2.2 Solving Equations Using the Multiplication Principle

Scott McDaniel

2.2 Solving Equations Using the Multiplication Principle

Learning Objectives

After completing this section, you will be able to:

Solve equations using the Multiplication Property of Equality and the Division Property of Equality.
Handle equations with fractional or negative coefficients.
Translate verbal statements involving products and quotients into equations.
Check solutions by substitution.

Launch – Balancing by Scaling

In Section 2.1, you learned how to keep equations balanced by adding or subtracting the same number on both sides.

Now let’s return to the idea of a balance scale.

A black-and-white illustration of a balance scale with two pans. The left pan holds a single 5-unit weight, and the right pan holds two weights labeled 3 and 2. The scale is level, showing both sides are balanced and equal in weight. — **Figure 2.2A** (“Addition Scale” by Florida Center for Instructional Technology)

Because both sides have the same total weight, the scale is balanced. This situation represents the equation

$5 = 3 + 2$

If both sides of a scale are equal, then doubling or halving each side keeps the scale balanced. For example, if we multiply the left side by 2, then to keep the scale balanced, we must also multiply the right side by 2.

Balanced scale with two 5-unit weights on the left and two 3-unit weights plus two 2-unit weights on the right, showing equal totals on both sides. — **Figure 2.2B**

Multiply both sides by 2:

$2\cdot 5 \;=\; 2\cdot(3 + 2)$

The equality is preserved because both sides are multiplied by the same number.

This idea is called the Multiplication Property of Equality.

Multiplication and Division Properties of Equality

You may multiply or divide both sides of an equation by the same nonzero number without changing the solution.

This property allows us to clear fractions, isolate variables, and simplify equations while keeping them balanced—just like the scale.

The Multiplication and Division Properties of Equality

Multiplication Property of Equality

If $a = b$ , then $a \cdot c = b \cdot c$ .

Division Property of Equality

If $a = b$ and $c \ne 0$ , then $\frac{a}{c} = \frac{b}{c}$ .

These properties allow us to “undo” multiplication or division in order to isolate the variable.

Example 2.2.1

Solve for $x$ : $\frac{x}{5} = 3$

Solution

Multiply both sides by 5:

$x = 15$

✅ Check: $15 \div 5 = 3$

Example 2.2.2

Solve for $y$ : $6y = 30$

Solution

Divide both sides by 6:

$y = 5$

✅ Check: $6 \cdot 5 = 30$

Example 2.2.3

Solve for $n$ : $-4n = 12$

Solution

Divide both sides by –4:

$n = -3$

✅ Check: $-4(-3) = 12$

Example 2.2.4

Solve for $x$ : $\frac{1}{3}x = 9$

Solution

Multiply both sides by 3 (the reciprocal of $\frac{1}{3}$ ):

$x = 27$

✅ Check: $\frac{1}{3}(27) = 9$

Example 2.2.5

Solve for $p$ : $\frac{p}{8} = -5$

Solution

Multiply both sides by 8:

$p = -40$

✅ Check: $-40 \div 8 = -5$

Example 2.2.6

Solve for $t$ : $\frac{-t}{6} = 2$

Solution

Multiply both sides by –6:

$t = -12$

✅ Check: If $t = -12$ , then $\frac{-(-12)}{6} = \frac{12}{6} = 2$ ✓

Final Answer: $t = -12$

Note: As shown in the video above, we could have also done this one in 2 steps: (1) Multiply both sides by 6; (2) divide (or multiply) both sides by -1.

Example 2.2.7

Solve for $a$ : $-\frac{3}{4}a = 9$

Solution

Multiply both sides by $-\frac{4}{3}$ (the reciprocal of $-\frac{3}{4}$ ):

$a = 9\left(-\frac{4}{3}\right) = -12$

✅ Check: $-\frac{3}{4}(-12) = 9$

Common Mistakes

🚩 Multiplying or dividing only one side of the equation.

🚩 Forgetting to change the sign when dividing by a negative number.

🚩 Multiplying by the wrong reciprocal.

🚩 Dividing by 0 (undefined).

Try It

Translating Products and Quotients into Equations

Sometimes equations are written in words rather than symbols. To solve them, first translate the words into algebra.

Key Vocabulary

Product → multiplication

Keywords: times, twice, double, multiplied by, the product of

Quotient → division

Keywords: divided by, the quotient of, per, ratio of

When translating, identify key words and use a variable such as $x$ , $y$ , $n$ , or $k$ .

Example 2.2.8

Translate and solve: “The product of $x$ and 6 is 54.”

Solution

Translation: $6x = 54$

Divide both sides by 6:

$x = 9$

✅ Check: $6 \cdot 9 = 54$

Example 2.2.9

Translate and solve: “The quotient of $y$ and 5 is –7.”

Solution

Translation: $\frac{y}{5} = -7$

Multiply both sides by 5:

$y = -35$

✅ Check: $-35 \div 5 = -7$

Example 2.2.10

Translate and solve: “The product of –4 and $n$ is 20.”

Solution

Translation: $-4n = 20$

Divide both sides by –4:

$n = -5$

✅ Check: $-4(-5) = 20$

Example 2.2.11

Translate and solve: “The product of $\frac{1}{2}$ and $m$ equals 9.”

Solution

Translation: $\frac{1}{2}m = 9$

Multiply both sides by 2:

$m = 18$

✅ Check: $\frac{1}{2}(18) = 9$

Example 2.2.12

Translate and solve: “The quotient when $k$ is divided by –9 is 4.”

Solution

Translation: $\frac{k}{-9} = 4$

Multiply both sides by –9:

$k = -36$

✅ Check: $-36 \div (-9) = 4$

Try It

Key Takeaways

Multiplying or dividing both sides of an equation by the same non-zero number keeps it balanced.
The reciprocal of a coefficient can be used to isolate the variable.
“Product” means multiplication; “quotient” means division.
Translate verbal statements carefully and check solutions by substitution.
Watch for negative signs when multiplying or dividing.

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Launch – Balancing by Scaling

Multiplication and Division Properties of Equality

The Multiplication and Division Properties of Equality

Multiplication Property of Equality

Division Property of Equality

Common Mistakes

Translating Products and Quotients into Equations

Key Vocabulary

License

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