Do there exist further assumption on $Ax=b, x\geq 0$ that imply the solvability?

Question

For a given matrix $A$ and a vector $b$ one wants to solve the system $Ax=b, x\geq 0$.

I am interested in criterias that give the solvability of the above system. I am aware of the famous Farkas Lemma that states that either $Ax=b, x\geq 0$ or $A^Ty\geq 0, b^Ty<0$ has a solution.

I am curious: Are there more direct results? E.g. if the matrix $A$ or the vector $b$ satisfy some special conditions, we know the system is solvable.

Answer 1

Vector $b$ must be in the column space of the matrix $A$ is the general condition for solvability.