Common Core Algebra II.Unit 1.Lesson 1.Variables, Terms and Expressions

All right. Well, the first thing that we want to take a look at is a variable. A variable is a quantity that's represented by a letter or assembled that's unknown, unspecified, or can change within the context of a problem. So you've been working with variables for a while, a lot of students just think of them as letters, which is understandable. You know, X, Y, Z, and T are probably three of the most common that we have. But even in this course, we're going to introduce some symbols like theta and alpha from the Greek from the Greek alphabet. And we have these things because there's a lot of times when we when we want to be able to manipulate things that have value in other words that have numbers attached to them. But we either don't know what those numbers are or even more important in this course, those numbers can change. All right? Now from a variable, we build up to what's known as a term. A term is a single number or a combination of numbers and variables using exclusively multiplication or division. So in other words, something like this, which is three times X, that's a term. 

Something like this, X plus three, that's not a term, because there's no additional loud in a term. Now we can have more than one variable more than one number. We could have 5 X, Y squared. And that's all multiplication. Because that's 5 times X times Y times Y we could even have something like this. X, Y to the 5th, divided by three Z cubed. That would also be a good term. All multiplication all division. Now we can even make more interesting things by combining terms into what are known as expressions. An expression is simply a combination of terms using addition or subtraction. So for instance, if we had our term from up above three X and we added to it, the other term, 5 X, Y squared, this would now be an example of an expression. Don't get me wrong, an expression can definitely just contain one term. So for instance, 5 X that's a pretty, pretty good expression. It's just fairly boring. There's not a lot going on. So you have variables. If you have terms and you have expressions. 

So what we're going to now do is we're going to start using these three concepts in a variety of different review problems. Pause the video now if you need to and write down anything that's on the screen before we move on. All right, here we go. Exercise number one. Let's get into it. Consider the expression two X squared plus three X -7. Letter a asks us how many terms does this expression contain? Just consider this for a moment. How many terms? All right. Well, terms are all about multiplication and division. So we've got that two X squared. That's a term. It separated from the next term by that plus, or that addition. And that three X is then separated from the last term by the subtraction. So we have one, two, three, three terms that are combining in this expression to create the overall structure of it. Now, the point about expressions is that they have values that depend on the variables that are within them. So letter B says a value this expression evaluate, which means to find the value of you see the word value in that word right there it is. Right there without the E but find the value of. 

So evaluate this expression without your calculator when X is negative three, show your calculations. Now the little comment because this is our first video for common core algebra two. You'll see many times that I'll say do the following without your calculator. Now many students will still grab their calculator and do it. But when I say to do something without your calculator, it's almost like it's a piece of advice. It's saying to you, look, there's something within this that you're going to review if you do it without your calculator. But if you do it with your calculator, you'll miss out. So pause the video right now and see if you can figure out what the value of this expression is, what's known as a trinomial, because there's three terms. What the value of this expression is, when you substitute X equals negative three. 

All right. Well, what you miss when you do this on your calculator is really thinking a little bit about order of operations, signed numbers, things like that. So let's just go through the evaluation. I'm putting negative three and for each one of these things. Now remember my order of operations insists that I do this exponent first. So a negative times a negative, as we all know is a positive, so negative three times negative three positive 9. I'm going to leave the rest of it unevaluated for right now, and then we'll go back. In order of operations, then these two products will come next. These two multiplications will come next, so two times 9 is obviously 18. Three times negative three is negative 9. And then we have -7. Now everything from here on out is addition and subtraction. So we just have to do it left to right. 18 plus negative 9 is positive 9. And then 9 -7 is a positive two. So that expression has a value of two, when X equals negative three. All right, last little piece letter C what is the sum of this expression with the expression 5 X squared -12 X plus two. See if you can remember how to do that. 

