I have a linear programming problem that already has the slack variables defined and is in convenient matrix form. There are no artificial variables. I've been told that $x_1$ and $x_3$ are basic variables, and I've been asked to determine whether that set of basic variables is feasible or optimal for the problem.
Thankfully, all of the problems give exactly one basic variable for each constraint, so I don't need to try to guess at interior point solutions / guess which basic variables will resolve to zero.
I'm not giving the problem, because the specific problem doesn't matter- what I need to know is the process by which I can determine whether a given set of basic variables would lend themselves to a feasible, optimal, or infeasible solution. I know how to find an optimal solution using the simplex method (in tabular or matrix form), but I have no idea how to complete this problem. I'm sure it's a relatively simple algorithm, but I'm not seeing it.
(I'm also not giving the problem because it's homework, and because I'm not sure it would be the best example for learning the algorithm.)