Solving Equations That Contain Fractions Using the Multiplication Property of Equality
Learning Outcomes
- Use the multiplication and division properties to solve equations with fractions and division
Solve Equations with Fractions Using the Multiplication and Division Properties of Equality
We will solve equations that require multiplication or division to isolate the variable. First, let's consider the division property of equality again.The Division Property of Equality
For any numbers [latex]a,b[/latex], and [latex]c[/latex],[latex]\text{if }a=b,\text{ then }\Large\frac{a}{c}=\Large\frac{b}{c}[/latex].
If you divide both sides of an equation by the same quantity, you still have equality.Example
Solve: [latex]10q=44[/latex] Solution:| [latex]10q=44[/latex] | ||
| Divide both sides by [latex]10[/latex] to undo the multiplication. | [latex]\Large\frac{10q}{10}=\Large\frac{44}{10}[/latex] | |
| Simplify. | [latex]q=\Large\frac{22}{5}[/latex] | |
| Check: | ||
| Substitute [latex]q=\Large\frac{22}{5}[/latex] into the original equation. | [latex]10\left(\Large\frac{22}{5}\right)\stackrel{?}{=}44[/latex] | |
| Simplify. | [latex]\stackrel{2}{\overline{)10}}\left(\Large\frac{22}{\overline{)5}}\right)\stackrel{?}{=}44[/latex] | |
| Multiply. | [latex]44=44\quad\checkmark[/latex] | |
Try It
[ohm_question]146141[/ohm_question]The Multiplication Property of Equality
For any numbers [latex]a,b[/latex], and [latex]c[/latex],[latex]\text{if }a=b,\text{ then }ac=bc[/latex].
If you multiply both sides of an equation by the same quantity, you still have equality.Example
Solve: [latex]\Large\frac{x}{7}\normalsize=-9[/latex].Answer: Solution:
| [latex]\Large\frac{x}{7}\normalsize=-9[/latex] | ||
| Use the Multiplication Property of Equality to multiply both sides by [latex]7[/latex] . This will isolate the variable. | [latex]\color{red}{7}\cdot\Large\frac{x}{7}\normalsize=\color{red}{7}(-9)[/latex] | |
| Multiply. | [latex]\Large\frac{7x}{7}\normalsize=-63[/latex] | |
| Simplify. | [latex]x=-63[/latex] | |
| Check. Substitute [latex]\color{red}{-63}[/latex] for x in the original equation. | [latex]\Large\frac{\color{red}{-63}}{7}\normalsize\stackrel{?}{=}-9[/latex] | |
| The equation is true. | [latex]-9=-9\quad\checkmark[/latex] | |
Try It
[ohm_question]146147[/ohm_question]Example
Solve: [latex]\Large\frac{p}{-8}\normalsize=-40[/latex]Answer: Solution: Here, [latex]p[/latex] is divided by [latex]-8[/latex]. We must multiply by [latex]-8[/latex] to isolate [latex]p[/latex].
| [latex]\Large\frac{p}{-8}\normalsize=-40[/latex] | ||
| Multiply both sides by [latex]-8[/latex] | [latex]\color{red}{-8}\Large(\frac{p}{-8})\normalsize=\color{red}{-8}(-40)[/latex] | |
| Multiply. | [latex]\Large\frac{-8p}{-8}\normalsize=320[/latex] | |
| Simplify. | [latex]p=320[/latex] | |
| Check: | ||
| Substitute [latex]p=320[/latex] . | [latex]\Large\frac{\color{red}{320}}{-8}\normalsize\stackrel{?}{=}-40[/latex] | |
| The equation is true. | [latex]-40=-40\quad\checkmark[/latex] | |
Try It
[ohm_question]146336[/ohm_question]Solve Equations with a Coefficient of [latex]-1[/latex]
Look at the equation [latex]-y=15[/latex]. Does it look as if [latex]y[/latex] is already isolated? But there is a negative sign in front of [latex]y[/latex], so it is not isolated. There are three different ways to isolate the variable in this type of equation. We will show all three ways in the next example.Example
Solve: [latex]-y=15[/latex]Answer: Solution: One way to solve the equation is to rewrite [latex]-y[/latex] as [latex]-1y[/latex], and then use the Division Property of Equality to isolate [latex]y[/latex].
