Welcome back, everyone. So let's take a look at this example. Again, we've got these three system of equations, and we want to solve this, meaning we want to find solutions for x, y, and z. The way we're going to do this is by writing a matrix in row echelon form. Remember, that's just that specific format where we have ones along the diagonal and zeros everywhere else. Let's go ahead and get started here. The first thing we have to do is convert this into a matrix just by pulling out all the numbers and coefficients. So this is going to be a 2, 4, 6, and then 24. This will become a 1, 5, 12, and then 60, and this will become 3, 6, 15, and then 20. So remember, all we have to do is just work the equations from top to bottom, focus on one number at a time. And then, also, every time we get ones, we're going to try to focus on everything else, getting them to be zeros. So let's take a look at this first number here. I just want a one that's in this position. I've got a 2 there, so that's bad. So can I use any of my operations like swapping, multiplying, or adding to solve this? And, actually, I can. So what you'll see here is that I can actually swap these 2 rows in order to get one in that position. So the first thing you can do is just swap. So we'll swap row 1 and row 2, and they'll basically just become each other. Alright? So in other words, I'm just gonna rewrite this matrix, and this is going to be 1, 5, 12, and then I've got 60. And this is going to become 2, 4, 6, and then 24. And then I've got 3, 6, and then 15. And then finally, I've got 20. Alright? So I've gotten one of my numbers already. I've got 1. Now remember, rather than focusing on these other numbers and getting them to be ones, the next thing you want to do is focus on getting everything underneath those ones to be zeros before you move on to the next one. So the way we do that, remember, is by adding. So the way we what we're going to do is we're going to take something to this row over here. And in order to cancel out this 2 to become a 0, I have to add it to something else. I can't add it to 3 because those things aren't going to cancel out to 0. But what I can do is I can add row 2, and I can add it to a multiple of row 1 because I have this thing as 1 over here. So I can do this by multiplying row 1 by negative 2. And that will become my new row 2. Let's work this out real quick. So my row 2 is equal to, again, 2, 4, 6, and then 24. And then if I multiply negative 2 times row 1, that equals remember, I just multiply all these things by negative 2. So this is going to be negative 2. This would be negative 10. This will be negative 24, and this will be negative 120. Alright? Now, once you add those two things, what do you get? These things will cancel, and you'll just end up with 0. Negative 6, this would be negative 18, and this would be negative 96. Alright? So now that becomes my new row 2, and so you can just rewrite this matrix. So, really, what happens is I just get 1, 5, 12, and 60, then the 3, 6, 15, and 20 because those things remain completely unchanged. And now the only thing that changes is row 2, which is now 0, negative 6, negative 18, and then negative 96. Alright? Notice how we got a 1, and now I've got one of my zeros. Now I'm just going to focus on getting this thing to be 0, and I'm just going to do a very similar step. Instead of adding something to row 2, I'm going to add something to row 3. And so I'm going to add row 3. And just like we added a multiple of row 1, we're going to do the exact same thing. But instead of negative 2, all we're going to do is we're going to multiply or add to a multiple of negative 3 of row 1. Alright. So it's the same exact procedure, just trying to cancel out that 3 there. So what does this become? Well, row 3 again is just equal to I've got 3, 6, 15, 36, 15, and 20, and then negative 3 times row 1 equals I'm going to have negative 3, This becomes negative 15, negative 36, and then negative 180. Alright? So if you add all these things together, this becomes your new row 3, which should equal the 3 and the negative 3 should cancel out to 0, then this should become negative 9, this should become negative 21, and this should become negative a100 I'm sorry. This should be this is you this is gonna be negative 160. Alright? So that's that's the little barrier. It's not a one. Alright? So just be careful there. So this is what your new, row 3 should look like. So let's rewrite this. Alright? So we've got 1, 5, 12, and 60, and we've got 0, negative 6, negative 18, negative 96. And now we've got, 0, negative 9, negative 21, and then negative 160. Alright. So So, again, making some progress here. I've got ones and I've got zeros underneath. That's good. Now I can focus on the next one. So the easiest way to get this row over here and this entry to be a negative one or sorry, a positive one is I can actually take this whole entire equation, and I don't have to add anything to it. I can just multiply it. Right? Because it's not like trying to get this number to be 0. I can multiply by something to get this to be positive 1. So So what we're gonna do is we're gonna multiply. Now how do I get negat