Find the unique solution for the systems of linear equations
w +2x+2y-z=5
2w-x-y+4z=6
-w+3x+4y-2z=5
3w+x+y-4z=4
The first step is to write the augmented matrix for the system of linear equations:
[1 2 2 -1 | 5]
[2 -1 -1 4 | 6]
[-1 3 4 -2 | 5]
[3 1 1 -4 | 4]
Next, we perform row operations to simplify the matrix and put it in reduced row-echelon form. We want to get zeros in the bottom-left three entries of the matrix. Here are the steps:
1. Swap R1 with R3:
[-1 3 4 -2 | 5]
[2 -1 -1 4 | 6]
[1 2 2 -1 | 5]
[3 1 1 -4 | 4]
2. Multiply R1 by 2 and add it to R2:
[-1 3 4 -2 | 5]
[0 5 7 2 | 16]
[1 2 2 -1 | 5]
[3 1 1 -4 | 4]
3. Multiply R1 by -1 and add it to R3:
[-1 3 4 -2 | 5]
[0 5 7 2 | 16]
[0 -1 -2 1 | 0]
[3 1 1 -4 | 4]
4. Multiply R1 by -3 and add it to R4:
[-1 3 4 -2 | 5]
[0 5 7 2 | 16]
[0 -1 -2 1 | 0]
[0 -8 -11 2 | -11]
5. Multiply R2 by -1/5:
[-1 3 4 -2 | 5]
[0 -1 -7/5 -2/5 | -16/5]
[0 -1 -2 1 | 0]
[0 -8 -11 2 | -11]
6. Multiply R2 by -1 and add it to R3:
[-1 3 4 -2 | 5]
[0 -1 -7/5 -2/5 | -16/5]
[0 0 3/5 -3/5 | 16/5]
[0 -8 -11 2 | -11]
7. Multiply R2 by -3/5 and add it to R4:
[-1 3 4 -2 | 5]
[0 -1 -7/5 -2/5 | -16/5]
[0 0 3/5 -3/5 | 16/5]
[0 0 -2/5 16/5 | -37/5]
8. Multiply R3 by 5/3:
[-1 3 4 -2 | 5]
[0 -1 -7/5 -2/5 | -16/5]
[0 0 1 -1 | 32/15]
[0 0 -2/5 16/5 | -37/5]
9. Multiply R3 by 1 and add it to R1:
[0 3/5 1 -3/5 | 107/15]
[0 -1 -7/5 -2/5 | -16/5]
[0 0 1 -1 | 32/15]
[0 0 -2/5 16/5 | -37/5]
10. Multiply R3 by 7/5 and add it to R2:
[0 3/5 1 -3/5 | 107/15]
[0 0 0 0 | 27/15]
[0 0 1 -1 | 32/15]
[0 0 -2/5 16/5 | -37/5]
11. Multiply R3 by 2/5 and add it to R4:
[0 3/5 1 -3/5 | 107/15]
[0 0 0 0 | 27/15]
[0 0 1 -1 | 32/15]
[0 0 0 10/5 | -13/5]
Finally, we have the reduced row-echelon form of the augmented matrix. To find the solution to the system of linear equations, we can write the corresponding system of equations:
0w + (3/5)x + y - (3/5)z = 107/15
0w + 0x + 0y + 0z = 27/15
0w + 0x + z - 1z = 32/15
0w + 0x + 0y + 10/5z = -13/5
From the second equation, we can see that it is simply 0 = 27/15, which is not true. Therefore, there is no solution to the system of linear equations.