Suppose that in the final simplex system for a dual maximum problem of a given minimum problem, there is a degenerate basic slack variable \(x_{j}\). In the equation to which \(x_{j}\) belongs is some non-basic variable \(x_{k}\) with a positive coefficient; and furthermore in the objective row, \(x_{k}\) has a strictly negative coefficient \(c_{k}\). Show that if \(x_{k}\) is made basic, the resulting vector \(\mathbf{y}\) of negatives of slack coefficients also achieves the minimum of the objective function of the minimum problem. Need \(\mathbf{y}\) be feasible for the minimum problem?