>> Another technique though that we can use and I really favor this because you're going to see how easy it is in just a second, is to actually continue in the matrix format and cause the entries above the ones to become zeros, and then we can just read our answer right off the matrix. Here, I'll show you what I'm talking about. We're going to start with -- we're going to start with this, alright and then operate on this. Let's come up about out here, I've just taking this and I've repeated it over here. Alright, now here's what we're doing. We're trying to cause these three entries to become zero, alright those three, and we're going to operate on these two first. Now, because the bottom row is in good shape that is it's in the condition that we wanted to be, we bring it down. And now we're kind of working from the bottom of the matrix up. So, we're deciding how to cause that entry to be a zero and to do it we play it off of the bottom row. You see so, we're going to operate on this row and add to that one, so it will be negative 3 times 1 then plus 3, so it's negative 3 times row 3, then plus row 2. So, negative 3 times row 3 plus row 2. And I know you must be thinking, gee this is going to be an awful lot of work, but it turns out that to go from these echelon form into this new form called Reduced Echelon Form, it's really, really easy because all the action is way over here with the constants. That's the only activity that is meaningful in this whole process. Why, I'll show you what I mean. It's negative 3 times row 3, so negative 3 times zero you see plus zero is zero, a trivial operation. An operation that we could skip in when we see this in the future, then negative 3 times zero plus 1 is 1, again, a trivial situation. Negative 3 times 1 is negative 3 plus 3 is zero, that's the way we set this all up. So, those operations were trivial and all the action is over at the constants. Alright, negative 3 times 2 is negative 6, negative 6 plus 5 is negative 1. Alright, now we decide, see we're kind of working our way up. Now, we decide what we need to do with this rascal and see and operating against the third row. And once again it's the same operation, it's negative 3 times row 3 plus row 1. So, negative 3 times row 3 plus row 1. You see and once again it's the idea that everything is kind of trivial because negative 3 times row 3, negative 3 times zeros you see would be zero and then add those guys and you get the same things back. Now, with these we set it up to get the zero in that position and now the important part, negative 3 times 2 is negative 6, negative 6 plus 9 is 3. Alright, in one more step we'll have our answers. Now, notice the third row is in great shape, so I'll just bring it down and the second row is in good shape, too. And the object here is to cause that entry to become a zero. So, let's see, to do that we'll play these two rows off of one another, so we're going to operate on row 1 -- excuse me, we're going to operate on row 2 and add to row 1. So, it's 2 times row 2 plus row 1, 2 times row 2 plus row 1. Now, just as an observation here in all of our notations for operations we show an operation on some other row and then we add the row we're in, that's kind of the way this notation that always works and it works well this way. We're just adding the row that we're in. We're operating on some other row. Alright, anyway two times row 2, so it's 2 times all of the items in row 2 plus the items in row 1. Now, once again most of the, some of the items are trivial, 2 times zero is zero plus 1 is 1, 2 times 1 is 2 with negative 2 is zero as expected that's why we set it up this way. Two times zero is zero plus zero is zero, trivial. Now, 2 times row 2 that's 2 times negative 1 plus 3 is 1. Alright, this is Reduced Echelon Form. When we have this main diagonal of 1, so zero is below, zero is above. And now, we can just read our answers. This is, remember in the equation format this is 1x equals 1, this is 1y equals negative 1, this is 1z equals 2. So, x is 1, y is negative 1 and z is 2. We can just read off the ordered triplet like this, right from the constants of the matrix in Reduced Echelon Form.