Solve this equation using the Gauss Jordan elimination method.
4x - y + 2z = 13
x + 2y - 2z = 0
-x + y + z = 5
1. Write the augmented matrix for the system of equations:
[ 4 -1 2 | 13 ]
[ 1 2 -2 | 0 ]
[ -1 1 1 | 5 ]
2. Perform row operations to get the matrix into reduced row-echelon form:
R2 -> R2 - (1/4)R1
[ 4 -1 2 | 13 ]
[ 0 9/4 -9/2 | -13/4 ]
[ -1 1 1 | 5 ]
R3 -> R3 + R1
[ 4 -1 2 | 13 ]
[ 0 9/4 -9/2 | -13/4 ]
[ 0 0 3 | 18 ]
R2 -> (4/9)R2
[ 4 -1 2 | 13 ]
[ 0 1 -2 | -3 ]
[ 0 0 3 | 18 ]
R1 -> R1 + R2
[ 4 0 0 | 10 ]
[ 0 1 -2 | -3 ]
[ 0 0 3 | 18 ]
R3 -> (1/3)R3
[ 4 0 0 | 10 ]
[ 0 1 -2 | -3 ]
[ 0 0 1 | 6 ]
R2 -> R2 + 2R3
[ 4 0 0 | 10 ]
[ 0 1 0 | 9 ]
[ 0 0 1 | 6 ]
R1 -> R1 - 2R3
[ 4 0 0 | -2 ]
[ 0 1 0 | 9 ]
[ 0 0 1 | 6 ]
3. Read off the solutions from the reduced row-echelon form:
x = -1
y = 9
z = 6
Therefore, the solution to the system of equations is x = -1, y = 9, z = 6.