8.2 Solve equations using the division and multiplication properties

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.

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.

Solve: $4x=\mathrm{-28}.$

Solution

We use the Division Property of Equality to divide both sides by $4.$

Divide both sides by 4 to undo the multiplication.
Simplify.
Check your answer. Let $x=\mathrm{-7}$ .

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

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In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?

Solve: $\frac{a}{\mathrm{-7}}=\mathrm{-42}.$

Solution

Here $a$ is divided by $\mathrm{-7}.$ We can multiply both sides by $\mathrm{-7}$ to isolate $a.$

Multiply both sides by $\mathrm{-7}$ .
Simplify.
Check your answer. Let $a=294$ .

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Solve: $-r=2.$

Solution

Remember $-r$ is equivalent to $\mathrm{-1}r.$

Rewrite $-r$ as $\mathrm{-1}r$ .
Divide both sides by $\mathrm{-1}$ .
Check.
Substitute $r=\mathrm{-2}$
Simplify.

In Solve Equations with Fractions , we saw that there are two other ways to solve $-r=2.$

We could multiply both sides by $\mathrm{-1}.$

We could take the opposite of both sides.

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

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 $\frac{2}{3}.$

Multiply by the reciprocal of $\frac{2}{3}$ .
Reciprocals multiply to one.
Multiply.
Check your answer. Let $x=27$

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

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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