When you add slack variables, the dual program isn't affected.
In the original primal program, there are two variables $x_1, x_2$, corresponding to two constraints in the dual; there are two constraints, corresponding to two dual variables which we'll call $u_1$ and $u_2$. Because the constraints are inequalities, $u_1$ and $u_2$ will be nonnegative variables. Here is the primal-dual pair:
\begin{array}{rrll rrll}
\text{maximize } & 6x_1 + 10x_2 & & & \text{minimize} & 60u_1 + 45u_2 \\
& 2x_1 + 8x_2 & \le 60 & (u_1) & & 2u_1 + 3u_2 & \ge 6 & (x_1) \\
& 3x_1 + 5x_2 & \le 45 & (u_2) & & 8u_1 + 5 u_2 & \ge 10 & (x_2) \\
& x_1, x_2 & \ge 0 & & & u_1, u_2 & \ge 0
\end{array}
When we add slack variables in the primal, two things change. First, the constraints become equations, which means $u_1, u_2$ are now unrestricted variables (with no nonnegativity constraints). Second, there are now two more dual constraints corresponding to the slack variables $s_1$ and $s_2$. Those dual constraints turn out to be exactly the constraints $u_1 \ge 0$ and $u_2 \ge 0$ which we lost from the first change. So we get an identical dual:
\begin{array}{rrll rrll}
\text{maximize } & 6x_1 + 10x_2 & & & \text{minimize} & 60u_1 + 45u_2 \\
& 2x_1 + 8x_2 + s_1 & = 60 & (u_1) & & 2u_1 + 3u_2 & \ge 6 & (x_1) \\
& 3x_1 + 5x_2 + s_2 & = 45 & (u_2) & & 8u_1 + 5 u_2 & \ge 10 & (x_2) \\
& x_1, x_2, s_1, s_2 & \ge 0 & & & u_1 & \ge 0 & (s_1) \\
& & & & & u_2 & \ge 0 & (s_2)
\end{array}
If you like, you can write both dual programs in matrix notation as $\mathbf u^{\mathsf T}\!A \ge \mathbf c^{\mathsf T}$. This gives us
$$
\begin{bmatrix}u_1 & u_2\end{bmatrix} \begin{bmatrix}2 & 3 \\ 8 & 5\end{bmatrix} \ge \begin{bmatrix}6 & 10 \end{bmatrix}
$$
in the first dual, and
$$
\begin{bmatrix}u_1 & u_2\end{bmatrix} \begin{bmatrix}2 & 3 & 1 & 0 \\ 8 & 5 & 0 & 1\end{bmatrix} \ge \begin{bmatrix}6 & 10 & 0 & 0\end{bmatrix}
$$
in the second dual. The change is because in the first dual, we're thinking of $u_1 \ge 0$ and $u_2 \ge 0$ as nonnegativity constraints (treated as a special case), while in the second dual, they are just two linear constraints like any others.
We don't typically add slack variables to the dual, but if we did, it would have an identical (but slightly sillier) effect on the primal.
I think the cleanest way to do it is to add a nonpositive slack variable: turn $2u_1 + 3u_2 \ge 6$ into $2u_1 + 3u_2 + w_1 = 6$, where $w_1 \le 0$. A nonpositive slack variable in the dual will correspond to a $\ge$ constraint in the primal, which will replace the nonnegativity constraint on $x_1$. We'll get:
\begin{array}{rrll rrll}
\text{maximize } & 6x_1 + 10x_2 & & & \text{minimize} & 60u_1 + 45u_2 \\
& 2x_1 + 8x_2 + s_1 & = 60 & (u_1) & & 2u_1 + 3u_2 + w_1 & = 6 & (x_1) \\
& 3x_1 + 5x_2 + s_2 & = 45 & (u_2) & & 8u_1 + 5 u_2 + w_2 & = 10 & (x_2) \\
& x_1 & \ge 0 & (w_1) & & u_1 & \ge 0 & (s_1) \\
& x_2 & \ge 0 & (w_2) & & u_2 & \ge 0 & (s_2) \\
& s_1, s_2 & \ge 0 & & & w_1, w_2 & \le 0
\end{array}
Note that $s_1, s_2$ still have special-cased nonnegativity constraints, just like $w_1, w_2$ still have special-cased nonpositivity constraints.
If we really wanted to do terrible, unnatural things to our primal-dual pair, we could add slack variables to the constraints $x_1 \ge 0, x_2 \ge 0, u_1 \ge 0, u_2 \ge 0$. This would turn the nonnegativity constraints on $s_1, s_2$ and the nonpositivity constraints on $w_1, w_2$ into ordinary constraints, make the matrix even bigger, and add more variables to both programs. You can keep going like this indefinitely, creating bloated primal-dual pairs that are ultimately equivalent to the original.
Best Answer
Let $x_{ij}$ the amount schnuppels of duckwheat which are delivered from location $i$ to location $j$, where $k,n,c,m$ denote the corresponding locations Kansas, New York, California and Mexico. Kansas can deliver at most 15. Thus the tranportation constraints (supply) is $x_{kn}+x_{kc}\leq 15$. Similar for Mexico: $x_{mn}+x_{mc}\leq 8$. For both constraints you need a slack variable each to obtain equalities. For New York and California you have demand constraints, thus at-least-constraints. I agree with your constraints.
$$x_{kn}+x_{mn}\geq 10, \quad x_{kc}+x_{mc}\geq 13$$.
For both constraints you need a surplus variable each and probably artificial variables if you apply the simplex algorithm. I agree with your objective function as well.