What is the solution using gauss jordan Elimination..
2x-4y+5z=10
6x+8y+3z=12
X-7y-9z=63
Pls.help me to solve this problem
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
To solve the given system of equations using Gauss-Jordan elimination, we will perform a series of row operations to transform the augmented matrix into reduced row-echelon form. Here are the steps to follow:
Step 1: Set up the augmented matrix
The augmented matrix for the system of equations is:
[ 2 -4 5 | 10 ]
[ 6 8 3 | 12 ]
[ 1 -7 -9 | 63 ]
Step 2: Begin by creating a leading 1 in the first row, first column (known as the pivot):
- Divide Row 1 by 2 (the coefficient of x in the first equation).
[ 1 -2 2.5 | 5 ]
[ 6 8 3 | 12 ]
[ 1 -7 -9 | 63 ]
Step 3: Use row operations to create zeros below the leading 1 in the first column:
- Multiply Row 1 by -6 and add it to Row 2.
- Multiply Row 1 by -1 and add it to Row 3.
[ 1 -2 2.5 | 5 ]
[ 0 20 -12 | -18 ]
[ 0 5 -11.5 | 58 ]
Step 4: Create a leading 1 in the second row, second column (pivot):
- Divide Row 2 by 20 (the coefficient of y in the second equation).
[ 1 -2 2.5 | 5 ]
[ 0 1 -0.6 | -0.9 ]
[ 0 5 -11.5 | 58 ]
Step 5: Use row operations to create zeros above and below the leading 1 in the second column:
- Multiply Row 2 by 2 and add it to Row 1.
- Multiply Row 2 by -5 and add it to Row 3.
[ 1 0 1.3 | 3.2 ]
[ 0 1 -0.6 | -0.9 ]
[ 0 0 -4.5 | 4.5 ]
Step 6: Create a leading 1 in the third row, third column (pivot):
- Divide Row 3 by -4.5 (the coefficient of z in the third equation).
[ 1 0 1.3 | 3.2 ]
[ 0 1 -0.6 | -0.9 ]
[ 0 0 1 | -1 ]
Step 7: Use row operations to create zeros above the leading 1 in the third column:
- Multiply Row 3 by -1.3 and add it to Row 1.
- Multiply Row 3 by 0.6 and add it to Row 2.
[ 1 0 0 | 4 ]
[ 0 1 0 | 1 ]
[ 0 0 1 | -1 ]
In this final matrix, each row represents the coefficients of the variables (x, y, and z) in the system of equations. Therefore, the solution to the system is:
x = 4, y = 1, z = -1.