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.

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

Accepted Answer

Welcome back. I am so glad you're here. We are given this system of equations and we're asked to solve for it. Well, we recall from previous lessons that when we have a system of equations, we can create a matrix using the coefficients and constant terms from our equations. So the first equation will become our first row of our matrix and the coefficients are 21 negative 11 and the constant term is three. And now looking at the second equation that's going to become our second row, that will be 12 negative 711. Our third equation will be our third row, one negative two, negative 515. And our fourth equation will be our fourth row two, negative three, negative 12 and nine. And we are asked to use either Gaussian elimination or Gauss Jordan elimination. I'm going to demonstrate Gauss Jordan elimination. And that means we want to convert this matrix into one that has for the first four columns, a diagonal row of ones surrounded by zeros. So the first column would be 1000. The second column would be 0100. The third column would be 0010 and the fourth column would be 0001. And then this fifth column, that's where it all happens because in this case, it will give us our WXY and Z solutions. So it's a fair bit of a journey to get from the one on the left to the one on the right. But we can do it. What we want to start off by doing is getting this one in the first column. So we're going to convert this to, to row one. How do we do that? Well, we're going to take this row and multiply it by the reciprocal of that two. So we're going to multiply the row by one half. So keeping track of our steps here to replace row one, we're going to take one half of row one. And because we're replacing row one, we can copy down rows 23 and four. So that is 12, negative 7111 negative two, negative 5152, negative three, negative 129. Now replacing row 12 times one half is one. Yay, one's done one times, one half is one half, negative, one times one half is negative one half, one times one half is still one half and three times one half is three halves. Oh This is gonna be fraction. All right. Now we've got a one in position for our first column and now we want to get zeros in the rest of column one. So let's work on row two here to replace row two to get a zero there. We're going to work with the column or excuse me with the row that has a one in the desired position. That's row one, we got that one. And in this case, we will multiply row one times negative one and then add that to row two because we're replacing row two, we can copy over rows 13 and 411 half, negative one half, one half, three halves, skipping row two, one negative two, negative 515 and two negative three, negative 129. And now doing the work to replace row two, taking row one times negative one and adding it to row two, negative one times one is negative one plus one is the zero that we want. Yay negative one times one half is negative one half plus two is three halves, negative one times negative one half is positive one half plus negative seven is negative 13 halves, negative one times one half is negative one half plus one is one half and negative one times three halves is negative three halves plus one is negative one half. All right. One of our zeros are in place. Now let's get the zero in place for row three. In order to get a zero. There, we are going to work with the row that has the one in the desired place. That's row one again. And again, this time, we're going to multiply row one times a negative one and add that now to row three because we're replacing row three, we can copy down rows 12 and four. So that will be 11 half, negative one half, one half, three halves, 03 halves, negative 13 halves, one half, negative one half, skipping row three, two, negative three, negative 129. Now replacing row three, negative one times one is negative one plus one is the zero that we want. Yay. Negative one times one half is negative one half plus a negative two is negative five halves, negative one times negative one half is positive one half plus a negative five is negative nine halves, negative one times one half is negative one half plus one is one half, negative one times three halves is negative three halves plus five is seven halves. So fraction. OK. Now that final zero in column one will be replacing row four. We'll work with the row that has the one in the desired spot. Still row one, we'll multiply row one times negative two and add that to row four. We can copy over rows one through 311 half, negative one half, one half, three halves, 03 halves, negative 13 halves, one half, negative one half and zero, negative five halves, negative nine halves, one half, seven halves. And here we go row four, negative two times one is negative two plus two is the zero that we want. Yay. Negative two times one half is negative two plus negative three or excuse me, negative two times one half is negative one plus negative three is negative four, negative two times negative one half is a positive one plus negative one is zero, negative two times one half is negative one plus two is one negative two times three halves is