Linear Programming - Motivation behind the Dual Simplex Method I am trying to understand the motivation behind the Dual Simplex Method. However, I have run into some roadblocks while understanding the rationale behind the Dual Simplex Method. This is my current understanding of the Simplex, the Primal and Dual problem:

$1$. For a minimization problem, the Simplex Algorithm proceeds with first a basic feasible solution; then it replaces individual basis columns with an external column until $c_j - C_B B^{-1} A_j >0~\forall~j$ where $c_j$ is the $jth$ cost tuple ; $C_B$ is the cost corresponding to the feasible basis $B$ and $A_j$ is the $jth$ column external to $B$.
$2$. If $x_0$ be the primal feasible solution and $y_0$ be the dual feasible solution and both satisfy the complementary slackness conditions, then $x_0$ is the primal optimal solution and $y_0$ is the dual optimal solution.
$3$. If $c^T x_0 = b^T y_0$ where $B$ is the feasible basis, then $x_0$ is the primal optimal solution and $y_0$ forms the dual optimal solution.

Using this, my professor has tried to implement the Dual Simplex Algorithm by first accounting for a tuple of the RHS $:b_r < 0$ and then proceeding ahead.
However, I do not quite understand the need to consider $b_r < 0$ nor the algorithm ahead. My confusion is: - The simplex method is applied to the dual formulation without explicitly finding the dual. How is that done?Could someone help me build the dual simplex algorithm from here?

After adding the slack variable, consider a standard-form linear programming problem:
$$
\begin{align*}
&\text{max} \quad &&c^Tx \\
&\text{subject to}\quad &&Ax = b \\
&&& x\geq 0
\end{align*}
$$
where the matrix $A$ is a partitioned-matrix form of $A = [B\ \ N]$. We can write
$$
x = \begin{pmatrix} x_1 \\
\vdots \\
x_n \\
w_1 = x_{n+1} \\
\vdots \\
w_m = x_{n+m}
\end{pmatrix} \quad \text{and}\quad z = \begin{pmatrix} z_1 \\
\vdots \\
z_n \\
y_1 = z_{n+1} \\
\vdots \\
y_m = z_{n+m}
\end{pmatrix}
$$
where $x_i$ are the primal variables, $w_i$ are the primal slacks, $z_i$ are the dual slacks, and $y_i$ are the dual variables. Let the subscript $N$ indicate the non-basis, and $B$ indicate the basis. Now the primal dictionary updates as:
$$
\begin{align*}
\zeta &= \bar{\zeta} - z_N^Tx_N \\
x_B &= \bar{x}_B - B^{-1}Nx_N
\end{align*}
$$
and the dual dictionary actually updates as
$$
\begin{align*}
-\zeta &= -\bar{\zeta} - \bar{x}_B^Tz_B \\
z_N &= \bar{z}_N + (B^{-1}N)^Tz_B
\end{align*}
$$
Notice that here $(B^{-1}N)^T$ is the negative transpose of $-B^{-1}N$.
Take a look at https://vanderbei.princeton.edu/542/lectures/lec6.pdf or Chapter 6 of the book Linear Programming by Vanderbei for a detailed explanation of the negative transpose property.
