**5.4 Solve Equations with Decimals**

The topics covered in this section are:

- Determine whether a decimal is a solution of an equation
- Solve equations with decimals
- Translate to an equation and solve

**5.4.1 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.

**HOW TO: 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 1**

Determine whether each of the following is a solution of $x-0.7=1.5$.

- $x=1$
- $x=-0.8$
- $x=2.2$

**Solution**

Part 1. | |

$x-0.7=1.5$ | |

Substitute $1$ for $x$. | $1-0.7 \stackrel{?}{=} 1.5$ |

Subtract. | $0.3 \neq 1.5$ |

Since $x=1$ does not result in a true equation, $1$ is not a solution to the equation.

Part 2. | |

$x-0.7=1.5$ | |

Substitute $-0.8$ for $x$. | $-0.8-0.7 \stackrel{?}{=} 1.5$ |

Subtract. | $-1.5 \neq 1.5$ |

Since $x=-0.8$ does not result in a true equation, $-0.8$ is not a solution to the equation.

Part 3. | |

$x-0.7=1.5$ | |

Substitute $-0.8$ for $x$. | $2.2-0.7 \stackrel{?}{=} 1.5$ |

Subtract. | $1.5=1.5$✓ |

Since $x=2.2$ results in a true equation, $2.2$ is a solution to the equation.

**5.4.2 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 EqualityFor any numbers $a,b$, and $c$, If $a=b$, then $a-c=b-c$. | Addition Property of EqualityFor any numbers $a,b$, and $c$, If $a=b$, then $a+c=b+c$. |

The Division Property of EqualityFor any numbers $a,b$, and $c$, and $c \neq 0$ If $a=b$, then $\frac{a}{c} = \frac{b}{c}$. | The multiplication Property of EqualityFor any numbers $a,b$, and $c$, If $a=b$, then $ac=bc$. |

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

**Example 2**

Solve: $y+2.3=-4.7$.

**Solution**

We will use the Subtraction Property of Equality to isolate the variable.

$y+2.3=-4.7$ | |

Subtract $2.3$ from each side, to undo the addition. | $y+2.3-2.3=-4.7-2.3$ |

Simplify. | $y=-7$ |

Check. | $y+2.3=-4.7$ |

Substitute $y=-7$. | $-7+2.3 \stackrel{?}{=} -4.7$ |

Simplify. | $-4.7=-4.7$✓ |

Since $y=-7$ makes $y+2.3=-4.7$ a true statement, we know we have found a solution to this equation.

**Example 3**

Solve: $a-4.75=-1.39$.

**Solution**

We will use the Addition Property of Equality.

$a-4.75=-1.39$ | |

Add $4.75$ each side, to undo the subtraction. | $a-4.75+4.75=-1.39+4.75$ |

Simplify. | $a=3.36$ |

Check. | $a-4.75=-1.39$ |

Substitute $a=3.36$. | $3.36-4.75 \stackrel{?}{=} -1.39$ |

Simplify. | $-1.39=-1.36$✓ |

Since the result is a true statement, $a=3.36$ is a solution to the equation.

**Example 4**

Solve: $-4.8=0.8n$.

**Solution**

We will use the Division Property of Equality.

Use the Properties of Equality to find a value for $n$.

$-4.8=0.8n$ | |

We must divide both sides by $0.8$ to isolate $n$. | $\frac{-4.8}{0.8} = \frac{0.8n}{0.8}$ |

Simplify. | $-6=n$ |

Check. | $-4.8=0.8n$ |

Substitute $n=-6$. | $-4.8 \stackrel{?}{=} 0.8(-6)$ |

$-4.8=-4.8$✓ |

Since $n=-6$ makes $-4.8=0.8n$ a true statement, we know we have a solution.

**Example 5**

Solve: $\frac{p}{-1.8} = -6.5$.

**Solution**

We will use the Multiplication Property of Equality.

$\frac{p}{-1.8} = -6.5$ | |

Here, $p$ is divided by $-1.8$. We must multiply by $-1.8$ to isolate $p$. | $-1.8 ( \frac{p}{-1.8} ) = -1.8(-6.5)$ |

Simplify. | $p=11.7$ |

Check. | $\frac{p}{-1.8} = -6.5$ |

Substitute $p=11.7$. | $\frac{11.7}{-1.8} \stackrel{?}{=} -6.5$ |

$-6.5=-6.5$✓ |

A solution to $\frac{p}{-1.8} = -6.5$ is $p=11.7$.

**5.4.3 Translate to an Equation and Solve**

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

**Example 6**

Translate and solve: The difference of $n$ and $4.3$ is $2.1$.

**Solution**

Translate. | |

Add $4.3$ to both sides of the equation. | $n-4.3+4.3 = 2.1+4.3$ |

Simplify. | $n=6.4$ |

Check. | Is the difference of $n$ and $4.3$ equal to $2.1$? |

Let $n=6.4$: | Is the difference of $6.4$ and $4.3$ equal to $2.1$? |

Translate. | $6.4-4.3 \stackrel{?}{=} 2.1$ |

Simplify. | $2.1=2.1$✓ |

**Example 7**

Translate and solve: The product of $-3.1$ and $x$ is $5.27$.

**Solution**

Translate. | |

Divide both sides by $-3.1$. | $\frac{-3.1x}{-3.1} = \frac{5.27}{-3.1}$ |

Simplify. | $x=-1.7$ |

Check. | Is the product of $-3.1$ and $x$ equal to $5.27$? |

Let $x=-1.7$: | Is the product of $-3.1$ and $-1.7$ equal to $5.27$? |

Translate. | $-3.1(-1.7) \stackrel{?}{=} 5.27$ |

Simplify. | $5.27=5.27$✓ |

**Example 8**

Translate and solve: The quotient of $p$ and $-2.4$ is $6.5$.

**Solution**

Translate. | |

Multiply both side by $-2.4$. | $-2.4 ( \frac{p}{-2.4} ) = -2.4(6.5)$ |

Simplify. | $p=-15.6$ |

Check. | Is the quotient of $p$ and $-2.4$ equal to $6.5$? |

Let $p=-15.6$: | Is the quotient of $-15.6$ and $-2.4$ equal to $6.5$? |

Translate. | $\frac{-15.6}{-2.4} \stackrel{?}{=} 6.5$ |

Simplify. | $6.5=6.5$✓ |

**Example 9**

Translate and solve: The sum of $n$ and $2.9$ is $1.7$.

**Solution**

Translate. | |

Subtract $2.9$ from each side. | $n+2.9-2.9=1.7-2.9$ |

Simplify. | $n=-1.2$ |

Check. | Is the sum $n$ and $2.9$ equal to $1.7$? |

Let $n=-1.2$: | Is the sum $-1.2$ and $2.9$ equal to $1.7$? |

Translate. | $-1.2+2.9 \stackrel{?}{=} 1.7$ |

Simplify. | $1.7=1.7$✓ |

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*Marecek, L., Anthony-Smith, M., & Mathis, A. H. (2020). Use the Language of Algebra. In Prealgebra 2e. OpenStax. https://openstax.org/books/prealgebra-2e/pages/5-4-solve-equations-with-decimals*.*License: CC BY 4.0. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction*