Hi all, I need help with solving a linear system in a finite field, Z_7 and Z_11.

The original given linear system is

2x + 4y - 2z = 1

2x + y + z = 3

5x - 3y + 7z = 0

Under positive representation for Z_7 and Z_11, I get the following linear systems respectively

2x + 4y + 5z = 1

2x + y + z = 3

5x + 4y = 0

and

2x + 4y + 9z = 1

2x + y + z = 3

5x + 8y +7z = 0

I did the usual gaussian elimination, keeping all the coefficients in Z_7 and Z_11 respectively, and I got

Z_7: x = 0, y = 0, z = 3, determinant = 6

Z_11: x = 9, y = 5, z = 2, determinant = 6

The solutions my teacher gave were

Z_7: x = 0, y = 0, z = 4, determinant = 6

Z_11: x = 10, y = 8, z = 1, determinant = 6

However, I substituted his solutions into the respective linear systems, and they don't seem to be correct.

For example, for Z_7, for the equation

2x+y+z = 3, I get 2(0) + (0) + (4) = 4 != 3

For Z_11, for the equation

2x+y+z = 3, I get 2(10) + (8) + (1) = 7 != 3

Where did I go wrong?

Thank you.

Regards,

Rayne