Solving Linear Equations Involving Fractions

Need some practice without fractions first?

When solving equations involving fractions, it's usually easiest to clear fractions first by multiplying by the least common denominator of all the fractions involved.

The procedure is illustrated in the examples below. Once the fractions are gone, the equations are much simpler!

Examples

Solve: $\displaystyle\frac{2}{3}x + 6 = 1$

Solution: Write a nice clean list of equivalent equations:

$\displaystyle\frac{2}{3}x + 6 = 1$	original equation
$\displaystyle3\left(\frac{2}{3}x + 6\right) = 3(1)$	multiply both sides by $\,3\,$
$2x + 18 = 3$	simplify; all fractions are gone
$2x = -15$	subtract $\,18\,$ from both sides
$\displaystyle x = -\frac{15}{2}$	divide both sides by $\,2\,$

Solve: $\displaystyle -3x -\frac{8}{9} = \frac{5}{6}$

Solution:

$\displaystyle -3x -\frac{8}{9} = \frac{5}{6}$	original equation
$\displaystyle 18\left(-3x -\frac{8}{9}\right) = 18(\frac{5}{6})$	multiply both sides by $\,18\,,$ which is the least common multiple of $\,9\,$ and $\,6\,$
$-54x - 16 = 15$	simplify; all fractions are gone
$-54x = 31$	add $\,16\,$ to both sides
$\displaystyle x = -\frac{31}{54}$	divide both sides by $\,-54\,$

Practice

For more advanced students, a graph is available. For example, the equation $\,\frac{2}{3}x + 6 = 1\,$ is optionally accompanied by the graph of $\,y = \frac{2}{3}x + 6\,$ (the left side of the equation, dashed green) and the graph of $\,y = 1\,$ (the right side of the equation, solid purple).

Notice that you are finding the value of $\,x\,$ where these graphs intersect. Click the ‘Show/Hide Graph’ button to toggle the graph.