Common Core Algebra I.Unit 2.Lesson 1.Equations and Their Solutions.by eMathInstruction

So it's important to understand what an equation is. Let's take a look at the definition. The definition of an equation. An equation is simply a statement about the equality of two expressions. That statement might be true. It might be false, or it may be open. It simply takes the form expression number one equals expression number two. Now again, we should know what expressions are, right? Expressions or any combinations of numbers we know and numbers we don't know. All right? An equation is simply saying, hey, this expression equals this expression. It doesn't mean that they're actually equal. All right, it's just expressing inequality. Very, very important. Now, very often, you get a very easy question on this kind of topic, right? Take a look at exercise one. Which of the following is not an equation? All right, number one, three plus one equals four plus zero. Absolutely an equation. We're expressing equality between this expression and this expression. Is it true equality? I don't care, by the way, it is. But I don't really care. 

In exercise two we're saying X squared minus two X equals 8. Is that true? I actually don't even know, but that's definitely an equation. This expression is equal to this expression. Number four, one plus three equals 6. Definitely an equation. That's an expression. That's an expression, not a complicated expression. Now, by the way, that's a false equation, but that's not the point, right? The point is we're expressing some kind of equality between two expressions. Finally, number three is the correct choice. It's not an equation, right? All this is an expression. But it in no way is trying to establish equality between two things. All right? So the very, very important that you can distinguish between an expression, which is just a combination of numbers we know and numbers we don't know with addition subtraction multiplication and division, and when we set two expressions equal to each other. Okay? So I am going to clear that out. Make sure you know the definition of an equation. For a lot of people, it boils down to something as simple as this. If there's an equal sign, it's an equation. If there's no equal sign, it's not. But still. 

Yeah. Let's get some practice. Okay. Consider the equation two X -8 equals ten minus X why can't you determine whether this equation is true or false? Think about that for a moment. Pause. Right? Okay. Well, we can't determine whether this equation is true or false because we don't. Have a value. For X we don't have a value for X you know, so it might be true. It might be false. Right now we would call this an open equation. It's an open equation. Its true or falsehood can't be determined. Why it's called open? Eh. Eh, not very interesting. But let's take a look at letter B, it says if X equals 5 will the equation be true, how can you tell? Well, I'd like you to pause the video right now and think about how you can tell whether this equation is true or false if X is 5. 

Take a moment. Well, hopefully, what you did is you said, look. You know, if X is 5, then you can actually evaluate these two expressions. You can actually just evaluate them. So for instance, two times 5 -8, that almost looks like a two. But not quite. So I'm going to get rid of it. So two times 5 -8, well, is that equal to ten -5? I'm going to put a little question mark there. I'll order of operations. We get ten -8 equals ten -5, two is equal to 5. Well, no. All right? It's actually a false equation. There's nothing wrong with that, by the way. It's false. Whatever. So it's not true. It's a false equation because when we put 5 in there, we get something that is false. It's not true. Let her C says show that X equals 6 makes the equation true. 

Remember to think very carefully always about your order of operations. Pause the video for a moment and go ahead and do this. All right, let's work through it. Well, the way that we can always test to see if an equation is true or not, is to see if the expression on the left-hand side is equal to the expression on the right-hand side. So two times 6 is 12 -8, and ten -6, 12 -8 is four. Ten -6 is four. So true. And again, it's true because the expression on the left is equal to the expression on the right when X is 6. It's as simple as that. It may be equal for other values of X, it may not be. But it is certainly equal when X is equal to 6. All right, I'm going to clear this out, but I really want you to understand this. A lot of what we talk about in this unit is going to boil down to things being true or things being false. 

So, you know, you're going to have all these algebraic procedures adding, subtract and multiply and divide and rearrange in factoring all sorts of things. But at the end of the day, it's all about true and false. So I'm going to clear this out, copy down or anything you need to. And it's gone. All right. So let's talk about solutions to equations. Solutions to equations. A value for a variable is called a solution to an equation if, when substituted into both expressions, the results in the equation result in the equation being true. All right? In other words, if a value of X makes the equation true, it's called a solution. If a value of X makes the equation false, it's not. Okay. So let's play around with that. 

Now, I can't even possibly explain how important that is. I mean, I can't even begin to explain. Okay? So much of algebra is about finding the solution or solutions to equations. But ultimately, I should be able to give you any value of X and ask you, is this a solution to the equation? And as long as you know order of operations and arithmetic, you should be able to tell. So watch. Let's see if 7 is a solution to the equation two X plus three equals 17. All right, so I'm going to see if it makes this equation true. Two times 7 is 14. 14 plus three is 17, this is true, so yes. And by the way, the answer is yes. The answer is not true. True tells us the answer is yes. All right. Let's see if ten is a solution to this equation. All right, I'm going to get ten -20 divided by 5, and I want to see if this expression is now equal to negative four. Well, ten -20 is negative ten. 

