2 Equations & 3 Unknowns

I have picked $3$ natural numbers $x, y, z$. You may ask me for $2$ linear combinations of these numbers with integer coefficients. That is, you give me integers $a$, $b$ and $c$, and I tell you the value of $ax + by + cz$. After seeing the answer, you give me a second triplet $(a, b, c)$ and I tell you $ax + by + cz$ for it. Is it possible to always determine $x$, $y$ and $z$ with at most $2$ such questions? If yes, think about the algorithm; if no, prove why.