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Variables and Constants in Algebra

by Ron Kurtus (10 January 2008)

Algebra is the branch of mathematics that uses letters to represent numbers. Usually, the letters are called variables, since they are unknown and can change or vary. Some letters can be designated as constants, since their values remain constant, once assigned. Often letters are used as constants simply for the sake of convenience over using a large or complex number. Typically, numbers toward the end of the alphabet—such as x, y, z—are designated as variables and those toward the beginning of the alphabet—such as a, b, c—are designated as constants.

Questions you may have are:

• How are variables used?
• How are constants used?
• When do constants replace large numbers?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Variables

A variable is a letter that can represent a range of numbers, depending on its usage. The most common designation of variables are x and y, since they also represent the axes on a graph.

As an example of variables, consider the equation 2x + 3y = 12, where x and y are variables. That means that x and y can vary over a wide range of numbers. Solutions to the equation include x = 0 and y = 4, x = 3 and y = 2, and x= 6 and y = 0.

Special case situation

In a trivial or identity equation, such as x = 3, x would be a constant. But if the equation was taken in the context of a graph with x and y axes, solutions would include x = 3 and y = 0, x = 3 and y = 1, etc. This is a special case, and although x has a constant designation, x and y are still considered variables.

Constants

A constant is a letter that represents a fixed number. Typically, letters in the first half of the alphabet are used as constants. It is a good idea to let the reader know if a given letter is a constant, so that there is no confusion.

A good example of using constants in an equation is the general form of a quadratic equation:

y = ax² + bx + c

where

• x and y are variables
• a, b and c are constants

This means that you could insert any values for a, b and c and this equation would still be a quadratic equation and follow the rules for its solution.

Complex constant

In some cases, a constant is used for the sake of convenience if it represents a large or complex number.

Boltzmann constant example

For example, the scientific equation for the pressure of an ideal gas is:

p = kNT/V

where

• p is the pressure of the gas (a variable)
• N is the number of molecules in the gas (a variable)
• T is the temperature of the gas (a variable)
• k is the Boltzmann constant

N and T can vary over a wide range, giving the solution to the equation, p, also a varying range of values. Thus, they are all variables. Meanwhile, k is a fixed or constant value of k = 1.38*10^-23. You can see it is convenient to represent this complex number by the constant k in the equation.

Speed of light example

Another example is Einstein's famous E = mc² equation, which shows the relationship between the maximum energy for a given mass. In this equation:

• E is the energy variable
• m is the mass variable
• c is the speed of light, which is a constant or fixed number
• c² is c-squared or c times c

Since the speed of light is 186,000 miles per second or 300,000 kilometers per second, it is much more convenient to simple designate it as c in the equation.

Summary

Since Algebra uses letters to represent numbers that can vary, those letters are called variables. Some letters can be designated as constants, since their values remain constant, once assigned. Often letters are used as constants simply for the sake of convenience over using a large or complex number. Typically, numbers toward the end of the alphabet are designated as variables and those toward the beginning of the alphabet are designated as constants.

Answers to Readers' Questions

Vary your outlook on things

Resources

The following resources provide information on this subject:

Websites

Algebra Resources

Books

Top-rated Algebra books

Miscellaneous

Mini-quiz to check your understanding

1. Why would x and y be variables in the equation y = 2x + 1?

2. Why should you make sure you let people know that a letter designates a constant in an equation?

3. Do you have to use a constant to represent a large number in an equation?

If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.

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