# Cannot apply Complementary Stackness to Dual Linear Problem

I have the next primal problem.I have to solve It obtaining the Dual problem and up to here everything is nice,but the problem appears when I have to apply the Complementary Slackness:

Primal problem is: Maximize $$2x_1+3x_2-4x_3$$,
Subject to: ,
$$3x_1+5x_2+2x_3=15$$ ,
$$2x_1+3x_2-4x_3=8$$ ,
$$x_1\geq0, x_2\geq0,x_3\geq0$$

I obtain the Next Dual Problem: Minimize $$15\pi_1 +8\pi_2$$

Subject to:
$$x_1:3\pi_1+2\pi_2\geq2$$
$$x_2:\ 5\pi_1+3\pi_2\geq3$$ ,
$$x_3:\ 2\pi_1-4\pi_2\geq-4$$

When I solve this dual problem I have that the Dual solution is (0,1).The issue here is when I put it in the equations the condition of $$>$$ doesnt satisfy,It only satiesfies the condition of =,so I cannot apply Complementary Slackness and say that any primal variable is equal to zero.If anybody knows how to proceed now,I'd very greatful.

The best way to deal with your problem is to use the strong duality. It says that the the primal optimal objective is equal to the dual optimal objective: $$f(x^*)=g(\pi^*)=8$$. That means you have three equations with 3 unknowns.
$$\begin{eqnarray} &2x_1+3x_2-4x_3=&8 \\ & 3x_1+5x_2+2x_3=&15\\ & {2x_1+3x_2-4x_3}=&8\end{eqnarray}$$
It is obvious that one equation is redundant. So for the optimal solution you have to express two variables by the remaining variable, i.e. $$x_1$$. And also you have to regard the non-negativity condition. Can you proceed?