Equations with Brackets & Fractions

Equations with Brackets & Fractions

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 $2\left(x-3\right)=10$

  • Expand the brackets 

$2x-6=10$

• This now has the same form as before

  • Solve by adding 6 then dividing by 2

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

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

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

    • 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

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

• Simplify the fractions

  • Imagine the 10 is in the numerator and cancel

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

• This now has the same form as before

  • Solve by subtracting 40 then dividing by 2

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

• 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

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

• Simplify the fractions, and expand any brackets

$4=3\left(x-2\right)$

$4=3x-6$

• This now looks like a standard linear equation

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

$$\begin{gathered} 10=3x \\ \frac{10}{3}=x \end{gathered}$$