| [latex]-y=15[/latex] | |
| Rewrite [latex]-y[/latex] as [latex]-1y[/latex] . | [latex]-1y = 15[/latex] |
| Divide both sides by [latex]−1[/latex]. | [latex]\Large\frac{-1y}{\color{red}{-1}}=\Large\frac{15}{\color{red}{-1}}[/latex] |
| Simplify each side. | [latex]y = -15[/latex] |
| [latex]y=15[/latex] | |
| Multiply both sides by [latex]−1[/latex]. | [latex]\color{red}{-1}(-y)=\color{red}{-1}(15)[/latex] |
| Simplify each side. | [latex]y=-15[/latex] |
| [latex]y=15[/latex] | |
| Substitute [latex]y=-15[/latex] . | [latex]-(\color{red}{-15})\stackrel{?}{=}(15)[/latex] |
| Simplify. The equation is true. | [latex]15=15\quad\checkmark[/latex] |
Try It
[ohm_question]146153[/ohm_question]Solve Equations with a Fraction Coefficient
When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to [latex]1[/latex]. For example, in the equation:[latex]\Large\frac{3}{4}\normalsize x=24[/latex]
The coefficient of [latex]x[/latex] is [latex]\Large\frac{3}{4}[/latex]. To solve for [latex]x[/latex], we need its coefficient to be [latex]1[/latex]. Since the product of a number and its reciprocal is [latex]1[/latex], our strategy here will be to isolate [latex]x[/latex] by multiplying by the reciprocal of [latex]\Large\frac{3}{4}[/latex]. We will do this in the next example.Example
Solve: [latex]\Large\frac{3}{4}\normalsize x=24[/latex]Answer: Solution:
| [latex]\Large\frac{3}{4}\normalsize x = 24[/latex] | ||
| Multiply both sides by the reciprocal of the coefficient. | [latex]\color{red}{\Large\frac{4}{3}}\cdot\Large\frac{3}{4}\normalsize x = \color{red}{\Large\frac{4}{3}}\cdot\normalsize 24[/latex] | |
| Simplify. | [latex]1x=\Large\frac{4}{3}\cdot\Large\frac{24}{1}[/latex] | |
| Multiply. | [latex]x=32[/latex] | |
| Check: | [latex]\Large\frac{3}{4}\normalsize x=24[/latex] | |
| Substitute [latex]x=32[/latex] . | [latex]\Large\frac{3}{4}\normalsize\cdot 32\stackrel{?}{=}24[/latex] | |
| Rewrite [latex]32[/latex] as a fraction. | [latex]\Large\frac{3}{4}\cdot\Large\frac{32}{1}\stackrel{?}{=}24[/latex] | |
| Multiply. The equation is true. | [latex]24=24\quad\checkmark[/latex] | |
Try It
[ohm_question]146156[/ohm_question]Example
Solve: [latex]-\Large\frac{3}{8}\normalsize w=72[/latex]Answer: Solution: The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.
| [latex]\Large\frac{3}{8}\normalsize w=72[/latex] | ||
| Multiply both sides by the reciprocal of [latex]-\Large\frac{3}{8}[/latex] . | [latex]\color{red}{-\Large\frac{8}{3}}\Large(-\frac{3}{8}\normalsize w\Large)=(\color{red}{-\Large\frac{8}{3}})\normalsize 72[/latex] | |
| Simplify; reciprocals multiply to one. | [latex]1w=-\Large\frac{8}{3}\cdot\Large\frac{72}{1}[/latex] | |
| Multiply. | [latex]w=-192[/latex] | |
| Check: | [latex]-\Large\frac{3}{8}\normalsize w=72[/latex] | |
| Let [latex]w=-192[/latex] . | [latex]-\Large\frac{3}{8}\normalsize(-192)\stackrel{?}{=}72[/latex] | |
| Multiply. It checks. | [latex]72=72\quad\checkmark[/latex] | |
Try It
[ohm_question]146158[/ohm_question]In the next video example you will see another example of how to use the reciprocal of a fractional coefficient to solve an equation.
https://youtu.be/Ea5eW8rZxEILicenses & Attributions
CC licensed content, Original
- Solving One Step Equations Using Multiplication and Division (Basic). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex: Solving One Step Equation in the Form: -x = a. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Ex: Solve a One Step Equation by Multiplying by Reciprocal (a/b)x=-c. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Question ID: 146141, 146147, 146336, 146153, 146156, 146158. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
