  Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, and let
${\mathcal S}$ denote the set of all nonnegative integer solutions of the
equation $x_1a_1+\ldots +x_na_n=y_1b_1+\ldots +y_mb_m$. A solution
$(x_1,\ldots,x_n,y_1,\ldots,y_m)$ in ${\mathcal S}$ is called
$\textit{minimal}$ if it cannot be expressed as the sum of two nonzero
solutions in ${\mathcal S}$. For each pair $(i,j)$ with $1\leq i\leq n$ and
$1\leq j\leq m$, the solution whose only nonzero coordinates are $x_i=b_j$ and
$y_j=a_i$ is called a $\textit{generator}$. Our main result shows that every
minimal solution is a convex combination of the generators and the
zero-solution. This proves a conjecture of Henk-Weismantel and, independently,
Ho\c{s}ten-Sturmfels.
