Hey, everyone. Welcome back. So in this problem, we're going to take this system of equations that we have given to us right here, and we're going to solve it using row operations. And the key thing here is, remember, when we use row operations, we're trying to get a matrix in row echelon form. And remember what that looks like. Row echelon form means that we have a matrix with all ones on the diagonal, and we have all zeros under the diagonal. And remember that these numbers over here that are sort of above and to the right of the diagonal can be anything. There's no restriction on them. So, we're going to have to take this system of equations and turn it into a matrix first so that we can start doing that. Let's get started here. The first thing is we're going to convert this into a matrix. And, really, we've done this before. We just have to pull out the coefficients. So this is going to be 1, 3, 4, and 2. This will be 2, 5, 7, 9, and then 4, 8, 10, and then this is going to be 14. Alright? So if you look here, I've already actually gotten one of the numbers that I need. I've gotten one of the ones on the diagonal, so it's a little bit of a head start, which is good. Now, what we have to do is we have to focus on these numbers. I want these to be ones, and then I want these numbers, the 2, 4, and the eights. I want those things to become zeros. And how do I do that? I'm going to have to use all the row operations that we've learned in order to get that system of equations or that matrix in row echelon form. Let's go ahead and get started here. Now you might think that you should focus on any number. Like, for example, you could focus on this 5 or the 10. But remember that second tip that we discussed in the video. Obviously, you always want to work down sort of from top to bottom in your equations, but you also every time you get a one, you want to make sure that you get all these numbers to be 0 underneath the diagonal, or underneath that one before you start focusing on the next one. What happens is if you try to make this one, then you're going to affect this cell over here, this number. And then later on, you're going to have to sort of get this to be 0, and then you're going to have to sort of mess up the one that you've already gotten. So it's always better to get this thing to be 0 or these numbers to be 0. Let's go ahead and do that. Alright? So, how do I get this number to be 0? Well, I can't swap because nothing was going to get me a 0 in that place, and I can't multiply this thing. I have to multiply this whole entire equation by 0. So, The only thing I have to do is the only thing I can do is I can add. So I'm going to have to add something to row 2 in order to make it 0. So, how do I do that? Well, the only number that's going to make this 0 is if I multiply or if I add it to negative 2. So I'm going to have to add so row 2 to some multiple of some equation in order to get a new row 2, and that's going to give me a 0. Now, one of the reasons it's really nice to get ones in these equations is because then you could just multiply them by a number, to get something to cancel out with this number over here. So, for example. So I've got this 2. I need it to cancel out by becoming by adding it to negative 2. So what I can do is I can take this whole entire row, and I can multiply it by negative 2. So I'm going to do negative 2 times row 1 and then add it to row 2, and that's going to become my new row 2. Let's work it out real quickly and just see how this works. So, this row 2 that I have is just equal to 2, 5, 7, and 9. Right? That's what that row is. What about negative two times row 1? Take all the numbers that you see over here and multiply them by negative 2. What do you get? I'm going to get negative 2, and I'm going to get negative 6, and I'll get negative 8, and then negative 4. Alright. So you multiply all those numbers. Now what happens is when you add these 2 rows, what you'll see is that the 2 and negative 2 will cancel, leaving you with just 0. The negative 6 and 5 becomes negative 1. Negative 8 + 7 becomes negative 1, and the negative 4 + 9 becomes 5. So this over here is actually what your new row 2 is. Alright? So now let's go ahead and rewrite this matrix. And remember, the only thing that gets affected here is just this row 2. So let's rewrite this. So this is going to be it's going to be 1, 3, 4, 2. And then remember, the 3rd row is going to be unaffected, so 4, 8, 10, and 14. But now the 2nd row gets rewritten. So 0, -1, -1, 5. Alright? Now if you look here, we've actually made some progress. I've got a one here, and then I've got a 0 over here. So it's making some progress. And if you actually look, this number is really close to being the one really close to being the one that we need along the diagonal, but it's negative. But we'll focus on that later. Remember, what we also want to do is we want to keep on working down the equations and getting all the numbers underneath the ones to be zeros. So now let's take a look at this third equation. You actually see that we're going to do something very similar. I'm going to have to get this to be a 0, so I can't swap it. Otherwise or I could swap it, but then I'm going to have to deal with the row that I just messed up, and so that's not going to be a good idea. I can't multiply this, so I'm going to have to add it to something. I'm going to have to add something to row 3 in order to get these numbers to cancel. So I'm going to add row 3 to something. Now just like I multiplied the first equation by negative 2 to cancel out the 2 that was here, I can do the exact same thing. I can instead multiply row 1 by negative 4, so then I'll get a negative 4 that cancels out with a positive 4.
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
7. Systems of Equations & Matrices
Introduction to Matrices
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