5.2 Using Equations to Represent Situations

Learning Objectives

You will be able to write addition and subtraction phrases as single-variable expressions.
You will be able to write multiplication and division phrases as single-variable expressions.
You will be able to write single variable equations from verbal models.
You will be able to model real-world situations with simple equations.

Introduction

Real Life Example: Riding the T

Kara and her twin brother Marc are going to be spending one month in Boston with their grandparents. They are very excited! Not only is it summer in Boston and the weather will be terrific, but Boston has a fantastic subway system called the T and they will be able to ride it to get all around.

Kara is very excited about visiting the museums, but Marc, who is a huge baseball fan, is hoping for a trip to Fenway Park before the month is up. Seeing the Red Sox play would be a huge bonus!

On the first day, the twins’ Grandpa tells them that they are going to spend the day learning how to ride the subway. Given that the twins are 14 and that they will be traveling together, Grandpa feels comfortable that they will be fine. After a quick discussion on safety, they are ready to go.

Grandpa says that each ride on the subway will cost $0.85 for a teenager. They will be riding to the Boston Common and back, and they may take another ride too.

Kara goes upstairs to get her money and tries to figure out how much they will need in all. She is having a tough time because she doesn’t know how many train rides they will take in all. She tells Marc what she is trying to figure out.

“Well until we know the number of rides, you can’t figure out the total,” he says. “But you can write an expression.”

An expression? How can Kara do this? What does an expression mean? Why would you want one?

This lesson is all about expressions and equations. Learning how to write expressions can be very helpful when you have a variable that can change, such as the number of train rides. Pay attention and at the end of the lesson you will be able to help Kara write an expression to show the amount of money needed for multiple train rides.

Guided Learning

Write Addition and Subtraction Phrases as Single-Variable Expressions

An expression shows how numbers and/or variables are connected by operations, such as addition, subtraction, multiplication, and division. Notice in the examples that an expression does not have an equal sign. This is because the value of the variable in each expression can change, or you could say that we can evaluate an expression using different given values for the variable.

$& 50-2 && 4-a && 12z && \frac{4x}{3}$

Three of the expressions above include variables, such as a, z , and .

An expression that includes one or more variables is called an algebraic expression. Each variable in an algebraic expression can have a different value. Once again, you will not see an equal sign with an algebraic expression. We can use algebraic expressions to represent words or phrases.

Often in word problems or in other situations in math, you will be given a set of words or a phrase that you will need to rewrite as an expression. When you do this, you will be looking for words that mean different operations or things in math. This way you can write an expression that correctly represents the words or phrase.

We are going to start with addition and subtraction phrases. Take a look at this chart.

Addition Expressions		Subtraction Expressions
1 plus		4 less
2 and		6 less than
3 more than		fewer than 7

The bolded key words in the phrases above provide clues about whether or not you should write an addition or a subtraction expression. While key words can be a helpful guide, it is important not to rely on them totally. It is always most important to think about which operation makes the most sense for a particular situation.

Example A

Abdul has $5 more than Xavier has. Let stand for the number of dollars Xavier has. Write an algebraic expression to show the number of dollars Abdul has.

The phrase is “$5 more than Xavier.” Use a number, an operation sign, or a variable to represent each part of that phrase.

$& \$ \underline{5} \ \underline{more \ than} \ \underline{Xavier}\\& \downarrow \qquad \quad \downarrow \qquad \quad \ \downarrow\\& \ 5 \qquad \ \ + \qquad \quad x$

Notice that in this phrase, the key words “more than” means you should write an addition expression.

So, the expression 5+x represents the number of dollars Abdul has. We also could have written this as. x+5 because addition is commutative. That means, the order in which numbers are added does not matter.

Our answer is x+5 .

Example B

6 less than a number

Notice the key phrase “less than” which means subtraction.

"A number" means a variable.

Because it says “6 less than a number,” the 6 will follow the variable.

x-6

Our answer is x-6 .

Remember, sometimes writing an expression is not as simple as relying on keywords.

Example C

Lian is inches shorter than Hannah. Hannah is 65 inches tall. Write an algebraic expression to show Lian's height in inches.

