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(1 point) Solve the system+4x5 +3x6 +2x5 +2x6 8x5 +x6-x -2x1372...

Question

(1 point) Solve the system+4x5 +3x6 +2x5 +2x6 8x5 +x6-x -2x1372

(1 point) Solve the system +4x5 +3x6 +2x5 +2x6 8x5 +x6 -x -2x1 372



Answers

Solve each system in Exercises $1-4$ by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure described in this section.
Find the point $\left(x_{1}, x_{2}\right)$ that lies on the line $x_{1}+5 x_{2}=7$ and on the line $x_{1}-2 x_{2}=-2 .$ See the figure.

Okay, so this problem are given these two equations and asked to find the point of intersection just basically just asking us to solve the system. We're gonna be using elementary grow operations for this. First things first, we're going to try to get rid of this three export in the second room. Did you that we see the one x one in the first row and we can figure out that to get rid of it, we can add negative three times were one to row two, and that's going to give us a new road two of eight. X two is equal to two, and we're still gonna have our original road one right here. Next we can solve for X two by dividing by eight on both sides. And that gives us a new row. Two of X two is equal to one over four, and we're still gonna have our original first row right here. So next we're going to get rid of this negative five. Excuse yourself for X one, and to do that, we can add five times wrote to t grow one, and that's going to give us a new one of X one is equal to nine over four. And our second row, of course, we found to be ex Cheek was one over four. And then our answer in X one comma X two is gonna be 904 comma, one over four.

All right, so on number 42, I'm going to start off by getting rid of this top equations fractions by multiplying each term by the greatest common factor in this case, it will be a 14. So that's gonna get me x -2, Y is equal to seven. Then I'm gonna go ahead and distribute this and simplify this. I'm gonna have to x minus four. Y plus three is equal to 20 I'm gonna go ahead and subtract the three here, so I end up having to X -4 Y. So four is equal to 17. So I want to try and eliminate um the method of elimination. So I need to multiply this top uh equation by negative two to try and eliminate the excess when I do that distribution and it was negative two, X Plus four y equals -14. Well, unfortunately because these will cancel, these will cancel, I end up with zero on the left and I end up with a three on the right and saying zero goes three. That is a false statement, zero does not equal three. So this is an empty solution set. Or you can say that there is no solution for

We're gonna solve this systems of equations by elimination. So we have 1/4 x plus y equals 11 over for and then we have X minus 1/2 y equals two. So first off, we want to get rid of these fractions. So what we can do is we can multiply everything by four on this first equation and then multiply everything by two on the second question. So that will do. X plus four y equals 11 and then two x minus y equals four. So now done away with my fractions. But I don't have any. Coefficients are the same. So let's make this into two over here, so I'm gonna multiply this by too. So I have two eggs plus eight y equals 22. Well, now I have to access here. So two eggs minus y equals four. Gonna subtract the bottom one from the top one. So that will leave me with nine. Why? 22 minus four is 18 the white off my line. And why will equal to now I'm gonna plug in wides. Another equation. I happen plugging into this one. So two X minus two equals four at tunable sides. to x equal. Six. Divide off my tube and X equals three. So my equation So the solutions and sequester will be three common too, where X equals three and y equals two.

