Solve Equations Using the Division and Multiplication Properties of Equality

Solve equations using the Division and Multiplication Properties of Equality

We introduced the Multiplication and Division Properties of Equality earlier, modeling how these properties worked using envelopes and counters and then applying them to solving equations. We restate them again here as we prepare to use these properties again.

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. Solve: $4x = -28$.

Use the Division Property of Equality to divide both sides by $4$:

Simplify:

Check: substitute $x = -7$ into $4x = -28$:

Since this is a true statement, $x = -7$ is a solution to $4x = -28$.

In that example, to “undo” multiplication, we divided. When the variable is divided by a number, we “undo” division by multiplying.

Example. Solve: $\tfrac{a}{-7} = -42$.

Here $a$ is divided by $-7$. We can multiply both sides by $-7$ to isolate $a$:

Simplify:

Check: substitute $a = 294$ into $\tfrac{a}{-7} = -42$:

The solution checks.

Example. Solve: $-r = 2$.

Remember $-r$ is equivalent to $-1r$. Rewrite $-r$ as $-1r$, then divide both sides by $-1$:

Check: substitute $r = -2$ into $-r = 2$:

We saw earlier that there are two other ways to solve $-r = 2$: we could multiply both sides by $-1$, or we could take the opposite of both sides. All three approaches lead to the same solution.

Example. Solve: $\tfrac{2}{3}x = 18$.

Since the product of a number and its reciprocal is $1$, our strategy will be to isolate $x$ by multiplying by the reciprocal of $\tfrac{2}{3}$. Multiply both sides by $\tfrac{3}{2}$, the reciprocal of $\tfrac{2}{3}$:

Reciprocals multiply to one, so the left side becomes $1x$:

Multiply:

Check: substitute $x = 27$ into $\tfrac{2}{3}x = 18$:

Notice that we could have divided both sides of the equation $\tfrac{2}{3}x = 18$ by $\tfrac{2}{3}$ to isolate $x$. While this would work, multiplying by the reciprocal requires fewer steps.

Solve equations that need to be simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible.

Example. Solve: $8x + 9x - 5x = -3 + 15$.

Start by combining like terms to simplify each side:

Divide both sides by $12$ to isolate $x$:

Check: substitute $x = 1$ into the original equation:

The solution checks.

Sometimes the variable ends up on the right side of the equation after combining like terms.

Example. Solve: $11 - 20 = 17y - 8y - 6y$.

Simplify each side by combining like terms:

Divide both sides by $3$ to isolate $y$:

Check: substitute $y = -3$ into the original equation:

Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side.

Some equations have parentheses that must be distributed before we can combine like terms.

Example. Solve: $-3(n - 2) - 6 = 21$.

Remember — always simplify each side first. Distribute, then combine like terms:

Divide both sides by $-3$ to isolate $n$:

Check: substitute $n = -7$ into the original equation:

The solution checks.

Key terms

Division Property of Equality — for all real numbers $a$, $b$, $c$, and $c \neq 0$, if $a = b$, then $\tfrac{a}{c} = \tfrac{b}{c}$. Multiplication Property of Equality — for all real numbers $a$, $b$, and $c$, if $a = b$, then $ac = bc$.

This section is adapted from Prealgebra 2e, Section 8.2: Solve Equations Using the Division and Multiplication Properties of Equality by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the Be Prepared quiz, Links to Literacy and media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.