Algebra: An Introductory Note

Many people found themselves enjoying math right up until they encountered algebra. So, if algebra seems scary to you, then know that you are not alone. That said, you need to realize something: you already know how to do algebra. You may not believe it, but, really, you do.

Consider someone asking for the answer to 3+1. Certainly, this should not be an intimidating problem. It is probably second nature for you to add the numbers together and arrive at what they equal. An algebra problem with letters is not necessarily any more complex. Indeed, often they are very similar problems.

In fact we can ask the exact same question from above like this 3+1=x. The letter x on the right side of that equation is simply asking what the left side of the equation equals. It’s asking a question you can confidently answer, so don’t let it scare you simply because it formulates a simple question into algebra-speak (read: algebraically). “What does some variable (i.e. letter) equal?” is the quintessential question in algebra. All it really asks of you is to take what you know (in this case, 3+1) and make it relate appropriately (in this case, equal) to what you do not know (in this case, x).

If that is not startling enough news for you, there is more: that is the only question algebra ever actually asks: “What makes what you do not know relate to what you do know in the prescribed way?”

Incidentally, that brings us to a fine place to start – with a few definitions to help clarify algebra-speak.

A variable is something that has an unknown value. Variables are most often represented in mathematical equations with italicized letters. In algebraic equations, you try to determine the value of variables using other values that you know and mathematical calculations. In functions, variables are what you can assign values to in order to calculate some other dependent value. In expressions, variables represent the possibility for any one number in a given set. The letters x and n are both variables in the following equation: x + 10n = x + nx

An expression is an algebraic statement or definition. That is to say, it does not claim any definite relation between values and therefor has no “solution”; it simply says something. x + 10 is an expression that means “ten more than some number x”. Expressions do not make equality assignments like “equal to” or “less than”.

A function is a way to calculate some dependent value given other values. Variables act as place holders for the values necessary for computation. For instance, if you made ten dollars an hour, you could formulate your earnings with a function like f(h) = 10h. The left side of the equals signifies that you are encountering a function that takes some value that you give it and refers to it as h. In our example, the variable h is meant to be replaced by the number of hours you worked. 10h is the function body that would give you the amount of money made after working h hours. So, the function can only have a definite value if h is given a definite value, that makes the value of the function dependent on h.

An equation establishes a definite relation between some values. For example, 20 = 10x means that some number x times ten has some fixed relationship (namely, equals) to the number twenty. The unknown value of x can (usually) be calculated to satisfy the relation. (If no number could possibly satisfy the relationship, then it would have no solution.)