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Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality

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

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

 Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality
Subtract \(\frac{9}{16}\) from each side to undo the addition.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality
Simplify on each side of the equation.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality
Simplify the fraction.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality
Check:Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality 
Substitute \(y=-\frac{1}{4}\).Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality 
Rewrite as fractions with the LCD.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality 
Add.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality 

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

 Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality
Add \(\frac{5}{9}\) from each side to undo the addition.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality
Simplify on each side of the equation.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality
Simplify the fraction.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality
Check:Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality 
Substitute \(a=-\frac{1}{3}\).Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality 
Change to common denominator.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality 
Subtract.Solving Equations with Fractions Using Addition, Subtraction, and Division Properties of Equality 

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.

Optional Video: Solve One Step Equations With Fractions by Adding or Subtracting

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