2.15: Solving Single-Step Equations

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Learning Outcome

Use the addition, subtraction, multiplication, and division properties to solve single-step equations

We will begin solving equations below, but first we need to understand what a solution to an equation looks like. 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 — 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.

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 $y=\Large\frac{3}{4}$ is a solution for $4y+3=8y$ .

Solution:

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

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

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

TRY IT

An interactive or media element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=94

Let’s use a model to help understand how the process of solving an equation is like solving a puzzle. An envelope represents the variable – since its contents are unknown – and each counter represents one.

Suppose a desk has an imaginary line dividing it in half. We place three counters and an envelope on the left side of desk, and eight counters on the right side of the desk. Both sides of the desk have the same number of counters, but some counters are hidden in the envelope. Can you tell how many counters are in the envelope?

The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 8 counters.
What steps are you taking in your mind to figure out how many counters are in the envelope? Perhaps you are thinking “I need to remove the $3$ counters from the left side to get the envelope by itself. Those $3$ counters on the left match with $3$ on the right, so I can take them away from both sides. That leaves five counters on the right, so there must be $5$ counters in the envelope.”

What algebraic equation is modeled by this situation? Each side of the desk represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope $x$ , so the number of counters on the left side of the desk is $x+3$ . On the right side of the desk are $8$ counters. We are told that $x+3$ is equal to $8$ so our equation is $x+3=8$ .

The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 8 counters.
$x+3=8$

Let’s write algebraically the steps we took to discover how many counters were in the envelope.

	$x+3=8$
First, we took away three from each side.	$x+3\color{red}{--3}=8\color {red}{--3}$
Then we were left with five.	$x=5$

Now let’s check our solution. We substitute $5$ for $x$ in the original equation and see if we get a true statement.

$x+3=8$

$\color{red}{5}+3=8$

$8=8\quad\checkmark$ Our solution is correct. Five counters in the envelope plus three more equals eight.

example

Write an equation modeled by the envelopes and counters, and then solve the equation:

The image is divided in half vertically. On the left side is an envelope with 4 counters below it. On the right side is 5 counters.

Solution

On the left, write $x$ for the contents of the envelope, add the $4$ counters, so we have $x+4$ .	$x+4$
On the right, there are $5$ counters.	$5$
The two sides are equal.	$x+4=5$
Solve the equation by subtracting $4$ counters from each side.

We can see that there is one counter in the envelope. This can be shown algebraically as:

$x+4=5$

$x+4\color{red}{--4}=5\color{red}{--4}$

$x=1$

Substitute $1$ for $x$ in the equation to check.

$x+4=5$

$\color{red}{1}+4=5$

$5=5\quad\checkmark$

Since $x=1$ makes the statement true, we know that $1$ is indeed a solution.

try it

Write the equation modeled by the envelopes and counters, and then solve the equation:

The image is divided in half vertically. On the left side is an envelope with one counter below it. On the right side is 7 counters.

[reveal-answer q=”7845″]Show Answer[/reveal-answer]
[hidden-answer a=”7845″]

$x+1=7$

$x=6$

[/hidden-answer]

Write the equation modeled by the envelopes and counters, and then solve the equation:

The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 4 counters.

[reveal-answer q=”9034″]Show Answer[/reveal-answer]
[hidden-answer a=”9034″]

$x+3=4$

$x=1$

[/hidden-answer]

Our puzzle has given us an idea of what we need to do to solve an equation. The goal is to isolate the variable by itself on one side of the equation. In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality.

Subtraction Property of Equality

For all real numbers $a,b$ , and $c$ , if $a=b$ , then $a-c=b-c$ .

In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable.

But suppose we have an equation with a number subtracted from the variable, such as $x - 5=8$ . We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides.

We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.

Addition Property of Equality

For all real numbers $a,b$ , and $c$ , if $a=b$ , then $a+c=b+c$ .

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 $x+3=8$ .

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” $3$ from both sides of the equation and found the solution $x=5$ .

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 examples 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 original equation to make sure you have a true statement.