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In Exercises 19 and $20,$ choose $h$ and $k$ such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part.$\begin{aligned} x_{1}+3 x_{2} &=2 \\ 3 x_{1}+h x_{2} &=k \end{aligned}$

(a) $h=9$ and $k \neq 6$(b) $h \neq 9$(c) $h=9$ and $k=6$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 2

Row Reduction and Echelon Forms

Introduction to Matrices

Missouri State University

Baylor University

University of Michigan - Ann Arbor

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this question asks us to solve the given land your system, using any methods. So we're going to use the Gaussian elimination method, which means we need to put our Matrix and to reduce national on form and then solve for the variables from there. So the equations that were given we can use to make a matrix, which from the first equation the first row is gonna read. X one plus three x two plus zero x three Since there's no x three term plus four x four equal zero, the next one is going to give us x one plus four x two plus two x three is equal to zero then our third equation gives us negative two x two minus two x three minus x four equals zero. After that, we have two x one minus four x two plus x three plus Explores equal to zero and finally we have Excellent, but it's two x two, but his ex Terry Plus at four is equal to zero. So now we work on this matrix to put it in, reduce special informed. So the first step we're gonna do is we're gonna switch Row one and road for So are one of those are for and our four equals. Our current are one. We do that we get this matrix. Okay, once we have that matrix, what we're going to do is we're going to try to knock out this term up here to make it a zero. So in order to do that, we need to take the second row and said it equal to our tumor in his 1/2 are one. So once we do that, we worked that out and we get this matrix so the first row stays the same. And then the second row changes because of the previous row operation we just did on it and all the other roser going states name cause we're not doing any row operations on them. Okay, Now our next up is we want Thio make this entry is zero. And so, in order to do that, we take the fourth row and we subtract 1/2 our one from it. When we do that, we get this nature. So first rose, the same second room's the same. Third row is gonna be the same. And then our fourth row is going to become zero guys. Negative 1/2 seven house and zero. And then our last road stays the same. Okay, so now what we want to do is we want to make this entry a zero, and we're knocking all of zeroes out and this column or not, knocking all the non zero entries out in this column and making them zero. So we're left with this as the only non zero element in the column, because it's gonna be a pivot. So in order to knock out that one right there, we have to take the fifth row. So are five and subtract 1/2 are one from it. When we do that, you got this matrix. First Row stays the same as just a second. That's just third and the fourth. And in the fifth room becomes 00 Negative three. House. What? Huh? And zero. So now we've successfully knocked out all the zeros, the non zero elements in that column except the first pivotal one, which is the two. And our next step is going to be to work on the second column. So what we want to do is we want to take our or three term right here, and we want to make this a zero. So to do that, we say or three is equal to 1/3 or three. Uh, 1/3 or two. Plus our three. Let me do that. We get this matrix, our third row is going to become 00 Negative. Three halfs. Negative. 76 zero, and then the other roads, we're gonna stay the same. Okay? So as you probably guessed, the next thing that we want to do is get rid of this five and make it a zero entry. So to do that, we want to take the fourth row. It's a tract from that 56 of the second row. When we do that, we get this matrix here. We'll see that are five got knocked out. Not our fifth roses. Okay, so the next thing that we're gonna do is we're going to switch are three with our four, and we're just doing that. Thio help the steps for getting into recession form. So the pivots fall nicely with each other in that staggering format. Okay, so now after that, what we want to dio is we want to make this element zero So in order to do that, we take or fourth row and subtract 67 or three from it. Once we do that, we get the following matrix, and then our fourth row becomes this. We see that we've knocked out that determine its zero now. Okay, Now our next step is to make this a zero. So in order to do that, we take our fifth row. So are five. And from our five, we subtract 67 are three. Once we do that, we get this new Matrix. But this is zero. And then negative. 95 over 12. And then our last road becomes 000 and then negative 20 over seven zero. All right, so now what we want to do is we want to get rid of this tournament. We want to make it a zero. So in order to do that, we take or five. We said that equal to or five one is 12 over 19 are for still, once we do that, we had this matrix, which is to native four warned 10 063 House Fate of 1/2 0 00 Negative. 7/4. 47. 12 zero 000 Negative. 95 over 21 0 And then finally, our row of all zeros. Now what we want to dio is we want to make sure that each of these pivot points are all equal to one because that's a requirement bird. There do special informants in threes and gassy and elimination. Uh, well, put it and reduce social inform before we do any any more solving. So we can do these all the same stuff we can say our one is equal to 1/2 are one are two is equal to 1/6 are too are three is equal to negative 4/7 are three and or four is equal to negative 21 95th are for It's me do that. We get this new matrix and then we start calling for our variables x one x two x three The next four So we'll start with this road right here and work our way up. So this first row So we're selling for explore the streets Explore is equal to zero. We moved to the third row appear which reads x three waas negative 47 over 21 times X four is equal to zero. Since we know from here that exportable zero, you know, that cancels, which means that X three has to equal zero. Okay, Still, for X two, we do exactly the same thing. X two plus 1/4 x three minus 1 12 x four equals zero. We know that X four is equal to zero. No exterior is equal to zero. So these will cancel, Which leaves us with x two is equal to zero And for our last one x one we see x one minus two x two plus 1/2 x three was 1/2 We're running out of room a little bit, so all right, a little smaller one minus two x two 1st 1/2 x three. 1st 1/2 that's four equals zero. And since we have X for X three index to all equal zero, he's all cancel. Which leaves us with X one equals zero. So that is gonna be our solution. It's the trivial solution. And we have that x one x two x three and four is equal to the zero doctor. So all these equal zero

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