# Using the Division and Multiplication Properties of Equality for Multi-Step Equations

### Learning Outcomes

- Solve a linear equation that requires simplification before using properties of equality
- Solve a linear equation that requires a combination of the properties of equality

- simplify by combining like terms
- isolate x by using the division property of equality

### Example

Solve: [latex]8x+9x - 5x=-3+15[/latex] Solution: First, we need to simplify both sides of the equation as much as possible Start by combining like terms to simplify each side.[latex]8x+9x-5x=-3+15[/latex] | |

Combine like terms. | [latex]12x=12[/latex] |

Divide both sides by 12 to isolate x. | [latex]\Large\frac{12x}{\color{red}{12}}\normalsize =\Large\frac{12}{\color{red}{12}}[/latex] |

Simplify. | [latex]x=1[/latex] |

Check your answer. Let [latex]x=1[/latex] | |

[latex]8x+9x-5x=-3+15[/latex] | |

[latex]8\cdot\color{red}{1}+9\cdot\color{red}{1}-5\cdot\color{red}{1}\stackrel{\text{?}}{=}-3+15[/latex] | |

[latex]8+9-5\stackrel{\text{?}}{=}-3+15[/latex] | |

[latex]12=12\quad\checkmark[/latex] |

### Try it

[embed]### example

Solve: [latex]11 - 20=17y - 8y - 6y[/latex]Answer: Solution: Simplify each side by combining like terms.

[latex]11-20=17y-8y-6y[/latex] | |

Simplify each side. | [latex]-9=3y[/latex] |

Divide both sides by 3 to isolate y. | [latex]\Large\frac{-9}{\color{red}{3}}\normalsize =\Large\frac{3y}{\color{red}{3}}[/latex] |

Simplify. | [latex]-3=y[/latex] |

Check your answer. Let [latex]y=-3[/latex] | |

[latex]11-20=17y-8y-6y[/latex] | |

[latex]11-20\stackrel{\text{?}}{=}17( \color{red}{-3})-8(\color{red}{-3})-6(\color{red}{-3})[/latex] | |

[latex]11-20\stackrel{\text{?}}{=}-51+24+18[/latex] | |

[latex]-9=-9\quad\checkmark[/latex] |

### try it

[embed]### example

Solve: [latex]-3\left(n - 2\right)-6=21[/latex] Remember—always simplify each side first.Answer: Solution:

[latex]-3(n-2)-6=21[/latex] | |

Distribute. | [latex]-3n+6-6=21[/latex] |

Simplify. | [latex]-3n=21[/latex] |

Divide both sides by -3 to isolate n. | [latex]\Large\frac{-3n}{\color{red}{-3}}\normalsize =\Large\frac{21}{\color{red}{-3}}[/latex][latex]n=-7[/latex] |

Check your answer. Let [latex]n=-7[/latex] . | |

[latex]-3(n-2)-6=21[/latex] | |

[latex]-3(\color{red}{-7}-2)-6\stackrel{\text{?}}{=}21[/latex] | |

[latex]-3(-9)-6\stackrel{\text{?}}{=}21[/latex] | |

[latex]27-6\stackrel{\text{?}}{=}21[/latex] | |

[latex]21=21\quad\checkmark[/latex] |

### try it

[embed]## Licenses & Attributions

### CC licensed content, Original

- Solving an Equation with One Set of Parentheses.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Solve Linear Equations in One Variable with Simplifying (One-Step Mult/Div).
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Question ID 141884, 141901, 141911.
**Authored by:**Lumen Learning.**License:**CC BY: Attribution.**License terms:**IMathAS Community License, CC-BY + GPL.

### CC licensed content, Specific attribution

- Prealgebra.
**Provided by:**OpenStax**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected].