# Algebra

Algebra is a branch of mathematics in which arithmetic is extended to deal with unknown numbers or relationships using letters and other symbols. It is a broad branch that is used to some degree in almost all other areas of mathematics. As such, for those who seek to pursue virtually any higher study of mathematics, it is important to have a strong grasp of algebra.

The letters or symbols used in algebra are called variables. As a basic example:

? + 2 = 4

If the "?" is replaced with 2, we get 2 + 2 = 4, which we know to be true. In this example, ? is the variable. By convention, the most common symbol used to represent a variable in algebra is "x," though theoretically, it can be any symbol. Letters and other symbols that can easily be related to each other are commonly used in algebra because equations can involve the use of many variables. As the problems we are trying to solve get more complicated, it helps to have simple variables like x, y, and z that we can easily keep track of, rather than symbols like ?, &, %, that can also have various other meanings other than the variable they are used to represent.

## Elementary algebra

Elementary algebra includes concepts such as variables and algebraic notation, simplifying expressions, equations, properties of equality and inequality, substitution, solving algebraic equations, and more. These form the foundation that will eventually allow us to approach more difficult algebraic and mathematical topics.

Briefly, a large part of elementary algebra involves learning how to manipulate equations in such a way that allows us to solve for the variables in the equation. In the above example, we used a simple addition fact just to demonstrate the use of variables, but the problems we can solve using algebra can be signficantly more involved. A key aspect of algebra is learning about the many different symbols used in mathematics, and the rules for manipulating them.

Getting comfortable with using variables to represent unknown numbers, or numbers that can take on many values, allows us to tackle more interesting, complex, and relevant problems in a manner that is clearer and more concise than having to describe everything in words. As an introduction to the types of problems you may encounter in algebra, and the process necessary to solve them, below is an example of a word problem solved using algebra and equality.

Example

Tyler counted the empty spaces in his stamp book. He found that he needs to collect only 37 more stamps to fill his book, which holds 500 stamps. How many stamps does he have already?

To solve this problem using algebra, first write an equation that describes the problem situation. Let x represent the number of stamps that Tyler already has. We are adding 37 stamps to whatever number of stamps Tyler has (x) to fill his book of 500 stamps, so:

x + 37 = 500

We can solve for x by subtracting 37 from each side of the equation. This is because an equation is a statement of equality, so both sides must be equal. We want to isolate x, because we know that if we have a statement of x = some number, that that is our solution. Subtracting 37 from one side of the equation isolates x, but we must also subtract 37 from the other side of the equation, or the equation will no longer be equal.

x + 37 - 37 = 500 - 37

x = 463

Thus, Tyler has 463 stamps.