Solve each system of equations using matrices. Use Gaussian el...

Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}3w - 4x + y + z = 9 \w + x - y - z = 0 \2w + x + 4y - 2z = 3 \-w + 2x + y - 3z = 3\end{cases}$

Accepted Answer

Welcome back. I am so glad you're here. We are given the system of equations and we're asked to solve for it. We recall from previous lessons that when we have a system of equations we can rewrite it as a matrix with the coefficients and the constant term terms. So the first equation is going to be our first row will take the coefficients which are two negative two, negative 31 and the constant term six. And then the second equation will be the second row 11113. The third equation will be the third row, 12, negative 318. And the fourth equation, the fourth row 31722. And we can choose Gaussian elimination or Gauss Jordan elimination. I'm going to demonstrate Gauss Jordan elimination, which means that we are going to get this matrix here in black to look like a matrix that has a diagonal row of ones for the first four columns and they'll be surrounded by zeros. So the first column will be 1000. The second column will be 0100. The third column will be 0010. And the fourth column will be 0001. And then The 5th column will give us our solutions. It will give us WX, So there are quite a few steps in order to get our matrix looking like that blue one and I will record the steps here on the left side. The first step we're going to take is to get this to in the upper left hand corner to be the one. And so in order to get that to to be one. We're going to multiply that rho times the reciprocal of two times one half. So we're going to take row one And we're going to multiply it by 1/2 because we're replacing row one, we can copy down rose to three and four. So that's negative 31831722. And now multiplying row one by one half, two times one half is the one that we want negative two times one half is a negative one, negative three times one half is negative three halves, one times one half is one half and six times one half is three. All right. We've got row one replaced. Now we need to get zeros and the rest of the positions of column one. So let's start here with row two. And to get a zero in that position in row two. We need to work with the road that has the one in the desired position which is row one. We're going to take row one in this case times a negative one and add that to road to because we're replacing road too. We can copy down rose 13 and four. So that's one negative one, negative three halves, one half, three skipping row negative 318 and row 43172 and two. And now replacing row two negative one times one is negative one plus one is the zero that we want, ye negative one times negative one is a positive one plus one is two negative one times negative three halves is positive. Three halves plus one is five halves negative one times one half is negative one half plus one is one half negative one times three is negative three plus three is zero. Alright, we've got row two replaced, we have the zero in position. Now looking at column one, we want to get a zero in row three. So to replace row three to get a zero, there we're going to work with the row that has the one in the desired position which is row one And again we'll multiply that by a negative one And this time add that to row three because we're replacing row three we can copy down rows 12 and four. So that'll be one negative one, negative three halves, one half, 30 to 5 halves, one half, zero and 31722. And now for row three will take negative one times one which is negative one plus one is the zero that we want, ye negative one times negative one is positive one plus two is three negative one times negative three halves is positive, three halves plus negative three is negative three halves negative one, times one half is negative one half plus one is positive one half and negative one times three is negative three plus eight is positive five. Alright, one more position in column one. We want to get row four where this three is in order to turn that three into a zero. We're going to work with the road that has the one in the desired position which is row one. This time we'll multiply row one times negative three And add that to row four. Because we're replacing row four, we can copy over rows 12 and 31 negative one negative three halves one half, 30 to 5 halves, one half, zero, 03 negative three halves one half, five and row four negative three times one is negative three plus three is the zero that we want. Ye negative three times negative one is positive three plus one is four, negative three times negative three halves is a positive nine halves plus seven, gives us 23 halves negative three times one half is negative three halves plus two, gives us one half. Look at all those