 So we've already gone through some of the rules of, you know, how you add and subtract. If you have two things added, subtract it together with equal sign, how you would solve for a variable. We've done multiplication division. We've done exponents and radicals, right? And we talked about cross multiplication, where it's just sort of a tool that you can use to help you out in solving equations, when you get larger equations, more complicated equations. One thing we really haven't talked about is what's the big deal about solving equations. Why are we learning this? And this is something that is rarely talked about before you get into this stuff. Usually, you know, in most curriculums that I've seen, they actually, you know, teach you how to solve equations. And they tell you what it is that you're solving for, why you learned this technique, right? It's like adding and subtracting, you know, think about solving equations as, you know, the first time you learn how to add and subtract or multiply and divide, right? You had to, you know, you're pretty rusty at it at first, and then slowly, once you got it down, it was, you know, just like riding a bicycle, you don't forget how to add, you don't forget how to subtract. You might forget how to multiply and divide, but adding and subtracting, you never forget, right? Solving equations is the same thing. This is sort of the core of mathematics, when you're going to analyze functions, where you have to solve equations. So solving equations, the reason we're going to learn this technique is, you know, if you can think about every equation as basically a function. For example, we've already talked about simple equations like 2x plus 1 is equal to 0, right? And we solve for this. You bring the 1 over, it becomes negative 1 divided by 2, so x is equal to negative a half. But what is it exactly here that you're solving for? This isn't unknown. It could be anything. But the way it works is, every equation that you see, when you see 0 on this side, think of this as another variable. This is your function that you're forcing for it to be equal to 0. Now some of you guys, some of you haven't gotten into functions yet. Don't worry about it. We will get into it at some point. But basically, functions are an equation you come up with where you say, this thing varies based on this. So this 0 here, you can think of as y. Okay. Better yet, better symbol than y is f of x because it gives you more information. So for those of you who've gotten into functions where you're using f of x, stop using y, this is way more powerful. And later on, we will talk about this. But for now, while we're solving equations, the way you should think about this is, you're solving this equation for when y is equal to 0. So what you're doing is, you're forcing this thing to be equal to 0 and then solving for when, what x has to be for y to be equal to 0. And in the first series, we talked about x and y. And x and y are just a coordinate system. So this is your Cartesian coordinate system. You have your x-axis here. You have your y-axis here. And your y-axis is really f of x. Hopefully you can see this. And your y-axis is f of x. So what you're doing here, if your function was y is equal to 2x plus 1, usually you put the y on this side, by the way, but we're doing it this way for now. So y is equal to 2x plus 1. This is a linear equation. A linear equation means a line for those of you who've gotten into linear equations. One question that I've asked a lot of students in the past is, what's a linear equation? A linear equation is a line. So this graphs a line that we will get into as well later on. But what this does is, you're forcing y to be equal to 0. We've talked about before in the first series, when y is equal to 0, you're on the x-axis. If this is a coordinate system, if you take a point here, that's your x and your y. Well, y is positive here going up, negative here going down, x is negative this way, positive that way. At the pivot here, you're at 0, 0. This 0 means the y. When you move this way, the y doesn't change, it's still 0. So when you're solving for equations, what you're doing is, forcing y to be 0, and you're asking yourself, when is this function, when is this equation, equal to 0 for what values of x? And what that gives you is, where you cross the x-axis for your function. Now, this is a line, the y-intercept is 1, and your slope is going to be 2 over 1, 1, 2 over 1. Your y-intercept, your x-intercept is going to be here, which is, if we solve this equation for y is equal to 0, we're forcing y to be 0. That means we're forcing y to be 0, and we're trying to find out what x is when y is equal to 0. And when we solve for this, we're going to get a negative of half, and that's where the x-intercept is. So for all of these questions that we're getting, when they're asking you to solve an equation, what they're asking you to do is solve for the x-intercepts. And there are numerous, numerous ways, later on in high-level mathematics, all they're going to say is solve for this equation. But there are different terms that mean the same thing. Solving is the same thing as finding the x-intercepts, finding solutions, finding roots when it comes to quadratic or higher-degree functions or higher-degree equations. So solve, let's see, do we have enough room here? Solve means x-intercept, and finding the roots means finding the zeros. All of these things are asking you the same thing, and solving is getting a solution for something. All of these things are asking you the same question. They're asking you to find out when you're crossing the x-axis, no matter how complicated the function gets. This is just a linear equation that could give you a quadratic equation. So when they give you a function like this, this represents a quadratic function. The quadratic function is, for this it would be a problem, because that thing is positive. What you're doing is, when they give you a question like this and they say solve for this, and in general functions put over here, when they actually want you to solve for this, they'll put the zero on this side. They mean the same thing, it doesn't make a difference if you put equal zero here or equal zero here. So what they would do is say, solve this equation, and what you're doing is, try to find out what x is when y, when the function is equal to zero. So when you're crossing the x-axis. And that's what we're doing when we're solving equations, and the tools that we learned of how to move things around when they're added, subtracted together, multiplied, divided together, or exponents and radicals. Those rules that we learned are going to help us, you know, solve equations which are going to be later on when we're going to start talking about functions. They're going to have meaning for us when we're trying to interpret the functions. Now, that's why we're learning this stuff, that's why we're about to go into this whole series of solving different types of equations. And what I'm going to do is, I'm going to go through this stuff really fast, okay, because there's a lot of material. When I decided to talk about this, I opened up a gigantic can of worms. Because, you know, you can start with linear functions, go to quadratic functions and cube functions and higher degree functions. And, you know, every level is just a little bit more complicated. A few more rules get added on for you to be able to solve for the equations, because it's not as simple as just getting x by itself on this side, right? Because if you remember your exponents and radicals, you can't add this guy and this guy together. So, it's just by, you know, moving things around, you're not going to be able to get x equals the number. This has two solutions, or a maximum of two solutions. Or, yeah, maximum of two solutions. There's different techniques for solving this equation than there is for linear functions, such as, you know, two x plus one is equal to one, or is equal to zero when you're solving for it, right? So, this one is really straightforward. All you do is just move this stuff around and divide. You're done. For this one, you can't do that. You have to factor this equation. And the way we're going to deal with that is, use a property of zero where it tells us the only way that you can get any type of equation equal to zero if things are being multiplied together, if one of the terms is zero. But we'll talk about that as well in this series, or in these batch of videos that's coming up. And that's the reason why we're solving equations. It's sort of a precursor for us to be able to deal with functions. Super powerful, super important. So, basically, one of the first steps we get into when we're analyzing functions, when we're creating models, when we're, you know, trying to make predictions or trying to understand some kind of system.