1. (5 points) -2 Classify the phase-plane portrait for the linear planar system x' = 8)x...

Write the coefficient matrix and the augmented matrix for each system. $$\left\{\begin{aligned} 8 x-8 y &=5 \\ x-y+z &=1 \end{aligned}\right.$$

Officially this timing with you. Um, check our solution. So let's check it out directly so we can plug in our Exeter. Why? Value into equation one and two on If this is a switching, um, or equations should be true. So let's start with equation one. So our X is too, so that's negative. Two plus seven is equal to five. So we get that five of his you crucified. So this is true. Now, let's check our equation too. So here we're gonna have one half times two plus seven is equal to eight. So this gives us that one plus seven. Is it eight? So we have that bte aids. So this year is also true.

In this problem, we have to find vortex focus and direct grease off the parabola and sketched X grab. Now we are given two y square is equals toe by tex and like, this s y squared equals toe fight over two X Now, comparing this equation with the standard equation That is why square Cipel's toe four p x we get b Ys equals toe 5/8 and the focus off the parabola is 5/8 0. Therefore, the education off the direct tricks becomes exist, equals toe minus five or eight, and vortex will be at origen 00 Now we'll draw the graph now in the graph This one here is why exist This is here is X exist this parable of vice principals fight over two x. This point here will be five or 80 And this is the direct tricks exit polls toe minus five or eight. So that's a solution

All right, we're going to solve this system by graphing. And so that means we're going to graft both lines and see where they intersect. And the intersection point will be the solution of the system. So we're starting with the first equation, and we see that the line has a slope of 1/5 and a Y intercept of four. The 40.4 So we can start by plotting that, Why intercept and then using our slope, we can go up one and over five. That takes us to right about there. We can also go from our Y intercept down one and over five the other way. That takes us to somewhere around here. And then we can connect those points and we have our line. And now we move on to the second equation. Let's go ahead and divide everything in this equation by two. When we get why equals 1/5 X plus four, we'll notice that's the same as the other equation. So it's the same line all over again. This is just the same line on top of itself, and so we have infinitely many solutions because they're infinitely many points of intersection

So this system, we're at eight close. Let's make it here First we have to find, like, unveiling. Today we take the primitive. They led by yet language. Close my three most field. Find this idea vector When the but the I C zero We have my A B minus five. Yeah, by we have 25 00 Let's reduce this yet. My five by your your Here You're here. You're here. And yet one minus 100 We get our first hiding vector. You can say X one year extra. Extra three variable right here. The variable. Next you can write. I didn't see if you will. You go to 11 You at one. You get our first. I'd like to be one. Me or the 11 We have only one idea, Inspector were given a two by two matrix. This is not diagonal vote. In order to find second Eigen Vector, we're gonna do something called generalized vectors. We're gonna do equals three I where the He was a hero. So it will be I weird. If you do it, you should get 00 This is very all the zeros you choose. Generalized in Vector Be not to be equal to 01 Were we take the Might it be I I mean, yes, my five minus 55 Be not one. And yet by what meeting? Get, uh, the solution corresponds. Generalize like inspector Recall X p you get Excuse me by doing three d. Is there one? Both. He I was five here. You should get three d time. 01 five p one, actually. Choose one common zero to get my five Everything skill weaken by. We have a generalized vector. You mean you appear three d? I was 10 He Why five here. So you get you could increase. I was 15 He and by our first solution responding to the idea though you 11 I give you three p. I was 11 going generally have, uh, the 12311 what you see by five p and that solution