We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > Mathematics for the Liberal Arts Corequisite

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 a,ba,b, and cc,

if a=b, then ac=bc\text{if }a=b,\text{ then }\Large\frac{a}{c}=\Large\frac{b}{c}.

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 1010 to get 4444, and we can use division to find out.

Example

Solve: 10q=4410q=44 Solution:
10q=4410q=44
Divide both sides by 1010 to undo the multiplication. 10q10=4410\Large\frac{10q}{10}=\Large\frac{44}{10}
Simplify. q=225q=\Large\frac{22}{5}
Check:
Substitute q=225q=\Large\frac{22}{5} into the original equation. 10(225)=?4410\left(\Large\frac{22}{5}\right)\stackrel{?}{=}44
Simplify. )102(22)5)=?44\stackrel{2}{\overline{)10}}\left(\Large\frac{22}{\overline{)5}}\right)\stackrel{?}{=}44
Multiply. 44=4444=44\quad\checkmark
The solution to the equation was the fraction 225\Large\frac{22}{5}. We leave it as an improper fraction.

Try It

[ohm_question]146141[/ohm_question]
Now, consider the equation x4=3\Large\frac{x}{4}\normalsize=3. We want to know what number divided by 44 gives 33. So to "undo" the division, we will need to multiply by 44. 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 a,ba,b, and cc,

if a=b, then ac=bc\text{if }a=b,\text{ then }ac=bc.

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 x7=9\Large\frac{x}{7}\normalsize=-9.

Example

Solve: x7=9\Large\frac{x}{7}\normalsize=-9.

Answer: Solution:

x7=9\Large\frac{x}{7}\normalsize=-9
Use the Multiplication Property of Equality to multiply both sides by 77 . This will isolate the variable. 7x7=7(9)\color{red}{7}\cdot\Large\frac{x}{7}\normalsize=\color{red}{7}(-9)
Multiply. 7x7=63\Large\frac{7x}{7}\normalsize=-63
Simplify. x=63x=-63
Check. Substitute 63\color{red}{-63} for x in the original equation. 637=?9\Large\frac{\color{red}{-63}}{7}\normalsize\stackrel{?}{=}-9
The equation is true. 9=9-9=-9\quad\checkmark

Try It

[ohm_question]146147[/ohm_question]

Example

Solve: p8=40\Large\frac{p}{-8}\normalsize=-40

Answer: Solution: Here, pp is divided by 8-8. We must multiply by 8-8 to isolate pp.

p8=40\Large\frac{p}{-8}\normalsize=-40
Multiply both sides by 8-8 8(p8)=8(40)\color{red}{-8}\Large(\frac{p}{-8})\normalsize=\color{red}{-8}(-40)
Multiply. 8p8=320\Large\frac{-8p}{-8}\normalsize=320
Simplify. p=320p=320
Check:
Substitute p=320p=320 . 3208=?40\Large\frac{\color{red}{320}}{-8}\normalsize\stackrel{?}{=}-40
The equation is true. 40=40-40=-40\quad\checkmark

Try It

[ohm_question]146336[/ohm_question]
In the following video we show two more examples of when to use the multiplication and division properties to solve a one-step equation. https://youtu.be/BN7iVWWl2y0

Solve Equations with a Coefficient of 1-1

Look at the equation y=15-y=15. Does it look as if yy is already isolated? But there is a negative sign in front of yy, 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: y=15-y=15

Answer: Solution: One way to solve the equation is to rewrite y-y as 1y-1y, and then use the Division Property of Equality to isolate yy.

y=15-y=15
Rewrite y-y as 1y-1y . 1y=15-1y = 15
Divide both sides by 1−1. 1y1=151\Large\frac{-1y}{\color{red}{-1}}=\Large\frac{15}{\color{red}{-1}}
Simplify each side. y=15y = -15
Another way to solve this equation is to multiply both sides of the equation by 1-1.
y=15y=15
Multiply both sides by 1−1. 1(y)=1(15)\color{red}{-1}(-y)=\color{red}{-1}(15)
Simplify each side. y=15y=-15
The third way to solve the equation is to read y-y as "the opposite of y."y\text{."} What number has 1515 as its opposite? The opposite of 1515 is 15-15. So y=15y=-15. For all three methods, we isolated yy is isolated and solved the equation. Check:
y=15y=15
Substitute y=15y=-15 . (15)=?(15)-(\color{red}{-15})\stackrel{?}{=}(15)
Simplify. The equation is true. 15=1515=15\quad\checkmark

Try It

[ohm_question]146153[/ohm_question]
In the next video we show more examples of how to solve an equation with a negative variable. https://youtu.be/FJmNpHeOcpo

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 11. For example, in the equation:

34x=24\Large\frac{3}{4}\normalsize x=24

The coefficient of xx is 34\Large\frac{3}{4}. To solve for xx, we need its coefficient to be 11. Since the product of a number and its reciprocal is 11, our strategy here will be to isolate xx by multiplying by the reciprocal of 34\Large\frac{3}{4}. We will do this in the next example.

Example

Solve: 34x=24\Large\frac{3}{4}\normalsize x=24

Answer: Solution:

34x=24\Large\frac{3}{4}\normalsize x = 24
Multiply both sides by the reciprocal of the coefficient. 4334x=4324\color{red}{\Large\frac{4}{3}}\cdot\Large\frac{3}{4}\normalsize x = \color{red}{\Large\frac{4}{3}}\cdot\normalsize 24
Simplify. 1x=432411x=\Large\frac{4}{3}\cdot\Large\frac{24}{1}
Multiply. x=32x=32
Check: 34x=24\Large\frac{3}{4}\normalsize x=24
Substitute x=32x=32 . 3432=?24\Large\frac{3}{4}\normalsize\cdot 32\stackrel{?}{=}24
Rewrite 3232 as a fraction. 34321=?24\Large\frac{3}{4}\cdot\Large\frac{32}{1}\stackrel{?}{=}24
Multiply. The equation is true. 24=2424=24\quad\checkmark
Notice that in the equation 34x=24\Large\frac{3}{4}\normalsize x=24, we could have divided both sides by 34\Large\frac{3}{4} to get xx 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.

Try It

[ohm_question]146156[/ohm_question]

Example

Solve: 38w=72-\Large\frac{3}{8}\normalsize w=72

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.

38w=72\Large\frac{3}{8}\normalsize w=72
Multiply both sides by the reciprocal of 38-\Large\frac{3}{8} . 83(38w)=(83)72\color{red}{-\Large\frac{8}{3}}\Large(-\frac{3}{8}\normalsize w\Large)=(\color{red}{-\Large\frac{8}{3}})\normalsize 72
Simplify; reciprocals multiply to one. 1w=837211w=-\Large\frac{8}{3}\cdot\Large\frac{72}{1}
Multiply. w=192w=-192
Check: 38w=72-\Large\frac{3}{8}\normalsize w=72
Let w=192w=-192 . 38(192)=?72-\Large\frac{3}{8}\normalsize(-192)\stackrel{?}{=}72
Multiply. It checks. 72=7272=72\quad\checkmark

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/Ea5eW8rZxEI

Licenses & Attributions