All right now we're going to solve for ay ay ay ay ay ay. Right there. And the equation here is d ay ay ay ay ay ay ay ay ay over. deesis. And all we do is, is we substitute. So, we saw for D ay ay ay ay ay ay and a previous step.

Right there. The CES, we saw that the very beginning and we do the math. 4500 divided by 3750. So da is one point, I'm sorry, is 1.2. That's it. Let's go on to the next one.

Okay, now we're looking for IB. And we know that IB is di b divided by the service. Well, again, there's the CES. We saw For the IB previously there it is there. So when I do this math, i b equals zero dot four. So here's a and IB right there.

Okay, let's go to the next one. Okay, and once again, we're solving for IC, di C we solved that a previous step this D says, we do the math 1500 over 3750 and we get IC equals zero dot four. Okay, we can check our answers now and what we can do is use the one of the equation that has that has all three variables i a, I be nice see and basically all we do is plug in So is 21.2. So I say a minus five times 1.2 plus 20 times zero dot four, plus a minus five times zero dot four. And by doing the math, I get minus 20 plus eight minus two equals zero. So checks.

All right, we're gonna do one more. And then at the end of that one, I'm going to use say, that website that I showed you previously, and plug some variables in there and make sure our answers check also, but here we got, we're good on this one now. So let's, let's go on to the next slide. Okay, um, why don't we do this one here. And I have the solutions for this in the upcoming slides. But why don't you hit the pause button and try to do them all right, and go all the way through Okay, because even though you may have the correct order, you may mess up your signs.

I do that. So I've checked the solutions on the next next one, thoroughly, least, hopefully thoroughly, and I believe them to be correct. And then after we do this, we'll just go over to that, the website that I gave you at the beginning, and we'll check them there and everything should match pretty much. All right. So hit the pause button, do the exercises. And then now on the next slides, we'll talk about the solutions, okay.

All right, here are the here's the solution for that problem. Or for these problems, I should say. And again, we set them up is um, we have to put them in a nice order. So we can set up our matrix, which we're showing you right here. Hey, And if you look 620282026 All right, and then what do we do? We start over again with these two.

And we just bring these two over, right? All right, and I just copied this matrix down here. Okay, the first thing we want to do is find the system determinant. All right, and let me clear the slide off. And all we do is go from the upper left hand corner down to the right, and we multiply 682. I mean 68668620 to zero to, let's see two to zero.

And then the last one right here, zero to three, zero to three. So here you go here, here, and they're all right. All right, they do the math. Okay? 288 plus zero plus zero is 288. Okay, now what do I do, I want to find what's in the denominator.

So now I go from the lower left hand corner up, and 080, multiply that plus two times two times six right here. And then six times two times two. Here, I do the math. Okay zero plus 24 plus 24 is 48. And then what I need to do is I need to subtract the first portion, which in this example is 288 minus the second portion in this case is minus 48. And I get my system determinant as 240.

Let me clear the slide. All right 240 Okay. Okay, now we want to find IAA. And what we, if you remember with our matrix when I'm looking for this unknown, okay, in my matrix, I substitute these coefficients for this. So if you look at my matrix, The first column is 12 dot zero dot 812 dot zero dot eight. All right, then I just continue on minus 282026.

And then again, I repeat these right here. I'm sorry, I repeat these two here for this, Alrighty. And it looks like I've got a little bit of an hour here on the matrix. I forgot them but on the on the actual computation that correct they should be negatives here. Okay. So when we do the math here, okay, we go down 12 eight, 612 eight, six, minus two times minus two times minus eight.

Right here. All right. And then this one here, zero, times zero, times 02. So that becomes zero. So if I do the math, all right, 576 minus 32 plus zero is this number here 544. And then I do the bottom and if you remember, we go from the lower left up, like I show you here.

So now we have eight times eight is zero, times 02 times two times 12. And plus six times zero times two. So when I do the math here, it's 48. Okay, so when I do the math, I get 544 minus 48 right here. And if you look at the previous No, you're not going nuts. I noticed this mistake on the slide and I stopped and I changed that.

So, so now we subtract 544 minus minus 48. We get 4096 divided by 242 40. Well, my do the math, I get 2.067. Okay, so let's go on Diaby. Okay, we're going to solve for IB now and the only thing we have to remember to do was solving in the B column, the IB column. So when I do my matrix, I need to substitute What was these numbers with with these here?

Okay, again, we've done them you do the math, there's a solutions and down here you do the same thing. All right. So I B is 48, which we solved here, divided by the system determinant, which was the very thing we saw so far. We do the math and we get zero dot two. So I B is zero dot two. Okay, we're finding ice now.

And basically all we have to do is remember, we're finding this row so substitute this is the equation, we end up with a minus 336 is the bottom equation and not the bottom means up the number we're going to subtract from, and we end up with a minus 32. So here's what we get. Minus 304. Remember a minus times a minus, all right, changes to a plus subtract and so forth. So we end up with a minus 1.27. All right, and that's my answer.

All right. So that's pretty much it. We kind of repeat, repeat, repeat. The only thing you got to remember is when we substitute those, those columns of numbers or those that number, call them to the variable that we want to walk. Find. Okay, let's stop.

Okay, and here's our check. We just plug in I took we check this one usually we I mean, they all should work. But we did this one because it had all three variables. And so we just plug in here, here and here and you can do the math. So when we do this, we come out with zero. So the checks.

Okay, here's the website that I showed you at the beginning of this lecture series. And here's the URL right there. All right, we're going to use up this one here called Kramer's rule. So let me just bring this up a bit. And we're going to use the standard matrix dimensions three by three. And you'll see what I mean when I say set the matrix Okay, you My three variables, and those are the answers down here.

All right, um, you can select if you want fractional a decimal, I'm going to leave it on decimal. And I'm going to minimize this now and set it up. Well, it's got to be there somewhere. There we go. Okay, and I'm going to move it over there. And what I'm going to do is just plug in these numbers.

So six, minus two, zero, up here, minus two, eight minus two, zero minus two 612 zero and minus eight and we're going to hit the solve button. And there are my answers right here. Okay, let me just move this over here. All right, 2.0666. Okay, I went to two decimal points 2.0670 dot zero to a minus 1.2 triple sixes, their quadruple sixes, I went to do decimal points, minus 1.27. So it checks a lot faster.

A lot nicer if you got an F you got a lot of these to do. Go to the website, plug them in just feel comfortable about how mechanically doing them. All right. You'll get an idea. Alright, well, I think we're gonna end this here. And this will end this, this this section on linear equations, the next section.

And the last will be an introduction to trigonometry. And that will be the last section in the series. And at the end, there'll be a little bit of a note from me and we'll see you over there.