Write x + y = 1 + 7k , x + z = 2 + 7m , y + z = 3 + 7n , k,m,n integers.

Substract now the 3rd. one from the 1st.:

x - z = -2 + 7(k-n) , and now add this to the second one

2x = 7(k-n+m) ==> x = 0 (mod 7), and from here y = 1 (mod 7), z = 2(mod 7), and the general solution is:

x = 7k

y = 1 + 7m

z = 2 + 7n

k,m,n integers.

Tonio