## Solving Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

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In As we saw in Mathematics 102 and Mathematics 103, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.

### Addition, Subtraction, and Division Properties of Equality

For any numbers \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\)

\(\text{if}\phantom{\rule{0.2em}{0ex}}a=b,\text{then}\phantom{\rule{0.2em}{0ex}}a+c=b+c.\) | Addition Property of Equality |

\(\text{if}\phantom{\rule{0.2em}{0ex}}a=b,\text{then}\phantom{\rule{0.2em}{0ex}}a-c=b-c.\) | Subtraction Property of Equality |

\(\text{if}\phantom{\rule{0.2em}{0ex}}a=b,\text{then}\phantom{\rule{0.2em}{0ex}}\frac{a}{c}=\frac{b}{c},c\ne 0.\) | Division Property of Equality |

In other words, when you add or subtract the same quantity from both sides of an equation, or divide both sides by the same quantity, you still have equality.

## Example

Solve: \(y+\frac{9}{16}=\frac{5}{16}.\)

### Solution

Subtract \(\frac{9}{16}\) from each side to undo the addition. | ||

Simplify on each side of the equation. | ||

Simplify the fraction. | ||

Check: | ||

Substitute \(y=-\frac{1}{4}\). | ||

Rewrite as fractions with the LCD. | ||

Add. |

Since \(y=-\frac{1}{4}\) makes \(y+\frac{9}{16}=\frac{5}{16}\) a true statement, we know we have found the solution to this equation.

We used the Subtraction Property of Equality in the example above. Now we’ll use the Addition Property of Equality.

## Example

Solve: \(a-\frac{5}{9}=-\frac{8}{9}.\)

### Solution

Add \(\frac{5}{9}\) from each side to undo the addition. | ||

Simplify on each side of the equation. | ||

Simplify the fraction. | ||

Check: | ||

Substitute \(a=-\frac{1}{3}\). | ||

Change to common denominator. | ||

Subtract. |

Since \(a=-\frac{1}{3}\) makes the equation true, we know that \(a=-\frac{1}{3}\) is the solution to the equation.

The next example may not seem to have a fraction, but let’s see what happens when we solve it.

## Example

Solve: \(10q=44.\)

### Solution

\(10q=44\) | ||

Divide both sides by 10 to undo the multiplication. | \(\frac{10q}{10}=\frac{44}{10}\) | |

Simplify. | \(q=\frac{22}{5}\) | |

Check: | ||

Substitute \(q=\frac{22}{5}\) into the original equation. | \(10\left(\frac{22}{5}\right)\stackrel{?}{=}44\) | |

Simplify. | \(\require{cancel}\stackrel{2}{\cancel{10}}\left(\frac{22}{\cancel{5}}\right)\stackrel{?}{=}44\) | |

Multiply. | \(44=44\phantom{\rule{0.2em}{0ex}}✓\) |

The solution to the equation was the fraction \(\frac{22}{5}.\) We leave it as an improper fraction.