**Introduction To Variables**

A variable is a symbol for a number whose value is unknown. A variable might represent quantities at different times. For example if you are paid by the hour for your job and you earn $10 per hour, letting *x* represent the number of hours worked would allow you to write your earnings as “10 *x* .” The value of your earnings *varies* depending on the number of hours worked. If an equation has one variable, we can use algebra to determine what value the variable is representing.

Variables are treated like numbers because they are numbers. For instance 2 + *x* means two plus the quantity *x* and 2 *x* means two times the quantity *x* (when no operation sign is given, the operation is assumed to be multiplication). The expression 3 *x* + 4 means three times *x* plus four. This is not the same as 3 *x* + 4 *x* which is three x’s plus four x’s for a total of seven x’s: 3 *x* + 4 *x* = 7 *x* .

**Checking Your Answer**

If you are working with variables and want to check whether the expression you have computed is really equal to the expression with which you started, take some larger prime number, not a factor of anything else in the expression, and plug it into both the original expression and the last one. If the resulting numbers are the same, it is very likely that the first and last expressions are equal. For example you might ask “Is it true that 3 *x* + 4 = 7 *x* ?” Test *x* = 23: 3(23) + 4 = 73 and 7(23) = 161, so we can conclude that in general 3 *x* + 4 ≠ 7 *x* . (Actually for *x* = 1, and only *x* = 1, they are equal.)

This method for checking equality of algebraic expressions is not foolproof. Equal numbers do not *always* guarantee that the expressions are equal. Also be careful not to make an arithmetic error. The two expressions might be equal but making an arithmetic error might lead you to conclude that they are not equal.

Practice problems for this concept can be found at: Algebra Introduction to Variables Practice Test.

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