Find all the whole solutions of the equation: $15x+12y+30z=24$
I know there is a very similar post on this that is the following: Prove that the Diophantine equation $ax+by+cz=e$ has a solution if and only if $(a,b,c)\mid e$. And I also know that $mcd(15,12,30)=3$ with which one could reduce this to the problem $5x+4y+10z=8$ but I do not know how to reach a solution or find all the solutions, could someone help me please? Thank you very much
Denote $y = 3u-5v$ and $z = 2v-u$ so that the equation reduces to $$15x+12(3u-5v)+30(2v-u)=24$$ $$ \Rightarrow 5x+2u=8$$ Solving gives infinite solutions, namely $x=2-2k$ and $u=5k-1$ for some $k\in\mathbb{Z}$.
Can you find something similar to work out $y$ and $z$?