Negative ten divided by 5 is negative two. Is negative two equal to negative four? No. That's false. Right? That tells us the answer is no. Right? A little more complicated. Let's try this one. Actually, let me pause. Why don't you try C and D? I know they're a little bit more complicated, but we've certainly evaluated expressions of this difficulty level. So evaluate the left, evaluate the right if they turn out to be equal, the equation is true, and if it's true, the net value of X is a solution. So pause the video now and take as much time as you need to on letter C and letter D. All right. Let's go through it. Let's see if four is a solution. All right, I'm going to put it in everywhere I see an X. We'll have two times 9 on this side. 6 times three on that side, two times 9 is 18, 6 times three is 18. That is true. And so the answer is. Yes. It is a solution. All right, let's try this one. 

Being very carefully. Negative one squared minus one. Two times negative one plus two. Negative one times negative one is one. Be careful on that. Try not to use your calculator. Two times negative one is negative two. One minus one is zero. Negative two plus two is zero. That is true. So yes, it is a solution. Very, very important to understand this. Now, in previous years, your teacher probably called this a check. Press it out. Okay, we'll learn this algebraic procedure. And then you got your final answers check. Okay? This isn't just a check. This is what it means for something to be a solution. A value of X, Y, Z, T, whatever is a solution to an equation, if it makes the left-hand side equal to the right-hand side if it makes the equation true. And it's not a solution if it makes the equation false. Very, very important. So I'm going to clear this out, pause the video if you need to. All right, here we go. It's gone. 

And moving on. All right, determine whether each of the following values for the given variable is a solution to the given equation. Show the calculations that lead to your final conclusions. All right, a little bit dice you're here. We got some fractions involved. But I know you can do this. So what I'd like you to do is try E and F on your own, and then we'll go through them. All right, let's do it. Let's see if two is a solution to this equation. All right, put it into the side. Let's see if it's equal to 5. All right, the numerator, two plus two is four. Minus one. C three times four is 12 divided by four minus one. 12 divided by four is three. Uh oh. Actually, I shouldn't say. Two is not equal to 5. 

That's a false statement. And all that means is that no. X equals two is not a solution to that equation. However, we saw a better no matter how ugly it looks. Two is not a solution. Let's see if a is a solution to this equation. At the same time, I'm going to talk to you a little bit about how to multiply by a fraction because it may have been a little while. I think it's a good skill to be able to do without your calculator. All right, let's talk about doing three-fourths times 8. Remember how you can do this. You can either do three times 8 and then divide by four, or you can do 8 divided by four and then times three. A lot of people will do that first because then we can do three times two and get 6. Smaller number. 

Now negative one-half times 8. That's shouldn't be that hard, the negative one times four is negative four. 6 minus one is 5. Negative four plus 9 is 5. That is a true equation. And that says that yes. X equals 8 is a solution to that equation. All right? That's it. If the left-hand expression is equal to the right-hand expression for a value of X, then the equation is true, and that value of X is a solution. Pause the video now if you need to. All right, we're going to scrub some text. Here we go. All right, last problem. Kirk was checking to see if X equals 7 was the solution to the equation four X minus three equals two X plus 11. He concluded that it was not a solution based on the following work. Was he correct? So take a look. What did Kurt do? Says no. But there is a problem. 

Now, it doesn't mean that he wasn't correct. It could easily not be a solution. But there's something wrong with the work. Right? What's wrong with the work? What's wrong with the work happens right here? And happens right here, but at least Kirk is consistent. Right? The problem is, four times 7 minus three, and two times 7 plus 11. The multiplication has to come first. We have to get 28 there. Right? We can't do this traction first, then do the multiplication because the subtraction is not in parentheses. Order of operations says we must do that multiplication first, and then look what happens. 28 minus three is 25. 14 plus 11 is 25. Yes, see, that's true. That means the answer. Is actually yes. All right, so that is a solution to the equation. 

But Kirk doesn't think it is because he doesn't know his order of operations very well. All right? So watch those. I'm going to clear this out. So copy down what you need to pause the video. All right, here we go. All right, let's wrap this up. So in this lesson, we saw something that was amazingly important. Actually, a few things. First, we saw just the idea of an equation. And the idea that an equation can be either true, false, or open, open, you just want to think about it as I can't tell whether it's true or false. It doesn't really even have value. Of true or false. But then we also learn maybe something way more important. What does it mean for something to be a solution to an equation? A value of X or whatever variable you're working with is a solution to an equation if when it's substituted into the equation, it makes that equation true. All right? So thank you for joining me for yet another common core algebra one lesson by eMac construction. My name is Kirk Weiler. And until next time, keep taking. And keep solving problems.