Pause the video for a moment. All right. Well, what I'm going to do is I'm going to write my trinomial. Two X squared plus three X -7 immediately underneath this trinomial, because really this is no different than adding three or sorry, two, three-digit numbers. Like if I had something like this. Right? I'm going to do it exactly the same way. Two plus negative 7 is negative 5. Negative 12 plus positive three is negative 9, X and 5 X squared plus two X squared is 7 X squared. All right, so that's what I get when I add those two expressions together. In the past, you would have called this combining light terms combining like terms. And we'll talk about that a little bit more in a second. Pause the video now if you need to and write down any of this text. All right. Clearing it out. Light terms. Two or more terms that have the same variables. Raised to the same powers. All right? And we want to be able to combine like terms and we want to be able to do them quickly. 

Take a look at exercise too. It says most students learn that to add two light terms, they simply add the coefficients and leave the variables and powers unchanged. So take a look just real quick before we finish reading the rest of the text. These two are like terms. They both have X and Y in them. They both have X to the second and Y to the first, so they're light terms, same variables, same powers. And to combine them, we simply say, oh, I got four, 6, so I have ten of them. Well, why does it work? Below is an example of the technical steps to combine two like terms. What real number property? Oh, let's see if you can remember this. Justifies the first step. What justifies the fact that we can sort of take that X squared times Y, which is multiplying both the four and the 6. And bring it out and just add the four and the 6. Do you remember what property that is? It's what's known as the distributive property. 

And we're going to review all three of the major real number properties in just a little bit. I can't write distributive apparently. Oh, let me erase that a little bit. It's been a little while since I've written on the pad. The distributive property. All right, well, look at that property as well as two other important ones in just a moment. All right, so pause the video if you need to for a moment, and then we're going to move on. Okay, here we go. The real number properties. Everything about algebra starts from these three properties. Because these three properties tell us how real numbers the numbers that you're used to using. Interact when it comes to addition. Really and subtraction. Multiplication and division. So here we go. Let's review them. If a, B, and C are any real numbers, then the following properties are true. All right? The commutative property. A plus B is the same as B plus a, we all know this, right? Three plus four is the same as four plus three. They're both 7. Write a times B this one's a little bit trickier, I think, for most kids to really understand. But four times three is 12. Three times four is 12. They're the same thing. All right? 

Number two, the associative properties of addition and multiplication. This is what happens when we add three or more numbers together or multiply three or more numbers together. Basically the associative property says, it doesn't matter which order you do them in. So if I was doing three plus four plus 5, I could add the three and the four together first get 7, add 5, get 12. Or I could add the four and the 5 together first. Get 9. Then add three and get 12. And the same is true with multiplication. Finally, the one that we just saw the distributive property of multiplication and division over addition and subtraction, wow, that's a word you want. Basically just says if we're adding or subtracting two numbers. And then we multiply that sum, we can multiply each portion first and then add or subtract. That's what we used in that last property, but we kind of went in this direction instead of this direction. 

The same also works for division, and that's very, very important, because we'll eventually have some expressions even in this lesson, where we have to distribute division. We have to divide each term in an expression, and that's really what we're doing. Each term in an expression by whatever the divisor is. All right, these properties, these three properties. The commutative associative and distributive properties lay the groundwork for almost every algebraic manipulation that we do. All right? So I'm going to clear out the screen. There's a lot to write down here. And let's get back into some problems. Let's see what we have. Exercise number three, the procedure for simplifying the linear expression 8 times two X plus three plus 5 times three X plus one is shown below. State the real number property that justifies each step. Well, you've got your three properties. You've got your commutative, your associative, and your distributive. 

Why don't you pause the video right now? Take a look at each little manipulation and ask yourself what did we do? That justifies each step in this manipulation. All right, let's go through them. So in the first step, what we've got is we've got our original expression. And it looks like we took the two X and we multiplied it by 8, and the three and we multiplied it by 8. Same way over here. That is our friend, the distributive property. The distributive property gets used a lot. The distributive property. Now, when we're looking for the next step, what I want to do is I want to compare this to this. Now, what exactly did we do? Well, this says, look, I know that I had two times X and then times 8. But what I'm going to first do is multiply the 8 and the two first. And I'm going to multiply that 5 and three first. Okay? And by grouping those together instead of grouping the two in the X and the three in the X, what we're doing is we're using the associative property. Okay, we're allowing us ourselves to do that multiplication first. 

