Variable

Pronunciation: /ˈvɛər.i.ə.bəl/ Explain

A variable represents a value that can change (vary). A variable can also represent an unknown value. A letter of the alphabet is usually used to represent a variable.

Article Index

What is a Variable
Representation of Variables
Independent Variables
Dependent Variables
Function Notation and Variables

What is a variable?

In algebra, a variable is used for two reasons:

1. To represent an unknown quantity; and
2. To represent a quantity that can change.

Take the following addition problem: 22 + 17 = ____

When we start, we know that we are adding 22 to 17. Until we do the addition, we do not know the value of the sum of 22 and 17. If we wrote this addition problem with a variable, it might look like this:

22 + 17 = a

In this case, the variable 'a' represents an unknown quantity.

Using variables like this allows us to write algebraic equations such as:

3 + 2 · a = 13

In this case, the unknown quantity is in the middle of the equation, not all by itself.

Look at the following equation. Guess what part of the equation is a variable. Now move the mouse cursor over the equation. When the cursor is over the variable, the variable will change colors.

3 · x + 2 = 23

For this equation, how do you think we can find out what x represents?

Understanding Check 1

Click on the check box by the best answer.

1. In algebra, a variable is usually used to represent what?
   How the weather changes.
   No, a variable does have to do with changes, but can include changes in things that are not the weather, such as the price of candy bars.
   A quantity that changes.
   Yes, a variable can represent any quantity that changes.
   Something that does not change.
   No, a variable is used to represent something that does change.
2. Which of the following represents variables? Click on all that apply.
   a in the expression 3a + 2
   Yes, the value of a is unknown, so a is a variable.
   5 in the expression 5x - 3
   No, 5 does not change. It is always 5.
   x in the equation y = -3x + 4
   Yes, in this equation x can take on any value.

Representation of Variables

In math, a single letter of the alphabet is used to represent a variable. This letter is usually lower case. By convention the letters 'i', 'j', and 'k' are used to represent values that must be integers (not decimals or fractions).

Sometimes, Greek letters are used for variables. Three of the Greek letters are α, β, and γ. For more information on the Greek alphabet, see Greek Alphabet.

Independent Variables

An independent variable is a variable whose value does not depend on any other variable. Use for an example the price of gasoline at a gas station. There are three values involved: the number of gallons of gas pumped, the price per gallon of the gas, and the total price for all the gas pumped. The price per gallon is a constant. It can not be the independent or dependent variable. The number of gallons pumped is independent of the total price of the gas pumped. The number of gallons pumped is the independent variable.

By convention, the independent variable in an equation is on the right hand side of the equals sign. For example, in the equation y = 3x + 2, x is taken to be an independent variable. Its value does not depend on the variable y. However, the variable y does depend on the value of x. Since y is defined by the equation to take the value of 3x + 2. y is not an independent variable.

Dependent Variables

A dependent variable is a variable whose value depends on other variable(s). By convention, the dependent variable in an equation is all by itself on the left hand side of the equals sign. In the equation r = t - 1, the value of r is defined to be one less than t. So r is a dependent variable. Since the variable t does not depend on the value of r. This means that t is not a dependent variable.

Now look at the expression p - q + 7. Is p a dependent or independent variable? The answer is, we can't tell. If there is no equal sign, we can't tell if a variable is a dependent or independent variable. If we can't tell whether a variable is dependent or independent. The status of the variables is unknown.

Understanding Check 2

Given each equation, determine the independent and dependent variables. Click the (Click to Select Answer) text until the answer you choose is shown in the box. The click the 'Check Answer' button to check your answer.

g = a2 + 3a	a is (Click to select answer) the independent variable. the dependent variable. unknown. Check Answer That is correct. Since the variable a is on the right hand side of the equal sign, it is the independent variable. That is incorrect. Since the variable a is on the right hand side of the equal sign, it is the independent variable. Please click the (Click to Select Answer) text above until the correct answer appears. Then click the Check Answer button.	g is (Click to select answer) the independent variable. the dependent variable. unknown. Check Answer That is correct. Since the variable g is on the left hand side of the equal sign, it is the dependent variable. That is incorrect. Since the variable g is on the left hand side of the equal sign, it is the dependent variable. Please click the (Click to Select Answer) text above until the correct answer appears. Then click the Check Answer button.
y = 3x - 2	x is (Click to select answer) the independent variable. the dependent variable. unknown. Check Answer That is correct. Since the variable x is on the right hand side of the equal sign, it is the independent variable. That is incorrect. Since the variable x is on the right hand side of the equal sign, it is the independent variable. Please click the (Click to Select Answer) text above until the correct answer appears. Then click the Check Answer button.	y is (Click to select answer) the independent variable. the dependent variable. unknown. Check Answer That is correct. Since the variable y is on the left hand side of the equal sign, it is the dependent variable. That is incorrect. Since the variable y is on the left hand side of the equal sign, it is the dependent variable. Please click the (Click to Select Answer) text above until the correct answer appears. Then click the Check Answer button.
3t - v	t is (Click to select answer) the independent variable. the dependent variable. unknown. Check Answer That is correct. Since there is no equal sign, we can't tell if the variable t is dependent or independent. So it is unknown. That is incorrect. Since there is no equal sign, we can't tell if the variable t is dependent or independent. So it is unknown. Please click the (Click to Select Answer) text above until the correct answer appears. Then click the Check Answer button.	v is (Click to select answer) the independent variable. the dependent variable. unknown. Check Answer That is correct. Since there is no equal sign, we can't tell if the variable v is dependent or independent. So it is unknown. That is incorrect. Since there is no equal sign, we can't tell if the variable v is dependent or independent. So it is unknown. Please click the (Click to Select Answer) text above until the correct answer appears. Then click the Check Answer button.

Function Notation and Variables

When using function notation, such as f(x) = 3x + 4, it is easy to tell if a variable is a dependent or independent variable. Anything inside the parenthesis is an independent variable. The dependent variable is everything on the left hand side of the equal sign.

In the example f(x) = 3x + 4, the independent variable is x and the dependent variable is f(x).

References

1. McAdams, David E.. All Math Words Dictionary, variable. 2nd Classroom edition 20150108-4799968. pg 188. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pp 53-54. www.archive.org. International Textbook Company. January 1963. Last Accessed 1/12/2010. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n70/mode/1up/search/variable. Buy the book

More Information

• Understanding Algebraic Variables (video). dummies.com. Wiley. 1/23/2010. http://www.dummies.com/how-to/content/understanding-algebraic-variables.html.

Cite this article as:

McAdams, David E. Variable. 5/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/v/variable.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

5/13/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/17/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
11/25/2008: Corrected article title. (McAdams, David E.)
6/7/2008: Corrected spelling. (McAdams, David E.)
3/11/2008: Corrected invalid link. (McAdams, David E.)
9/7/2007: Initial version. (McAdams, David E.)