4.7 Solve equations with 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 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.

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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.

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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