1.1: Variable Expressions
What if you were at the supermarket and saw the price of a loaf of bread, but you weren't sure how many loaves you wanted to buy? How could you represent the total amount of money spent on bread? After completing this Concept, you'll be able to write an expression that is equal to this amount, regardless of the number of loaves you buy.
Guidance
When someone is having trouble with algebra, they may say, “I don’t speak math!” While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a second language that you must learn in order to be successful. There are verbs and nouns in math, just like in any other language. In order to understand math, you must practice the language.
A verb is a “doing” word, such as running, jumping, or driving. In mathematics, verbs are also “doing” words. A math verb is called an operation. Operations can be something you have used before, such as addition, multiplication, subtraction, or division. They can also be much more complex like an exponent or square root.
Example A
Suppose you have a job earning $8.15 per hour. What could you use to quickly find out how much money you would earn for different hours of work?
You could make a list of all the possible hours, but that would take forever! So instead, you let the “hours you work” be replaced with a symbol, like \begin{align*}h\end{align*}
\begin{align*}amount \ of \ money = 8.15(h)\end{align*}
A noun is usually described as a person, place, or thing. In mathematics, nouns are called numbers and variables. A variable is a symbol, usually an English letter, written to replace an unknown or changing quantity.
Example B
What variables could be choices for the following situations?
a. the number of cars on a road
b. time in minutes of a ball bounce
c. distance from an object
Solution: There are many options, but here are a few to think about.
a. Cars is the changing value, so \begin{align*}c\end{align*}
b. Time is the changing value, so \begin{align*}t\end{align*}
c. Distance is the varying quantity, so \begin{align*}d\end{align*}
Example C
Write an expression for 2 more than 5 times a number.
Solution: First we need to choose a variable for this unknown number. The letter \begin{align*}n\end{align*}
\begin{align*}5(n).\end{align*}
Now we need to express "2 more" than \begin{align*}5(n)\end{align*}
\begin{align*}5(n)+2.\end{align*}
Vocabulary
Variable: A variable is a symbol used to represent an unknown or changing quantity.
Operation: An operation is an action you perform on a number or variable, such as addition, subtraction, multiplication, or division.
Guided Practice
1. What variable would you use to represent the length in yards of fabric?
2. Suppose bananas cost $0.29 each. Write an expression for the cost of buying a certain quantity of bananas.
3. Suppose your bank account charges you a $9 fee every month plus $2 for every time you use an ATM of another bank. Write an expression for the charges every month.
Answers
1. We often use the first letter of the word that the variable represents. Since we want to represent length, we could use \begin{align*}l\end{align*}
2. First we must choose a variable for the quantity of bananas purchased. What variable would you choose? One good choice is \begin{align*}b\end{align*}
\begin{align*} $0.29(b)\end{align*}
3. The bank charges $2 for every ATM withdrawal from another bank. That means $2 times the number of times you use the ATM of another bank is the amount of money charged. What variable should you use to represent the number of ATM withdrawals from another bank? One good choice would be \begin{align*}A\end{align*}
\begin{align*}2(A)\end{align*}
But the bank also charges us a fixed $9 every month, so we have to add that to the expression:
\begin{align*}2(A)+9\end{align*}
Practice
In 1 – 5, choose an appropriate variable to describe each situation.
 The number of hours you work in a week
 The distance you travel
 The height of an object over time
 The area of a square
 The number of steps you take in a minute
In 6 – 10, write an expression to describe each situation.
 You have a job earning $2000 a month
 Avocados are sold for $1.50 each
 A car travels 50 miles per hour for a certain number of hours
 Your vacation costs you $500 for the airplane ticket plus $100 per day
 Your cell phone costs $50 a month plus $0.25 for each text message
In 11 – 15, underline the math verb(s) in the sentence.
 Six times \begin{align*}v\end{align*}
v  Four plus \begin{align*}y\end{align*}
y minus six  Sixteen squared

\begin{align*}U\end{align*}
U divided by three minus eight  225 raised to the \begin{align*}\frac{1}{2}\end{align*}
12 power
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operation
Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Exponent
Exponents are used to describe the number of times that a term is multiplied by itself.Operations
Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.Square Root
The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.Image Attributions
Here you'll learn how to decide which variables to use when representing unknown quantities and how to form expressions by using these variables.