A system of equations is
\begin{align} x_1+x_2+x_3+x_4 &= b_1 \\ x_1+x_5+x_6+x_7 &= b_2 \\ x_2+x_5+x_7+x_8 &= b_3 \\ x_3+x_6+x_8+x_{10} &= b_4 \\ x_4+x_7+x_9+x_{10} &= b_5 \end{align}
where $b_1$, $\ldots$, $b_5$ and $x_1$, $\ldots$, $x_{10}$ are positive integers or zero. How can I determine all possible solutions from this additional information? There are necessarily finite solutions as there are only finite positive integers for each $x$ that are smaller than $b$. It is impossible to solve it by trying every possibility as the original system is way bigger (around 40 equations and 900 variables). I would appreciate every hint in the right direction.