Taking the Mystery Out of Unknowns

Some people have trouble with variables. For someone like me who's more comfortable with variables than actual numbers, that sounds pretty ridic, but it's actually not so weird. Variables are strange. Up until this point, there was nothing abstract about math at all. You had numbers and you did things with them to get other numbers. Now suddenly there's all these LETTERS and shit, what the fuck are you even supposed to do? It's not as complicated as it might seem, though. In fact, variables are pretty cool!

What is a variable? It's something that represents a number, but isn't a number. We usually use letters like x, y, or t (these are the most common ones), but we sometimes use funky symbols like θ. Don't get tripped up. The symbol being used doesn't matter, the principle of a variable is still the same. It's still just a number with an unknown or changing value. In fact, they're frequently referred to as "unknowns" when you have lots of them that you're trying to solve for.

Let's start with the basics. For equations with only one variable, you can think of the variable as a question mark. In x + 4 = 5, we don't know what x is. 4 is 4, 5 is 5, and x is ?. It's some number that makes the equation true. In this specific case, of course, it's obvious that there's only one value of x that makes the equation true. There is only ONE number that, when added to 4, equals 5, so there is only one possible value for x.

But we also have equations where there are multiple numbers that can make the equation true. For example, x² = 4. This x is still a question mark. What number equals 4 when you square it? The obvious answer is 2. The maybe-not-so-obvious answer is -2, because squaring a real number gives you a positive result. So here, one variable can be two different numbers while still making the equation true.

There are some equations where there are an infinite number of answers that will make the equation true. Mind-blowing, I know, but this is pretty much the entire basis of algebra, here. For a trivial example, 0*x = 0 has an infinite number of possibilities for x. x can be literally anything and the equation will still be true.

For a more useful example, 2x = y has an infinite number of possibilities for x. x can be anything and y will always equal 2x. There is no value you can put in here for x that will make that untrue. In this case, we call x the independent variable, and y the dependent variable, because the value of y depends on the value of x.

You can have equations with multiple independent variables, too. x + y = z for example. x can be anything, and y can be anything, and your values for x and y don't affect each other. z, however, changes based on the values of x and y, so it is the dependent variable, while x and y are both independent variables.

Okay, Guindo, that's cool and all. But, you ask, why are variables important? Why can't we just write the numbers there? Why do we need something that represents a number but isn't one?

Lots of reasons. The biggest reason is because we don't always have static numbers to use for these things. We don't always know what x is, or we might want to be able to plug in different values for x to see what we get.

Let's say, for example, that you need to hire ninjas to kill your enemies. These ninjas charge 5 kittens a kill (they really like kittens). That means one kill costs 5 kittens, two costs 10, etc. So an equation for figuring out how many kittens you'll have to hand over to the ninjas is 5x = y, where x is how many enemies you're paying to kill, and y is how many kittens it will cost. If you need to have 5 enemies killed, you better have 25 kittens to pony up.

Or you could calculate the cost of groceries or whatever, but who cares.

The really tricky part about variables, the part most people seem to have trouble with, is that you can't treat them like numbers because you don't know what value they have. How do you know what you can and can't do with variables? Like anything else in math, we have RULES for that.

First, most important thing to keep in mind is that different letters mean different values. x is x and y is y and t is t, and you can't mix them because they all mean different things. Think of it as "apples and oranges." One variable stands for apples, the other stands for oranges, and you can't add them together.

The second thing to keep in mind is that you CAN'T divide by a variable UNLESS you know for a fact that it cannot be zero. Unless your problem says x ≠ 0, YOU CANNOT DIVIDE BY X because it might be undefined.

Here, have a handy dandy list format:

1. x + 2y cannot be simplified because they're not the same variable; don't write 3xy, it's wrong. x + 2x is 3x. This is called "adding like terms." It helps to remember that 3x means x + x + x (less abstract: 3 * 2 = 2 + 2 + 2)

2. On the other hand, you can MULTIPLY different variables together. x*y = xy. But, and this is VITAL you guys, xy terms are their own term. x + xy + y cannot be simplified. You can only add an xy term to another xy term. Same thing for exponents on variables, x² is x*x and that's different from x.

I feel like I have droned on about variables for long enough. Hopefully that takes some of the mystery out of why we're using letters to stand in for numbers all the time.

This post was pretty boring, wasn't it? Sorry, I was having problems with laplace transforms of heaviside functions all weekend while studying for tomorrow's differential equations test. If that sounds intimidating: it is.

Next time, meh, idk. Graphs, maybe.