Operations on Variables

A variable is a letter or symbol that can hold any numeric value. It gets its name from the fact that it can vary across a set of numbers. For example, if the letter x is used as a variable, the letter x can stand for 1, 2.4, 3.1415, -17, 0, -1,000,000 or any other number. A variable can hold any value.

Another type of number besides the variable is the constant. A constant can also be a letter or symbol. If it is a letter it is usually capitalized. Unlike the variable, the constant can only be one value. For example, if C represents a contant then C could be 2.71828 but could not be any other number. In other words a variable can change from one number to another but a constant cannot change.

An expression is how numbers relate to one another. For example, 2x means 2 times the value contained in x. Also, x + 1 means that 1 is added to whatever x represents.

When we set one expression such as 2x to the same value as another expression such as x + 1, we have an equation. The equation of 2x is equal to x + 1 is written as

When a variable is set inside an equation, it is possible to determine what the value of this variable is. To do this, the variable has to be on one side of the equation by itself. Many times, this will create a new expression on the other side of the equation. In the case of 2x = x + 1, we subtract x from both sides to get x = 1. The steps are as follows:

Notice the -x cancels the x on both sides to leave the final answer. Since 2x = x + x, the steps can be written as:

One other way to cancel the x variables would be to divide both sides by x. For instance:

In this case, x is to one side but is not by itself. Sometimes cancellation has to be tried by trial and error. By subtracting both sides, x was solved faster. However, 2x = x + 1 can still be solved with cancellation by division. The steps are as follows: