Using the Subtraction and Addition Properties for Single-Step Equations

Learning Outcomes

Determine whether a number is a solution to an equation
Review the use of the subtraction and addition properties of equality to solve linear equations
Use the addition and subtraction properties of equality to solve linear equations with fractions or decimals

We began our work solving equations in previous chapters, where we said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.

Solution of an Equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.

Determine whether a number is a solution to an equation.

Substitute the number for the variable in the equation.
Simplify the expressions on both sides of the equation.
Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.

In the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.

EXAMPLE

Determine whether [latex]y=\Large\frac{3}{4}[/latex] is a solution for [latex]4y+3=8y[/latex].

Solution:

	[latex]4y+3=8y[/latex]
Substitute [latex]\color{red}{\Large\frac{3}{4}}[/latex] for [latex]y[/latex]	[latex]4(\color{red}{\Large\frac{3}{4}}\normalsize)+3\stackrel{\text{?}}{=}8(\color{red}{\Large\frac{3}{4}})[/latex]
Multiply.	[latex]3+3\stackrel{\text{?}}{=}6[/latex]
Add.	[latex]6=6\quad\checkmark[/latex]

Since [latex]y=\Large\frac{3}{4}[/latex] results in a true equation, [latex]\Large\frac{3}{4}[/latex] is a solution to the equation [latex]4y+3=8y[/latex].

Now it is your turn to determine whether a fraction is the solution to an equation.

TRY IT

We introduced the Subtraction and Addition Properties of Equality in Solving Equations Using the Subtraction and Addition Properties of Equality. In that section, we modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.

Subtraction Property of Equality

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

Addition Property of Equality

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

When you add or subtract the same quantity from both sides of an equation, you still have equality.

We introduced the Subtraction Property of Equality earlier by modeling equations with envelopes and counters. The image below models the equation [latex]x+3=8[/latex].

An envelope and three yellow counters are shown on the left side. On the right side are eight yellow counters.

The goal is to isolate the variable on one side of the equation. So we “took away” [latex]3[/latex] from both sides of the equation and found the solution [latex]x=5[/latex].

Some people picture a balance scale, as in the image below, when they solve equations.

Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is

The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.

In the following example we review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. You can check your solutions by substituting the value into the equation to make sure you have a true statement.

EXAMPLE

Solve: [latex]x+11=-3[/latex]

Show Solution

Now you can try solving an equation that requires using the addition property.

TRY IT

In the original equation in the previous example, [latex]11[/latex] was added to the [latex]x[/latex] , so we subtracted [latex]11[/latex] to “undo” the addition. In the next example, we will need to “undo” subtraction by using the Addition Property of Equality.

EXAMPLE

Solve: [latex]m - 4=-5[/latex]

Show Solution

Now you can try using the addition property to solve an equation.

TRY IT

In the following video, we present more examples of solving equations using the addition and subtraction properties.

You may encounter equations that contain fractions, therefore in the following examples we will demonstrate how to use the addition property of equality to solve an equation with fractions.

EXAMPLE

Solve: [latex]n-\Large\frac{3}{8}\normalsize =\Large\frac{1}{2}[/latex]

Show Solution

Now you can try solving an equation with fractions by using the addition property of equality.

TRY IT

Watch this video for more examples of solving equations that include fractions and require addition or subtraction.

You may encounter equations with decimals, for example in financial or science applications. In the next examples we will demonstrate how to use the subtraction property of equality to solve equations with decimals.

eXAMPLE

Solve [latex]a - 3.7=4.3[/latex]

Show Solution

Now it is your turn to try solving an equation with decimals by using the addition property of equality.

TRY IT

In this video, we show more examples of how to solve equations with decimals that require addition and subtraction.

		[latex]x+11=-3[/latex]
Subtract 11 from each side to “undo” the addition.		[latex]x+11\color{red}{-11}=-3\color{red}{-11}[/latex]
Simplify.		[latex]x=-14[/latex]
Check:	[latex]x+11=-3[/latex]
Substitute [latex]x=-14[/latex] .	[latex]\color{red}{-14}+11\stackrel{\text{?}}{=}-3[/latex]
	[latex]-3=-3\quad\checkmark[/latex]

		[latex]m-4=-5[/latex]
Add 4 to each side to “undo” the subtraction.		[latex]m-4\color{red}{+4}=-5\color{red}{+4}[/latex]
Simplify.		[latex]m=-1[/latex]
Check:	[latex]m-4=-5[/latex]
Substitute [latex]m=-1[/latex] .	[latex]\color{red}{-1}+4\stackrel{\text{?}}{=}-5[/latex]
	[latex]-5=-5\quad\checkmark[/latex]
		The solution to [latex]m - 4=-5[/latex] is [latex]m=-1[/latex]

		[latex]n-\Large\frac{3}{8}\normalsize =\Large\frac{1}{2}[/latex]
Use the Addition Property of Equality.		[latex]n-\Large\frac{3}{8}\normalsize\color{red}{+\Large\frac{3}{8}}\normalsize =\Large\frac{1}{2}\normalsize\color{red}{+\Large\frac{3}{8}}[/latex]
Find the LCD to add the fractions on the right.		[latex]n-\Large\frac{3}{8}\normalsize +\Large\frac{3}{8}\normalsize =\Large\frac{4}{8}\normalsize +\Large\frac{3}{8}[/latex]
Simplify.		[latex]n=\Large\frac{7}{8}[/latex]
Check:	[latex]n-\Large\frac{3}{8}\normalsize =\Large\frac{1}{2}[/latex]
Substitute [latex]n=\color{red}{\Large\frac{7}{8}}[/latex]	[latex]\color{red}{\Large\frac{7}{8}}\normalsize -\Large\frac{3}{8}\normalsize\stackrel{\text{?}}{=}\Large\frac{1}{2}[/latex]
Subtract.	[latex]\Large\frac{4}{8}\normalsize\stackrel{\text{?}}{=}\Large\frac{1}{2}[/latex]
Simplify.	[latex]\Large\frac{1}{2}\normalsize =\Large\frac{1}{2}\normalsize\quad\checkmark[/latex]
The solution checks.

		[latex]a-3.7=4.3[/latex]
Use the Addition Property of Equality.		[latex]a-3.7\color{red}{+3.7}=4.3\color{red}{+3.7}[/latex]
Add.		[latex]a=8[/latex]
Check:	[latex]a-3.7=4.3[/latex]
Substitute [latex]a=8[/latex] .	[latex]\color{red}{8}-3.7\stackrel{\text{?}}{=}4.3[/latex]
Simplify.	[latex]4.3=4.3\quad\checkmark[/latex]
The solution checks.

Module 9: Multi-Step Linear Equations