So, very hard to write and talk at the same time. But doable all right, so now what's the next thing I want to do? The next thing I want to do is really compare this to this. And what do I have here? I have 16 times X plus 24 plus 15 times X plus 5. And now what I've done is I've actually flip-flopped these two terms, so I have 15 X plus 24. Instead of 24 plus 15 X that is the commutative property, the commutative property. Technically of addition. But we're just going to call it the commutative property. Then I want to compare this line to this line and look what I did. I somehow took that X, which is multiplying both the 15 and the 16. And I brought it out here, if you will, allowing me to add these two. Now you might look at that and think, ah, that's the associative property, but that's actually the distributive property. The distributive property works both ways. The distributive property certainly, if we had the X sitting here and we multiplied it through, would give us 16 X plus 15 X, here we're just running it in reverse. 

Now, here to here, what did we really do? Well, technically speaking, we really have 31 X plus 24 plus 5. We should really do the 31 X plus 24 first. But there's not much we can do with that. So we're going to decide instead to take the 24 and the 25. And we're going to combine them together. And this is going to give us the associative property. Having a real hard time with that. All right. In other words, if I can just group those together, I will. Now, this last one didn't really need a line. Not really sure why I had one there. But we don't need one there. There is no property that justifies why 24 plus 5 is 29. It's just the way it is. All right? So each step in a simple manipulation, which you probably won't be thinking of as you do the manipulation though, does have some basis with just the way numbers behave. All right? And the more solid you are on those laws. It's not necessarily what their names are, but just how they work. 

The better you'll be at manipulating algebraic expressions. Anyway, pause the video now and write down anything you need to. All right, I'm clearing out the text. Let's take a look at exercise four. Because we reused real number properties to transform the expression 8 times two X plus three plus 5 times three X plus one into 31 X plus 29. These two expressions are equivalent. How can you test this equivalency show your test? Now let's just make sure we understand what expressions being equivalent means. That means that they'll have the same value for every. Input value. So it doesn't matter what X I choose. Those two expressions will have the same value. So technically, we can test that equivalency by grabbing any X value we want. Okay? So grab an X value. Make it simple. Don't use your calculator and figure out what each one of those expressions is equal to. 

And then I'll choose my own, okay? All right, let's go through it. I'm going to take the value X equals two. I wonder what you used. And I'll do the hard one first. The first thing I'm going to do is figure out what this expression is equal to when I put in X equals two. Now I know it's a little bit annoying. But here we go, right? I'm going to use a dot for multiplication right now. I hope that's okay. You should have seen that before. I have to remember my order of operations. My order of operations says work within the parentheses first, and within the parentheses what I have to do is that multiplication first, so two times two is four. Over here, I have three times two, which is 6. Now again, order of operations tells me I have really got to do what's in parentheses first. So four plus three is 7, and hey, I have another 7. Right, and then I'll have 8 times 7, which is 56. And then I'll have 5 times 7, which is 35. I'm doing all of this without a calculator. So let's figure out what we have. We have 91. 

All right. Let's try X equals two in the expression. 31 X plus 29, much simpler, right? I mean, if I had to evaluate a lot of these things, I would much rather use that simpler expression. Anyway, let me do 31 times two. That's going to give me 62. And then I'll add 29. And look at that. Now, of course, this doesn't prove that those two expressions are equivalent. Not in any way, shape or form. But if we grab the value of X and we substitute it into both and they gave us different values. Then we would know that we must have made a mistake manipulating the expression to simplify it. All right? Or we made a mistake in evaluating the expression. That's the kind of tricky thing there. All right. Well, pause the video and write down what you need to before we clear out and finish up. All right. Let's get rid of this text. 

Okay. So today, what we saw was we saw three very important ideas, variables, terms, and expressions. And we also reviewed the three fundamental rules of real numbers, right? Or three fundamental properties of real numbers. The commutative property, the associative property, and the distributive property. All of today's lesson really should have been a review from common core algebra one. But as I said, we do want to lay a firm foundation as we move forward in this course. All right. While I'm glad to be back, I'm glad to be recording more videos. I want to thank you for joining me today for another common core algebra two video by E math instruction. My name is Kirk Weiler, and until next time, keep thinking and keep solving problems.