Assuming finite optimal cost of a specific LP, find an optimal solution directly Minimize $\sum^n_{i=1} c_i x_i$ subject to $\sum^n_{i=1} a_i x_i = b$ (a single constraint), $x_i \ge 0$.

Derive a simple test for feasibility of this problem

Assuming the optimal cost is finite, develop a simple method for obtaining an optimal solution directly.



For the first part, I have  $\sum^n_{i=1} a_i x_i = b \iff \text {sgn}(b)=\text{sgn}(a_i)$ for some $i \in \{1,..,n\}$ or $b = 0$.
I have trouble obtaining this simple method for part two. Could someone help me out here?
I know $ -\infty &lt; \sum^n_{i=1} c_i x_i$ and $\sum^n_{i=1} a_i x_i = b$.
Hint: Since there are $n+1$ constraints on $n$ variables, there is a solution in which all but one constraint are tight. You can calculate all these solutions explicitly ("on paper"), and choose the best one.