EQ - Solve Equations with Fractions Lesson

Solve Equations with Fractions

In this lesson, you will learn to solve equations containing fractions. The goal when you are solving equations with fractions is no different than when working with other types of numbers - isolate the variable on one side of the equation.

Example 1

add and subtract Solve equations by adding or subtracting:

$LaTeX: x-\frac{1}{7}=\frac{3}{7}\\ \textcolor{green}{\text{Use the Addition Property of Equality}}\\ x-\frac{1}{7}+\textcolor{green}{\frac{1}{7}}=\frac{3}{7}+\textcolor{green}{\frac{1}{7}}\\ x=\frac{4}{7}\textcolor{green}{\text{ Add.}}$ $x-\frac{1}{7}=\frac{3}{7}\\ \textcolor{green}{\text{Use the Addition Property of Equality}}\\ x-\frac{1}{7}+\textcolor{green}{\frac{1}{7}}=\frac{3}{7}+\textcolor{green}{\frac{1}{7}}\\ x=\frac{4}{7}\textcolor{green}{\text{ Add.}}$

equation image indicator $LaTeX: \frac{3}{14}+y=-\frac{14}{21}\\ \textcolor{green}{\text{Use the Subtraction Property of Equality}}\\ \frac{3}{14}-\textcolor{green}{\frac{3}{14}}+y=-\frac{14}{21}\textcolor{green}{-\frac{3}{14}}\\ x=-\frac{28}{42}-\frac{9}{42}\textcolor{green}{\text{Find a common denominator}}\\ x=-\frac{37}{42}\textcolor{green}{\text{ Subtract (using the rules for integer subtractions- add the opposite).}}$ $\frac{3}{14}+y=-\frac{14}{21}\\ \textcolor{green}{\text{Use the Subtraction Property of Equality}}\\ \frac{3}{14}-\textcolor{green}{\frac{3}{14}}+y=-\frac{14}{21}\textcolor{green}{-\frac{3}{14}}\\ x=-\frac{28}{42}-\frac{9}{42}\textcolor{green}{\text{Find a common denominator}}\\ x=-\frac{37}{42}\textcolor{green}{\text{ Subtract (using the rules for integer subtractions- add the opposite).}}$

Example 2

multiply and divide Solve equations by multiplying or dividing:

Fraction equations often call upon the Multiplicative Inverse Property

Multiplicative Inverse Property
Words	Numbers	Algebra
The product of a nonzero number and its reciprocal is one.	$LaTeX: \frac{4}{5}\cdot\frac{5}{4}=1$ $\frac{4}{5}\cdot\frac{5}{4}=1$	$LaTeX: \frac{a}{b}\cdot\frac{b}{a}=1$ $\frac{a}{b}\cdot\frac{b}{a}=1$

$LaTeX: \frac{1}{3}x=\frac{3}{5} \\ \frac{1}{3}x\cdot \textcolor{green}{\frac{3}{1}}=\frac{3}{5} \cdot \textcolor{green}{\frac{3}{1} \text{Multiply by the reciprocal of} \frac{1}{3}\: \text{then simplify.}}\\ x=\frac{9}{5} or \:1\frac{4}{5}$ $\frac{1}{3}x=\frac{3}{5} \\ \frac{1}{3}x\cdot \textcolor{green}{\frac{3}{1}}=\frac{3}{5} \cdot \textcolor{green}{\frac{3}{1} \text{Multiply by the reciprocal of} \frac{1}{3}\: \text{then simplify.}}\\ x=\frac{9}{5} or \:1\frac{4}{5}$

equation image indicator $LaTeX: \frac{x}{4}=\frac{3}{7} \\ \frac{x}{4}\cdot \textcolor{green}{\frac{4}{1}}=\frac{3}{7} \cdot \textcolor{green}{\frac{4}{1} \:\:\text{Multiply by the reciprocal of} \frac{1}{4}\: \text{then simplify.}}\\ x=\frac{12}{7} or \:1\frac{5}{7}$ $\frac{x}{4}=\frac{3}{7} \\ \frac{x}{4}\cdot \textcolor{green}{\frac{4}{1}}=\frac{3}{7} \cdot \textcolor{green}{\frac{4}{1} \:\:\text{Multiply by the reciprocal of} \frac{1}{4}\: \text{then simplify.}}\\ x=\frac{12}{7} or \:1\frac{5}{7}$

Investigate

Erin needs $LaTeX: \frac{1}{4}$ $\frac{1}{4}$ cup of detergent for her laundry. She has $LaTeX: \frac{5}{8}$ $\frac{5}{8}$ cup of detergent in the bottle. Write and solve an equation to find out how much detergent she will have left in the box.

Strategy

Summarize the problem, using only key words or phrases.

Erin has $LaTeX: \frac{5}{8}$ $\frac{5}{8}$ cup of detergent and needs $LaTeX: \frac{1}{4}$ $\frac{1}{4}$ cup. How much will remain?

Next, translate the words into math. Use a variable to represent the remaining amount of detergent.

Total Detergent is the amount needed for laundry and the amount left.

$LaTeX: \frac{5}{8}=\frac{1}{4}+d$ $\frac{5}{8}=\frac{1}{4}+d$

Solution

$LaTeX: \frac{5}{8}=\frac{1}{4}+d \\ \frac{5}{8}-\textcolor{green}{\frac{1}{4}}=\frac{1}{4}-\textcolor{green}{\frac{1}{4}}+d\\ \frac{5}{8}-\frac{2}{8}=d\\ \frac{3}{8}=d \:\:\text{Erin will have }\frac{3}{8}\: \text{cup of detergent left.}$ $\frac{5}{8}=\frac{1}{4}+d \\ \frac{5}{8}-\textcolor{green}{\frac{1}{4}}=\frac{1}{4}-\textcolor{green}{\frac{1}{4}}+d\\ \frac{5}{8}-\frac{2}{8}=d\\ \frac{3}{8}=d \:\:\text{Erin will have }\frac{3}{8}\: \text{cup of detergent left.}$

Take a look at the video for some great examples of one-step equations using fractions, decimals, and integers!

Now that you have had a chance to see a few more examples, here is an opportunity for you to see if you are ready to move on. Complete the following activity before moving on to your assignments.

Equations with Fractions Practice

Equations with Fractions Homework

Now that you have spent some time learning strategies for solving equations with fractions, you are ready to complete your Equations: Solving Equations with Fractions Homework. Download your homework by CLICKING HERE. Links to an external site.

Once you have completed your homework, AND MAKE SURE YOU ATTEMPTED AND WORKED THE PROBLEMS OUT ON YOUR OWN, click here to download your homework key Links to an external site..

IMAGES CREATED BY GAVS