# 4.7 Solve equations with fractions

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By the end of this section, you will be able to:
• Determine whether a fraction is a solution of an equation
• Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality
• Solve equations using the Multiplication Property of Equality
• Translate sentences to equations and solve

Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

1. Evaluate $x+4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-3$
If you missed this problem, review Add Integers .
2. Solve: $2y-3=9.$
If you missed this problem, review Subtract Integers .
3. Multiply: $\frac{5}{8}·40.$
If you missed this problem, review Multiply and Divide Fractions .

## Determine whether a fraction is a solution of an equation

As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality , a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.

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, or a fraction.

## Determine whether a number is a solution to an equation.

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

Determine whether each of the following is a solution of $x-\frac{3}{10}=\frac{1}{2}.$

1. $x=1$
2. $x=\frac{4}{5}$
3. $x=-\frac{4}{5}$

## Solution

 ⓐ Change to fractions with a LCD of 10. Subtract.

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

 ⓑ Subtract.

Since $x=\frac{4}{5}$ results in a true equation, $\frac{4}{5}$ is a solution to the equation $x-\frac{3}{10}=\frac{1}{2}.$

 ⓒ Subtract.

Since $x=-\frac{4}{5}$ does not result in a true equation, $-\frac{4}{5}$ is not a solution to the equation.

Determine whether each number is a solution of the given equation.

$x-\frac{2}{3}=\frac{1}{6}\text{:}$

1. $x=1$
2. $x=\frac{5}{6}$
3. $x=-\frac{5}{6}$

1. no
2. yes
3. no

Determine whether each number is a solution of the given equation.

$y-\frac{1}{4}=\frac{3}{8}\text{:}$

1. $y=1$
2. $y=-\frac{5}{8}$
3. $y=\frac{5}{8}$

1. no
2. no
3. yes

## Solve equations with fractions using the addition, subtraction, and division properties of equality

In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality , 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.

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.

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