Topic C: Replacing letters and numbers
Let's say that I'm working in a restaurant, and I'm making $10 per hour. But on top of my hourly wage, I also get tips each hour. So this entire expression, you can view this as how much I might make in a given hour. Now, you might also realize that the number of tips or the amount of tips I might make in an hour can change dramatically from hour to hour. It can vary-- one hour it might be lunchtime, get a lot of tips, people might get some big-ticket items. The next hour, I might not have any customers. And then my tips might be really low. So the tips part right over here, we consider that-- the entire word, we consider that to be a variable. From scenario to scenario, it can change. So for example, in one scenario, maybe it's lunchtime. I'm getting really big tips. So tips is-- let's say it's equal to $30. And so the total amount I might make in that hour-- we can go back to this expression right over here-- it's going to be 10 plus-- instead of writing tips here, I'll write 30 because that's what my tips are in that hour. And so that is going to be equal to 40. Let me do it in that yellow color. It's going to be equal to $40. But let's say right after that, the restaurant slows down. We're out of the lunch hour for whatever reason. Maybe the restaurant next door has a big sale or something. And so the next hour, my tips go down dramatically. My tips go down to $5 for that hour. Now I go back to this expression. The total I make is my hourly wage plus the $5 in tips, which is equal to $15. As you see, this entire expression-- the 10 plus tips-- it changed depending on what the value of the variable tips is. Now, you won't see whole words typically used in algebra as variables. We get lazy. And so instead, we tend to use just easier-to-write symbols. And so in this context, instead of writing tips, maybe we could have just written 10 plus t, where t represents the tips that we get in an hour. And so then we would say, OK, what happens when t is equal to 30? Well, then, we have a situation. t is equal to 30. This evaluates to 10 plus 30, which would be 40. What would happen if t is equal to 5? Well, then, this would evaluate to 10 plus 5, which is equal to 15. Now, I want to be clear. We didn't even have to use t. We didn't even really have to use a letter, although in traditional algebra, you almost do use a letter. We could have written it as 10 plus x, where x is your tips per hour. x might not be as natural. It's not the first letter in the word tips. Or you could have even written 10 plus star, where you could say star represents the number of tips in an hour. But it just might have not made as much intuitive sense. But hopefully this gives you a general idea of just what a variable is. All it is is a symbol that represents varying values. And that's why we call it a variable.