Variables, coefficients, and constants, oh my! It's time to start talking a little bit about algebra. Welcome to Anywhere Math, I'm Jeff Jacobson, and today...
We're going to talk about algebraic expressions. Before we start going over some examples, let's talk about some of our terms here. So first, what is an algebraic expression? Now you should remember what a numerical expression is, because we've talked about that before.
An expression that includes numbers and operations. That's basically it. There's no equal sign.
Remember, expressions have no equal sign. That's equations. An algebraic expression is different than a numerical because it includes variables.
So the definition, and you should write this down, an expression that contains numbers, operations, so if it just has those, then it's just a numerical expression. But if it also has one or more variables, then all of a sudden we're talking about an algebraic expression. So some examples, x divided by 7. We've got a variable, we've got an operation, we've got a number. Same thing here, 3x plus 4, xy.
You don't see the operation, but when they're next to each other, that means multiplication, so there is an operation. So those are all examples of an algebraic expression. Some that aren't.
If you've got something like 3 plus 7, that's not an algebraic expression because there is no variable. That would be just a numerical expression. Same thing if you've got...
3x plus 4 equals 10. That also is not. Even though it has variable operations, it's got numbers, it has an equal sign. Which means this is actually an equation, not an expression at all. So make sure you understand what an algebraic expression is and what it includes.
Let's talk about some more vocabulary words. Here is an algebraic expression, 5p plus 4. We have operations, we've got addition, 5p, that means 5 times p, so we have multiplication. We have numbers, we have a variable, so it's an algebraic expression. Now, let's talk about the different parts of this algebraic expression. First, are the terms.
And the terms are just the parts of the expression. So in this case, we have two terms. We have 5p and we have 4. So if you had to write down and list what the terms were, you would write, well, 5p is a term and 4 is a term. Basically, you just take out the operations and you list just the parts of the expression. Next, the coefficient.
And you can see that I kind of color coordinated here. A coefficient is the numerical factor of a term. right, one of the parts that contains a variable.
Well, the only term here that contains a variable is the 5p. The coefficient is the numerical factor. So it's the part of the term, it's the number, basically, part of the term. So in this case, in this expression, the coefficient here is five, you can think of well, it's just the number in front of the variable.
Okay, just which it is, right? Next, a variable. Well, what is a variable?
A variable is just a symbol. Usually, we write it as a letter, and oftentimes, it's either X or Y because of the coordinate grid, right? You have the X axis and the Y axis, which is why oftentimes you see equations or expressions with X and Y. But it can be any letter, any letter you want. And it represents one or more numbers.
It's called a variable because it can vary. In this expression, I can say, well, let's make p equal to 5. Or let's say p is equal to 3. Or p is equal to a million. I can choose whatever I want for that p. That's why it's called a variable.
It can vary. It can change. And so in this case, our variable, blue with the blue, our variable is p. And last but not least is the constant.
Now when you think of constant, things that are constant are things that don't change. Okay? That's something that's constant. Gravity is a constant.
Gravity doesn't change unless you get out into space, right? So this constant, it's a term with no variable. So here, our only term with no variable was the 4. That 4, no matter what I make p, that 4 will always be 4. It's not going to change. It is a constant.
So these are the different terms and things you should know about an algebraic expression. Let's do an example. Example 1, identify the terms, coefficients, and constants of this algebraic expression. First, let's start with the terms.
Remember, those are the parts. And if you want to try this on your own, go for it. Go ahead and pause the video.
The terms are the parts of this, so we kind of take the operations out. And the terms, well, we've got a term here and a term here. So the terms, you have 5x and you have 13. Next. The coefficients.
Well, coefficients, we're only looking at the terms that have a variable. So coefficients, I'm not even going to bother with that 13. I'm just going to look at the 5x. And then remember, it's the numerical part of the term that has the variable. So out of this 5x, the coefficient is just the 5. And last but not least, the constants.
Remember that's something that doesn't change no matter what the variable is. So in this situation the constant is at 13. Okay, it doesn't include anything next to a variable, right? That's the constant. It stays 13 no matter what.
Okay, here's something to try on your own. Example 2, simplify the expression. So I have 1.5 times h times h times h. So hopefully you notice right off the bat you have repeated multiplication, h times h times h. Well, we know a shortcut, something that we can simplify that into, and that's using exponents.
I have h multiplied by itself three times, so that's the same as h cubed. And 1.5 times all of that, well, I can get rid of that, this multiplication symbol, and just put them next to each other. Right? That still means the same thing. So if I write 1.5h cubed, it's exactly the same thing, just simplified.
Right? So that is what it would look like if you simplify that expression. 1.5h cubed. And in this case, 1.5 would be your coefficient. Good.
Here's some more to try on your own. Here's our last example. Evaluate 3x minus 14 when x equals 5. So now in this situation, they're telling us exactly what the variable is going to be equal to, and that's going to be 5. So my first step is to... Substitute. To evaluate, I'm trying to find the value and in order to do that, I need to substitute that 5 in for x because it's saying x is equal to 5. So what I'm going to do is I have 3. Now a good habit to get into is when you substitute, use parentheses.
All I'm doing is replacing that 5 for the x. I'm substituting it in. So 3. times 5, remember 3x, that means 3 times x, so I have 3 times 5. I still have the minus 14. And now it's just order of operations. I have multiplication and I have subtraction.
Multiplication comes first, so 3 times 5 is 15, minus the 14 is 1. And that is my final answer. Here's some more to try on your own. Thank you so much for watching and as always if you like this video please subscribe.