The phrase is “ inches shorter than Hannah.” You also know that Hannah's height is 65 inches.

There are no keywords, so you need to think about what makes sense. If Lian is shorter than Hannah, her height will be less than 65 inches. So, write a subtraction expression. Use 65 to represent Hannah's height. Since Lian's height is less than Hannah's height, you will need to subtract inches from Hannah's height to represent Lian's height.

$& \underline{x} \ inches \ \underline{shorter \ than} \ \underline{Hannah}.\\& \Box \qquad \qquad \quad \downarrow \qquad \qquad \ \Box\\& \Box \qquad \qquad \quad \downarrow \qquad \qquad \ \Box\\& \Box \qquad \qquad \quad \downarrow \qquad \qquad \ \Box\\& 65 \qquad \qquad \quad - \qquad \qquad x$

The answer is 65-x .

Lesson Exercises

Write an expression for each phrase.

A number plus five
Six more than a number
Fifteen less than a number

Take a few minutes to check your answers with a friend.

Write Multiplication and Division Phrases as Single-Variable Expressions

Just as you can write addition and subtraction expressions from words or phrases, you can also write multiplication and division expressions. Once again, you can use keywords to help you with this. The more familiar you become with the keywords that identify a multiplication or division expression, the better you will become at writing expressions.

Here are some examples of how words or phrases can be translated into multiplication or division expressions.

Multiplication Expressions		Division Expressions
9 times	$9 \times k \ \ \text{or} \ \ 9k$	8 divided into groups	$8 \div n \ \ \text{or} \ \ \frac{8}{n}$
twice as much as	$2 \times m \ \ \text{or} \ \ 2m$	shared equally by 3 people	$q \div 3 \ \ \text{or} \ \ \frac{q}{3}$
		half of	$r \div 2 \ \ \text{or} \ \ \frac{r}{2}$
		one-third of	$p \div 3 \ \ \text{or} \ \ \frac{p}{3}$

The bolded key words in the phrases above provide clues about whether you should write a multiplication or a division expression. Remember, key words can be a helpful guide, but you should always think about which operation makes sense for a particular situation.

Example D

Write an algebraic expression to represent this phrase: 3 times a number, .

The phrase is “3 times a number, .” Use a number, an operation sign, or a variable to represent each part of that phrase.

$& \underline{3} \ \underline{times} \ a \ \underline{number, \ t}\\& \downarrow \quad \ \downarrow \qquad \quad \downarrow\\& \ 3 \quad \times \qquad \ \ t$

Notice that in this phrase, the key word “times” means you should write a multiplication expression.

So, the expression $3 \times t$ or represents the phrase. We also could have written this as $t \times 3$ because multiplication is commutative. That means that the order in which numbers are multiplied does not matter.

Example E

Mr. Warren separated 30 students into equal groups. Write an algebraic expression to represent the number of students in each group.

The phrase is “separated 30 students into equal groups.”

Think about what makes sense. Separating 30 students into equal groups means dividing 30 students into equal groups. So, write a division expression.

$& separated \ \underline{30 \ students} \ \underline{into} \ \underline{n \ equal \ groups}\\& \qquad \qquad \qquad \downarrow \qquad \qquad \downarrow \qquad \qquad \downarrow\\& \qquad \qquad \qquad 30 \qquad \quad \ \div \qquad \quad \ \ n$

Division is not commutative. The order in which we divide numbers matters. So, while $30 \div n$ represents the number of students in each group, $n \div 30$ does not.

The answer is $30 \div n$ .

Write a multiplication or division expression for each phrase.

Four times a number
Sixteen divided into a number of groups
The product of five and a number

Take a few minutes to check your answers with a neighbor.

Write Single Variable Equations from Verbal Models

What is the difference between an equation and an expression? Well, an expression is a phrase without an equal sign. This means that the variable in an expression can be changed and the expression can be evaluated differently. An equation has an equal sign. Therefore one side of an equation is equal to a value on the other side.

Now, let's examine how to write equations.

The same key words that helped you write expressions may also help you write equations. Here are some additional key words that you may find helpful.

Key Words for Addition or Multiplication Equations

Key Words for Subtraction Equations

Key Words for Division Equations

how many all together

how many in all

how many total

how many more

how many fewer

how many left

how much change

how many in each

Example F

4 times a number is twelve.