one halves in column, column four and negative three times three is negative nine plus two is negative seven. Okay, the first column is done. Ye still a lot more work to to go here. Okay now we want to get a one in the desired position for column two. So that's row two, column two. Where this too is Because we'll be replacing row two. We'll work with road to here to get the one there and we'll multiply it by the reciprocal of that to again multiplying this by one half. So we're taking one half times row two, replacing road too. So we can copy down Rose 13 and four. So that will be one negative one negative three halves one half, three -3/2 04, 1/2 -7. And now doing the work to replace road to zero times one half is 02 times one half is 15 halves, times one half is 5/4 one half times one half is 1/4 and zero times one half is zero. Now we want to get zeros in the rest of the positions for column two. So this negative one in row one can be the next one we focus on in order to get a zero there we're going to work with the road that has the one in the desired position which is road to. And in this case we'll just add road to to row one. That'll get r zero there because we're replacing row one we can copy over rose to three and 401 5/4 4th 0 negative three halves one half 504 halves one half negative seven. And now doing the work to replace row one one plus zero is one negative one plus one is the zero that we want. Ye negative three halves plus 5/4 gives us a negative 1/ one half plus 1/4 gives us 3/ and three plus zero is three. Alright continuing our work on Column two and we'll get a zero in this position in row three in order to get a zero. There will still work with row two and we'll multiply it by a negative three, Add that to row three because we're replacing row three. We can copy over 12 and negative 1/4 3 4th 0, 1, 5/4 1, 4th, 0 and 04 halves one half negative seven. Now replacing row three negative three times zero is zero plus zero is zero negative three times one is negative three plus three is the zero that we want negative three times 5/4 is negative 4th and plus negative three halves is negative 21 4th negative three times 1/4 is negative 3/4 plus one half is negative 1/4 a negative three times zero is zero plus five is still five. Okay, we're almost done with column to let's get this four in a row four into a zero. In order to do that. We'll continue to work with row two where we have that one where we want it and this time we'll multiply row two times negative four and add that to row for because we're replacing row four, we can copy over rows 12 and three. So we've got 10, negative 1/4 3 4th 301, 5/4 1 4th 000 negative 4th negative 1/4 and five. And now replacing row four negative four times 00 plus zero is zero, negative four times one is negative four plus four is the zero that we want, negative four times 5/4 is negative five plus 23 halves is 13 halves negative four times 1/4 is negative one plus one half is negative one half and negative four times 00 plus negative seven is negative seven. Okay, now we need to work on column three, getting the one in the desired position. So we can use that to get the rest of our zeros. So this will be in row three in order to get that to be one. We're going to multiply it this time by its negative reciprocal negative four 21st. So that that will become a positive one, negative 4 20 firsts times row three and we can copy over rows 12 and 410 negative 1/4 3 4th 301, 5/4 1 4th, 0, skipping row 300, 13 halves, one half negative one half and negative seven. Now replacing row 30 times negative four 21st zero. Both times negative 21 4th times negative four 21st is the one that we want and negative 1/4 times negative four 21st is going to be a positive. Oops. Yes there, sorry, positive one 21st and five times negative for 21st is going to be negative 2020 firsts. Okay, continuing with column three. We want a zero in this position in row one. To get easier. There will work with row three where we have the one where we want it and this time we'll multiply row three times 1/ And add that to real one. Because we're replacing the one, we can copy over rows 23 and 401, 5/4 1 4th 21st and negative 2020 firsts in 00 13 halves, negative one half and negative seven. Now replacing row one 1/ times zero is zero plus one is 1, 1/4 time 00 plus zero is 0 1/4 times one is 1/4 plus negative 1/4 is the zero that we want. A 1/4 times one, 21st is one 84th plus 3/ is 16. 20 firsts. Yes, I have all of these written down in a sheet of paper and 1 4th times negative 20 21st. No, sorry we're on 1/4 times one. 21st is going to be No, we did that one. I apologize. 1/4 times negative, 20 21st is negative 5, 20 firsts plus three is 58. 