 This is Chico. Now what we're going to do in this video is continue our discussion on how to study but this video is going to be a little bit different than the previous videos we've done because this video is going to be specifically geared towards mathematics. In the previous videos we sort of talked about some of the things we could do to optimize our studying abilities, to optimize our ability to absorb information, right? We basically talked about us figuring out why it is that we're studying what it is that we're studying, right? We put aside enough time that we could study for longer periods as opposed to shorter chunks, right? Because studying longer is better, right? We've got, you know, a good space, nice comfortable space to do whatever it is that we're doing for us right now being sitting at a table for you might be at the beach at the park. Wherever you're comfortable studying, do it, right? And we've got our schedule. We've got her to-do list. We've gone through our books, our textbooks, and basically we figured out what it is that we need to practice, what it is that we need to learn, right? For this video right now, what that entails is doing a little bit of mathematics, specifically doing some algebra, okay? And what we're going to do in this video is do some math problems and take a look at the pattern that emerges for a specific type of question, for a specific type of problem, okay? And what that's going to do or what we should keep in mind when we're doing this is when it comes to mathematics, certain questions, certain problems play out in a certain way. When it comes to algebra specifically, okay? There's certain things you do to solve equations. There's certain things you do to graph functions and that doesn't change based on what the numbers are in those equations or how complicated those equations look if they're the same type, right? So as long as we know a certain pattern associated with certain type of question, then that means we know how to solve all questions in that type, of that type, right? Mainly, anyway. There are certain variations, but in general, if we know a pattern for a certain question, we know how to solve those types of questions. Ask for what we're going to do. Now, we're going to take a look at a few different types of questions and I've covered some of this previously in the language of mathematics in series 3A and 3B and one of the first things we did was I showed you basically when we're solving equations, you know, when we're moving around an equal sign specifically and that's what's required here, right? To do algebra, you basically have to know how to move around an equal sign, right? And one of the things I showed you which was super powerful was cross multiplication, right? I said just imagine having one fraction equal to another fraction, right? To solve these types of problems, all you do, let's bring out a red pen for this, all you do, you cross multiply, right? You take the bottom over here, kick it up there, bottom over here, kick it up there and it becomes multiplication, right? So what you end up doing is, for example, let's say you have 2 over x is equal to 5 over 7, right? What you end up doing is grab this guy, kick it up there, grab this guy, kick it up there, right? You line up your equal sign as always, right? Whenever you're doing algebra, try to line up your equal sign, right? 7 times 2 is 14. x times 5 is 5x, right? Now all we've got to do is just divide by 5, divide by 5. I like writing my x's on this side, write it there. This is 14 over 5 and that's your answer, right? This was the most basic, one of the most basic patterns that I showed you in a while ago, series 3a, and this is something you should always keep in mind. This is something you're going to use always, right? So for example, let's say you had something more complicated. Let's say you had x plus 1 over 2x minus 5 is equal to 7x plus 2 over x minus 1, right? The pattern doesn't change. This is a fraction equals a fraction. So all you do is take this, kick it up there, take this and kick it up there, right? So cross multiplication is a pattern that you should always remember because it comes in super handy. And all we do for this one is line up our equal sign. This guy comes up here and multiplies this, right? Times x minus 1, right? This guy comes up here and multiplies this. So we've got 2x minus 5 times 7x plus 2, right? Sorry if this is a little bit too small, right? So all we do now is for this type of problem, we have to multiply this out and that's foiling, right? That's sort of another pattern that emerges where, you know, I never really understood the term foiling, right? But what it was for me, it was just a visual thing that I used to do, which is basically this multiplies this, this multiplies this, this multiplies this, this multiplies this, right? Same with this. This multiplies this, this multiplies this, this multiplies this, and this multiplies this. This is another pattern that emerges in mathematics. No matter what type of binomial you have multiplied by another type of binomial, this is exactly what you do, right? Now if we multiply this out, we're going to get x squared. x times negative 1 is negative x. 1 times x is x. 1 times negative 1 is negative 1. Over here we do the same thing. This multiplies this. 14x squared, this multiplies this, is going to be plus 4x. This multiplies this, is going to be minus 35x. This multiplies this, is going to be negative 10, right? So first pattern, cross multiplication, second pattern foiling, but you know multiplying two binomials together, right? When you do this, you combine the middle terms. For most, a lot of rudimentary simple binomials, that's what happens, right? So this becomes x squared. This kills this. Minus 1 is equal to 14x