SOLVING LINEAR EQUATIONS INVOLVING FRACTIONS

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,
as illustrated in the examples below.

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 $\,-54\,$)
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Solving Linear Equations, All Mixed Up

 
 

For more advanced students, a graph is displayed.
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 if you prefer not to see the graph.

Solve: