Answer:
To solve the system of equations using Gauss-Jordan elimination, we can write the augmented matrix and perform row operations to transform it into row echelon form.
The given system of equations:
2x + 3y = 9
4x + 6y = 7
Writing the augmented matrix:
[ 2 3 | 9 ]
[ 4 6 | 7 ]
Performing row operations:
1. Row 1 / 2 â Row 1:
[ 1 3/2 | 9/2 ]
[ 4 6 | 7 ]
2. Row 2 - 4 * Row 1 â Row 2:
[ 1 3/2 | 9/2 ]
[ 0 0 | -17 ]
3. Row 1 - (3/2) * Row 2 â Row 1:
[ 1 3/2 | 43/2 ]
[ 0 0 | -17 ]
4. Row 1 * 2/3 â Row 1:
[ 2/3 1 | 43/3 ]
[ 0 0 | -17 ]
5. Swap Row 1 and Row 2 for better readability:
[ 0 0 | -17 ]
[ 2/3 1 | 43/3 ]
6. Row 2 - (2/3) * Row 1 â Row 2:
[ 0 0 | -17 ]
[ 2/3 1 | 43/3 ]
7. (3/2) * Row 2 â Row 2:
[ 0 0 | -17 ]
[ 1 3/2 | 43/2 ]
8. Divide Row 2 by 3/2:
[ 0 0 | -17 ]
[ 1 1 | 43 ]
The augmented matrix is now in row echelon form. We can solve for the variables:
From Row 2, we have:
x + y = 43
Substituting this into Row 1 (or one of the original equations), we have:
0 = -17
This is contradictory, indicating the system has no solution. Thus, the system of equations is inconsistent and has no solution.