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.
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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Let's say you can spend d dollars on each gift. How do you describe the total cost of the gifts? We could add them together and say d + d + d +... well, 15 total ds. It would be simpler to just say 15 times d, or 15d. That's the cost of one gift times 15. So if you spent $10 per gift, you'd spend 15 times 10, which is $150.
Here's one that involves division. Let's say you're working with a study group. You all get hungry and order some pizza. There are three of you. If everyone's going to get the same amount, how much pizza can you eat? Well, you don't know how many slices there will be. So let's use s to stand in for the total number of slices. Since there are 3 of you, each of you gets s/3 slices. That's the total number of slices divided by 3. If there were 9 slices, you'd get 9 divided by 3, or 3 slices each.
So far, all of our expressions used just one variable. But life isn't always that simple. Let's say you're going on a date. It's dinner and a movie. You want to express how much this date will cost.
Entrees cost e and movie tickets cost t. How do you write this? Well, there are two of you, so two entrees will be 2e. And supposing your date doesn't bail on you before the movie, you'll need two tickets, or 2t. The total cost is 2e + 2t.
Sometimes we need to mix operators as well as variables. Don't worry. You can handle this. Let's say you don't have one date this week, but three. How do you show that? Just take our expression, 2e + 2t, and multiply the whole thing by 3 - so 3(2e + 2t).
In summary, an algebraic expression is a mathematical phrase that may include numbers, variables and operators. We use variables in place of the numbers we don't know. Then, using our operators, we explain the relationships between different values.
When this lesson is done, you should be able to:
Understand how to use algebra to express relationships in mathematics
Solve addition and subtraction expressions
Calculate the answers to problems involving multiple variables
Practice Problems: Expressing Relationships as Algebraic Expressions
Relationships: Associations between Real-World Quantities
Algebraic Expressions: Using variables and algebraic operations (addition, subtraction, multiplication, division) to express relationships
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.
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