squared minus 31x minus 10, right? And this brings us to another question, another type of question that comes along that you end up getting in mathematics, right? And what you end up doing is bringing everything to one side of the equation. Okay, so if we wanted to solve for this, for me I'd like everything to come to the left side, but I want my first x-squares to be positive, right? So what I'm going to do is I'm going to grab this and bring it over, and I'm going to grab that and bring it over. As we talked about previously, this becomes minus x squared, right? And this becomes plus 1 because the sign changes when we're moving them. So this becomes 14x squared minus x squared is 13x squared. And I like my numbers, my variables to be on the left side and my zero to be on the right side, right? So I'm just rewriting everything. I'm going to put the zero here. If I had more space here to put it all in, line up my equal sign, I would have fit it here, right? This becomes negative 31x minus 9 equals zero. And then we end up solving for this by factoring as something else we covered in series 3a and b, right? For this one it would be a complex trinomial factoring or we would use the quadratic equation, right? So this is one pattern, right? Cross multiplication Here's our second pattern, which is foiling and you can this process can occur when we have, you know, more than binomials multiplied by a trinomial, right? We could have a trinomial. Let's do another pattern here. Sure. Let's say we have a binomial 2x minus 1 times 3x squared plus x minus 4, right? Let's say we want to expand this. What do we do? Well, we do the same thing as the pattern here says, right? So we grab our orange marker. This is the same type of pattern, right? Associate this with orange if you want. All you do is every term here multiplies every term here, right? So this multiplies this, this multiplies this, this multiplies this, this multiplies this, this multiplies this, this multiplies this. Right? That's the same pattern as this. This just happens to be a binomial times a trinomial. This is a binomial times a binomial. Right? Let's say we have a trinomial times a trinomial. Pattern doesn't change. Right? Well, the main pattern doesn't change. It becomes a little bit more complicated, but the general just is the same thing, right? Let's say we have x squared minus x minus 1 times 2x squared plus 3x minus 4, right? Well, for this, it's the same deal as this, as the same deal as this. Every term here multiplies every term here, right? So this multiplies this, multiplies this, multiplies this, this multiplies this, multiplies this, oops, multiplies this, right? This guy, it becomes complicated now, right? It's not really complicated because you never leave these here, right? You go boink, boink, boink, boink, boink, boink. If you want, you go this this multiplies this, this multiplies this, this multiplies this, okay? And so on and so forth, right? So one pattern that we have is a cross multiplication pattern. You should always know this. Another pattern we have is when polynomials multiplied by polynomials, right? Binomial times a binomial. Simple. Binomial times a trinomial. Not bad. Trinomial times a trinomial. The lines become, you know, messy, but the process is the same, right? So let's take a look at some more complicated types of problems, questions that we may encounter. This is the side. Now, one type of problem we get is basically having a polynomial on one side of the equation and polynomial on the other side, right? When we get these types of problems, the name of the game is to combine like terms, bring everything to one side, set the other side equal to zero, right? And we talked about why it is that we have to set the other side equal to zero. And what happens when we do this? Usually we end up getting a certain type of v when we're solving for a polynomial, okay? Now, what we're going to do is we're going to do a single variable polynomial first. So you see how simple it is. And then we're going to do a more complicated one where we have a single variable, but we have powers. So let's say we had something like this. Now, this type of problem you're usually getting great at, or so. And the name of the game for this is, for these types of problems is line up your equal sign. And what you're going to do is combine like terms on either side first before you move around the equation. And what we're going to do is we're going to combine this guy and this guy. So 2x plus 5x is 7x. Negative 6 plus 4 is negative 2. 7x minus 4x is 3x. 3x plus 3x is 6x minus 1, right? And now what we're going to do is whenever you have one variable, you want your variable. In general, I like it on the left side, and I want the numbers on the right side. So I'm going to grab this guy, bring it over, change his size, becomes 6x. Grab that guy, bring it over, plus 2. Oops, this is minus 6x, right? If we bring a positive over, it becomes negative. So 7x minus 6x is x, and negative 2 plus, negative 1 plus 2 is 1. So your answer here becomes 1, right? And this is the pattern that emerges when you're solving these types of problems, when you're solving equations, which is basically a v, right? And then you get to your answer. So whenever you're solving these types of equations, if you're solving for a variable, or multi-variable equations, you want to bring the variables to one side and number to the other side possibly, right? So what we're going to do for, just to show you that this works for other types of questions, we're going to do a variable that, an equation, a question that ends up being a quadratic on one side, okay? So let's make this longer, bigger, right? 