Solve: $6x + 8y + 12z = 10$.
I have started to solve this by giving
$8y + 12z = 4u$.Then we have $6x + 4u = 10$. Solving this diophantine equation we get, $x = 5 + 2t$ and $u = -5 - 3t$.
Now we can have, $8y+ 12z = 4(-5-3t)$
We can have $4= 12(1) + 8(-1)$ as one representation and $4 =12(-1) + 8(2)$ as another. And two different solutions are obtained.
Are both the same or do we have a standard notion to choose the representation?
