I want to find the "optimal set of solutions" of the linear program
$$ \max \quad 3x_1 + 6x_2 + 2x_3$$ such that $$3x_1 + 4x_2 + 2x_3 \leq 200$$ $$ x_1 + 3x_2 + 2x_3 \leq 100 $$ $$x_j \geq 0, \quad j = 1, 2, 3 $$
WITHOUT using the simplex method knowing that the problem has an optimal solution with basic variables $x_1$ and $x_2$. So I tried the following: write the dual problem, and we have that, in a step of the simplex that we have $x_1$ and $x_2$ as basic variables, we know that the coefficients of the slack variables would be a solution for the dual problem, and then would use it to calculate the optimal solution for the primal problem in this basis. But that doesn't count as using the simplex method? And even worse, we don't really have any explicit method of finding all solutions without the simplex method, do we?
This is so confusing...