2x squared plus 5x squared minus 6x plus 4 minus 2 plus 1 is equal to 7x squared minus 4x squared plus 1x plus 3x squared minus 1. Now, this looks nasty, but the process is the same. We're going to combine like terms on either side first. Line up your equal sign. 2x squared, 5x squared is 7x squared. Negative 6x plus 4 is negative 2x. Negative 2 plus 1 is negative 1. 7x squared minus 4x squared is 3x squared, right? 3x squared plus 3x squared is 6x squared. 1x doesn't combine with any other axis. So we keep that as plus 1x and negative 1, right? And if you want to know how to do this, we talked a lot about these types of things combining like terms in series 3 and 3b, right? So we can't combine anything else anymore on this side, but, right? We can't combine anything on this side anymore. Not what we do is we bring all the values to one side because what we're going to recognize for these types of problems is we don't have, you know, we can't combine an x squared with an x. So we're going to have two terms here that are going to stick out. So we can't just automatically get isolated x. We're going to have to factor this thing, right? So that's something we're going to have to recognize, right? So we're going to bring this over. This becomes minus 6x squared. We're going to bring this guy over. This becomes minus x. And we're going to bring this guy over and this becomes plus 1, right? So on this side, we have zero left, right? On this side, we have 7x squared minus 6x squared is going to be x squared. We got negative 2x minus x is negative 3x and 1 minus 1, they kill each other, right? So we're down to here. What we're going to do now is factor out an x. So x comes out. We've got x minus 3 left here equal to zero, right? This so far is just what we had here with an addition process of factoring, right? So we still have our v, right? Our thing going like this, right? Because what we're trying to do is simplify, simplify, simplify. And to solve this right now is something that we've talked about again in series 3 and 3b, which is the power of zero, right? We've talked about the problems associated with zero being that we cannot divide by zero, right? When we divide by zero, we get undefined. We don't know what happens, right? The universe, our universe explodes, right? That's the problem associated with zero. But there's also a benefit associated with zero. The benefit with zero is the only way that you can multiply two or more things to give you a zero is if at least one of these things is zero, right? And since we don't know which one is equal to zero, we set both of them equal to zero, okay? So the power associated here is we can do a split with this. We can split this thing. And when we split it, what we end up having is we can set this equal to zero and x minus three equal to zero. And we got x is equal to three. And if you notice, this thing here is also a v. It's a mini version of this and a mini version of this, right? So when it comes to algebra, what we do for a specific type of problem can be embedded within the problem, right? You can think of these things as modules that you can add on to certain types of problems, powers that you have that sometimes you need to use, okay? So this is, I guess, if you want to think about it, the third type of pattern that emerges for us, let's take a look at, you know, other types of patterns that do emerge when we're doing problems like this, okay? So there are three types of patterns that we have right now, right? The first one is cross multiplication, where we can, that's one pattern. In English, we call this cross multiplication. In other languages, we call it different things, right? We have foiling, right? Plus or minus. Let's say we have polynomials multiplied together, right? Something in here or more. The pattern that emerges for this is, I guess some people call it the foiling pattern. This multiplies this, this multiplies this, this multiplies this, this multiplies this. And we saw more complicated versions of those, right? And we have equations that we end up solving, right? If we're given a certain type of problem, right? Where we have an equal sign in the middle, we've got something on one side and another thing on that side. What we end up doing is doing this, right? Reducing, simplifying, combining like terms until we get an answer where x or whatever variable it is equals that thing, right? And this was sort of the third pattern that we have. And these, these are things that's parts of algebra that show up everywhere. They're sort of modules. They're sort of things that we end up doing for all types of problems, right? These aren't specifically for this, right? We do a lot of questions like this initially to learn how to do this, right? We do a lot of questions like this initially to learn how to do this, right? We do this, right? Part of the process that developed from part of solving equations, right? Doing problems, doing algebra. But these themselves are embedded parts of other larger types of problems we end up getting, okay? So those are three patterns that we have. Let's take look at another one. Let's do, what we're going to do is solve system linear equations with two variables. And then after this, we're going to solve the system linear equations with three variables. And you'll see how they're similar and one builds on the other. And they actually end up using these things here, right? Or this thing here, okay? So let's do, let's say, you know, we get a system linear equations that are this. Let's do x plus 2y is equal to 6 and 2x minus y is equal to 3. Now, this is something that I haven't covered yet. It's called system linear equations, two variable, basically two dimension, right? And what these are is that's a line that we talked about in series one, the equation of a line. And that's a line as well. So whenever we get something like this, what we're doing is we're trying