### Learning Outcomes

- Solve equations with fractions using the Multiplication Property of Equality

## Solve Equations with Fractions Using the Multiplication Property of Equality

We will now solve equations that require multiplication to isolate the variable. Consider the equation [latex]\Large\frac{x}{4}\normalsize=3[/latex]. We want to know what number divided by [latex]4[/latex] gives [latex]3[/latex]. To “undo” the division, we will need to multiply by [latex]4[/latex]. 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 [latex]a,b[/latex], and [latex]c[/latex],

[latex]\text{if }a=b,\text{ then }ac=bc[/latex].

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 [latex]\Large\frac{x}{7}\normalsize=-9[/latex].

### Example

Solve: [latex]\Large\frac{x}{7}\normalsize=-9[/latex].

### Try It

### Example

Solve: [latex]\Large\frac{p}{-8}\normalsize=-40[/latex]

### Try It

In the following video we show two more examples of when to use the multiplication and division properties to solve a one-step equation.

## Solve Equations with a Coefficient of [latex]-1[/latex]

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

### Example

Solve: [latex]-y=15[/latex]

### Try It

In the next video we show more examples of how to solve an equation with a negative variable.

## 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 [latex]1[/latex].

For example, in the equation:

[latex]\Large\frac{3}{4}\normalsize x=24[/latex]

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

### Example

Solve: [latex]\Large\frac{3}{4}\normalsize x=24[/latex]

### Try It

### Example

Solve: [latex]-\Large\frac{3}{8}\normalsize w=72[/latex]

### Try It

In the next video example you will see another example of how to use the reciprocal of a fractional coefficient to solve an equation.