I have a doubt in the following question:
Consider the linear program
$\min\ \ 5x_1+12x_2 + 4x_3\ \ $, $\ \ $subject to
$x_1+2x_2+x_3 = 10$
$x_1,x_2,x_3 \geq 0$
You are given the information that $x_2$ and $x_3$ are positive in the optimal solution. Use the complementary slackness conditions to find the optimal dual solution.
I'm referred to Problem 1 here to get an idea on how to solve a dual linear program using complementary slackness. Notice that over there, the constraints are all inequalities, so they allow for the possibility of slackness. In fact, the slackness of some constraints is used to conclude that some of the dual constraints are tight, and vice versa.
The problem in the above question is that all constraints are equalities, which means the constraints must be tight! So the procedure detailed in the linked homework solution doc can't be used.
So far what I've done is to use the constraints to express the objective function only in terms of $x_1$ (that's because nothing is specified for $x_1$ of the optimal solution). It turns out to be an increasing function of $x_1$, so I can set $x_1 = 0$ to obtain the primal solution. But I can't understand how exactly to use complementary slackness to get the dual solution in this case. A hint would be appreciated.