to find out where these two lines cross. And there's three things that can happen with this thing. They could cross, they could have an intersection where there's one solution. They might be parallel or they might be lines on top of each other. And I'll get into detail of solving these types of equations later in the future. Right now, we're more interested in the pattern that emerges when we're solving this. So what we do with these types of problems is we try to eliminate one of these variables. So we only have one variable left. And for this, what we're going to do is we're going to make a decision to eliminate, let's say, the y. So to eliminate the y from this, because this is a system, which means they're together, right? What I'm going to do is combine these two equations. But the way I'm going to combine it is I want to combine it in a way that this guy will kill that guy. Now, for this guy to take out that guy, there needs to be two of these guys here, right? So if I end up adding two of these equations to this, two ys to this, that's a negative y. So two y plus negative two y is zero. They kill each other, right? So what I end up doing is I number my equations for these guys always. My equation one is going to come down here again, right? x plus two y is equal to six. My second equation, what I'm going to do is multiply it by two, the whole equation. And when you do that, this becomes four x. That becomes minus two y is equal to six, right? And what I'm going to do now is I'm going to add this equation with this equation. So if I add this, I'm going to get five x. This kills this, and this is going to be 12. And then I'm just going to divide by five, and I'm going to divide by five. So x is going to be equal to 12 over five. Okay, this is a system of linear equations with two variables. And the pattern that emerges here is to a certain degree, and this is, you're going to do this a lot if you're doing these things, is you multiply it down here. You move the equations down if you need to, multiplying by whatever it is that you need to, combining the like terms, right? Add these things, whatever ends up dropping off, drops off, and you have x is equal to 12 over five. Now what you end up doing is taking the x and plugging it back either into this equation or this equation, because you still need to solve for the y, right? So when you're solving these types of problems, it's a good idea to plug them into both of those to make sure it becomes a check to make sure that this is correct, right? So what I'm going to do is I'm going to plug it into equation one, and I'm going to plug it into equation two. So what we have here is this is going to be 12 over 5 plus 2y is equal to 6, plugging it to the first equation, right? And this is going to be 2 times 12 over 5 minus y, right? Is equal to 3. Now we've talked about what the best way to do, what the simplest way to do this is, you can multiply the whole equation by 5 to get rid of your fractions, right? So multiply this whole thing by 5. So this becomes 12 plus 10y is equal to 30. Multiply this whole thing by 5, right? 2 times 12 was 24, right? So this 5 kills this 5, 24 minus 5y is equal to 15. So what I'm going to do is solve for y here. I'm going to grab the 12, bring it over minus 12. So this is 10y is equal to 30 minus 12 is going to be 18, right? And I'm going to divide by 10, divide by 10. So y is going to be equal to 2 goes into both of those 9 over 5, right? Hopefully this is the same here. Oops, this should be 15 not 150. So I'm going to grab this guy, bring it over, becomes minus 24. So I have negative 5y is equal to 15 minus 24 is negative 9, divide by negative 5, divide by negative 5 and get y is equal to 9 over 5. The same answer, right? So what are the patterns that emerge? Multiply whatever you need to multiply, right? You solve your equation, which is a V, really, right? It doesn't look like it, but it is because it's so simple, right? We split it up to do a check. Again, this is this guy, right? This is again a V. That's a V as well, okay? And we end up getting our answer. And we end up getting our answer, right? So this is a more complicated, right? Pattern that emerges when we're solving system of linear equations. And it's sort of built. There's three of these guys in this, right? So if you know how to do this, the only additional thing you need to do is sub this back into this, right? So you would have to know that there's substitution involved here. And the only extra thing you need to do was this guy, right? So this is a system of equations using two variables, right? Let's do a system of equations using three variables. And you'll see this pattern emerge. This is going to be embedded within the other one, right? So let's take this aside. What we'll do in this example in this equation is do a system of linear equations using three variables, right? Basically, meaning it's a three-dimension. And I'll get into detail about this in the language of mathematics in future videos, right? Because that this is a topic I haven't covered yet, right? So let's assume we had the following three equations. We had x minus y plus z is equal to two. We got negative x minus y plus z is equal to four. And let's say we had 2x plus y plus 2z is equal to one. Now, the name of the game for this is, we want to find out what each one of those variables are, right? We're trying to find out where these three three-dimensional lines cross, right? So what we're going to do is number these equations. Let's call this equation one, two, and three. So what we're going to do is we're going to try to eliminate one of the variables in the first step to solve this, to solve the system, right? So let's assume, because this is going to be easy eliminating the x, the x's from equation one and