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Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.
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 $x-\frac{3}{10}=\frac{1}{2}.$
ⓐ | |
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.
$x-\frac{2}{3}=\frac{1}{6}\text{:}$
Determine whether each number is a solution of the given equation.
$y-\frac{1}{4}=\frac{3}{8}\text{:}$
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.
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: $y+\frac{9}{16}=\frac{5}{16}.$
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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