I want to prove the following:
Let $j_1, \ldots, j_8 \in \mathbb{Z}_{\ge 0}$. If \begin{align*} j_1 + j_2 + 2 j_3 - j_4 - j_6 -2 j_8 &= 0 \\ -j_1 + j_2 + j_4 + 2j_5 - j_6 -2 j_7 &=0 \end{align*} show that there exist $i_1, \ldots, i_{12} \in \mathbb{Z}_{\ge 0}$ such that \begin{align*} j_1 &= i_1 + i_5 + i_6 + 2 i_9 \\ j_2 &= i_2 + i_6 + i_7 + 2 i_{10} \\ j_3 &= i_3 + i_8 + i_{11} + i_{12} \\ j_4 &= i_1 + i_7 + i_8 + 2 i_{12} \\ j_5 &= i_4 + i_5 + i_9 + i_{11} \\ j_6 &= i_2 + i_5 + i_8 + 2 i_{11} \\ j_7 &= i_4 + i_7 + i_{10} + i_{12} \\ j_8 &= i_3 + i_6 + i_9 + i_{10} \end{align*}
This statement is the final step in a wider proof of a fact that I believe is true, so I'm hoping for a proof rather than a counterexample.
The problem is trivial if there are no restrictions on each $i_k$, but I'm struggling to solve it with the constraint that they are non-negative. One can reduce the second system to reduced row echelon form, though it's not clear how to choose each unconstrained variable to ensure all the $i_k$ are non-negative.
I would appreciate if someone knows how to solve the problem, or a reference to learn about techniques which may be useful. I believe Farkas' lemma may be of assistance, though I'm unsure of how to apply it here.