I have a doubt about an application of the Complementary Slackness theorem. I want to show that $\mathbf{x}^{\ast} = (7.25, 0 , 3.25 , 0.75)$ is an optimal solution of the linear program (which is, checking on a software)
\begin{align*} \max \quad x_1 + x_2 - 2x_3 + 2x_4 \\ x_1 - x_2 - x_3 - 2x_4 = 2.5 \\ x_1 + x_2 + x_4 = 8 \\ x_1 + 2x_2 - x_3 = 4 \\ x_1, x_2, x_3, x_4 \geq 0 \ . \end{align*} What I did was write the dual problem, test the feasibility of $\mathbf{x}^{\ast}$, and it happens that $\mathbf{x}^{\ast}$ satisfies all constraints with equality, so, by Complementary Slackness, there should be a $\mathbf{y} ^{\ast} = (y_1^{\ast}, y_2^{\ast}, y_3^{\ast})$ such that equality holds on the dual problem constraints, that is
\begin{align*} y_1^{\ast} + y_2^{\ast} + y_3^{\ast} = 1 \\ -y_1^{\ast} + y_2^{\ast} + 2y_3^{\ast} = 1 \\ y_1^{\ast} + y_2^{\ast} = 2 \\ -2y_1^{\ast} + y_2^{\ast} = 2 \ , \end{align*}
and... the system has no solution.
EDIT: One of the problems I was having was finding the proper dual problem, which is
\begin{align*} \min \quad 2.5y_1 + 8y_2 + 4y_3 \\ y_1 + y_2 + y_3 \geq 1 \\ -y_1 + y_2 + 2y_3 \geq 1 \\ -y_1 - y_3 \geq -2 \\ -2y_1 + y_2 \geq 2\ , \end{align*}
so, by the version of the Complementary Slackness Theorem I am familiar with, a solution for the primal is optimal if and only if the dual problem constraints are satisfied as an equality for some $y^{\ast} = (y_1^{\ast}, y_2^{\ast}, y_3^{\ast})$, everytime a constraint is satisfied with equality (and in this case, ALL constraints are satisfied with equality). So, either this version of the theorem is false, or it doesn't hold when all constrains are satisfied with equality in the primal problem, and that is my doubt.