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.

${\begin{matrix} 3 x_{1} + 5 x_{2} - 8 x_{3} + 5 x_{4} = - 8 \\ x_{1} + 2 x_{2} - 3 x_{3} + x_{4} = - 7 \\ 2 x_{1} + 3 x_{2} - 7 x_{3} + 3 x_{4} = - 11 \\ 4 x_{1} + 8 x_{2} - 10 x_{3} + 7 x_{4} = - 10 \end{matrix}$

Accepted Answer

Welcome back. I'm so glad you're here. We are about to embark on a journey here because we are going to use elimination to provide a solution here. We're going to first convert it to matrix using its coefficient. So let's just get that first one done. We know that the coefficient of X is one. So we've got one coefficient of Y is to the second row. 2312 negative one. The third row negative one. There's no Y value. So that zero to negative and the fourth row, there's no X value. So we've got 0 to 114. Okay, that's our matrix that we're going to be using to start and what we want it to look like Using elimination is we want this lovely diagonal row of ones and zeros filling in around the ones. And we will then have our X value answer our Y value answer because they correspond with the ones our Z value answer and are w value answer as our fifth column. So we're going to turn this into this and it's going to be an absolute trip to get there. The basic pattern we're going to use is start with a column. We're going to get the one into place and then all the zeros. Then we move on to the next column. The second column we get the one into place and use that to fill in all the zeros continuing on all the way to the right. That's the basic pattern that we're going to use here if that's all you needed to know how to do it and you know how to convert all of those. I mean go ahead and fast forward quite a bit and I'll see you in just a second with the answer. But if you need help doing the practice of getting those ones and zeros buckle in because this is we've got a trip here. So the first thing we're going to do is look at the first column and the ones in the right place. Great. Okay, so now we can work on converting those other 3 to 0 as well, the other two because the fourth row already has a zero. So let's start working on that too. And that means I'm going to label this number one, our first step, We're going to replace row two. And to do that, we are going to take row one, the one with the one which is the one we're going to be working with. Usually we're going to take row one and we're going to multiply it by negative two and then add that to the second row. So we're going to take negative two times row one plus row two. And that will get us our first matrix here as we move along. So because we're replacing road to we can copy down Rose 13 and four. So 13412, skip road two negative 10 to negative 23, 02114. And going back up and looking at our original. So we take negative two times one which is negative two plus two. So that's how we get our zero in that first spot negative two times three is negative six plus three is going to be negative three negative two times four is negative eight plus one is negative seven negative two times one is negative two plus two. Oh there's like a bonus zero and negative two times two is negative four plus negative one is negative five. So we have updated our row two. Now we want to work on this spot right here getting a zero there. So we're working on row three, row three is going to be replaced and again we're going to work with our row one because it's got the one in that column and we're going to multiply that by actually we don't have to multiply it we just have to add them together. So we're going to take row one and add it to row three. We can say row one plus row three. We'll replace row three. So this is our second matrix. And because we're replacing row three we get to copy all those others. So 134120 negative three negative seven zero and -5. Can you tell this is going to be quite the journey 02114. And now we take the first row and add it to the third. Row one plus negative one is zero. We got our zero there three plus zero is 34 plus two is 61 plus negative two is negative one and two plus three is five. And also there are so many opportunities for error in here. You want to just go as slow and steady and carefully as you can handle it. So now we've got our first column is done. Great. Now we want step two for column to, we want to get our one in this spot here. So for step three, we're replacing row two. We want to get a one in that second spot. So we're going to multiply row two by negative one third. We're taking row two times negative one third. That'll get us a one in our second spot because we're replacing row two. We can copy these other. Rose negative 15 and 0 to negative three times zero or negative one third times zero is zero negative one third, times negative three is going to get us our one negative one third, times negative seven is going to give us seven thirds negative one third time zero, another zero negative one third times negative five is going to get us five thirds. Cool. We've got our one in that spot and now we've got our one in place and it will help us fill in our zeros. So let's get a zero at this top spot here for our step four, Step four. We're going to replace row one and we'll do that by utilizing our row that has the one in it. And we're going to take that and multiply it by negative three and add it to row one. So negative three times row two plus row one will get us our zero in that second spot. So now we will take and copy everything except road one, which we're replacing 017 3rd, 05 3rd. Yeah, we're going to be doing this for a while. 