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# Variables and Constants in Algebra

by Ron Kurtus (revised 17 August 2012)

** Algebra** is the branch of mathematics that uses letters to represent numbers. Usually, the letters are called

*, since they are unknown and can change or vary.*

**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—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 include:

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

This lesson will answer those questions.

## 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 = ax ^{2} + 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)**k**is the Boltzmann constant**N**is the number of molecules in the gas (a variable)**T**is the temperature of the gas (a variable)**V**is the volume of the gas (a variable)

**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.

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## Variables and Constants in Algebra