I am current studying LP and also the simplex method. However, the slides from my professor are not that detailed so I was trying to search for online resources. Particularly about how to convert an LP to standard form (In our definition, all equations need to be in equal signs)
However, I have been getting different answers regards what should you do when you encounter a $\geq$ sign.
One book (Larson, Elementary Linear Algebra) explained that all you have to do is to add a surplus variable. Ex:
$$ 5x_1+3x_2 \geq 10 \to 5x_1 + 3x_2-x_3 = 10 $$
However, another book (Intro to OR by Hillier) and some other resources stated that not only a slack variable is needed, but we also need to introduce an artificial variable. (And thus using either "Big-M method" or the "two-phase method" to solve it)
$$ 5x_1+3x_2 \geq 10 \to 5x_1 + 3x_2-x_3 + \overline{x}_5 = 10 \\ \text{where } \overline{x}_5 \text{ is an artificial variable} $$
I am not really sure which way is the correct way
They're both correct, however the first book forgot$^{(1)}$ to mention that you should add an artificial variable as if we took the following:
$$ 5x_1+3x_2 \geq 10 \to 5x_1 + 3x_2-x_3 = 10 $$
and put it into a tableau we would get:
\begin{array} {|c|c|} \hline BV & x_1 & x_2 & x_3 & RHS\\ \hline ? & 5 & 3 & -1 & 10 \\ \hline \end{array}
since we have no starting basic variable for the Simplex Method, we'd be forced to make one like so:
\begin{array} {|c|c|} \hline BV & x_1 & x_2 & x_3 & RHS\\ \hline x_1 & 1 & \frac{3}{5} & -\frac{1}{5} & 2 \\ \hline \end{array}
$$\text{or}$$
\begin{array} {|c|c|} \hline BV & x_1 & x_2 & x_3 & RHS\\ \hline x_2 & \frac{5}{3} & 1 & -\frac{1}{3} & \frac{10}{3} \\ \hline \end{array}
$$\text{or}$$
\begin{array} {|c|c|} \hline BV & x_1 & x_2 & x_3 & RHS\\ \hline x_3 & -5 & -3 & 1 & -10 \\ \hline \end{array}
The third tableau is invalid if $x_3$ has the non-negativity constraint $x_3\ge0$. The above only really works if we have only one constraint in the model, or if $x_1$ or $x_2$ only appears in this constraint and no other.
Suppose we have the following constraints of a model: $$5x_1+3x_2 \ge 10$$ $$x_1+x_2 \le 5$$
Since we cannot simply add just a negative slack to the first constraint as we'll have no starting basic variable in that row, and we can't just modify $x_1,x_2$, or $x_3$ to be a starting basic variable, we'll have to add an artificial variable like so:
$$5x_1+3x_2 -x_3+a_1= 10$$ $$x_1+x_2 +x_4 = 5$$
Where $a_1$ is the artificial variable. Which will look like the following in the initial tableau: \begin{array} {|c|c|} \hline BV & x_1 & x_2 & x_3 & x_4 & a_1 & RHS\\ \hline a_1 & 5 & 3 & -1 & 0 & 1& 10 \\ \hline x_4 & 1 & 1 & 0 & 1 & 0 & 5 \\ \hline \end{array}
However, the introduction of an artificial variable does impact our objective function in a significant way. Thus, you should look into the Big-M and the Two-Phase methods for more information on how to add artificial variables into the objective function.
$(1)$ Forgot is a loaded term, they might have meant for the reader to be creative on trying to derive a basic variable without trying to use the artificial variable technique like what was shown, as it is often better for a model to not ever have to deal with artificial variables.