036, negative 1502114. And we got negative three times zero plus one. So negative three times 00 plus one. So one stays there negative three times one is negative three plus three, gives us zero, negative three times seven thirds, gives us negative seven plus four will give us negative three, negative three times 00 plus one is one, negative three times five thirds will give us negative five plus two is negative three. And we've got that top row replaced. So now looking at the second column, let's get our zero here, replace that three. So this is step five and four are step five. We're replacing row three and to get that will work with our row that has the one in it and we'll take it times a negative three and add it to row three negative three times row two plus row three. This is really similar to Step four, we are replacing row three. So that means we can copy every other row over 10, negative 31 negative 3017 3rd 05 3rd replacing row 302114 in row four. Okay, negative three times zero is zero plus zero is zero, negative three times one is negative three plus three is zero. We got our zero. Their negative three times seven thirds is negative seven plus six is negative one, negative three times 00 plus negative one is still negative one negative three times five thirds is going to be negative five plus five is going to be zero. Alright, we've got our zero in that spot. And now let's work on getting a zero here. The last zero in our second column. So let's bring this down a little bit. We are on step six And step six were replacing Row four and we're going to do that by working with the row that has the one in it. And we're going to multiply that row times negative two. So negative two, row two plus, row four will replace row four, replacing row four means we can copy over rows 123, So 10, negative 31, negative three 017 3rd 3rd 00, negative one, negative 10. And now we take negative two times zero is zero plus zero, negative two times one is negative two plus 20. We got our zero, Their negative two times seven thirds is going to give us out plus one is so negative 14 3rd plus one is going to give us negative 11 3rd negative two times zero is zero plus one is one and negative two times five thirds is negative 13th plus four is two thirds. And yeah, I did that with a calculator beforehand. I can't do it that fast in my head. So we've got our zero and we've got our second column. All set. We are making progress. So now step seven, You can skip this down a little more step seven. We want to get our one in place in the third column. So that's this negative one. And so we'll just multiply row three times negative one. We're going to replace row three And we're going to multiply by -1. We're replacing row three. So that means we can copy over rows 12 and negative 31 negative 3rd 05 3rd skip row three, row 400, negative 11 3rd one and two thirds. Okay, row three negative one times 00 negative one times 00. These do get easier because they fill in with a lot of zeros negative one, times negative one is one. We got our one in its spot negative one times negative one is one. Again. They're negative one times zero is zero. We've got step seven done. So now what do we do? We're going to start filling in our zeros. So we will replace this negative three with the zero for step eight. And to do that we're going to use our row with the one in it. We're going to multiply that by three and then add it to row one. So to replace row one we take three times row three plus row one. And because we're replacing row one we skip down to row two and start copying that over 017 3rd 05 3rd negative 11 3rd 12 3rd. And we take row three times three and add it to row 13 times 00 plus one is times 00 plus zero is 03 times one is three plus negative three is zero. We got our zero there three times one is three plus one is four. Three times 00 plus negative three is still negative three. Now we've got to get our zero into this spot replacing that 7/3 and again we're going to work with the row that has the one in it. So our step nine Is going to be taking row or replacing road to we're going to take row eight and multi r sorry step eight row three and multiply by negative seven thirds. And so we'll take negative seven thirds times. Row three plus row two. To replace row two. Because we're replacing road to we can copy down all these other rows so 1004 - replacing row two. So skip 00110 and 00 negative 11 3rd one and two thirds. Multiplying row three negative seven thirds times 00 plus negative seven thirds plus times zero is zero plus one is one negative seven thirds times one is negative seven thirds plus seven thirds. Gives us our zero in the spot. We wanted negative