# Using the Subtraction and Addition Properties for Multi-Step Equations

### Learning Outcomes

- Solve a linear equation that needs to be simplified before using the subtraction and addition properties of equality
- Check your solution to a linear equation to verify its accuracy

### Example

Solve: [latex-display]3x - 7 - 2x - 4=1[/latex-display] Solution: The left side of the equation has an expression that we should simplify before trying to isolate the variable.[latex]3x-7-2x-4=1[/latex] | |

Rearrange the terms, using the Commutative Property of Addition. | [latex]3x-2x-7-4=1[/latex] |

Combine like terms. | [latex]x-11=1[/latex] |

Add [latex]11[/latex] to both sides to isolate [latex]x[/latex] . | [latex]x-11\color{red}{+11}=1\color{red}{+11}[/latex] |

Simplify. | [latex]x=12[/latex] |

Check.Substitute [latex]x=12[/latex] into the original equation. [latex-display]3x-7-2x-4=1[/latex-display] [latex-display]3(\color{red}{12})-7-2(\color{red}{12})-4=1[/latex-display] [latex-display]36-7-24-4=1[/latex-display] [latex-display]29-24-4=1[/latex-display] [latex-display]5-4=1[/latex-display] [latex-display]1=1\quad\checkmark[/latex-display] The solution checks. |

### TRY IT

[embed]### example

Solve: [latex]3\left(n - 4\right)-2n=-3[/latex]Answer: Solution: The left side of the equation has an expression that we should simplify.

[latex]3(n-4)-2n=-3[/latex] | |

Distribute on the left. | [latex]3n-12-2n=-3[/latex] |

Use the Commutative Property to rearrange terms. | [latex]3n-2n-12=-3[/latex] |

Combine like terms. | [latex]n-12=-3[/latex] |

Isolate n using the Addition Property of Equality. | [latex]n-12\color{red}{+12}=-3\color{red}{+12}[/latex] |

Simplify. | [latex]n=9[/latex] |

Check.Substitute [latex]n=9[/latex] into the original equation. [latex-display]3(n-4)-2n=-3[/latex-display] [latex-display]3(\color{red}{9}-4)-2\cdot\color{red}{9}=-3[/latex-display] [latex-display]3(5)-18=-3[/latex-display] [latex-display]15-18=-3[/latex-display] [latex-display]-3=-3\quad\checkmark[/latex-display] The solution checks. |

### TRY IT

[embed]### example

Solve: [latex]2\left(3k - 1\right)-5k=-2 - 7[/latex]

Answer: Solution: Both sides of the equation have expressions that we should simplify before we isolate the variable.

[latex]2(3k-1)-5k=-2-7[/latex] | |

Distribute on the left, subtract on the right. | [latex]6k-2-5k=-9[/latex] |

Use the Commutative Property of Addition. | [latex]6k-5k-2=-9[/latex] |

Combine like terms. | [latex]k-2=-9[/latex] |

Undo subtraction by using the Addition Property of Equality. | [latex]k-2\color{red}{+2}=-9\color{red}{+2}[/latex] |

Simplify. | [latex]k=-7[/latex] |

Check.Let [latex]k=-7[/latex]. [latex-display]2(3k-1)-5k=-2-7[/latex-display] [latex-display]2(3(\color{red}{-7}-1)-5(\color{red}{-7})=-2-7[/latex-display] [latex-display]2(-21-1)-5(-7)=-9[/latex-display] [latex-display]2(-22)+35=-9[/latex-display] [latex-display]-44+35=-9[/latex-display] [latex-display]-9=-9\quad\checkmark[/latex-display] |

### TRY IT

[embed]## Licenses & Attributions

### CC licensed content, Original

- Solve Linear Equations in One Variable with Simplifying (One-Step Add/Subtract).
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Question ID 141735, 141737, 141739.
**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].