Prove that for any integers $x,y,z$ there exist $a,b,c$ such that $ax+by+cz=0$ It is rather obvious that for any 3 coprime integers $x,y,z$ there exist 3 non-zero integers $a,b,c$ such that:
$$ax+by+cz=0$$
Any simple argument to prove it?

I am going to change the language of the problem for clarity: Let a, b and c be the constants that satisfy the property of being coprime. You asked what the solutions are to satisfy:
$$ ax + by = cz $$
For variables x, y, and z in $ \mathbb Z $
Note that in your form, one merely needs make z the opposite sign to yield a solution, then flip the variable names as I did.
We say that $ ax + by $ is a linear combination of cz. If the gcd(x,y) | cz, we have an easy case to do all the solutions with the extended euclidean algorithm. Note that you specified x and y coprime, so the gcd(x, y) = 1. See extended euclidean algorithm.
Otherwise, there are infinitely many solutions for large enough cz. This is evident in the fact that if x and y are coprime, then there exists a linear combination for any value larger than xy - x - y, a fact that can be found here. In fact this is almost a duplicate problem.
If x and y are not coprime, divide both sides by their greatest common divisor and adjust z to be appropriate for the problem. Tada! Now they are coprime.
