4x - y + 2z = 13
x + 2y - 2z = 0
-x + y + z = 5
Solve the above equation using the Gauss Jordan elimination method.
To solve the system of equations using Gauss-Jordan elimination method, we will first write the augmented matrix:
[ 4 -1 2 | 13 ]
[ 1 2 -2 | 0 ]
[-1 1 1 | 5 ]
Step 1: Make the element at row 2, column 1 to be zero.
Multiply row 1 by -1/4 and add it to row 2:
[ 4 -1 2 | 13 ]
[ 0 9/4 -3/2 | -13/4 ]
[ -1 1 1 | 5 ]
Step 2: Make the element at row 3, column 1 to be zero.
Multiply row 1 by 1/4 and add it to row 3:
[ 4 -1 2 | 13 ]
[ 0 9/4 -3/2 | -13/4 ]
[ 0 0 3/4 | 18/4 ]
Step 3: Make the element at row 2, column 2 to be one.
Multiply row 2 by 4/9:
[ 4 -1 2 | 13 ]
[ 0 1 -2/3 | -3/4 ]
[ 0 0 3/4 | 18/4 ]
Step 4: Make the element at row 1, column 2 to be zero.
Multiply row 2 by 1 and add it to row 1:
[ 4 0 0 | 7 ]
[ 0 1 -2/3 | -3/4 ]
[ 0 0 3/4 | 18/4 ]
Step 5: Make the element at row 3, column 2 to be zero.
Multiply row 2 by -3/4 and add it to row 3:
[ 4 0 0 | 7 ]
[ 0 1 -2/3 | -3/4 ]
[ 0 0 0 | 9 ]
Step 6: Make the element at row 3, column 3 to be one.
Divide row 3 by 9:
[ 4 0 0 | 7 ]
[ 0 1 -2/3 | -3/4 ]
[ 0 0 1 | 9 ]
The simplified augmented matrix corresponds to the system of equations:
4x = 7
y - (2/3)z = -3/4
z = 9
Solving these equations, we get:
x = 7/4
y = 3/4
z = 9
Therefore, the solution to the system of equations is:
x = 7/4, y = 3/4, z = 9