# 1.5.1: Expressions and Variables

- Page ID
- 4280

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

## 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 without knowing the amount of loaves?

**Converting Words to Math**

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**.

**Let's 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 h for hours, and write an equation such as:

amount of money=8.15(h)

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.

**Now, let's determine what variables could be choices for the following situations:**

- The number of cars on a road

The number of cars is the changing value, so c is a good choice.

- Time in minutes of a ball bounce

Time is the changing value, so t is a good choice.

- Distance from an object

Distance is the varying quantity, so d is a good choice.

**Finally, let's practice writing expressions and write an expression for 2 more than 5 times a number:**

An **algebraic expression **is a mathematical phrase combining numbers and/or variables using mathematical operations.First we need to choose a variable for this unknown number. The letter n is a common choice, so we'll use that. To write the expression, first express 5 times the number by

5(n).

Now we need to express "2 more" than 5(n)Two more means that we should add two.

5(n)+2.

**Examples**

Example \(\PageIndex{1}\)

Earlier, you were asked how to represent the total amount of money spent on loaves of bread if you didn't know how many loaves you wanted to buy.

**Solution**

Let's say that a loaf of bread costs $2. The number of loaves is the unknown number. Since we want to represent loaves, we could use l. You can use a different variable if you would like. Since it costs $2 for each loaf, we multiply that by the number of loaves purchased (l):

$2(l)

Example \(\PageIndex{1}\)

What variable would you use to represent the length in yards of fabric?

**Solution**

We often use the first letter of the word that the variable represents. Since we want to represent length, we could use l. We could also use y for yards, to be more specific. Either choice would be good in this case.

Example \(\PageIndex{1}\)

Suppose bananas cost $0.29 each. Write an expression for the cost of buying a certain quantity of bananas.

**Solution**

First we must choose a variable for the quantity of bananas purchased. What variable would you choose? One good choice is b, for banana. Now, it costs $0.29 for each banana, so we multiply that by the number of bananas purchased:

$0.29(b)

Example \(\PageIndex{1}\)

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.

**Solution**

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 A, for ATM. So the charges for the ATM are represented as follows:

2(A)

But the bank also charges us a fixed $9 every month, so we have to add that to the expression:

2(A)+9

**Review**

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 v
- Four plus y minus six
- Sixteen squared
- U divided by three minus eight
- 225 raised to the 1/2 power

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.1.

### Vocabulary

Term | Definition |
---|---|

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. |

### Additional Resources

PLIX: PLay, Learn, Interact, eXplore: **Climbing Mountains**

Video: **The Language of Algebra: A Sample Application**

Practice: **Expressions and Variables**