Find the general integer solution (x, y, z) to the pair of equations

3x + 4y + 7z = 1

5x + 9y + 2z = 3

Put t = x + y + 2z. Then we have;

3t + y + z = 1

5t + 4y - 8z = 3:

Eliminating y, we get 7t + 12z = 1.

I can't solve this last part. I don't know why, it should be so easy... It should be...

(t, z) = (-5 + 12u, 3 - 7u), u an arbitrary integer but I'm nowhere near that. I end up doing about 4 substitutions but it all goes in a mess.