First, notice that we have the word “times” which means to multiply. We also have the word “is” which is a keyword for equals.

$4 \ times \ x=12$

The equation is 4x=12 .

Example G

Seven less than a number is fourteen.

First, notice we have the words “less than” which means subtraction. Then we have the word “is” which is a keyword for equals.

x-7=14

The answer is x-7=14 .

Write an equation for each verbal phrase.

Six and a number is twenty.
Eighteen divided by a number is three.
Five times a number is twenty-five.

Take a few minutes to check your answers.

Model Real-World Situations with Simple Equations

We can also apply this to word problems that represent real life situations.

Example H

Kelvin has twice as many stamps in his collection as Murray has in his. If Kelvin has 60 stamps in his collection, write an equation to represent , the number of stamps in Murray's collection.

Use a number, an operation sign, a variable, or an equal sign to represent each part of the problem. Since Kelvin has 60 stamps in his collection, represent the number of stamps in Kelvin's collection as 60.

$& \underline{Kelvin} \ \underline{has} \ \underline{twice \ as \ many \ stamps ... as \ Murray}.\\& \downarrow \qquad \quad \downarrow \qquad \qquad \qquad \downarrow\\& 60 \qquad \ = \qquad \qquad \quad \ 2m$

This equation 60=2m , represents , the number of stamps in Murray's collection.

Example I

Carrie made 3 liters of lemonade for a party. After the party, she had 0.5 liter of lemonade left. Write an equation to represent , the number of liters of lemonade that her guests drank.

Use a number, an operation sign, a variable, or an equal sign to represent each part of that problem. Since the question tells us how many liters of lemonade were left after the party, write a subtraction equation.

Since she had 0.5 liter of lemonade left, is the number of liters that were drunk at the party. For this problem, it may help to write an equation in words and then translate those words into an algebraic equation.

$& (\text{number of liters made}) \ - \ (\text{number of liters guests drank}) \ = \ (\text{number of liters left})\\& \qquad \downarrow \qquad \qquad \qquad \qquad \ \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \qquad \ \downarrow \qquad \qquad \quad \downarrow\\& \qquad \ 3 \qquad \qquad \qquad \qquad - \qquad \qquad \ \ \ \ \ n \qquad \qquad \qquad \quad \ \ = \qquad \qquad \ 0.5$

This equation, 3-n=0.5 , represents , the number of liters of lemonade that Carrie's guests drank during the party.

Real Life Example Completed

Riding the T

Here is the original problem once again. Reread it and underline any important information.

Kara is very excited about visiting the museums, but Marc, who is a huge baseball fan, is hoping for a trip to Fenway Park before the month is up. Seeing the Red Sox play would be a huge bonus!

Grandpa says that each ride on the subway will cost $0.85 for a teenager. They will be riding to the Boston Common and back, and they may take another ride too.

“Well until we know the number of rides, you can’t figure out the total,” he says. “But you can write an expression.”

An expression? How can Kara do this? What does an expression mean? Why would you want one?

Now that you know what an expression is, you can write one to help Kara figure out the amount of money based on the number of train rides.

The amount of money per ride does not vary. It costs $0.85 per ride for a teenager.

The number of train rides does vary. This is where a variable is very useful. It can change according to the number of train rides. Let’s use .

The expression is .85x .

If Kara changes the variable according to the number of train rides that she and Marc take, then she can figure out the cost per day of riding the train.

If they go on four train rides for example, here is the expression.

.85(4)

Evaluating this expression, the cost would be $3.40.

Expressions are very helpful for figuring out different problems with changeable parts like this one.

Review

Keep in mind the key terms from addition, subtraction, multiplication, and division when translating words into expressions.
When “a number” is talked about in a problem, it should be represented with a variable in an expression.
Expressions do not have equal signs.

Expression

An expression shows how numbers and/or variables are connected by operations, such as addition,

subtraction, multiplication, and division.

Algebraic Expression

An expression that includes one or more variables is called an algebraic expression.

Equation

An equation is a mathematical sentence with an equal sign.

Variable

A variable is a letter that represents a number.