# Using the Properties of Equality to Solve Equations With Decimals

### Learning Outcomes

- Determine whether a decimal is a solution to an equation
- Solve an equation that contains decimals using the addition and subtraction properties

## Determine Whether a Decimal is a Solution of an Equation

Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals. Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We’ll list these steps here again for easy reference.### Determine whether a number is a solution to an equation

- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- Determine whether the resulting equation is true.
- If so, the number is a solution.
- If not, the number is not a solution.

### example

Determine whether each of the following is a solution of [latex]x - 0.7=1.5[/latex] 1. [latex]x=1[/latex] 2. [latex]x=-0.8[/latex] 3. [latex]x=2.2[/latex] Solution1. | |

[latex]x-0.7=1.5[/latex] | |

Substitute [latex]\color{red}{1}[/latex] for x. | [latex]\color{red}{1} - 0.7\stackrel{?}{=}1.5[/latex] |

Subtract. | [latex]0.3\not=1.5[/latex] |

2. | |

[latex]x-0.7=1.5[/latex] | |

Substitute [latex]\color{red}{0.8}[/latex] for x. | [latex]\color{red}{0.8} - 0.7\stackrel{?}{=}1.5[/latex] |

Subtract. | [latex]-1.5\not=1.5[/latex] |

3. | |

[latex]x-0.7=1.5[/latex] | |

Substitute [latex]\color{red}{2.2}[/latex] for x. | [latex]\color{red}{2.2} - 0.7\stackrel{?}{=}1.5[/latex] |

Subtract. | [latex]1.5=1.5[/latex] |

### try it

[ohm_question]146375[/ohm_question]## Solve Equations with Decimals

In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.### Properties of Equality

Subtraction Property of Equality
For any numbers [latex]a,b,\text{and }c[/latex]
If [latex]a=b[/latex], then [latex]a-c=b-c[/latex]. |
Addition Property of Equality
For any numbers [latex]a,b,\text{and }c[/latex]
If [latex]a=b[/latex], then [latex]a+c=b+c[/latex]. |

The Division Property of Equality
For any numbers [latex]a,b,\text{and }c,\text{and }c\ne 0[/latex]
If [latex]a=b[/latex], then [latex]{\Large\frac{a}{c}}={\Large\frac{b}{c}}[/latex] |
The Multiplication Property of Equality
For any numbers [latex]a,b,\text{and }c[/latex]
If [latex]a=b[/latex], then [latex]ac=bc[/latex] |

### example

Solve: [latex]y+2.3=-4.7[/latex]Answer: Solution We will use the Subtraction Property of Equality to isolate the variable.

[latex]y+2.3=-4.7[/latex] | ||

Subtract [latex]\color{red}{2.3}[/latex] from each side, to undo the addition. | [latex]y+2.3\color{red}{- 2.3}=-4.7\color{red}{- 2.3}[/latex] | |

Simplify. | [latex]y-7[/latex] | |

Check: |
[latex]y+2.3=-4.7[/latex] | |

Substitute [latex]y=\color{red}{-7}[/latex]. | [latex]\color{red}{-7}+2.3\stackrel{?}{=}-4.7[/latex] | |

Simplify. | [latex]-4.7=-4.7[/latex] |

### try it

[ohm_question]146377[/ohm_question]### example

Solve: [latex]a - 4.75=-1.39[/latex]Answer: Solution We will use the Addition Property of Equality.

[latex]a-4.75=-1.39[/latex] | ||

Add [latex]4.75[/latex] to each side, to undo the subtraction. | [latex]a-4.75+\color{red}{4.75}=-1.39+\color{red}{4.75}[/latex] | |

Simplify. | [latex]a=3.36[/latex] | |

Check: |
[latex]a-4.75=-1.39[/latex] | |

Substitute [latex]a=\color{red}{3.36}[/latex]. | [latex]\color{red}{3.36}-4.75\stackrel{?}{=}-1.39[/latex] | |

[latex]-1.39=-1.39[/latex] |

### try it

[ohm_question]146379[/ohm_question]### example

Solve: [latex]-4.8=0.8n[/latex]Answer: Solution We will use the Division Property of Equality. Use the Properties of Equality to find a value for [latex]n[/latex].

[latex]-4.8=0.8n[/latex] | ||

We must divide both sides by [latex]0.8[/latex] to isolate n. |
[latex]{\Large\frac{-4.8}{\color{red}{0.8}}}={\Large\frac{0.8n}{\color{red}{0.8}}}[/latex] | |

Simplify. | [latex]-6=n[/latex] | |

Check: |
[latex]-4.8=0.8n[/latex] | |

Substitute [latex]n=\color{red}{-6}[/latex]. | [latex]-4.8\stackrel{?}{=}0.8(\color{red}{-6})[/latex] | |

[latex]-4.8=-4.8[/latex] |

### try it

[ohm_question]146382[/ohm_question]### example

Solve: [latex]{\Large\frac{p}{-1.8}}=-6.5[/latex]Answer: Solution We will use the Multiplication Property of Equality.

[latex]{\Large\frac{p}{-1.8}}=-6.5[/latex] | ||

Here, p is divided by [latex]−1.8[/latex]. We must multiply by [latex]−1.8[/latex] to isolate p |
[latex]\color{red}{-1.8}({\Large\frac{p}{-1.8}})=\color{red}{-1.8}(-6.5)[/latex] | |

Multiply. | [latex]p=11.7[/latex] | |

Check: |
[latex]{\Large\frac{p}{-1.8}}=-6.5[/latex] | |

[latex]{\Large\frac{\color{red}{11.7}}{-1.8}}\stackrel{?}{=}-6.5[/latex] | ||

Substitute [latex]p=\color{red}{11.7}[/latex]. | [latex]-6.5=-6.5[/latex] |

### try it

[ohm_question]146383[/ohm_question]## Licenses & Attributions

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- Solving One Step Equations Using Addition and Subtraction (Decimals).
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. - Ex: Solve a One Step Equation With Decimals by Multiplying.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. - Ex: Solve a One Step Equation With Decimals by Dividing.
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