20 firsts. Okay, Now continuing with Column three. We want to get a zero in row two. In order to do that. We are going to work with the road that has the one where we want it, which is still row three and we'll take row three times a negative 5/4 and add that two, row two to replace road to. We can copy down Rose 13 and four. So we've got 100 16 firsts, 58 20 firsts and skipping road 21st and negative 2020 firsts and 00 13 halves negative one half and negative seven. And now to replace road to negative 5/4 times plus 00, negative 5/4 times zero is still zero plus one is one negative 5/4 times one is negative 5/4 plus positive 5/4 is the zero that we want. Now we've got negative 5/4 times one, 21st is negative five 84th plus a positive 1/4 is 4, 20 firsts And -5 4th times negative 2021st is a positive 25/21. Okay, we're almost done with column three. Now we've got this 13 halves in row four that we want to become a zero. So we will take row three times negative 13 halves and add that to row four. And we can copy over rows 12 and 3100, 16 20 firsts 58 firsts, 0104, 21st 25 20 firsts and 21st and negative 2020 firsts. Okay, now replacing row four negative 13 halves times 00 plus zero is zero both times negative halves, times one is negative 13 halves plus 13 halves. Is the zero that we want negative 13 halves, times one, 21st is negative 13 40 seconds plus negative one half is negative 20 firsts and negative 13 halves times negative 2020 firsts is 130 20 firsts plus negative seven is negative 17 firsts. Okay. One column left. We can do this. We've got this alright. For column four we first want to get our one in place. So that's for row four, row four, column four. To get that to be a one, we're going to multiply it by its negative reciprocal, So that will become a positive one. So multiplying it by -21/17s. And because we're replacing row four, we can copy over rows 12 and 3100, 16, 21st, 58 21st, 0104, 21st, 25 20 firsts and 0011 21st, negative 2020 firsts. And now replacing row four. Well the 1st 3 or zeros and negative seventeenth's times negative 17 21 is the one that we want 21st 20 once. Not even sure anymore. And again, that's the same thing. We've got a one both times. We've got row four done. Let's finish up with column for and solve for W. X. Y. And z. Remember what we started doing 300 years ago. Alright, let's work on row one. So to get a zero in that fourth column of row one, we're going to work with row four and we'll multiply it by a negative 16 20 firsts negative 16 21st times row four plus row one. And we'll copy over Rose to three and four. So 21st, 25 21st. 0011 21st negative 20 21st. And this lovely looking 100011. Okay, Row one, negative 16 21st, times zero is zero plus one is still the one negative 16 21st time 00 plus zero is zero. Both times here, negative 16 21st, times one is negative 16 20 firsts plus positive. 16 21st is the zero that we want and negative 21st times one is negative 16 21st plus 58 21st is two. All right. Two more. Two more. We've got this. Let's work on road to To get a zero in that position in row two. Let's multiply row four times negative for 21st, so negative four 21st, times, row four plus row two copying over Rose 13 and 410002 skipping row 20011 21st negative 2020 firsts and 00011. Now replacing road to negative four 21st times zero is zero plus zero is zero these two times and will be a one there negative four 21st times one is negative four 21st plus four. 21st is the zero that we want and negative four 21st times one is negative four. 21st plus 25. 21st equals positive one. Just one more. We've got this all right in row three to get a zero. There will work with row four and we'll multiply it by a negative 20 firsts And then we'll add that to row three. We can copy over rows 12 and four. And look how much nicer. This is and 00011. Oh my goodness, just lovely. Alright, negative one. 21st time zero is zero plus zero is zero here one here negative one. 21st times one is negative one. 21st plus one. 21st is the zero that we want and negative one 21st times one is negative one. 21st plus negative 20 21st is negative one. That's it. We did it. This is the matrix that we want with our diagonal ones for the first four columns and then the fifth column corresponds with our W, X, Y and Z. So R W X, Y and Z are 21 negative one and one. And let's bring that solution way back up to the top where we started doing this would be a mess for a second, and we'll see which answer that matches with. Okay, 21 negative one and one matches with answer choice B. Well done, we'll catch you on the next one.

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$\begin{cases}\)3w - 4x + y + z = 9 \\(\w\) + x - y - z = 0 \\2w + x + 4y - 2z = 3 \\-w + 2x + y - 3z = 3\(\end{cases}$

Key Concepts

Systems of Linear Equations

Matrix Representation of Systems

Gaussian Elimination and Gauss-Jordan Elimination

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