two, we're going to try to eliminate the x's, and then that way we have two equations with y and z, right? So first thing we're going to do is generate two new types of equations, right? So what I'm going to do, I'm just going to add equation one plus equation two, okay? So if I end up adding these, I'm going to rewrite these. x minus y plus z is equal to two, and negative x minus y plus z is equal to four, okay? So what I can do right now is I can add this equation with this equation and this is going to kill this, right? So this becomes negative two y plus two z is equal to six, and I'm going to number this equation equation four, because it's a new equation, right? That we derive from combining equation one and two. Now I need to get rid of x in my next process as well, right? So what I'm going to do, I'm going to combine equation two and three, but I'm going to have to multiply equation two by two because I need a negative two x here to cancel out two x, right? So two times equation two plus equation three. That's what the algorithm, what I'm going to be doing, right? So this is going to be negative two x minus two y plus two z is equal to eight, and I'm just going to write down equation three by itself, two x plus y plus two z is equal to one, and I'm going to combine these two guys. If I combine them, this guy kills that guy. This is negative y plus four z is equal to nine, okay? Now what I'm going to do, I'm going to number this equation five, right? And what we need to do now is this is the system that I need to solve. Well, this is what we had in the previous example, right? So from now on, all it is is the previous system. So all we have is this process, right? That we're going to do here. So that's the pattern that emerges. So what I have to do is make a decision of what variable I'm going to get rid of, right? For me, I'm going to get rid of the z, right? For me to get rid of the z here, I need this guy to be negative four z because when I add them together, the negative four z plus four z, they'll illuminate each other, right? So I'm going to multiply equation four by negative two, okay? And equation five, I'm just going to bring down by itself. So equation four, if I multiply by negative two, I'm going to have negative four y minus, oh, I'm going to have positive four y, my bad, positive four y minus four z is equal to negative 12, okay? And I'm just going to bring this guy down, which is negative y plus four z is equal to nine. And one of the things that I've mentioned before, which is super important is try to line up your equal signs, right? Whenever you're doing mathematics, whenever you're doing algebra. Now, what's going to happen here is when I add this guy and this guy, this is my equation four, this is my equation five, right? This is going to kill that guy. So that guy's gone and that guy's gone. So this is going to be three y is equal to negative three, right? So all that happens now is I divide by three, I divide by three. So y is equal to negative one. So what we have right now is the y value. We figured out what y is here. What we need to do is find x and find x and z. Now, we can't go directly from here to here. We need to do one step in between. We need to figure out what z is, right? So what we're going to do is bring this guy up here, sub and y is equal to negative one here. And that's going to give us the answer. Now, to make sure that we did this correctly, I'm going to also going to do it here as well. I'm going to sub and y is equal to negative one here as well to make sure I end up getting the same answer for z before I do the next step, right? So this becomes negative one and negative is one plus four z is equal to nine. I'm going to bring the one over. It's negative one. So this is four z is equal to eight. And then I'm going to divide by four. So z is going to be equal to two, right? That's my z value for this. I'm just going to have to make sure that that's correct, right? So I'm going to bring in negative one here. So that's going to be two y plus two z is equal to six. Oops, not negative two y. It's just two y because I subbed the negative one for y, right? So negative two times negative one is two. And I'm going to grab the two here, bring it over minus two. So I have two z is equal to four. And I'm going to divide by two, divide by two. So z is equal to two. Same answer. So so far, I know that I've done this question correctly, right? Because I have the same, most likely anyway, same value for z. Now all I have to do is figure out what the x is, right? So what I'm going to do is I'm going to pick one of these equations and plug in the values for y and z and find out what the x is. So let's do it for number one, right? Let's plug z is equal to two and y is equal to negative one in the first equation, right? In equation one. So what we end up having is this is x minus negative one plus two is going in there. Two is equal to two. Okay. Negative and negative is positive. So that's going to be x plus one plus two, which is going to be three. So this is going to be x plus three is equal to two. So x is equal to negative one when I bring this over, right? Let's do the same thing, but plug it into equation two or three just to make sure we have the right answer, right? So let's bring in y is negative one. Let's bring in x is two into equation number two. So we're going to have over here, if you take a look at it, negative x minus negative one plus two is going to be equal to four. Negative x, negative and negative is positive. So one plus two is three is equal to four. I'm going to bring this guy over. So that's negative three. So we got negative x is equal to one. So x is equal to negative one, right? The same answer. Okay. So I know I've done this question correctly. So the final answer for this