Expressing Relationships as Algebraic Expressions

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  • 0:05 Expressions and Variables
  • 0:58 Algebraic Expressions
  • 1:36 Addition and Subtraction
  • 2:56 Multiplication and Division
  • 4:06 Multiple Variables
  • 4:53 Lesson Summary
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Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Expert Contributor
Robert Ferdinand

Robert Ferdinand has taught university-level mathematics, statistics and computer science from freshmen to senior level. Robert has a PhD in Applied Mathematics.

What do you do when you don't know what a number is but you do know how it relates to something else? You use an algebraic expression. In this lesson, we'll learn how to express relationships as algebraic expressions.

Expressions and Variables

Do you ever have trouble putting what you want to say into words? Maybe you're trying to explain something and it's like you're speaking in another language that the other person can't understand. I think this is how my high school chemistry teacher felt while teaching me. Or maybe you're trying to tell someone you, you know, like them, or like like them, but he or she just isn't hearing you. It's frustrating, I know. Or, I mean, you know, I've heard.

Life would sometimes be simpler if we could just use math to speak for us all the time. Fortunately, there are endless real-life situations we can express using algebra. We just need a few tools. First, of course, are numbers. This is math, after all. We also need variables. A variable is a symbol that represents an unknown number. Then we'll need some operators, like addition and division. With these tools, we can say all kinds of things with ease.

Algebraic Expressions

We use the tools to build algebraic expressions. An algebraic expression is a mathematical phrase that may include numbers, variables and operators. It's basically like a sentence. But you're substituting these numbers, variables and operators for words.

Algebraic expressions can look like x + 1, 17y, 4a - 3 or q/6.

The most common and useful application of this idea is in solving word problems. We need to take the real situations that are in regular language in the problem and translate them into algebraic language to better understand them. These can involve several different types of operations.

Addition and Subtraction

Let's start with addition and subtraction expressions. Here's one: There are two competing lemonade stands run by siblings April and Mike. We want to describe the relationship between the prices for a cup of lemonade between the two stands. April is selling her lemonade for 50 cents less than Mike's. What do we do?

First, we need a variable for the cost of April's lemonade. Let's call that a. We use that in place of a number we don't know. Now, if Mike's lemonade is 50 cents more than April's, we can describe his as a + 50. That a + 50 describes the cost of Mike's lemonade relative to the cost of April's.

We can test our expression by substituting a number for our variable. Let's say a = 75 cents. a + 50 = $1.25. Is $1.25 50 cents more than 75 cents? Yes! So we know that we have the correct expression.

We could also use subtraction here. We could use m as the cost of Mike's lemonade. Since April's is 50 cents less than Mike's, hers would be m - 50. Again, we could test this. Let's say m = 80 cents. So m - 50 = 30 cents. Is 30 50 cents less than 80? It is. It's also a very cheap cup of lemonade.

Multiplication and Division

Next, let's look at multiplication. Let's say it's the holiday season and you're buying gifts for your siblings. Since you live in an algebra problem, let's say you have 15 siblings. That's a lot, so you have to keep these gifts small.

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Additional Activities

Practice Problems: Expressing Relationships as Algebraic Expressions

Key Terms

  • Relationships: Associations between Real-World Quantities
  • Algebraic Expressions: Using variables and algebraic operations (addition, subtraction, multiplication, division) to express relationships

Materials Needed

  • Paper
  • Pencil

Practice Problems (Show your work):

(a) Ray's age is x years. Eugene is 7 years older than Ray. Ken's age is 8 years less than twice Eugene's age. Express Eugen's age and Ken's age as functions of x.

(b) A box has chocolate, vanilla, and strawberry cookies. The number of chocolate cookies is three times the number of vanilla cookies. The number of strawberry cookies is five more than four times the number of chocolate cookies. Assume the number of vanilla cookies is A. Write down the number of chocolate cookies and the number of strawberry cookies in terms of A.

(c) A movie ticket costs $p. A ticket to the rodeo costs $q. How much do 7 rodeo tickets and 9 movie tickets cost in terms of p and q?

(d) There are 16 students in a class. All 16 students buy one hamburger and one soda each. The class teacher pays the vendor a total of $U for the sodas and a total of $V for the hamburgers. Find the total cost incurred by each student for their one hamburger and one soda.

Answers (To Check Your Work):

(a) Eugene's Age: (x + 7) years; Ken's age: (2x + 6) years.

(b) The number of Chocolate Cookies: 3A; Number of Strawberry Cookies: 12A + 5.

(c) $(9p + 7q).

(d) $(U/16 + V/16).

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