4.7 Solve equations with fractions  (Page 2/5)

We used the Subtraction Property of Equality in [link] . Now we’ll use the Addition Property of Equality.

Solve: $a-\frac{5}{9}=-\frac{8}{9}.$

Solution

Add $\frac{5}{9}$ from each side to undo the addition.
Simplify on each side of the equation.
Simplify the fraction.
Check:
Substitute $a=-\frac{1}{3}$ .
Change to common denominator.
Subtract.

Since $a=-\frac{1}{3}$ makes the equation true, we know that $a=-\frac{1}{3}$ is the solution to the equation.

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The next example may not seem to have a fraction, but let’s see what happens when we solve it.

Solve equations with fractions using the multiplication property of equality

Consider the equation $\frac{x}{4}=3.$ We want to know what number divided by $4$ gives $3.$ So to “undo” the division, we will need to multiply by $4.$ 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.

Let’s use the Multiplication Property of Equality to solve the equation $\frac{x}{7}=\mathrm{-9}.$

Solve: $\frac{x}{7}=\mathrm{-9}.$

Solution

Use the Multiplication Property of Equality to multiply both sides by $7$ . This will isolate the variable.
Multiply.
Simplify.
The equation is true.

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Solve: $\frac{p}{\mathrm{-8}}=\mathrm{-40}.$

Solution

Here, $p$ is divided by $\mathrm{-8}.$ We must multiply by $\mathrm{-8}$ to isolate $p.$

Multiply both sides by $\mathrm{-8}$
Multiply.
Simplify.
Check:
Substitute $p=320$ .
The equation is true.

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Solve equations with a coefficient of $\mathrm{-1}$

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

Solve: $-y=15.$

Solution

One way to solve the equation is to rewrite $-y$ as $\mathrm{-1}y,$ and then use the Division Property of Equality to isolate $y.$

Rewrite $-y$ as $\mathrm{-1}y$ .
Divide both sides by −1.
Simplify each side.

Another way to solve this equation is to multiply both sides of the equation by $\mathrm{-1}.$

Multiply both sides by −1.
Simplify each side.

The third way to solve the equation is to read $-y$ as “the opposite of $y\text{.”}$ What number has $15$ as its opposite? The opposite of $15$ is $\mathrm{-15}.$ So $y=\mathrm{-15}.$

For all three methods, we isolated $y$ is isolated and solved the equation.

Check:

Substitute $y=\mathrm{-15}$ .
Simplify. The equation is true.

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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 $1.$

For example, in the equation: