# Solving Linear Equations Involving Fractions

Need some practice without fractions first?

- Solving Simple Linear Equations with Integer Coefficients
- Solving More Complicated Linear Equations with Integer Coefficients

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

$\displaystyle\frac{2}{3}x + 6 = 1$ | 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 $\,2\,$ |

$\displaystyle -3x -\frac{8}{9} = \frac{5}{6}$ | 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.