negative three plus nine is six. OK. We've got one column done. It's like a quarter of the way there. Nice. OK. Now let's work on column two. Next, we are going to get the one in the position we need for column two, which is in row two where this three halves is. So we'll be replacing row two and we're going to multiply by the reciprocal of that target target number. So we're going to multiply everything times two thirds. So we're taking two thirds times row two and now we'll copy over rows 13 and four. So we've got 11 half, negative, one half, one half, three halves. So many fractions skipping row two and zero, negative five halves, negative nine halves, one half, seven halves and row 40, negative 4016. That's my favorite row. It doesn't have fractions yet. OK? Replacing row 20 times two thirds is 03 halves, times two thirds is the one that we want. Yay. Negative 13 halves times two thirds, the twos cancel out. So that's negative 13 3rd, one half times two thirds is one third and negative one half times two thirds is negative one third. OK? Now we've got that one in place and we can use it to put zeros into position for the rest of column two. Let's work on row one of column two. We'll replace that by multiplying row two by negative one half and adding that to row one. Now we can copy over rows 23 and 401, negative 13 3rd, one third, negative one third, zero, negative five halves, negative nine halves, one half, seven halves and zero, negative 4016. Now replacing row one negative one half times zero is zero plus one is still one negative one half times one is negative one half plus one half is the zero that we want negative one half times a negative 13 3rd is 13 6th plus negative one half is five thirds, negative one half times one third is negative 1/6 plus one half is one third and negative one half times negative one third is positive 1/6 plus three halves is five thirds. All right. Continuing our work on column two. Let's replace this negative five halves. Let's get that to be a zero. So we will work with row two. We'll multiply it by a positive five halves and we will add that to row three. We can copy over rose one, two and 4105 3rd, one third, five thirds, 01, negative 13 3rd, one third, negative one third, skipping row 30, negative 401 and six. Now to replace row three, we've got five halves times zero is zero plus zero is still 05 halves times one is five halves plus negative five halves is the zero that we want. Yay. Now we've got five halves times negative 13 3rd is negative 65 6th plus negative nine halves is negative 46 thirds, five halves times one third equals 5/6 plus one half equals four thirds. I definitely did all of these fractions in advance. I did not memorize these. No, I can't. I do it that fast in my head. Five halves times negative one third equals negative 5/6 plus seven halves gives us eight thirds. OK, one more in column two, this negative four in row four. We'll work with row two and we'll multiply row two times four and add that to row four and we can copy over rows 12 and three. So we've got 105 3rd, one third, five thirds, 01 negative 13 3rd, one third and negative one third and 00, negative 46 3rd, four thirds and eight thirds. And now replacing row four, four times zero is zero plus zero is still 04 times one is four plus negative four is zero. Yay. Four times negative 13 3rd is negative 52 3rd plus zero is negative 52 3rd. Oh, there goes those nice whole numbers and four times one third is four thirds plus one is seven thirds. Four times negative one third is negative. Four thirds plus six is 14 3rd. Ok. Halfway nice. All right. Now we've got to get a one in the desired position in column three. That's row three of column three. So we'll be replacing row three. And to do that, we'll multiply row three times this time times the negative reciprocals times negative 3 46. So that will be a positive one and we can copy over rows 12 and 4105 3rd. Excuse me, one third, five thirds, 01, negative 13 3rd, one third, negative one third and row 400, negative 52 3rd, seven 3rd, 14 3rd. And now replacing row 30 times negative 3 46th is zero both times and negative 46 3rd times negative 3 46th is the one that we want. Yay. Four thirds times negative 3, 46th equals negative 2, 23rd and negative 3 46 times eight thirds is negative 4, 23rd. Ok. Now let's get zeros in the rest of column three. Let's work on row one where we have five thirds, we will work with the row that has the one in the desired spot. So row three, let's multiply row three times negative five thirds and then add that to row one copying over rows 23 and 401 negative 13 3rd, one third, negative one third, 001 negative 2 23rd and negative 4 23rd. Then 00, negative 52 3rd and seven thirds and 14 3rd. We're getting there, we're getting there. All right now, row one negative five thirds times zero is zero plus one is one negative five thirds times 00 plus zero is zero, negative five thirds times one is negative five thirds plus five thirds is the zero that we want. Yay. Negative five thirds times one third or excuse me, negative five thirds times negative 2, 23rd equals 10 69th plus one third is 11 23rd and negative five thirds times negative 4, 23rd is 20 69th plus five thirds is 45 23rd. Ok? Now getting the next zero into position, we'll work on row two of column three and we'll use row three to get us where we need. Let's multiply row three times 13 3rd and add that to row two. Ok. Copying down rows 13 and four, we've got 100, 11. That's not even 11 by any stretch of the imagination, 11 23rd, 45 23rd replacing row two. So skipping it 001, negative 2 23rd, negative 4 23rd and 00. Negative 52. 