2 Equations & 3 Unknowns

I guessed 3 natural numbers - $x, y, z$. You can ask me 2 sums of these numbers with any integer coefficients - $(a, b, c)$. That is, you give me a, b and c and I tell you the result of the expression $a*x+b*y+c*z$. Seeing the answer, you then give me the 2nd triplet of $(a,b,c)$ & I will tell $a*x+b*y+c*z$. Give me the algorithm to find $x$, $y$ and $z$.