4 and 2 is 4 Multiply both sides by 4

$\begin{gathered} 4 \times \frac{5 x}{4} & = 4 \times 1 \frac{1}{2} \end{gathered}$

Simplify (cancel) the fractions

$\begin{gathered} 5 x & = 2 \end{gathered}$

To solve this equation, divide both sides by 5

$x=\frac{2}{5}$