Determine Whether a Number is a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!

To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of $x+2=7?$ If you said $5,$ you’re right! We say $5$ is a solution to the equation $x+2=7$ because when we substitute $5$ for $x$ the resulting statement is true.

Since $5+2=7$ is a true statement, we know that $5$ is indeed a solution to the equation.

The symbol $\stackrel{?}{=}$ asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign $\text{(=)}$ or not-equal sign $\text{(≠).}$

Example 2.28

$\text{Determine whether}x=5\text{is a solution of}6x-17=16.$

Solution

Multiply.
Subtract.

So $x=5$ is not a solution to the equation $6x-17=16.$

Example 2.29

$\text{Determine whether}y=2\text{is a solution of}6y-4=5y-2.$

Solution

Here, the variable appears on both sides of the equation. We must substitute $2$ for each $y.$

Multiply.
Subtract.

Since $y=2$ results in a true equation, we know that $2$ is a solution to the equation $6y-4=5y-2.$

Model the Subtraction Property of Equality

We will use a model to help you 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 as in Figure 2.3. 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?

Figure 2.3

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.” Figure 2.4 shows this process.

Figure 2.4

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$x+3=8.$

Figure 2.5

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

First, we took away three from each side.
Then we were left with five.

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

Our solution is correct. Five counters in the envelope plus three more equals eight.

Example 2.30

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

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.

Illustrates the process of setting up and partially solving a linear equation, x + 4 = 5, using a step-by-step description and its mathematical representation.

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

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

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

Try It 2.59

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

Try It 2.60

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

Solve Equations Using the Subtraction Property of Equality

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 equations. 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.

Think about twin brothers Andy and Bobby. They are $17$ years old. How old was Andy $3$ years ago? He was $3$ years less than $17,$ so his age was $17-3,$ or $14.$ What about Bobby’s age $3$ years ago? Of course, he was $14$ also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages $3$ years ago.

Example 2.31

Solve: $x+8=17.$

Solution

We will use the Subtraction Property of Equality to isolate $x.$

Subtract 8 from both sides.
Simplify.

Since $x=9$ makes $x+8=17$ a true statement, we know $9$ is the solution to the equation.

Example 2.32

Solve: $100=y+74.$

Solution

To solve an equation, we must always isolate the variable—it doesn’t matter which side it is on. To isolate $y,$ we will subtract $74$ from both sides.

Subtract 74 from both sides.
Simplify.
Substitute $26$ for $y$ to check.

Since $y=26$ makes $100=y+74$ a true statement, we have found the solution to this equation.

Solve Equations Using the Addition Property of Equality

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.

Remember the $17\text{-year-old}$ twins, Andy and Bobby? In ten years, Andy’s age will still equal Bobby’s age. They will both be $27.$

We can add the same number to both sides and still keep the equality.

Example 2.33

Solve: $x-5=8.$

Solution

We will use the Addition Property of Equality to isolate the variable.

Add 5 to both sides.
Simplify.

Example 2.34

Solve: $27=a-16.$

Solution

We will add $16$ to each side to isolate the variable.

Add 16 to each side.
Simplify.

The solution to $27=a-16$ is $a=43.$

Translate Word Phrases to Algebraic Equations

Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean equals. Some words that translate to the equal sign are:

• is equal to
• is the same as
• is
• gives
• was
• will be

It may be helpful to put a box around the equals word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.

We will practice translating word sentences into algebraic equations. Some of the sentences will be basic number facts with no variables to solve for. Some sentences will translate into equations with variables. The focus right now is just to translate the words into algebra.

Example 2.35

Translate the sentence into an algebraic equation: The sum of $6$ and $9$ is $15.$

Solution

The word is tells us the equal sign goes between 9 and 15.

Locate the “equals” word(s).
Write the = sign.
Translate the words to the left of the equals word into an algebraic expression.
Translate the words to the right of the equals word into an algebraic expression.

Example 2.36

Translate the sentence into an algebraic equation: The product of $8$ and $7$ is $56.$

Solution

The location of the word is tells us that the equal sign goes between 7 and 56.

Locate the “equals” word(s).
Write the = sign.
Translate the words to the left of the equals word into an algebraic expression.
Translate the words to the right of the equals word into an algebraic expression.

Example 2.37

Translate the sentence into an algebraic equation: Twice the difference of $x$ and $3$ gives $18.$

Solution

Locate the “equals” word(s).
Recognize the key words: twice; difference of …. and ….	Twice means two times.
Translate.

Translate to an Equation and Solve

Now let’s practice translating sentences into algebraic equations and then solving them. We will solve the equations by using the Subtraction and Addition Properties of Equality.

Example 2.38

Translate and solve: Three more than $x$ is equal to $47.$

Solution

		Three more than x is equal to 47.
Translate.
Subtract 3 from both sides of the equation.
Simplify.
We can check. Let $x=44$.

So $x=44$ is the solution.

Example 2.39

Translate and solve: The difference of $y$ and $14$ is $18.$

Solution

		The difference of y and 14 is 18.
Translate.
Add 14 to both sides.
Simplify.
We can check. Let $y=32$.

So $y=32$ is the solution.