In Exercises 1 - 24, use Gaussian Elimination to find the comp...

Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}w - 2x - y - 3z = -9 \w + x - y = 0 \3w + 4x + z = 6 \2x - 2y + z = 3\end{cases}$

Accepted Answer

welcome back. I am so glad you're here. We are given this system of equations and we are asked to find the complete solution using Gaussian elimination. Alright. We need to recall from previous lessons that when we have a system of equations, we can create a corresponding matrix with its coefficients and constant terms. So going with the first equation there, the coefficients are too three -2 -3 38 for that constant term. And the second equation, the coefficients are one, 24, nice of them to leave that space where the Z's are, that's a zero and then 95 for the constant term. The third equation will be one negative 201 57. And the fourth equation will be 01 negative four negative 1 38. So there is our matrix. And what we need to do for Gaussian elimination is we need to get this looking like a matrix that starting in the upper left hand corner has a one and then below that all zeros. And then for the second column there will be something at the top but then another one because we're creating dying the ones and below that zeros for the third column, it will be something and something and then that one that we need and a zero below that. And the fourth column will be something something something one, there's our four ones and then that final column will be something something something and then it will be our Z. What this final one equals will be our Z solution. So we can put a Z there. So once we get it looking like that, then we're going to use substitution to go back and solve, we'll have what Z is and we'll figure out what Y and x and w are. Now if you haven't done these for a while um this will be quite a journey, this will have quite a few steps but we can do this. So what we're going to do first, the first thing we want is we want a one In this upper left hand position. So I'm going to write off to the left what steps we're taking. So we want to replace row one And to do that. I am going to there's a few ways to do it. I'm going to take row two And I'm going to multiply that by a -1. So negative row two and add that to row one. Now because we are replacing row one, I'm going to skip writing that row and start copying down row two and I'm not going to include all of the brackets. I'm just going to write the numbers because we're going to do this a few times. So we have 95 for row 21, negative 201 57 for row three, 01, negative four, negative 1 38 for row four. And now doing that work taking row two, multiplying it by negative one and then adding that to row one. So negative one, times one is negative one, negative one plus two is one. That's what we want there. Now, moving across the route negative one, times two is negative two plus three is one negative one. Times four is negative four plus negative two is negative six, negative one, times zero is zero plus negative three is still negative three and negative one times 95 is negative 95 plus 38 is negative 57. All right, We've done that first step a little bit more to go. Next thing we want. We're going to work on the columns. So, we want a zero in this second position here. Working our way down. And the way we get those zeroes is that we're going to do the work on the road that has the one in the desired position. So to replace row two, we are going to take row one and we're going to again multiply it by a negative one. So a negative row one and then add that to row two. Because we are replacing row two, I'm going to copy down Rose 13 and four. So, we've got 11 negative six, negative three, negative 57 skipping row 21, negative 201 01 negative four negative 38. Now doing the work to replace road to. So working across row one, multiplying everything by negative one and adding that to row two negative one, times one is negative one plus one is the zero that we want negative one, times one is negative one plus two is one negative one. Times negative six is positive, six plus four is 10 negative one. Times negative three is positive, three plus zero is positive three negative one. Times negative 57 is positive, 57 plus 95 is 152. I did do all of these ahead of time. I'm not doing these in my head. So the next thing we want is to get a zero in this position here in row three, we're replacing row three. We're going to do the work on the road that has the one in the desired spot. So that's row one. We're going to take row one and again, we're going to multiply that by a negative one And then add that now to row three, scooch ng down, we are going to use this whole thing. Okay to replace row three, we're going to copy down rows 12 and four. So I'm going to do that 11, negative six, negative three, negative, 57 01, 10 3, 152 skip row three and negative four negative 1 38. Now to replace row three, taking row one, multiplying it by negative one, negative one, times one is negative one plus one is the zero that we wanted, negative one, times one is negative one plus negative two is negative three negative one, times negative six is positive, six plus zero is a positive six negative one times negative three is positive three plus one is four, negative one. Times negative, 57 is positive. 57 plus 57 is 114. Okay, that whole first column is done. Yay. And now the next thing we want to do is get a one in the desired position in the second column. We already got that. Yes, a little bit of work done for us. Now. We want zeros below it. So let's work on replacing this negative three. Excuse me, In order to get a zero there, we're going to do the work on the road that has the one in the desired position. So we're going to take, to replace row three again, we're going to take row two. We're going to multiply that by three. So three times row two and add that to row three. Because we're replacing row three, I'm going to copy down rows 12 and 411 negative six negative three negative 01, 10 3, 152 and row 4101, negative four negative 1 38. Okay, now doing the work three times 00 plus zero is 03 times one is three plus negative three is the zero that we want. Oops, I wanted to do this in a different color. I apologize. So 00, three times 10 is 30 plus six is 36 times three is nine plus four is 13 and three times 152 is 456 plus 114 is 570. Okay, we've got the zero in the spot. We want it there now. We want another zero below it in this position. In row four, replacing row four to get the zero there. We're going to do the work on the road that has the one in the desired position. That's row too. And so in this case we'll just need to multiply row two times in negative one and add that to row four, scooting down a little bit because we are replacing row four. We can copy down rows 12 and three. So that's 11 negative six, negative three, negative 57 01 3. 