Solve Equations Using the Division and Multiplication Properties of Equality
We introduced the Multiplication and Division Properties of Equality in Solve Equations Using Integers; The Division Property of Equality and Solve Equations with Fractions. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Solve Equations Using Integers; The Division Property of Equality). We restate them again here as we prepare to use these properties again.
Definition: Division and Multiplication Properties of Equality
Division Property of Equality: For all real numbers a, b, c, and c ≠ 0, if a = b, then \(\dfrac{a}{c} = \dfrac{b}{c}\).
Multiplication Property of Equality: For all real numbers a, b, c, if a = b, then ac = bc.
When you divide or multiply both sides of an equation by the same quantity, you still have equality.
Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by 4, so we will divide both sides by 4 to ‘undo’ the multiplication.
Example \(\PageIndex{1}\):
Solve: 4x = −28.
Solution
We use the Division Property of Equality to divide both sides by 4.
Divide both sides by 4 to undo the multiplication. 
$$\dfrac{4x}{\textcolor{red}{4}} = \dfrac{28}{\textcolor{red}{4}}$$ 
Simplify. 
$$x = 7$$ 
Check your answer. Let x = −7. 
$$\begin{split} 4x &= 28 \\ 4(\textcolor{red}{7}) &\stackrel{?}{=} 28 \\ 28 &= 28\; \checkmark \end{split}$$ 
Since this is a true statement, x = −7 is a solution to 4x = −28.
Exercise \(\PageIndex{1}\):
Solve: 3y = −48.
 Answer

y = 16
Exercise \(\PageIndex{2}\):
Solve: 4z = −52.
 Answer

z = 13
In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?
Example \(\PageIndex{2}\):
Solve: \(\dfrac{a}{−7}\) = −42.
Solution
Here a is divided by −7. We can multiply both sides by −7 to isolate a.
Multiply both sides by −7. 
$$\textcolor{red}{7} \left(\dfrac{a}{7}\right) = \textcolor{red}{7} (42)$$ 
Simplify. 
$$a = 294$$ 
Check your answer. Let a = 294. 
$$\begin{split} \dfrac{a}{7} &= 42 \\ \dfrac{\textcolor{red}{294}}{7} &\stackrel{?}{=} 42 \\ 42 &= 42\; \checkmark \end{split}$$ 
Exercise \(\PageIndex{3}\):
Solve: \(\dfrac{b}{−6}\) = −24.
 Answer

b = 144
Exercise \(\PageIndex{4}\):
Solve: \(\dfrac{c}{−8}\) = −16.
 Answer

c = 128
Example \(\PageIndex{3}\):
Solve: −r = 2.
Solution
Remember −r is equivalent to −1r.
Rewrite −r as −1r. 
$$1r = 2$$ 
Divide both sides by −1. 
$$\dfrac{1r}{\textcolor{red}{1}} = \dfrac{2}{\textcolor{red}{1}}$$ 
Check.
Substitute r = −2. 
$$r = 2$$ 
Simplify. 
$$\begin{split} (\textcolor{red}{2}) &\stackrel{?}{=} 2 \\ 2 &= 2\; \checkmark \end{split}$$ 
In Solve Equations with Fractions, we saw that there are two other ways to solve −r = 2.
 We could multiply both sides by −1.
 We could take the opposite of both sides.
Exercise \(\PageIndex{5}\):
Solve: −k = 8.
 Answer

k = 8
Exercise \(\PageIndex{6}\):
Solve: −g = 3.
 Answer

g = 3
Example \(\PageIndex{4}\):
Solve: \(\dfrac{2}{3}\)x = 18.
Solution
Since the product of a number and its reciprocal is 1, our strategy will be to isolate x by multiplying by the reciprocal of \(\dfrac{2}{3}\).
Multiply by the reciprocal of \(\dfrac{2}{3}\). 
$$\textcolor{red}{\dfrac{3}{2}} \cdot \dfrac{2}{3} x = \textcolor{red}{\dfrac{3}{2}} \cdot 18$$ 
Reciprocals multiply to one. 
$$1x = \dfrac{3}{2} \cdot \dfrac{18}{1}$$ 
Multiply. 
$$x = 27$$ 
Check your answer. Let x = 27. 
$$\begin{split} \dfrac{2}{3} x &= 18 \\ \dfrac{2}{3} \cdot \textcolor{red}{27} &\stackrel{?}{=} 18 \\ 18 &= 18\; \checkmark \end{split}$$ 
Notice that we could have divided both sides of the equation \(\dfrac{2}{3}\)x = 18 by \(\dfrac{2}{3}\) to isolate x. While this would work, multiplying by the reciprocal requires fewer steps.
Exercise \(\PageIndex{7}\):
Solve: \(\dfrac{2}{5}\)n = 14.
 Answer

n = 35
Exercise \(\PageIndex{8}\):
Solve: \(\dfrac{5}{6}\)y = 15.
 Answer

y = 18
Practice Makes Perfect
Solve Equations Using the Division and Multiplication Properties of Equality
In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.
 8x = 32
 7p = 63
 −5c = 55
 −9x = −27
 −90 = 6y
 −72 = 12y
 −16p = −64
 −8m = −56
 0.25z = 3.25
 0.75a = 11.25
 −3x = 0
 4x = 0
In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.
 \(\dfrac{x}{4}\) = 15
 \(\dfrac{z}{2}\) = 14
 −20 = \(\dfrac{q}{−5}\)
 \(\dfrac{c}{−3}\) = −12
 \(\dfrac{y}{9}\) = −6
 \(\dfrac{q}{6}\) = −8
 \(\dfrac{m}{−12}\) = 5
 −4 = \(\dfrac{p}{−20}\)
 \(\dfrac{2}{3}\)y = 18
 \(\dfrac{3}{5}\)r = 15
 \(− \dfrac{5}{8}\)w = 40
 24 = \(− \dfrac{3}{4}\)x
 \(− \dfrac{2}{5} = \dfrac{1}{10} a\)
 \(− \dfrac{1}{3} q = − \dfrac{5}{6}\)
Solve Equations That Need to be Simplified
In the following exercises, solve the equation.
 8a + 3a − 6a = −17 + 27
 6y − 3y + 12y = −43 + 28
 −9x − 9x + 2x = 50 − 2
 −5m + 7m − 8m = −6 + 36
 100 − 16 = 4p − 10p − p
 −18 − 7 = 5t − 9t − 6t
 \(\dfrac{7}{8} n − \dfrac{3}{4} n\) = 9 + 2
 \(\dfrac{5}{12} q + \dfrac{1}{2} q\) = 25 − 3
 0.25d + 0.10d = 6 − 0.75
 0.05p − 0.01p = 2 + 0.24
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) After reviewing this checklist, what will you do to become confident for all objectives?