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Repeat Problem 51 for the electric circuit shown in the figure. $$ \begin{aligned} I_{1}-I_{2}+I_{3} &=0 \\ I_{1}+2 I_{2} &=V_{1} \\ 2 I_{2}+2 I_{3} &=V_{2} \end{aligned} $$ (FIGURE CANNOT COPY)

So this is an electrical circuit problem where we know that two I won Um which we're gonna call X two, X plus six times X plus y two is going to be Y equals 12. So we know that's the case and then we're going to have that four I too, which is going to be for Y plus six times I won plus I chew is going to be X plus Y. And that right there is going to be equal to 12 as well. Based on that, we want to find the values of X and Y. So zooming in, we see, Do we get x equals 1.091 and y equals zero 545. So rounding to the nearest 10th, we see that I one is going to equal zero or 1.1 night too Is going to equal 0.5. And those are both in amps for the final answer.

Here. The gration is asking about the graph. So first drove the X axis and why access? So this is our I X is And this is our ex sexist. So this is 1234 white 6789 10 Sin the same on why Axes 1234567 And Israelis in the negative direction. So the first point of it is given That is minus one minus four. So we can draw that here. So this is our first point. The second point which is given that days 10 So the second point Well, come here. The third point is 23 This is here and four Tonys five and one. This will come here the 6th 1 and the last one is 10 and seven. So this is 2468 and then finally 10. And this is selling. And the fifth point, which is 85 So it and here five Just connect these point and you will get your graph. So this is your final graph according toa these points. So this is final graph and solution off our question

Weather today we're going to sketch the vector field f of X y equals minus 1/2 I plus Why minus x times J. Now, if you notice that in the eye direction this is independent off X and Y so will always move in this case to the left. Where left indicates the guy direction the negative I direction by half. But however much removing the J direction will be dependent on both our X and R Y, there are some lines that we can look out to simplify our sketching. So if we look at why equals zero then f of x and zero is just minus half I minus X j Along this line, we can sketch similarly on the line X equals zero F zero. Why is minus 1/2 I plus why J along the line y equals X, um f of x and X Because if we insert Wake was X, we just have minus 1/2 I because they cancel. But this is an example of lines of the form. Why minus X you call some constant along these lines. We have a vet field off the form, um, minus off I plus C J So let's begin sketching X. Why? So the line y equals zero is just the X axis and along the X axis we're always going to go, right? So if we start at X equals zero, we're going to go Sorry to the left by 1/2. And then if we go in the negative X direction, we will. We have my sex so we'll have a positive y. So if we just go, we always have to the left by 1/2. Then we'll start increasing. Then the further out we go, the steeper this gets. Now this hope happens inthe e opposite direction as we go to positive X. In fact, if we take a point here, we still go to the left by 1/2 but then will go down instead by an ever increasing amount. And this will cause lines to get longer. Um, because the length of a vector field, if we have a I plus B J, the length is a squared plus B squared square root. And in this case, a is minus 1/2 and B is just getting bigger. So now we've done this line, okay. And look at the line X equals zero, which is just the y axis. And now this happens in very much the same way. So at the origin of the 1st 2 um, here agree, have two lines. So if we increase, why a bit we're still going to go left by half, But then we're going to go up in our why. And in fact, if these two points of the same point, um which we can make, like so I like it. No there. So if we choose this point so than these two are directly related, they say this is X equals one and this is why it was one thes will be parallel lines on the reason we know these are parallel lines because the Y minus X equals c their line along this line. And then if we go up a bit more, it will become of it longer and a bit steeper. And similarly, if we go down at that, we still got the left. But because why is negative? We will go down a bit. I turned down a bit more. They would have a point down here that was, um on the same line as this vector stuff we consider the line. Why equals X? Uh, draw this in here. Likely along the line Y equals X. We can see that vector field is just constant off minus behalf in the eye direction. So what it looks like a vector field is doing is that it's so off rotating. If you can see that it's getting steeper as we get here, we can add in more points. If we were to consider, um, the line why equals minus X then, uh, f of X and minus X would be 1/2 I plus two y j. So if we draw in this line, well, im y equals minus X. And we were to put some points so again at the origin disagrees with what we had before. Um, safe, which is a point up here, will go to the left by half, but then we'll go up twice amount. So if we are, um, higher up then Wyche was ah, half. Then this will actually be above this line somewhere like this. They will get steeper and steeper. So here is our drawing of our vector field for the vector field f of x and y equals minus half I plus why minus x times J Thank you

Further to say we're going to sketch the vector field f of X y equals ah half X I plus why j. In order to get a feel for how respect field will look, let's plug in a few values. We can see that at the origin the vector field vanishes. So 00 returns 00 Now let's plug in some easy points. So if we look at 10 we get 1/2 0 If we, um so this will be our point here. Then we might want to go up. So 11 we get 1/2 1 then we'll go across, and very one which will remain is 01 more go across. So minus 11 will get minus 1/2 1 If we extend a table a bit, we go down, um, to have minus 10 you have minus 1/2 0 Then if we go down minus one minus one gives minus 1/2 minus one right to get zero minus one to give Syria minus one. Then finally one minus one, giving 1/2 minus one. Let's plot that on some axes. So the zero is a vector of their length says effectively. Just a point. Now, at 10 we go to the right half on up zero. So we attach that Arab now the 0.11 we go to the right 1/2 we go up one se attack scenario like so 01 we go to the right zero, but up one. Then at minus 11 we go to the left 1/2 hand up one. Then if we go down to the point minus 10 we dio to the right by half led to the left by half we go down again to the point minus one minus one. We get to the right by half on down by one. On that zero minus one, we just get down by one and then only here we go to the right 1/2 and down one so she can see, um, it will be the full vector field. Um, will be like so if we, um smoothly continue. We can see it's pushing out from the origin. However, because it is a factor of 1/2 in front of X, it means that it's squashed as it goes out along the x axis. Um, and so, yeah, if we were to add more points, would get a more fuller description of our vector field. But here's a sufficient idea of ah, vector field f of x and Y equals half eggs. I plus why j thank you.


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