 Hi there everyone. It's still this wonderful time between Christmas and New Year, so it's still nice and quiet especially after work and our late afternoon, quite peaceful. Now, before we get to these two more systems of equations and go to Mathematica just to learn a bit more, get a bit more intuition about this, maybe one or two things. Now, if you're one of the medical students at my university, which you're obviously not, you would know that I always try to get to know my students and I want the students to get to know me. I don't know how you can be a lecturer, if you don't connect, you have to connect with another human being if you're going to share knowledge, if you're going to impart knowledge, if you're going to learn from someone, you've got to connect with them. So let's let's learn a little bit about each other. I'll tell you about myself. First this environment, this is obviously my office here at Khridiskyr Hospital and what you're going to hear is a lot of noise. There's always this background noise. Now Khridiskyr is built right next to a huge highway, multi-lane highway, so there's always going to be the traffic noise, but there's always also this constant companion called the wind. Now, there's this little city in Saudi Arabia called Port Elizabeth. It's called the windy city, but whoever called that the windy city has never lived here. Now Khridiskyr is in a suburb called Observatory and I live in Observatory and I can tell you there's plenty windy. The wind just blows outside all the time and if I put a microphone by my door, this is what it sounds like. So you know about my office now afternoon, especially late afternoon, very windy and noisy and there's not much I can do about that. This wind is really a constant companion. The other thing is something about myself, I jog five and a half years now. I jog almost every day and for the last year and a half it's been trail running. I love trail running. I usually run in a place called Newlands Forest, which is at the back of Table Mountain. And yesterday for the first time in five and a half years, I came down. I fell down hard. I came downhill on the trail, little roots sticking out of a big tree and I tripped over it and it was just going at such a speed that I couldn't save it. So this is what I look like. And I also came to work in shorts and that's a whole different story there. That's a different thing. Anyway, so I came down and hop along and bit saw and stuff. Now, what do you call these in your countries? Yeah, locally we call these Rostes. You know, these grass burns, but I didn't fall on grass, but some people call them grass burns, but we call them Rostes. What do they call them locally? I'm wondering. Anyway, so I have a system of equations here and I have a system of equations there. Let's have a look at these 2y equals minus x plus 2 and 2y equals minus 2x plus 4. You can see there's something fishy going on here. The second one is just the first one multiplied throughout by 2. So if I were to write it out that the way that we do in linear algebra, I'm going to have 2x plus 2y equals 4 and you can see this is just twice times that one. If I have the first one, I don't actually have two equations here. I just have a single equation because if I were to simplify this one, I'm just going to get back to that one. So I have not learned anything new. If I do my augmented matrix 2 to 4, there's my augmented matrix. You can clearly see that nothing new is added to this first line. And if I were just to multiply this up by a half or negative of a half and add the two, you can see where that's going to go. It's very intuitive to see that there is not two equations here. There's a single equation and the same would go for three equations and three unknowns. If one of them is just a constant multiple of the other, then you don't have three equations. You actually just have two. Let's have a look at this. I have two equations and they have the exact same slope. They have different y-intercepts, of course. And if I were to write these out, it would be let's make them negative 2x plus y equals 3 and negative 2x plus y equals 1. And my augmented matrix is going to be negative 2 or 1 and a 3 and a negative 2 and a 1 and a 1. And there's my augmented matrix. And you can clearly see here, from this, not so easy, from this, not so easy, but the intuition from school would be very easy that this is 2 times 2. The slope here is exactly the same. The y-intercept is different. So two different systems here. Let's go to Mathematica and just draw some nice graphs of this and do row-reduce on these two and see where we end up. But again, as I say, it's not about the row operations that you are going to do. You have to do them. You're going to get them in a test, so be it. It's not what it is. I want you just to understand intuitively what is going on here. Let me show you lovely pictures and teach you how to code this on your own. And then it will make so much sense. You get it and you can just carry on with studying linear algebra. Let me hop along to my computer and I'll show you. Here we are in Mathematica. Let's have a look at these two equations. Before we get there, maybe I told you about the cells in the previous video. So if I click that little plus there, I'm going to go to other styles. Let me bring it in here from my second monitor. And the one that I'm going to choose is display formula numbered. So this is going to give me this. If I do these, I see the number one there. And now I can just type normally. And this would be as it would be in a textbook. Let's do minus six plus two. And I have, what did we have? Two y and we had minus two x plus four. That's what we had. So I'm not typing any code yet. This is actual text