Mathematics » Introducing Fractions » Solve Equations with Fractions

# Solving Equations with Fractions Using the Multiplication Property of Equality

## Solving 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.

### The Multiplication Property of Equality

For any numbers $$a,b,$$ and $$c,$$

$$\text{if}\phantom{\rule{0.2em}{0ex}}a=b,\text{then}\phantom{\rule{0.2em}{0ex}}ac=bc.$$

If you multiply both sides of an equation by the same quantity, you still have equality.

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

## Example

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

### Solution

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

## Example

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

### Solution

Here, $$p$$ is divided by $$-8.$$ We must multiply by $$-8$$ to isolate $$p.$$

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

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 the example below.

## Example

Solve: $$-y=15.$$

### Solution

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

 Rewrite $$-y$$ as $$-1y$$. Divide both sides by −1. Simplify each side.

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

 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 $$-15.$$ So $$y=-15.$$

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

Check:

 Substitute $$y=-15$$. Simplify. The equation is true.

### Solving 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:

$$\frac{3}{4}x=24$$

The coefficient of $$x$$ is $$\frac{3}{4}.$$ To solve for $$x,$$ we need its coefficient to be $$1.$$ Since the product of a number and its reciprocal is $$1,$$ our strategy here will be to isolate $$x$$ by multiplying by the reciprocal of $$\frac{3}{4}.$$ We will do this in the example below.

## Example

Solve: $$\frac{3}{4}x=24.$$

### Solution

 Multiply both sides by the reciprocal of the coefficient. Simplify. Multiply. Check: Substitute $$x=32$$. Rewrite $$32$$ as a fraction. Multiply. The equation is true.

Notice that in the equation $$\frac{3}{4}x=24,$$ we could have divided both sides by $$\frac{3}{4}$$ to get $$x$$ by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.

## Example

Solve: $$-\frac{3}{8}w=72.$$

### Solution

The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.

 Multiply both sides by the reciprocal of $$-\frac{3}{8}$$. Simplify; reciprocals multiply to one. Multiply. Check: Let $$w=-192$$. Multiply. It checks.