3rd, seven 3rd and 14 3rd. Ok. Replacing row two, 13 3rd times zero is zero plus zero is zero and 13 3rd 3rd is zero time, 13 3rd times zero is zero plus one is still 1, 13 3rd times one is 13 3rd plus negative 13 3rd is the zero that we want. Yay, 13 3rd times negative 2, 23rd is negative 26 69th plus one third is negative 1 23rd, 13 3rd times negative 4, 23rd is negative 52 69th plus negative one third is negative 25 23rd. Ok. Now let's finish off column three. Let's get this negative 52 3rd to a zero. We'll use row three to do that. We'll multiply row three times 52 3rd. But when you woke up today, you thought I'm gonna multiply row three times 52 3rd. Sorry, we're replacing row four. So we've got 52 3rd times. Row three plus row four is the work we're doing and we can rewrite rows 12 and 3100, 11, 23rd, 45 23rd. 010, negative 1 23rd, negative 25 23rd and 001 negative 2, 23rd negative 4, 23rd. And now replacing row four, we have got zero times. 52 3rd is zero plus zero is zero. Both of these times 52 3rd times one is 52 3rd plus negative 52 3rd is the zero that we want. 52 3rd times negative 2, 23rd is negative 104 69th plus seven thirds is 19 23rd. 52 3rd times negative 4, 23rd equals negative 208 69th plus 14 3rd is 38 23rd. Ok, we're close 3/4 of the way done. We're almost there. Now, let's work on getting that final one in into position for column four. That's row four, column four. We want a one there. So we're going to multiply this row by the reciprocal of that number. So that'll be times 23 19th times row four and copying over rows 12 and three, 100, 11, 23rd, 45 23rd, 010, negative 1 23rd, negative 25 23rd and 001, negative 2, 23rd negative 4, 23rd. Oops, I wrote 24th third, didn't I? Well, obviously it's right there on the screen there now. Negative 4, 23rd. Ok. Replacing row four, we've got zero times. 23. 19th is 03 times. It is and 19 23rd times 23 19th is the one that we want. Yay. 20 excuse me, 38 23rd times 23 19th simplifies 22. Look at that lovely, nice number we have there. Yay. Ok. Now let's get the rest of column four. Let's work on row one up here. So we are going to take row one and replace it to do that. We will use the row that has the one in the desired position. That's row four. We're going to take row four times negative 11 23rd, negative 11 23rd times row four plus row one. Ok. Skipping row one. We've got zero. Oops, sorry. I thought I switched colors. A 010, negative 1 23rd, negative 25 23rd and 001 negative 2 23rd and negative 4 23rd. And then this lovely row 400012. Ok. And now replacing row one negative 11 23rd times 00 plus one is one and negative 11 23rd times zero is zero plus zero is zero, both times negative 11 23rd times one is negative 11 23rd plus 11 23rd is the zero that we want. And now negative 11 23rd times two is negative 22 23rd plus 45 23rd is one. Yay. All right. Two more. Two more. We have got this right now. Let's replace the fourth column. Row two. We're going to use row four to do it. We're going to multiply row four times 1 23rd and add that to row two. So we can copy over rows 13 and 4100010. Oops, that's the one we're replacing three is 001, negative 2 23rd, negative 4, 23rd and 00012. Now replacing row 21 23rd times zero is zero plus zero is zero 1 23rd times 00 plus one is 11 23rd times one is 1 23rd plus negative 1 23rd is the zero that we want. Yay. And 1 23rd times two is 2 23rd plus negative 25 23rd is negative +11 more. We got this. We got this. All right. Row three needs to be replaced. We'll use row four to help. We'll multiply row four times 2, 23rd. And then we will add that to row three. We can rewrite rows 12 and 410001. Look how nice that is 0100, negative one. So nice and 00012. Now replacing row three, zero times 2, 23rd is zero plus zero is 00 times 2, 23rd is zero plus one is 11 times 2, 23rd is 2 23rd plus negative 2 23rd is the zero that we want. And drum roll, please. 2 23rd times two is 4 23rd plus negative. 4 23rd is zero. This is it. We've gotten our matrix looking the way we wanted. So I kind of squished the 4th and 5th columns a little bit. But recall from the beginning of this lesson 300 years ago that this fifth column is our WXY and Z solutions. So our answers are one negative 10 two. Let's bring this back up to the top and see what it matches and be a mess for a second while we grab it up here so long ago. Thanks for sticking with me or if you know you just fast forward it to the end. Welcome back. So we have one negative 102 that matches with answer choice. D 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.
$\begin{cases}w + x + y + z = 4 \\2w + x - 2y - z = 0 \\w - 2x - y - 2z = -2 \\3w + 2x + y + 3z = 4\end{cases}$

Key Concepts

Systems of Linear Equations

Matrix Representation of Systems

Gaussian Elimination and Gauss-Jordan Elimination

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