When a problem contains a word for an unknown number or quantity, we use a variable to represent that unknown number or quantity. We can replace any of the words or values with x. Or y, or z, or just about any other letter. We could also replace them with some other variable, but these are the three that get the most play. They must have great agents.
Here are some English phrases translated into mathematical symbols.
Sometimes a problem already contains a variable, and asks for some other quantity in terms of that variable. In that case, we don't need to plug in and solve anything. Instead, we only need to describe the quantity using an expression that has the variable in it. It comes in handy when your mother storms into your room demanding to know how many chocolate chip cookies you ate, and you can tell her, "x – 3." She can't possibly deduce from that remark how many you ate, and you'll be off the hook. We promise.
Alice has x dollars. Tanya has 3 dollars more than Alice. How many dollars does Tanya have?
Tanya has (the amount Alice has) + 3, which is x + 3 dollars.
This may actually be good news for Alice. She'll be in a lower tax bracket.
Sometimes a problem doesn't come right out and say there's a variable, but there is still one hiding in there somewhere. When all else fails, check under the bed and in the closet.
Dwight has some brownies, but Bill has three times as many brownies. Lucky Bill. Express the number of brownies Bill has in terms of the number of brownies Dwight has. Not to make Dwight feel bad about it or anything, just so's we know.
Bill has 3(the number of brownies Dwight has)
We need a variable to express the number of brownies that Dwight has. Since the problem didn't tell us what letter to use, we can use whatever letter we want. Let's use D for "Dwight." Bet you didn't see that coming. Then we can answer the question by writing "Let D be the number of brownies Dwight has. Then Bill has 3D brownies.'' Wow. They're so real-looking you could almost reach out and touch them.
When a problem doesn't tell us what letters to use for variables, we can pick our own. This situation is great for us, because we have control issues. When we do this, it's very, very, very important to write down—somewhere, anywhere—the meanings of the letters that we're using. Otherwise, we might arrive at the end of the problem and not remember if x is height, speed, or number of peanuts per mile. One would hope that you would never forget something as important as the variable that stands for peanuts per mile, but stranger things have happened. For example, there are times when even we don't know what we're talking about.
If a student answered the previous problem by saying "Bill has 3D brownies'' but never bothered to say what D was, someone could be confused. Either the teacher trying to read and grade the homework, or the student trying to use the homework to study for a test later. With enough time (and brownies) you can probably figure it out, but the idea is to not leave it to chance. This is one of those little details that can make math way easier than it would be otherwise. You'll be on your way to the next problem, while your friends are busy trying to figure out what x means. Man, there's nothing you love more than leaving your friends in the dust.
Be Careful: Whenever you introduce a new variable, write down what that variable means so you can remember when you come to the end of the problem. Don't trust your brain to remember. It has failed you before, and it will fail you again.