Solutions to $x+y+z=31$ and $x+2y+3z=41$ For the equations
$$x+y+z=31$$ $$x+2y+3z=41$$
is there a elegant way or method to find all the positive solutions in integers? Thus far, I have been using trial and error (which is time consuming). Another approach I tried was subtracting the first equation from the second to get $y+2z=10$. But I feel like there must be some faster way of doing this that I am not thinking of right now. Any input is appreciated.
No, I think the second approach you had is as quickly as this can be solved. Subtracting the first equation from the second gives $y + 2z = 10$, and the only possible ways that both $y$ and $z$ are positive integers is when $(y,z) = (8,1), (6,2), (4,3), (2,4)$. Plugging each set back into the first equation, we get that the four possible combinations for $(x,y,z)$ are $(22,8,1), (23,6,2), (24,4,3), (25,2,4)$.
Subtract the first equation from the second.

Express $y$ in terms of $z$

Now substitute into the first equation and you'll get and relation for $x$ with respect for $z$. 

Because $z \in \mathbb{R}$ there are infinite amount of solutions and all of them are of a closed form, you'll obtain. And this is the fastest way to compute solution, maybe you were confused, because there are infinite amount of them.