seven thirds times one is again negative seven thirds plus zero is going to give us negative seven thirds in this spot negative seven thirds times zero is zero plus five thirds is still five thirds. All right. We've only got one more zero needed in our third column. So let's work on this negative 11 3rd for our step 10, scoop this down Step 10. We want to replace row four and we're going to do that by working with our row that has the one in that column. And we're going to multiply it by 11 3rd and add it to row four. So 11 3rd times row three plus row four four, Step 10. Because we're replacing row four we can copy down the first three rows 1004, negative 3010 negative seven thirds five thirds. And keep copying row 300110. And now we'll start multiplying 11 3rd times zero is zero plus zero is 0, 11 3rd times zero is zero plus zero is 0, 11 3rd times one is 11 3rd plus negative 11 3rd gives us our zero that we wanted in that spot, 11 3rd times one is 11 3rd plus one. Which is going to give us 14/ and 11 3rd times zero is zero plus two thirds is still two thirds. Alright, our first three columns are all set. The next step is to get our one in the needed spot in the fourth column. So we look and that means we're tackling this 14 3rd and we want to turn that into one. So first step 11, we're dealing with row four again And we want to multiply it by its reciprocal to get one. So we're going to multiply this row by 33/14, row four gets multiplied by 3/14. And because we are replacing row four, we can copy over the first three rows negative three 010 negative seven thirds five thirds 00110. And now we're just going to multiply it 3/14 time, 00, 3/14 time 00, 3/14 time 003 14 times 14 3rd is one. We got our one in that spot in 3/14 times two thirds is going to be 1/7. No, I didn't do that in my head. I did that ahead of time And now we want to start filling in our zeros into the fourth column. So let's start dealing with this four for our step 12. And so we're replacing row one and we're going to be dealing with our row four because that has the one that we want. And we need to multiply row four times negative four and add that to row one. So negative four times 04 plus. Row one will replace rho one. Because we're replacing row one, we can copy over rose to three and -7/3 5/ 001100001, and negative four times zero is zero plus one. That's just one, negative four times 00 plus zero is zero, negative four times zero zero plus zero is zero, negative four times one is negative four plus four is r zero that we wanted in that spot. And now negative four times 1/7 plus three is going to give us negative 25/7. Yeah, I did that ahead of time, negative 25/7. Great. We've got our zero in that first spot. So now let's get a zero in this spot where negative seven thirds is so step 13. We're almost done guys, we're doing this step 13. We want to replace row two. We do that by working with the row that has the one in the spot we wanted. So row four we're going to multiply that times I wanted to be zero. So we're going to multiply our row by 7/3. So seven thirds times row four plus row two will give us zero in the spot we want because we're replacing row two we can copy over Rose 13 and 41000 negative 25 7th 010. We're replacing row two, skipping that 100110 for row three and 1/7 for row four. Okay, now multiplying row four by seven thirds. Seven third times zero is zero plus zero is 3rd times 00 plus one is one, seven thirds times zero is zero plus zero is zero and seven thirds times one is seven thirds plus negative seven thirds gives us +07 3rd times 1/7. Okay well the seventh cancel out. So that's one third plus five thirds. Okay, that's six thirds. Which is to that one. I did in my head and we've got our zero we're almost done. Look at this oh my goodness, we just need A zero right here and then we're going to have the pattern that we need. So step 14 we have got this guys. Okay, so we want to get a zero there which means we're going to take our row that has the one in the spot that we wanted And we're going to multiply it by the number needed to get to zero. So we're going to multiply row four times negative one replacing row three equals negative one times row four plus row three. We are replacing row three which means we copy over 12 and 41000 negative 25/ skip row 300011704. Okay negative one times zero is zero plus 00 negative one times 00 plus zero is zero negative one times zero is zero plus one is one negative one times one is negative one plus one is zero negative one times one. Seventh negative 1/7 plus zero is negative 1/7. We did it. We got this in the proper format which means that negative 25 7th is our X. Value two is our Y. Value negative one. Seventh is our Z value and 1/7 is R. W. Value. You guys We did it. Let's go back up here and see which answer choice that is alright matching our answer down below with the answer choices. That is answer choice. A Well done. We'll catch you on the next one

Key Concepts

Systems of Linear Equations

Gaussian Elimination with Back-Substitution

Gauss-Jordan Elimination

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