How to test if a feasible solution is optimal - Complementary Slackness Theorem - Linear Programming I have this linear program 
$$\begin{cases}
\text{max }z=&5x_1+7x_2-3x_3\\
&2x_1+4x_2-2x_3&\le8\\
&-x_1+x_2+2x_3&\le10\\
&x_1+2x_2-x_3&\le6\\
&x_1,\,x_2,\,x_3\ge0
\end{cases}$$
and a feasible solution of his dual $(D)$ is $y = {7/2,2,0}$.
I need to find an optimal basis of $(P)$ and an optimal basis of $(D)$ using the complementary slackness theorem. 
I thought about assuming that $y$ is an optimal solution of $(D)$ and find the optimal solution of $(P)$ and after using the corollary of this theorem to prove that $y$ is an optimal solution (then $x$ also). Is that the method to use? Is there a more appropriate methodology to solve this problem?
The dual problem is
$$\begin{cases}
\text{min }&8y_1+10y_2+6y_3\\
&2y_1-y_2+y_3&\ge5\\
&4y_1+y_2+2y_3&\ge7\\
&-2y_1+2y_2-y_3&\ge3\\
&y_1,y_2,y_3\ge0
\end{cases}$$
 A: Your formulation of the dual Problem is right-you forgot the negative sign at the third constraint only. As I said, you can eleminate the third constraint (primal problem).In this case the dual problem is: 
$$
\text{min } 8y_1+10y_2\\
 2y_1-y_2\ge 5\\
 4y_1+y_2\ge 7\\
-2y_1+2y_2\ge -3\\
y_1,y_2\ge0
$$
The solution of this problem is $y^T=(y_1,y_2)=(3.5;2)$ Your solution is right. The solution is $y_3=0$, because the third constraint is not necessary.
The compementary slackness condition is $X^TC^*=b^TY^*$
This gives: $\begin{pmatrix}5 & 7 & -3\end{pmatrix}\cdot \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix} 8 & 10 \end{pmatrix} \cdot \begin{pmatrix} 3.5 \\ 2 \end{pmatrix} \Rightarrow \begin{pmatrix}5 & 7 & -3 \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=48$ 
Additional
The following condition must hold $x_j \cdot z_j=0 \ \  \forall n$
$z_j$ are the slack variables of the dual problem. If you insert the solution for the dual problem, then you will see, that $z_1$ and $z_3$ are zero and $z_2$ is not zero. Thus the equations are (for all n):
$x_1\cdot 0=0 $
$x_2\cdot z_2=0$
$x_3\cdot 0=0$
Thus you know for sure, that $x_2=0$
And secondly this equation must hold: $s_i\cdot y_i=0 \ \ \forall m$
$s_i$ are the slack variables of the primal problem. Thus $s_1$ and $s_2$ are zero and so we have the equations:
$2x_1-2x_3=8$
$-x_1+2x_3=10 $
Remember, that $x_2=0$
This two equations you can solve.
