How to solve a diophantine equation pair?

Question

I know how to solve basic linear diophantine equations, but how would one solve this?: $$ \begin{equation} \begin{array}{rrrr} x &+& y &+& z &=& 31 \\ x &+& 2y &+& 3z &=& 41 \end{array} \end{equation} $$ I tried subistituting the $x$ from the lower one to the upper one but I don't know how I would then get the answers for $x$.

Answer 1

subtracting both equations we get $$y+2z=10$$ Can you proceed?

Answer 2

Hint: $y+2z=10$ should be easy to solve.