would be, if we're going to fit it down here, we'll put it here, is going to be x is negative one, y is negative one and z is two. That's my final answer to this question. Okay. Now, do you see the patterns that have come up? If you get any triple system here, any three variable system of equations with three equations that you need to solve, this is the pattern that you're going to see. Okay. Let's actually highlight this, show you what it looks like. Okay. Let's bring this guy over here. Let's do this guy in green. Yeah. Let's lay this out in green. Now what you're going to see here is the same pattern showing up for all questions involving system of equations involving three variables. Sometimes they're simpler because sometimes you can kill two variables in one shot, right? When you add or subtract the equations, which we could have done here. Okay. But this is the main pattern that appears, right? So what you're going to have initially is your questions, your three equations, right? And then what you're going to do is you're going to split them. And what you're going to do is combine your equations and come up with equation number four and equation number five. Okay. And then what you're going to do is you're going to combine equation number four and five into an answer for one of the variables. In our case, it happened to be Y, right? So, so far we've done this, we've done this, we've done this, we're here. And then what you're going to do is you're going to plug these back into here, right? And that's exactly what we did. And what you're going to do is you're going to solve if this is solving equations, right? If this is solving, right? If this is, well, not solving, but doing the algebra, calculating something. What we're going to do, we're going to do a whole bunch of calculations here, right? So we're going to do a bunch of calculations here. We're going to get our next variable. We're going to do a bunch of calculations here. We're going to confirm our next variable, right? This better match with this, right? And then what we're going to do is we're going to bring numbers from here, plug it into one of these equations, do more calculations and come up with our next value, right? We're going to combine, right? We're going to do our calculations and we're going to come up with our next variable, which hopefully should confirm this, should confirm with this, right? And this guy with that guy, okay? This is the main pattern that shows up for any type of question you get. For most of the questions you get with three variables. This is what you should keep in mind. And then you write down your answer here, right? A beautiful pattern. If you understand this, you know how to do all questions involving three variables, right? System of equations involving three variables. Should we do another one? Let's do another pattern. Now, let's do another pattern just to finish this off with, because the questions, the questions are infinite, the types of questions you can get. I just really wanted to make sure that you appreciated a certain type of pattern that emerges with certain types of problems, right? So, so far, let's do, let's do a little recap, right? So far, we've had cross multiplication where we're taking this guy and multiplying here, taking this guy and multiplying here. That's the first type of pattern we saw. The second type of pattern we looked at anyway was multiplying polynomials. I think we did that in orange. So, this is the second type of pattern. The third type of pattern that we looked at was just solving equations, right? Something like this. You end up you end up simplifying, simplifying, reducing, lining up your equal sign and getting an answer here, right? We did system of equations with two variables, right? We had our two equations here and we basically were running out of colors. I think we used orange. So, we multiplied one of the equations by something, another by something else, right? And we brought the calculation here, found an answer, we back substituted in one of the other equations, right? Where we basically did this, where we did this, right? And we did more calculations and we got an answer, right? And our answer would be your x and y, whatever they were, right? If they're different variables, their different variables is this guy and that guy and this was confirming this, right? That sort of pattern that emerged. For a system of three equations, right? We had one, two, three equations show up and let's do this in green, I guess, where we split this up, did calculations, brought it back, did calculations, kicked it up, did calculations, brought a number from here, brought a number from there, brought a number from there, brought a number from there, kicked it into one of the equations here or two of them just to confirm, right? So, we did calculations and we got our next answer, right? So, we had an answer here. Yes, I should do this in green. We got an answer here, we got an answer here, which was the same as this and we got an answer here, we got an answer here and those, this, this and this or this, this and this, same deal, would be our answers to this question, right? And these are some of the patterns that we have right now and what they should notice, what they should recognize, is that these patterns, the simpler patterns occur inside of the other more complicated patterns, right? This occurred here, this occurs here, right? Cross multiplying if you're dealing with fractions you end up using, we end up using here or here, right? This guy here, solving system two linear equations is embedded within this system, right? This part here, this guy and this guy are really this guy and this guy. We're multiplying one of these equations by value to come to here and when we solve for it, the back