For a homework problem I am asked to implement the dual simplex algorithm on the following linear program:
$$ \min -25x_1 -30x_2 \\ \\ s.t. 5x_1 +4x_2 \leq 120 \\ \\ 20x_1 +30x_2 \leq 690 \\ \\ x_i \geq 0$$
Now the issue that I am having is that once this problem is converted to standard form by adding the variables $x_3, x_4$, the initial solution using the third and fourth columns to obtain the basis matrix $I_2$ yields the "optimal" solution according to the algorithm since $B^{-1}b = I_2(120, 690)^T = (120, 690)^T \geq 0$.
I feel like there is something I am missing in the setup to this problem because clearly this is not a solution to the original LP, but according to the dual simplex algorithm I should terminate immediately.
Any help would be appreciated.