# Solving Linear Equations, All Mixed Up

This exercise mixes up problems from three earlier exercises:

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

## Examples

Solve:
$\,2x - 1 = 5\,$

Solution:

$2x-1=5$ | original equation |

$2x=6$ | add $\,1\,$ to both sides |

$x = 3$ | divide both sides by $\,3$ |

Solve:
$3 - 2x = 5x + 1$

Solution:

$3 - 2x = 5x + 1$ | original equation |

$3 = 7x + 1$ | add $\,2x\,$ to both sides |

$2 = 7x$ | subtract $\,1\,$ from both sides |

$\frac{2}{7} = x$ | divide both sides by $\,7\,$ |

$x = \frac{2}{7}$ | write in the most conventional way |

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 $\,3 - 2x = 5x + 1\,$ is optionally accompanied by the graph of $\,y = 3 - 2x\,$ (the left side of the equation, dashed green) and the graph of $\,y = 5x + 1\,$ (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.