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Find the general solutions of the systems whose augmented matrices are given in Exercises $7-14$ .$\left[\begin{array}{rrrrr}{1} & {-7} & {0} & {6} & {5} \\ {0} & {0} & {1} & {-2} & {-3} \\ {-1} & {7} & {-4} & {2} & {7}\end{array}\right]$

$\left\{\begin{array}{c}{x_{1}=7 x_{2}-6 x_{4}+5} \\ {x_{2} \text { is free }} \\ {x_{3}=2 x_{4}-3} \\ {x_{4} \text { is free }}\end{array}\right.$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 2

Row Reduction and Echelon Forms

Introduction to Matrices

Dk B.

December 10, 2021

the question and explanation is unrelated.

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for this question, you want to use gassy and Jordan elimination to solve built in your system of equation. So what that means is we want to take the Matrix that we're going to write for the equations they're given and put it in Reduced row echelon formed. So the first step that we're going to do is write out The Matrix to solve for the variables given the equations given, So our first equation read zero X one minus two x two plus three x three is equal to one three x one plus six x two minus three x three is equal to negative two and six x one plus six x two plus three x three is equal to five. So our first step and putting this in reduced Rochel inform is we want to switch these two rows so we're gonna set our one equal to our three and our three equal to our current are blond and what this is gonna give us is following matrix so we can see that road's been switched. The second row stays the same because we haven't done anything to it and the first row is become third row So our next step is we want to eliminate this entry. We want to make this entry zero so we could make sure that this pivot value is the only non zero entry in the road. So to do that, we're going to set our two equal to our two minus 1/2 or one and that gives us the following future. First row stays the same. Second row becomes 03 negative. About half negative, not half just for plugging and the values into this equation here. And our third room also stays the same because we didn't do anything to it. Next thing we want to dio is we want Thio eliminate this entry. So this leading entry of three is the only non zero elements in its column. So to do that, we set our three equal to or three waas. 2/3 are, too. When we do that, we get the following majors First row states to say second row stays the same and this third row become zero 00 and negative too. Again, from plugging in these values for our three and these feelings from or to into this equation that we wrote so our next up is we want to make this entry one. And once we do that, we can start eliminating the entries above it pretty easily, which will help us get it into producer special inform. So in order to make that entry born, we can see that we need to set our three equal to negative 1/2 times are three not gives us this matrix. First row stays the same. Second row, stays the same, and the third row become 0001 Okay, so now to knock out this entry, So to make it a zero, we want to set our two equal to are too. Was 9/2 our three. Because what that will do is that'll give us that this entry is equal to negative nine halfs, plus not half times one so not half plus negative. The house is equal to zero. So that gives us this matrix. Okay, so now our next up is we want to eliminate this country. So in order to do that, we do a very similar thing. Toe what we did right here. Well, you want to set? I'm gonna write and start reading over here. We want to set are one people Thio are one minus five or three. Okay, so we're going to do that. And that is gonna give us the following majors. First row stays the same. Sorry. First row doesn't say the same. And word we just knocked out that five entry. So that is going to become 6630 second rose days as it was before. And the third row also stays as it was before. Okay, so next up, we want to make this entry a one so we can start eliminating the non zero entries above it. In order to do that, we said our too well to 1/3 are too. And that gives us the fall of matrix. First row stays the same. Second row, become 01 negative, 3/2 0 and the last row stays the same. So now we want to eliminate this entry. So this is the only non zero entry and its road. So in order to do that, we have to set our one equal to our one. Minus six are, too. Once we do that, we get the following major 60 one. They're sorry. Six zero, 12 zero Kira. And the second worst is same. And the last roses say so. Now we want to make this entry one. So we want to divide the entire road by six, though, are one he was 16 or one, which gives us the following matrix. Okay, so now our matrix is in reduced. Groeschel inform, and we want to find a solution for it. But what we can notice really quickly is this last room right here. Reeves X one plus x two. But we're sorry. Uh, it breathes zero x one, but zero x two was zero x three is equal to one. All of these canceling to zero. Which means that we have an impression that reads zero is equal to one. This is obviously not true. Which means that there is no solution for this system up equations.

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