7.1.b – Using the Division and Multiplication Properties of Equality to Solve Equations

Learning Outcomes

Determine whether a number is a solution to an equation
Check your solution to a linear equation to verify its accuracy
Solve equations using the Division and Multiplication Properties of Equality
Solve equations that need to be simplified

Solve algebraic equations using the multiplication and division properties of equality

Just as you can add or subtract the same exact quantity on both sides of an equation, you can also multiply or divide both sides of an equation by the same quantity to write an equivalent equation. To start, let’s look at a numeric equation, [latex]5\cdot3=15[/latex], as an example. If you multiply both sides of this equation by [latex]2[/latex], you will still have a true equation.

[latex]\begin{array}{r}5\cdot 3=15\,\,\,\,\,\,\, \\ 5\cdot3\cdot2=15\cdot2 \\ 30=30\,\,\,\,\,\,\,\end{array}[/latex]

This characteristic of equations is generalized in the Multiplication Property of Equality.

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].

If two expressions are equal to each other and you divide both sides by the same number that is not equal to zero, the resulting expressions will also be equivalent.

Multiplication Property of Equality

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

If two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.

Stated simply, when you divide or multiply both sides of an equation by the same quantity, you still have equality. When the equation involves multiplication or division, you can “undo” these operations by using the inverse operation to isolate the variable.

Example

Solve [latex]3x=24[/latex]. When you are done, check your solution.

Show Solution

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]

Show Solution

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

Another way to think about solving an equation when the operation is multiplication or division is that we want to multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to [latex]1[/latex].

In the following example, we change the coefficient to [latex]1[/latex] by multiplying by the multiplicative inverse of [latex]\frac{1}{2}[/latex].

Example

Solve [latex]\frac{1}{2 }{ x }={ 8}[/latex] for [latex]x[/latex].

Show Solution

In the video below, you will see examples of how to use the Multiplication and Division Properties of Equality to solve one-step equations with integers and fractions.

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

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]

Show Solution

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

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.

Two-Step Linear Equations

If the equation is in the form [latex]ax+b=c[/latex], where [latex]x[/latex] is the variable, you can solve the equation as before. First “undo” the addition and subtraction, and then “undo” the multiplication and division.

Examples

Solve: [latex]4x+6=-14[/latex]

Solution:

In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.

Since the left side is the variable side, the 6 is out of place. We must “undo” adding [latex]6[/latex] by subtracting [latex]6[/latex], and to keep the equality we must subtract [latex]6[/latex] from both sides. Use the Subtraction Property of Equality.		[latex]4x+6\color{red}{-6}=-14\color{red}{-6}[/latex]
Simplify.		[latex]4x=-20[/latex]
Now all the [latex]x[/latex] s are on the left and the constant on the right.
Use the Division Property of Equality.		[latex]\Large\frac{4x}{\color{red}{4}}\normalsize =\Large\frac{-20}{\color{red}{4}}[/latex]
Simplify.		[latex]x=-5[/latex]
Check:		[latex]4x+6=-14[/latex]
Let [latex]x=-5[/latex] .		[latex]4(\color{red}{-5})+6=-14[/latex]
		[latex]-20+6=-14[/latex]
		[latex]-14=-14\quad\checkmark[/latex]

Solve: [latex]2y - 7=15[/latex]

Show Solution

Example

Solve [latex]3y+2=11[/latex]

Show Solution

Now you can try a similar problem.

Try It

In the following video, we show examples of solving two step linear equations.

Remember to check the solution of an algebraic equation by substituting the value of the variable into the original equation.

In the next section, we will learn how to solve equations that need to be simplified before they can be solved.

[latex]\Large\frac{a}{-7}\normalsize =-42[/latex]
Multiply both sides by [latex]-7[/latex] .	[latex]\color{red}{-7}(\Large\frac{a}{-7}\normalsize)=\color{red}{-7}(-42)[/latex] [latex]\Large\frac{-7a}{-7}\normalsize=294[/latex]
Simplify.	[latex]a=294[/latex]
Check your answer.	[latex]\Large\frac{a}{-7}\normalsize=-42[/latex]
Let [latex]a=294[/latex] .	[latex]\Large\frac{\color{red}{294}}{-7}\normalsize\stackrel{\text{?}}{=}-42[/latex]
	[latex]-42=-42\quad\checkmark[/latex]

[latex]-r=2[/latex]
Rewrite [latex]-r[/latex] as [latex]-1r[/latex] .	[latex]-1r=2[/latex]
Divide both sides by [latex]-1[/latex] .	[latex]\Large\frac{-1r}{\color{red}{-1}}\normalsize =\Large\frac{2}{\color{red}{-1}}[/latex]
Simplify.	[latex]r=-2[/latex]
Check.	[latex]-r=2[/latex]
Substitute [latex]r=-2[/latex]	[latex]-(\color{red}{-2})\stackrel{\text{?}}{=}2[/latex]
Simplify.	[latex]2=2\quad\checkmark[/latex]

[latex]2y-7[/latex] is the side containing a variable.[latex]15[/latex] is the side containing only a constant.
Add [latex]7[/latex] to both sides.		[latex]2y-7\color{red}{+7}=15\color{red}{+7}[/latex]
Simplify.		[latex]2y=22[/latex]
The variables are now on one side and the constants on the other.
Divide both sides by [latex]2[/latex].		[latex]\Large\frac{2y}{\color{red}{2}}\normalsize =\Large\frac{22}{\color{red}{2}}[/latex]
Simplify.		[latex]y=11[/latex]
Check:		[latex]2y-7=15[/latex]
Let [latex]y=11[/latex] .		[latex]2\cdot\color{red}{11}-7\stackrel{\text{?}}{=}15[/latex]
		[latex]22-7\stackrel{\text{?}}{=}15[/latex]
		[latex]15=15\quad\checkmark[/latex]

Module 7: Linear Equations