Suppose we have the following LP
\begin{align*} \min z = x_1 + 2x_2 \\ x_1 \geq 1 \\ x_1+x_2 \geq 2 \\ x_1,x_2 \geq 0 \\ \end{align*}
We can use simplex or a graphical method to find the optimal solution which is $2$. and the optimal feasile solution is $(2,0)$. now, we write the dual:
\begin{align*} \max z = y_1 + 2y_2 \\ y_1 + y_2 \leq 1\\ y_2 \leq 2 \\ y_1,y_2 \geq 0 \\ \end{align*}
but now we see that the optimal solution must occur at the point $(0,1)$ which gives optimal solution of $2$. As we can see, the optimal solutions are the same, but how can we relate the optimal feasible solutions?
Professor said it is easy to solve the dual, but in this case why do we need the dual if solving the primal was easy enough?