 Hi, this is Chih-chou. 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 this. 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? As 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 what 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 you cross multiply, right? You take the bottom over here, kick it up there, bottom over here, kick it up there, comes multiplication, right? 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 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 right 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. 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. 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, one times x is x, one times negative 1 is negative 1, over here we do the same thing, this multiplies this, 14x squared, this multiplies this, it's going to be plus 4x, this multiplies this, it's going to be minus 35x, this multiplies this, it's going to be negative 10. 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 movement entries, simple binomials, that's what happens, right? So this becomes x squared, this kills this, minus 1 is z2, 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. So if we want to solve for this, for me I'd like everything to come to the left side, but I want my first x squared 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 side change is where we're moving them. So this becomes 14x squared minus x squared is 13x squared. This becomes negative 31x minus 9 equals to 0. And then we end up solving for this by factoring this, something else we covered in series 3, 8 and 8, right? For this one it would be a complex time to move 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 binomial, right? We've got a trinomial. That's 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. 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. 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. Let's say we have x squared minus x minus 1 times 2x squared plus 3x minus 4, right? Well, for this, the same deal is this. That's the same deal is 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, you go. This multiplies this, this multiplies this, this multiplies this, okay. And so on and so forth, right? it. So one pattern that we have is a cross multiplication pattern. You should always notice. 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. 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. 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 equals 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 the 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. 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, okay? Now this looks nasty, but the process is the same. I'm going to combine like terms on either side first, line up your equal sign, okay? 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 x's, so keep that as plus 1x and negative 1, right? And if you want to know how to do this, we talk a lot about these types of things combining like terms in series 3 and 3b, right? So we can't combine anything else anywhere on the side, but we can't combine anything on the side anymore. Now what we do is we bring all the values to one side, so that's something we'll have to recognize, right? So we're going to bring this over and 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've 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 0, 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, we can split this thing. And when we split it, what we end up having is we can set this equal to 0 and x minus 3 equal to 0 and we got x is equal to 3. 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. 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. We have foiling. Let's say we have polynomials multiplied together. 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 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, 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 a look at a, look at another one. What we're going to do is solve system linear equations with two variables. And then after this, we're going to solve 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 I haven't covered yet. It's all the system linear equations, two variables, basically two dimensions, right? And what these are is, that's a line. I'm talking about a 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, 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 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 y's 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. 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 five plus two y is equal to six, plugging it into the first equation, right? And this is going to be two times 12 over five minus y, right? Is equal to three. Now we've talked about what the best way to do, what the simplest way to do this is. You multiply the whole equation by five to get rid of your fractions, right? So multiply this whole thing by five. So this becomes 12 plus 10y is equal to 30. Multiply this whole thing by five, right? Two times 12 is 24, right? So this five kills this five. 24 minus five y 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 two goes into both of those, nine over five, right? Hopefully this is the same here. Whoops, this should be 15, 950. So I'm going to grab this guy, bring it over, it becomes minus 24. So I have negative five y is equal to 15 minus 24 is negative nine, divide by negative five, divide by negative five, y is equal to nine over five. 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, okay? 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? Blue. 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 this is a topic I haven't covered yet, right? So let's assume we had the following three equations. 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 1, 2, and 3. Okay. 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 it's going to be easy to eliminate the x, the x is from equation 1 and 2, we're going to try to eliminate the x's. And then we have two equations, y and z, right? So first thing we're going to do is generate two new equations, right? So what do we do? I'm just going to add equation 1 plus equation 2. So if I end up adding these, I'm going to rewrite these, x minus y plus z is equal to 2 and negative x minus y plus z is equal to 4. So what I can do right now is I add this equation with this equation and this is going to kill this, right? So this becomes negative 2y plus 2z is equal to 6. And I'm going to number this equation, equation 4, because it's a new equation, right? That we try from combining equation 1 and 2. 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 2 and 3 but I don't have to multiply equation 2 by 2 because I need a negative 2x here to cancel out 2x, right? So 2 times equation 2 plus equation 3. That's what the algorithm what I'm going to be doing, right? So this is going to be negative 2x minus 2y plus 2z is equal to 8. And I'm just going to write down equation 3 by itself, 2x plus y plus 2z is equal to 1. And I'm going to combine these two guys. If I combine them this way, it kills that guy. This is negative y plus 4z is equal to 9. Now what I'm going to do, I'm going to number this equation 5, 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 on 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 4z because if I add them together, the negative 4z plus 4z will eliminate each other, right? So I'm going to multiply equation 4 by negative 2, okay? And equation 5, I'm just going to bring down by itself. So equation 4, if I multiply by negative 2, I'm going to have negative 4y minus, oh, I'm going to have positive 4y, my bad, positive 4y minus 4z is equal to negative 12, okay? And I'm just going to bring this guy down, which is negative y plus 4z is equal to 9. 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, you're going to read an algebra. Now, what's going to happen here is when I add this guy and this guy, this is my equation 4, this is my equation 5, right? This is going to kill that guy, so that guy's gone, and that guy's gone. So this is going to be 3y is equal to negative 3, right? So all that happens now is I divide by 3, I divide by 3, so y is equal to negative 1. So what we have right now is the y value. We figure out what y is here. What we need to do is 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 burn this guy up here, 7y is equal to negative 1 here, and that's going to give us the answer. Now to make sure that we did this correctly, I'm also going to do it here as well. I'm going to 7y is equal to negative 1 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 1 and negative is 1 plus 4z is equal to 9. I'm going to bring the 1 over, it's negative 1, so this is 4z is equal to 8, and I'm going to divide by 4. So z is going to be equal to 2, 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 1 here, so that's going to be 2y plus 2z is equal to 6. Oops, not negative 2y, it's just 2y because it's something negative 1 for y, right? So negative 2 times negative 1 is 2, right? And I'm going to grab the 2 here, bring it over minus 2, so I have 2z is equal to 4, and I'm going to divide by 2, divide by 2, so z is equal to 2. Same answer. So far, I know that I've done this question correctly, right? Because I haven't seen, 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 1, right? Let's plug z is equal to 2 and y is equal to negative 1 in the first equation, right? In equation 1. So what we end up having is this x minus negative 1 plus 2 is going in there, 2 is equal to 2, okay? Negative and negative is positive, so it's going to be x plus 1 plus 2, which is going to be 3, so it's going to be x plus 3 is equal to 2, so x is equal to negative 1 when I bring this over, right? Let's do the same thing, but plug it into equation 2 or 3, just to make sure we have the right answer, right? So let's bring in y is negative 1, let's bring in x is 2 into equation number 2. So we're going to have over here, take a look at it, negative x minus negative 1 plus 2 is going to be equal to 4, negative x, negative and negative is positive, so 1 plus 2 is 3, is equal to 4, I'm going to bring this guy over, so that's negative 3, so we got negative x is equal to 1, so x is equal to negative 1, right? The same answer, okay? So I know I've done this question correctly, so final answer for this would be, if we're going to put it down here, we're going to put it here, this is going to be x is negative 1, y is negative 1, as that is 2, 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. 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. 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're right 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. I just really wanted to make sure that you appreciated a certain type of pattern that emerges with certain types of problems, right? So far, let's do a little recap, right? So far, we've had cross multiplication where we're taking the sky and multiplying here, taking the sky 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 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, they're different variables as 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, right? 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, one thing you 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, system, solving system to linear equations is embedded within this system, right? This part here, this guy and this guy are really this guy and this guy. When we're multiplying one of these equations by value to get, 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 solving it into one of these guys, taking this value and this value and solving it 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 a test, 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.