00:01
In this question, we are asked to solve the given system of linear equations.
00:04
The first step would be to write down its augmented matrix, which is basically the, consists of the coefficients in front of t1, t2, t3 and t4, and the right -hand side.
00:18
So, the augmented matrix, the first row is going to be 4, negative 1, 0, negative 1, 35.
00:28
0 because there is no t3 in the first equation.
00:32
Whenever the variable is absent, we will put 0 in its place.
00:37
In the second row, we are going to get negative 1, 4, negative 1, 0 because t4 is absent in the second equation.
00:45
And 65.
00:48
The third row is going to be 0, negative 1, 0, negative 1, 4, negative 1 and 80.
00:58
And the last row is going to be negative 1, 0, negative 1, 4 ,000 ,000.
01:05
And 50.
01:07
So this is the augmented matrix of the system of equations.
01:14
Now we need to reduce it to a row echelon form or maybe even to the reduced row echelon form.
01:23
And the first step would be to interchange the first and the last rows.
01:33
So if we interchange the first and the last rows, we're going to get negative 1, 0, negative 1, 450 in the first row.
01:43
The second and the second then the third rows are going to be same.
01:56
And in the last, the first row will move to the last row.
01:59
4 negative 1, 0, negative 1, and 35.
02:14
Now we'll use the negative 1 in the first row to get rid of the entries below it.
02:26
So to get rid of the negative 1 in the second row, we'll need to multiply the first row by negative 1 and add it to the second row.
02:34
To get rid of the 4, we need to multiply the 1 by negative 4 and add it to the last row.
02:50
We will rewrite the first row.
02:56
In the second row, we are going to get 0.
03:00
0 times negative 1 equals to 0 plus 4 is going to be 4.
03:07
Negative 1 times negative 1 is 1.
03:10
And 1 minus 1 is 0.
03:15
4 times negative 1 is negative 4 plus 0 is negative 4.
03:22
And 50 times negative 1 is negative 50 plus 65 is going to be 15.
03:30
Now we will rewrite this, the third row because it's already 0 in the first column.
03:40
And the last row will become, we need to multiply negative 1 by 4 and add it to 4 in the last row.
03:48
We are going to get 0.
03:53
Right.
03:56
0 times 4 is 0.
03:58
Minus 1 is negative 1.
04:05
Negative 1 times 4 is negative 4 plus 0 is negative 4.
04:14
And 4 times 4 is 16 and 16 minus 1 is 15.
04:21
Finally, 50 times 4 is 200 and 200 plus 35 is 235.
04:31
All right.
04:37
Next, what we are going to do next, we will interchange the second and the third rows.
04:51
And what we are going to get is we will rewrite the first row.
04:57
Negative 1, 0, negative 1, 4 and 50.
05:02
In the second, the third row becomes the second row.
05:06
0, negative 1, 4, negative 1, 80.
05:11
The second row will become the third row.
05:14
04, 0 ,0 and 15.
05:18
And the last row will be same.
05:25
All right.
05:28
Now, we will.
05:30
We will use the second row, the first entry in the second row to get rid of the 4 and the negative 1 below it.
05:41
We will multiply the 2 row by 4 and add it to the 3rd row and we will multiply the 2nd row by negative 1 and add it to the last row.
05:58
And what we are going to get is we will rewrite the 1st row.
06:05
We will write the 2nd row, 0, negative 1, 4, 4.
06:09
Negative 1 and 80.
06:12
In the third row, we are going to get 0.
06:15
0.
06:17
4 times 4 is 16 plus 0 is 16.
06:20
Negative 1 times 4 is negative 4, minus 4 is negative 8.
06:25
And 80 times 4 is 320 plus 15 is 335...