Reading the primal solution from dual simplex tableau I was following ptrickJMT's video on how to solve a minimization linear programming problem and he did something that I did not understand why it works. He starts with the following minimization problem:
$$
\begin{matrix}\min &  C = 3x_1 + 9x_2 \\
s.t: \\
& 2x_1 + x_2 \ge 8 \\
& x_1 + 2x_2 \ge 8 \\
&x_1, x_2 \ge 0
\end{matrix}
$$
He proceeds to find the dual form of this example:
$$
\begin{matrix}\max &  P = 8y_1 + 8y_2 \\
s.t: \\
& 2y_1 + y_2 \le 3 \\
& y_1 + 2y_2 \le 9
\end{matrix}
$$
He proceeds to solve it using the regular simplex algorithm and arrives in the following matrix:
$$
\left[\begin{array}{ccccc|c}
2 & 1 & 1 & 0 & 0 & 3 \\
-3 & 0 & -2 & 1 & 0 & 3 \\
\hline
8 & 0 & 8 & 0 & 1 & 24 
\end{array}\right]
$$
Up to here, it's fine, I can see that the optimal solution is $C = P = 24$, and that $y_1 = 0$ and $y_2 = 3$. He also claimed that the optimal solution of the primal problem is $x_1 = 8$ and $x_2 = 0$, he did so by reading coefficients of the slack variables as seen below. I did not understand why this is true, can anyone clarify this to me?
$$
\left[\begin{array}{ccccc|c}
2 & 1 & 1 & 0 & 0 & 3 \\
-3 & 0 & -2 & 1 & 0 & 3 \\
\hline
8 & 0 & \color{red}{8} & \color{red}{0} & 1 & 24 
\end{array}\right]
$$
Thank you.
 A: He probably used the complementary slackness theorem. For this purpose I write the condition constraints with slack variables and therefore equalities.
$$
\begin{matrix}\min &  C = 3x_1 + 9x_2 \\
s.t: \\
& 2x_1 + x_2 -s_1= 8 \\
& x_1 + 2x_2 -s_2= 8 \\
&x_1, x_2 \ge 0
\end{matrix}$$

$$\begin{matrix}\max &  P = 8y_1 + 8y_2 \\
s.t: \\
& 2y_1 + y_2 +z_1= 3 \\
& y_1 + 2y_2 +z_2= 9
\end{matrix}
$$
The complementary slackness theorem states:
$x_j^*\cdot z_j=0 \ \forall \ \ j=1,2, \ldots , n$
$y_i^*\cdot s_i=0 \ \forall \ \ i=1,2, \ldots , m$
$s_i \text{ are the slack variables of the primal problem.}$
$z_j \text{ are the slack variabales of the dual problem.}$
From the dual optimal solution we know that $y_2^*=3$. That means that $3\cdot s_2=0\Rightarrow s_2=0$. That again means the second constraint of the primal is
$$x_1+2x_2=8\quad (1)$$
And last but not least we know that the optimal primal value is equal to the optimal dual value (strong duality theorem).  That means that $8\cdot y_1^*+8\cdot 0=y_1^*+8\cdot 3=24$. Therefore the equation for the optimal value of the primal is
$$3x_1 + 9x_2=24 \quad (2)$$
$$x_1 + 3x_2=8\quad (2a)$$
The solution follows straightforward if you subtract $(1)$ from $(2a)$. If you are more familiar with this theorems it is not very time-consuming. In this case more or less two minutes, I guess.
A: This link might help you: https://faculty.math.illinois.edu/~mlavrov/docs/482-fall-2019/lecture13.pdf 
The basic idea is that:
if $A_B$ is a submatrix of the original $A$ consisting of columns, corresponding to basis variables that give optimal solution $x^*$ of primal, then the optimal solution of dual is:
$$ y^*=c^T_B A_B^{-1} $$ where $ c_B $ - is a vector of coefficients, that correspond to objective function.
And from the revised simplex algorithm we have:
For any reduced cost in the final tableau we have the formula: $r_{d}= c^T_{d}-c^T_B A_B^{-1}A_{d}$
Where $d$ - indicates what variables we are interested in.
So if we chose d = {s1, s2, ... sm} - (so we look to original slak variables) then:
$A_{d}=I$
$c^T_{d}=0^T$
And so we have: $r_{d} = c^T_B A_B^{-1} = y^*$
That's how he read of the optimal solution of dual problem form final tableua