152 36 13, 570. Now to do the work to replace row four negative one times zero is zero plus 00 negative one times one is negative one plus one is zero negative one times 10 is negative 10 plus negative four is negative 14, negative one times three is negative three plus negative one is negative four negative one times 152 is negative, 152 plus 38 equals negative 114. Alright, we've got the first two columns done. Yay. Now we need a one in this Third row, third column where this 36 is. So we need to replace row three. And the way to get that 36 into a one is to divide everything in that row by 36. Or I'm just going to multiplying it by one. 36th. 36th. So this is taking row three times 1, And because we're replacing row three we can copy over rows 12 and four. So that's 11 negative six, negative three, negative 57 01, 10, 3, 152 00, negative 14, negative four and negative 114. And now replacing row 30 times 1 36 times 1, 36 0, 36 times one, 36th is one year, 13 times 1, 36 is 13, 36th and 570 times one, 36th is 95 6th. Okay, now we need to replace this negative 14. We need to get that to be a zero. So we're replacing row four and to do that to get a zero. We want to work with the row that has the one in the desired spot. So working with row three, we're going to multiply row three times 14. So 14 times row three and then add that to row four because we're replacing row four, we can copy down rows 12 and three. So I'll do that. We've got 11 negative six negative three negative 57 01 3 001 36th In 95/6. and now replacing row four we are taking 14 times zero is zero plus zero is 0. 14 times 00 plus zero is 0. 14 times one is 14 plus negative 14 is zero. Yay 14 times 36th is 91 months plus negative four. Gives us 1918 months in times 95 6th is 3rd plus a negative 114 is 323 3rd. Okay now we have got this thing almost all figured out. We just need a one in this last position and then we'll have our Z. Value and we can use substitution and figure out W. X and Y. So to get this last one to be a one we're replacing row four and we're going to multiply row four by the reciprocal of that 19 18th we're going to multiply it by 18 19th. So so we're going to take row four times 18 19 months. And because we're replacing row four we can copy down rows 12 and three. So that is 11 negative six negative three negative 57 01 3, 152 13 36 deaths and 6th. And now multiplying this whole row by 18 19 times 18 1900 times 18 19 0. I want to do the color. I'm sorry. So anyway, the first three or zero and 19 18th times 19th is one. He and now 323 3rd times 19th is the lovely round old number of 102. Okay, well we have got this in our proper desired format and we recall from previous lessons that this matrix that we have just solved for this is a system of coefficients in constant terms for our system of equations. So we can rewrite this now as a system of equations. So row one is the same thing as one. W plus one, X minus six, Y minus three. Z equals negative 57. Right, that's our first equation. Our second one is the second row. Disregard that. Zero for the W. And just start with our X. So we've got one X plus 10, Y plus three. Z equals 152. Ye two equations. Now the third one we've got zero for W and X. So now just going on with y, we've got one, Y plus 13 Z equals 6th. And our fourth equation, the fourth row is one, Z equals 102. Okay, well we have solved for Z. Z equals 102. Now we just need W. X and y. And we can use substitution. So if Z is 102. actually, I'll write that red up here for the part that's going to need to zoom back up to the top. So Z equals 102. And I'm going to substitute that in here for this third equation. So now this will read Why plus 13/36 times 102 equals 95 6th. Okay, so we've got Y plus 13 times 102 is 6th. That equals 95 6th. Well, we can subtract 6th from both sides and then we'll have why by itself. And why equals 95 6th minus 221 6 is negative 21 year. We've solved for Why Why equals negative 21. Okay. Now we can go back into the um second equation to solve for X. So we know that Y. Is going to be negative 21 Z is going to be 102. Let's write that over here. We've got X Still fits X plus 10, Not Why, but 10 times negative 21. No, that's not gonna fit. Alright, I'm bringing it down here. All right. See we're using up all the space today. Plus three times not z but Equals 152. Okay, so this is X. And 10 times negative. 21 is X minus 210 plus three times 102 is going to be 306 equals 152. Well negative. 210 plus 306 equals 96. So we've got x plus 96 equals 152. Subtracting 96 from both sides and we get an answer of X equals 56. 152 minus 96 is 56. Okay. We have solved for X and now we can substitute that back into the first equation along with Ry and Rz and we can solve for W. And then we've got this thing. Okay, where have I got room? Um Let's erase these. Okay, So we've got one W Plus our X is -6 times why? Which is negative -3 times the which is 102 and that Equals -57. Okay, so during the multiplication, we've got W plus 56 negative six times negative. 21 is plus negative three times 102 is minus 306 equals negative 57 combining the constant terms on the left hand side, 56 plus 126 minus 306 is negative. 124. So we've got w minus 124 equals negative 57. And then adding 124. Table sides gives us W equals 67. Yay, we've done it. Okay, now capturing the solutions and now I'm going to get them way back up to the top of the page and those will just be a mess for a second look. Remember where we started so many years ago. Okay. We've got our solutions and now looking at all of our answer choices, this matches with answer choice. B. Well done, You have done it. We'll catch you on the next one.

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}w - 2x - y - 3z = -9 \\w + x - y = 0 \\3w + 4x + z = 6 \\2x - 2y + z = 3\end{cases}$

Key Concepts

Systems of Linear Equations

Gaussian Elimination

Row Operations and Consistency of Systems

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