Using the Division and Multiplication Properties of Equality for Single-Step Equations

Learning Outcomes

  • Review and use the division and multiplication properties of equality to solve linear equations
  • Use a reciprocal to solve a linear equation that contains fractions

Let’s review the Division and Multiplication Properties of Equality as we prepare to use them to solve single-step equations.

Division Property of Equality

For all real numbers [latex]a,b,c[/latex], and [latex]c\ne 0[/latex], if [latex]a=b[/latex], then [latex]\Large\frac{a}{c}\normalsize =\Large\frac{b}{c}[/latex].

Multiplication Property of Equality

For all real numbers [latex]a,b,c[/latex], if [latex]a=b[/latex], then [latex]ac=bc[/latex].

Stated simply, 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 [latex]4[/latex], so we will divide both sides by [latex]4[/latex] to “undo” the multiplication.

example

Solve: [latex]4x=-28[/latex]

Solution:

To solve this equation, we use the Division Property of Equality to divide both sides by [latex]4[/latex].

[latex]4x=-28[/latex]
Divide both sides by 4 to undo the multiplication. [latex]\Large\frac{4x}{\color{red}4}\normalsize =\Large\frac{-28}{\color{red}4}[/latex]
Simplify. [latex]x =-7[/latex]
Check your answer. [latex]4x=-28[/latex]
Let [latex]x=-7[/latex]. Substitute [latex]-7[/latex] for x. [latex]4(\color{red}{-7})\stackrel{\text{?}}{=}-28[/latex]
 [latex]-28=-28[/latex]

Since this is a true statement, [latex]x=-7[/latex] is a solution to [latex]4x=-28[/latex].

Now you can try to solve an equation that requires division and includes negative numbers.

try it

In the previous example, to “undo” multiplication, we divided. How do you think we “undo” division? Next, we will show an example that requires us to use multiplication to undo division.

example

Solve: [latex]\Large\frac{a}{-7}\normalsize =-42[/latex]

Now see if you can solve a problem that requires multiplication to undo division. Recall the rules for multiplying two negative numbers – two negatives give a positive when they are multiplied.

try it

As you begin to solve equations that require several steps you may find that you end up with an equation that looks like the one in the next example, with a negative variable.  As a standard practice, it is good to ensure that variables are positive when you are solving equations. The next example will show you how.

example

Solve: [latex]-r=2[/latex]

Now you can try to solve an equation with a negative variable.

try it

In our next example, we are given an equation that contains a variable multiplied by a fraction. We will use a reciprocal to isolate the variable.

example

Solve: [latex]\Large\frac{2}{3}\normalsize x=18[/latex]

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

try it

The next video includes examples of using the division and multiplication properties to solve equations with the variable on the right side of the equal sign.