substitution here is really this, we're back substituting into one of these equations, right? And getting a value, right? And then the extra part is taking this value and this value and subbing into one of these guys, taking this value and this value and subbing into a different one, make sure we get the same answer, right? That way that confirms that we did this question properly, that we got the right answer, right? Especially if we're doing our tests, right? Because we don't want to throw marks away. So there's like five patterns that we've got here so far and this guy is really embedded within this. We could call this pattern four, I guess. What are we going to call this? What color? Let's call this black pattern four, right? So that's pattern four. So this is just basically algebra, some algebra patterns that show up whenever we're trying to solve equations. For the last sort of pattern, we're going to graph an absolute value function and we're going to do an absolute value of a linear function of a line, which is one of the most simple functions that you can have, okay? One of the most simple graphs that you can have. Now, for those of you that don't know and I will cover this at some point, the absolute value symbols. But absolute value symbols mean that inside the absolute value symbols, you could be positive or you could be negative. But once you use up the absolute value symbols, once you use this power, once you come out of that symbol, then you have to become positive, right? But when you're in the inside, you could be positive or negative. And that's the property, that's what we're going to do to solve this, to graph this function, okay? Because that's what we're going to need to do for this. So what we're going to do is we're going to take a look at this function when the inside is positive and we're going to take a look at this function when the inside is negative, right? Just to see what they look like, what this function looks like. So when it's positive, this becomes f of x is equal to positive 2x minus 1, right? And when it's negative, this becomes f of x is equal to negative 2x minus 1. Now, when we apply the positive, nothing changes. This thing stays what it is, right? So this becomes 2x minus 1. When we apply the negative, this becomes negative 2x plus 1, plus 1. So what we're going to do is graph this function. So we're going to draw our coordinate system here. And I think we talked about graphing lines in series 1, right? So the way you graph lines is you use y equals mxb, the concept of y equals mx plus b. And this is in the format of y is equal to mx plus b, where b is your y-intercept and m is your slope, right? So this is our first equation, right? This is our second equation. And whenever you have more than one equation to deal with in a question, number them. That way you know which one you're dealing with, right? So if we're graphing this one, our y-intercept is negative 1, and our slope is 2 over 1. So 1, 2, and 1. This is our first line. And whenever you're graphing multiple lines on a graph, multiple functions on a graph, number them. This guy is y-intercept as 1, and the slope is negative 2 over 1, so down 2 over 1. Here and here. And this is equation 2, graph 2. Now, this is not the graph of this, right? Because this is not a function, right? For a given x value, we have two different y's. Functions is something that we're going to talk about in series 4, right? Or we are talking about in series 4. So this is not a graph of this. This is the graph of this guy, and this is the graph of this guy. What we want to do is draw the graph of this. So all we need to do now is test the point on this graph for x and y to see if the equation is true. Now, the only point that we cannot test is where the two lines intersect, because they give us the same answer at that point. So what we're going to do is we're going to pick a point anywhere outside of that line, right? So what we're going to do is we're going to try to figure out what y f of x should be when x is somewhere either on this side or on this side of this point. Now, it's easy to figure out what it is. Pick a point on this side, because we know this is x is equal to zero, right? If this is x-axis, if that's y-axis, the y-axis is x is equal to zero. So what we're going to do is we're going to test x is equal to zero in the original equation. So what we're going to do is we're going to find f of zero. f of zero is going to be two times zero minus one, which is going to be zero minus one, and this is going to be absolute value of negative one, which is just one. So when x is zero, y has to be one, right? When x is zero, y is one. When x is zero, y is one. So we know this point is on the graph. That means this point is not. So what we end up doing is killing everything below the point of intersection where they intersect. And this graph, this guy right here, this function, graphs like this goes up. It bounces off the x-axis, and we'll talk a lot more about these types of functions in the future. And this process is something that you do. This method of solving absolute value equations is the same thing that you do no matter how complicated this polynomial becomes here. You split, right? You take it positive, you take it negative, you solve, right? You graph, you test the point, and you figure out, you test the point in the original function, you figure out which one it is, right? If we want to take a look at a more complicated one, here, let's do a more complicated one. Let's have this beside us while we do. So this pattern here is split, split, calculation, calculation, right? Graph it and test it, and then eliminate, kill whatever you need to kill, right? And it really doesn't make a difference how complicated those questions might be. So let's do another