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.
Let’s put this idea in practice with an example. We are looking for the number you multiply by [latex]10[/latex] to get [latex]44[/latex], and we can use division to find out.
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] |
|
The solution to the equation was the fraction [latex]\Large\frac{22}{5}[/latex]. We leave it as an improper fraction.
Now, consider the equation [latex]\Large\frac{x}{4}\normalsize=3[/latex]. We want to know what number divided by [latex]4[/latex] gives [latex]3[/latex]. So to “undo” the division, we will need to multiply by [latex]4[/latex]. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.
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.
Let’s use the Multiplication Property of Equality to solve the equation [latex]\Large\frac{x}{7}\normalsize=-9[/latex].
Example
Solve: [latex]\Large\frac{x}{7}\normalsize=-9[/latex].
Show Solution
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] |
|
Example
Solve: [latex]\Large\frac{p}{-8}\normalsize=-40[/latex]
Show Solution
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] |
|
In the following video we show two more examples of when to use the multiplication and division properties to solve a one-step equation.
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]
Show Solution
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] |
Another way to solve this equation is to multiply both sides of the equation by [latex]-1[/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] |
The third way to solve the equation is to read [latex]-y[/latex] as “the opposite of [latex]y\text{.''}[/latex] What number has [latex]15[/latex] as its opposite? The opposite of [latex]15[/latex] is [latex]-15[/latex]. So [latex]y=-15[/latex].
For all three methods, we isolated [latex]y[/latex] is isolated and solved the equation.
Check:
|
[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] |
In the next video we show more examples of how to solve an equation with a negative variable.
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]
Show Solution
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] |
|
Notice that in the equation [latex]\Large\frac{3}{4}\normalsize x=24[/latex], we could have divided both sides by [latex]\Large\frac{3}{4}[/latex] to get [latex]x[/latex] by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.
Example
Solve: [latex]-\Large\frac{3}{8}\normalsize w=72[/latex]
Show Solution
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] |
|
In the next video example you will see another example of how to use the reciprocal of a fractional coefficient to solve an equation.