So they want us to solve this by either using elementary grow operations or an augmented metrics. Um, so what should do both? So you can kind of see how this actually plays out in real time. Like how us applying these to the Matrix is also doing the same thing to this system of equations we have here. So first, let's set up the system of equations. So the first thing you need to make sure is that your variables are lined up, so x one x two lined up right on top of each other. That's good. And then we have our equal sign and all the constants on the right side, so that's good. And now all we're going to do is so in this first position here, we're going to put a two because that's the coefficient of our first variable. Then we put a four because that's the coefficient of our second variable. And instead of putting an equal side, we're going to put this line here and then on the right. We have negative for, and then we'll close it and then we'll do the same thing for the second row, so it would be 57 and then 11. Okay, Now, if I was going about trying to do this just by addition or subtraction, the first thing I would want to do is maybe divide this first row by two, because that way we'll just have X one. And then once we get X one, we could go ahead and clear out the X one below it. Five x one. Ah, lot easier. So let's do that. So I'm going to multiply this top room by one half. I'm going toe right that result down here. And so the notation. I don't think they write this notation in the book. But the notation I will normally uses one half row one because that's the first room. So let's go ahead and apply that here. So that's going to become X one plus two. X two is equal to negative two, and then the second equation states unchanged five x one plus seven x two is equal to 11. Okay, now let's do that same thing over here, so I'm going to multiply the first row by one half. So the only thing that's going to change is the first road, and we multiply each of the components here by one half. So two gives us one or gives us too. And then negative two gives us or negative forgives us. Negative two. And then again, we just right that equation down there exactly the same as we did before. So now if we were to go ahead and add negative five times Row one to Row two, that's going to cancel out that negative access. So what, this is going to be? So we'd multiply this top row by negative five, and then we add these and then that gives us our new road to. So let's just go ahead and do that really quickly. So that's going to give us zero x one plus s o negative 10 plus seven. So that would be negative. Three x two and then we'll have negative 10 plus 11, which is going to be equal toe What? So now this is our new equation, too, because that first one we didn't change it. Also, it's actually write out what we just did. So x one waas two x 20 tonight, too. So first equation still stays the same. But then this one here become zero x one minus three X two is equal toe one. So the way we would write that in this notation over here is going to be what we can't just kind of combine these two get this in the same way. So we write it instead. So we say what we multiply the first row by. So we multiply the first row by negative five row one, and we're going to add this to row two. So the notation is kind of weird. Um, but the operation you're applying is the first one, and what role you apply to is the second one. Er, least that's how I was taught to do. So the first row stays the same. We didn't do anything to it would be one too negative too. And now this second room. Remember? We did one times negative five. That gives us 02 times negative. Five negative. 10. Add that to three negative three and then negative, too. Times negative five gives us 10 at that to 11. That gives us 21 book actually over here. Seems like I even did this wrong here because that should have been 10. And when we added we got 21. Yeah. So good thing I was doing this on both sides because I was able to catch my error, right? But yeah. So now you can see how these two matrices kind of correspond to each other. Now, over here, what we would do. Well, I would multiply row Two by negative three. So at this point, let's just get rid of X one, because, I mean, it's just zero X. We really don't need that anymore. So we are that being gone and we're going to multiply this road bye. Negative one third. And if we do that so we get x one plus two, x 20 to negative two. First equation doesn't change. And then the second one is going to be X two is equal to negative seven now, to show what we did over here. So we're going to do line going down, and this is going to be negative one third row, because, remember, we want to somehow cancel out this to make it a one. So then we can go ahead and try toe, add it to the row above it. So let's go ahead and do that. So that's going to be so one too negative too. And then Or actually, let me do this in, uh, read to make it stand out from the rest of blue. So negative one third row two. So zero times negative. One third is zero negative. Three times negative. One third is one, and then 21 times negative. One third is negative. Seven. All right. And then at this point, if we want to cancel out the X two above this So if we come back over here, what we would do is what we would want to multiply this whole thing by Negative, too. And then add this. So in this case, the X one just drops down, so it would just be X one loss. So zero x two and then negative two times negative seven is 14. Add that to and we get 12. So if we were to rewrite this equation, it's going to be so x one plus zero x two is equal to 12. And in that second equation, there should stay unchanged. So it be X two is equal to negative seven. And so over here, if we repeat that same process. The way we write it is going to be so we want to cancel this to appear so we multiply rogue one. I mean wrote to buy negative too. And then we add that to road one. So negative two times zero is zero at that one. Still 11 times negative to negative. Two had that to to zero negative. Seven times negative too. Is 14 at that, too? We get 12 and in the second row stays the same. So 01 negative seven. And then if we come over here, we can erase that. And what this tells us is, well, X one is eager to 12 X two is equal to negative seven. And over here this tells us the same thing. Because if we write this out in that notation, this actually becomes lost zero x two plus zero x one. So this here is the system equivalent of what we have there. And so that just tells us x one is 12. Next two is equal to negative seven. So you really didn't need to do both ways, but I just wanted to kind of drilling the fact of what we were actually doing here since doing operations with matrices may be a new thing,


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