type of absolute value function, which is a lot more complicated than this, and we're not going to graph it. We're just going to look at the pattern that we have to follow. Now, this is a quadratic function if the absolute value symbols weren't there, right? So that's a quadratic function, and we want to graph the absolute value of a quadratic function. Now, the way this function is going to look like is, again, we have to do this. We have to take it positive, and we have to take it negative. So what we would do is we would take this function, the positive of this, and we would have to take the negative of this, okay? So the positive of this is just whatever it is, x squared plus 5x plus, oops, plus 6, and we would have to graph this guy as negative x squared minus 5x minus 6. Now, there's a process here to be able to put these in the form that we need to be able to graph them. So you would do all bunch of calculations here called completing the square, okay? Which is the simplest, you know, more complicated version of just rewriting this, right? Because when you're graphing a line, you just have to get in the form y equals an x plus b. When you're graphing a quadratic, you have to get in the form of y is equal to a x minus p squared plus q, right? And over here, you would have the same thing. So you would do something called completing a square through the whole process. And then you could graph it. For this one, the factors of this are 2 and 3. So x plus 2, x plus 3. So we know that x plus 2 and x minus 3 would be negative 2 and negative 3 would be the x intercepts. And then this thing, we know it opens up. So it would go like this, okay? Something like this. And this guy would do the same thing, but I think this way, right? So let's assume the graph, I'm pretty sure it's like that. But let's assume the graph of this looks like this, right? Okay. So this one would be our first equation. This one would be our second equation. So this would be our first equation. And this would be our second equation. And then all we would have to do is test an x value in the original function to find out which one of these it is, right? So since we're not going through x is equal to 0, what we would do is test the simplest x we can, which is x is equal to 0. So f of 0 would be 0 and 0. So it's just the absolute value of 6, which is just 6. So when x is 0, y would be 6 somewhere here. We know this guy goes through this, right? So we know that we're going to kill the bottom of the x-axis. That's the way it's going to work, which makes sense because f of x is y, y can never be negative. So that means y can never be negative. So what we would do is kill this, right? And the graph of this function would look like this, okay? And this pattern is the same pattern as this, right? We broke it into positive and negative. We did our calculations, whatever they may be, to get our functions in a form where we could graph it, right? Where we graphed it. We tested a point. It was easy for us. We tested x is equal to 0, right? We found out what this was. We plotted that point and we killed below wherever the intersection of the graphs was, right? So we killed these guys, right? And I'll get into these types of functions, these types of graphs a lot later in the language of mathematics, okay? And this is, I guess you could consider this to be the sixth type of pattern that we have, right? And if you see any absolute value functions that you have to graph, this is the main gist. These parts are modules that you have, patterns that you have from a simpler thing that you need to do. These would be something originally that you learned before you learned how to do this one, right? So it's a combination of things. You're bringing patterns from doing algebra, solving equation this way and you're taking one from this way. And if they happen to cross paths, you use them both. And that's the way you should think about solving equations, doing mathematics, if you're working on problems, okay? And it's nice to know how these patterns look, because it gives you powers, it gives you abilities. It allows you to remember how certain question should be solved, okay? And that's, you know, tip number six that I have for you and specifically math related is recognize the type of problem you have, right? And remember the pattern that you need to be able to solve that equation, to be able to answer that question, to be able to graph that function. I'll see you guys in the next video. Bye for now. Now before we begin taking a look at these patterns, okay, I got one more quick hint to give you when you're doing algebra, when you're solving equations, right? When you're doing problems in mathematics. And that's not to take your eyes off the question until you finish a specific type of question, right? If they're short enough that you can focus for that long, right? Because one thing that happens, one thing that I've seen students do is, when they're doing a certain type of problem, when they look away, right, and then can't look back at the problem again, sometimes they drop a negative sign. Sometimes they change a addition to a subtraction, right? Sometimes they forget a number or variable, right? And that basically means from that point on, they're making mistakes. The answer is not the correct answer, right? So keep this in mind when we're doing this. When you're doing, when you're doing algebra, when you're solving a problem, don't take your eyes off that problem until you're done. If you need a break, if you need to give your eyes a break, then once you finish, once you get the answer, you're finished with the algorithm, you're finished with the process was required to solve that question, right? Then look away before you move on to the next question or before you check your answer.