Answer:
To solve the system of linear equations using Gaussian elimination, we'll write the augmented matrix and perform row operations to transform it into row echelon form.
The given system of equations:
-2x_1 + 3x_2 + x_3 = 2
-3x_1 + 4x_2 + 2x_3 = 2
x_1 - 5x_2 + 4x_3 = -9
-2x_1 + 4x_2 - 4x_3 = 8
Writing the augmented matrix:
[ -2 3 1 | 2 ]
[ -3 4 2 | 2 ]
[ 1 -5 4 | -9 ]
[ -2 4 -4 | 8 ]
1. Row 1 Ã (-3) + Row 2 â Row 2:
[ -2 3 1 | 2 ]
[ 9 -13 -5 | -6 ]
[ 1 -5 4 | -9 ]
[ -2 4 -4 | 8 ]
2. Row 1 Ã (1/2) + Row 3 â Row 3:
[ -2 3 1 | 2 ]
[ 9 -13 -5 | -6 ]
[ -1 (3/2) 2 | -5/2]
[ -2 4 -4 | 8 ]
3. Row 1 Ã (-1) + Row 4 â Row 4:
[ -2 3 1 | 2 ]
[ 9 -13 -5 | -6 ]
[ -1 (3/2) 2 | -5/2]
[ 0 7 -3 | 6 ]
4. Row 2 Ã (-9/7) + Row 3 â Row 3:
[ -2 3 1 | 2 ]
[ 9 -13 -5 | -6 ]
[ -1 0 (59/7) | -(23/7)]
[ 0 7 -3 | 6 ]
5. Row 2 Ã (2/3) + Row 4 â Row 4:
[ -2 3 1 | 2 ]
[ 9 -13 -5 | -6 ]
[ -1 0 (59/7) | -(23/7)]
[ 0 0 (-11/7) | 0 ]
6. Row 3 Ã (-14/11) + Row 4 â Row 4:
[ -2 3 1 | 2 ]
[ 9 -13 -5 | -6 ]
[ -1 0 (59/7) | -(23/7)]
[ 0 0 1 | 0 ]
7. Row 3 Ã (1/2) + Row 1 â Row 1:
[ -1 3/2 3/2 | (4/7) ]
[ 9 -13 -5 | -6 ]
[ -1 0 1 | 0 ]
[ 0 0 1 | 0 ]
8. Row 3 Ã (9/7) + Row 2 â Row 2:
[ -1 3/2 3/2 | (4/7) ]
[ 0 -26/7 -14/7 | -42/7 ]
[ -1 0 1 | 0 ]
[ 0 0 1 | 0 ]
9. Row 3 Ã (1/2) + Row 4 â Row 4:
[ -1 3/2 3/2 | (4/7) ]
[ 0 -26/7 -14/7 | -42/7 ]
[ -1 0 1 | 0 ]
[ 0 0 1 |