The error is in the ratio computations. Although choosing the z-column is correct, the ratios are based off of the z- and v-values in each row. Thus, your ratios should be:
Column 1: v/z = 3/1 = 3
Column 2: v/z = -4/-1 = 4
Since Column 1 has the smaller ratio, it will be the pivot point. The rest should fall into suit.
To get an optimal solution, it is necessary to have a basic solution as a starting point. This means, that the number of constraints must be equal to the number of basic variables. Thus every constraint has to have a positive slack variable or a positive artificial variable. If the slack variable is negative or is not needed, then you add an artificial variable.
There are three cases:
$\color{blue}{\leq-\texttt{constraint}}$
If you have a $\leq$-constraint, then you have to add a slack variable for each constraint.
$2y+z \leq 2 \quad \Longrightarrow \quad 2y+z +s_1=2$
$\color{blue}{=\texttt{-constraint}}$
If you have a $=$-constraint, then you do not have to add a slack variable for each constraint. But you have to add an artificial variable for each constraint.
$x+y+z=4 \quad \Longrightarrow \quad x+y+z+a_1=4$
$\color{blue}{\geq-\texttt{constraint}}$
If you have a $\geq$-constraint, then you have to substract a slack variable for each constraint. Additionally you have to add an artificial variable for each constraint.
$x-2y+z \geq 3 \quad \Longrightarrow \quad x-2y+z-s_2+a_2 = 3$
The initial simplex tableau is
$\begin{array}{|c|c|c|c|c|c|c|c|} \hline x & y & z & s_1 & s_2 & a_1 & a_2 & RHS \\ \hline -2 & 1 & -3 & 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 2 & \color{red} 1 & 1 & 0 & 0 & 0 & 2 \\ \hline 1 & 1 & 1 & 0 & 0 & 1 & 0 & 4 \\ \hline 1 & -2 & 1 & 0 & -1 & 0 & 1 & 3 \\ \hline \end{array} $
The coefficients of the objective function have to be multiplied by (-1), because the objective function has to be maximized.
The initial basic solution is $(x, y, z, s_1, s_2, a_1, a_2)=(0, 0, 0, 2, 0, 4, 3)$
The first pivot element is the red One.
Best Answer
You are right, that you can´t get an Initial solution by using only slack variables, if you have $=$-constraints and $\geq$-contraints. I start with the three types of equations and then transform them.
$x+y\leq 8$
$2x-y=6$
$x+2y\geq 12$
Introducing slack variables to get equalities
$x+y+s_1= 8$
$2x-y=6$
$x+2y-s_2= 12$
The second and the third equation have no basic variable. In these cases you need artificial variables ($a_i$).
$x+y+s_1= 8$
$2x-y+a_1=6$
$x+2y-s_2+a_2= 12$
Now you are able to start the Simplex algorithm. Surely you need an objective function.