Welcome back, everyone. In previous videos, we saw how to solve problems like 2 times 3 plus 5, which involved numbers and operations only. These are what we call numerical expressions. In this video, we're going to talk about algebraic expressions, which are a little bit different, and this is, unfortunately, the part of math where we're going to start to mix up numbers with letters of the alphabet. I know that sounds scary at first, but I promise I'm going to break it down for you, and I'm going to show you that algebraic and numerical expressions are actually very similar. So first, let's get started with some basic vocabulary here.

What is an algebraic expression? Well, whereas something like 2 times 3 plus 5 was numbers and operations only, in this case, where we have 2x + 5, we have numbers, operations, and these things called variables. This letter x is called a variable, and so an algebraic expression is really just a combination of numbers, variables, and math operations. This variable here, this letter x, is just a letter that can represent any number. The idea here is that this x could be 3, but it also could be negative 2. It could even be 0, something like that. So it stands in place for any number, and the idea is that this value varies. That's why we call it a variable. And usually, the letter that we'll use in this course is the letter x, but later on, we'll see some other variables as well.

Now let's keep going here. This number 2 that sort of sits in front of this x is called the coefficient. So a coefficient is just a number that goes, let's say, attached to a variable, and when you see it in front, it means that it's multiplying the variable. Unlike the x, this 2 doesn't change. It can't be a 3 or a negative 2 or a 0, so the value does not change for coefficients. And usually what you'll see is that coefficients go at the beginning of your algebraic expressions.

Now last but not least, we have this 5, and this 5, unlike the 2, is a number that is without a variable. It's not attached to an x. And just like the 2, its value doesn't change. This 5 can't become a 2 or a negative 3 or something like that, and this is called a constant. Constants are usually seen at the end of your algebraic expressions, but that's basically all there is to it. It's numbers, operations, and variables. That's what makes up an algebraic expression.

Let's get some more practice here so we can see what kinds of things are expressions versus what aren't. So in our example here, we're going to determine which of the following are expressions, and we're going to identify coefficients and constants. Let's get started.

In part a, remember, what we're looking for here is numbers, operations, and variables. So here I have a number, which is 4. I've also got an operation, like a plus sign, and I have a variable, which is x. So is this an expression? It certainly is. It is an algebraic expression. Now remember, now we have to figure out coefficients and constants. Coefficients are numbers that multiply variables. So, which one do you think it is? Is it the 4 or is it the 8? This 4 is attached to this sort of square root symbol of x, but it goes in front and it multiplies, so this thing is going to be your coefficient. And this constant, or this 8 here that's by itself, is not attached to a variable. It's by itself, so this is going to be your constant.

So let's look at part b now, this 3 ( 14 + 5 ) / 6. So I definitely got numbers, and I definitely got operations like multiplication, addition, and division even. What I don't have here is a variable. So because I have no variable in this expression, it's actually not going to be an expression. This is just a good numerical expression. It's not algebraic.

So let's move on now to part c. Part c is 2 - 3xy. So we have numbers, we have symbols or operations like subtraction and even multiplication over here, and we've also got variables. In this case, we actually have 2 we have x, but we also have other letters like y that can also be variables. So this definitely is an algebraic expression. Now what's the coefficient? What's the constant? So do you think it's the 2 or the negative 3 or the 3 that's over here? Well, hopefully, you realize that the 3 is the one that's attached to the variable. So this is going to be your coefficient over here, and this 2 is going to be your constant because it's off by itself without a variable. Now you might be thinking, well, usually, constants will go at the end, and that's true, but this is actually a perfectly valid algebraic expression. Usually, constants do go at the end and coefficients go at the beginning, but you can actually just see them in any order, and this is perfectly fine.

Last but not least, we have 9 x = 18. I have a number over here, and I also have multiplication, so that's a symbol, and I also have a variable over here. So is this an algebraic expression? Well, it would be, except for this equals sign. Basically, what you need to know here is that when expressions have an equals symbol between them, it actually forms what's called an equation. And all you need to know for right now is that equations are not actually considered algebraic expressions. We'll talk about them much later on, but this is actually just going to be an equation, so it is not an expression. Anyway, that's the basics. Let's keep moving on, and thanks for watching.