and Mathematica is formatting it nicely for me. And what did we do? We just bought the variables on the same side. That was going to equal two and two x plus two y equals four. That's what we had. And that is our first system. Beautiful there. There's my cell. And it's a special kind of cell, which is just formatted text, formatted in the four mathematical equations. Beautiful. But let's plot these two. Remember the plot function. I'm going to pass it to argument. The first argument is going to be my set of, my list of two equations. And of course it's negative x plus two. And the second one, remember, I can't put in negative two x plus four. I can't do that because there's two y on the other side. And Mathematica is expecting simplified version of this equation. So it's actually just negative x plus two all over again. I've got to do that twice. And let's put the range in. So let's go from negative five to five. That's not my domain, say, on the x-axis. And let's just do this plot legends again. Plot legends. I'm going to use tab for auto-completion. I'm going to scroll down with my arrow key and inside the quotation marks the expression. So I could have just typed that myself. And remember that little arrow. You can get that arrow by just using the minus sign and then the greater than sign. I'm going to close there. And there they are. It's two of the exact same equations. There is, I've not learned something new. It's the exact same equation. The exact same line. So let's row reduce. Let's row reduce. And let's put those two in. It was one comma one comma two. And the other one was two comma two comma four. And close. The whole thing has got to go inside of curly braces. Two rows. And I'm going to close my square brackets. And I'm going to use this post text notation of matrix four. And there we go. I see something there. One, one, two, and zero, zero, zero. And that is what you're typically going to see if these two lines or the planes or whatever lie on top of each other. That second equation there, zero, zero, zero, it adds nothing new. So there's an infinite many solutions that will solve both equations. There are so many x values and so many y values that will solve both of these. Good. Let's go to the next one. And the next one, let's, let's put that in. Let's get used to typing that in. So I'm going to go to plus other styles. Let's bring that in. That's going to be this display formula number. And let's have a look. I'm looking on the board. The other one was y equals two x plus three and y equals two x plus one. Y equals two x plus one. That is what we wanted. If we were to write it out, that's going to be a two x plus y, well, a negative two x, I should say. Let's make that negative two x plus y equals three. And again, negative two x plus y equals one. Let's plot that out, plot function inside of curly braces, my two. Now I want to show you something. Instead of the multiplication, which is just the star, I'm going to just put a space, two x plus three. Mathematica will see that. That's convenience notation. I just put a space. It's easier to do than shift and eight. Mathematica will know if there's a space in there. That means it's multiplication. You can't just put two x without a space in between. Mathematica will not know that by that you mean multiplication. So two x plus three, that's the one. And the other one was two x plus one. Two space x plus one. Close my curly brace. Let's look at the domain on my x-axis. Let's go from negative five to five. That's fine with me. And let's do plot legends again and go down to expressions. That's what we want. And there we go. And we see that it is two parallel lines, just as we suspected. They will never intersect in the type of setup that we have here. And there will be no solutions. Now let me show you something. Instead of retyping my list all the time, I'm going to give it a computer variable name. And it's customary to use uppercase letters for matrices. So I'm going to say uppercase A equals, and then I'm going to do my augmented matrix, which was negative two comma one comma three. That was the one. And the second row was negative two comma one comma one. Close and close. And there we go. Shift-Enter, Shift-Return. There we go. Now I can say A and it is just there. I don't have to retype it. Or I can say A matrix form, matrix form, and that is a matrix form. Remember this was post-fix notation. So it was just shorthand for doing the following. Matrix form. I'm going to step down autocomplete. And I'm going to pass A as an argument inside of my square brackets. Exact same thing. So this post-fix notation with the two forward slashes. It's just shorthand for doing it this way. Now let's row-reduce A, row-reduce A. And let's do that in matrix form, matrix form. And there we go. Look at that second row, zero, zero one. So it says zero x's plus zero y's equals one. That makes no sense at all. You can't have zero times something plus zero times something and that equals one. It makes no sense. What this is telling you is that there are no solutions. This does not make sense. If you do your Gauss-Jordan elimination, you get to reduce row echelon form. You see this kind of thing. It makes no sense. There's nothing, that second line makes no sense. In other words, there aren't any solutions here. So this augmented matrix will end up looking like this. So there you go. Two sets of linear equations, two systems of linear equations. One with multiple solutions and one with no solutions at all. You've learned a few more things here in Mathematica. You've seen what it looks like. Intuitively, you understand what's going on here. I'm hoping that this helps you hit the subscribe button and I'll try to get more of these done in this period.