I have been learning the Simplex Method for solving minimization and maximization problems, but came across a small problem with every resource I have found online.
They all seem to imply that excluding my non-negativity constraint, I should have the same number of variables as constraints. For example: https://www.youtube.com/watch?v=8_D3gkrgeK8
Now let's say I had a problem with 20 variables and 7 constraints. For example: $$C=a_1X_1+a_2X_2 +...+a_{20}X_{20}$$ $$b_1X_1 + b_2X_2 +...+b_{20}X_{20} \ge q_1$$ $$c_1X_1 + c_2X_2 +...+c_{20}X_{20} \ge q_2$$ $$d_1X_1 + d_2X_2 +...+d_{20}X_{20} \ge q_3$$ $$e_1X_1 + e_2X_2 +...+e_{20}X_{20} \ge q_4$$ $$f_1X_1 + f_2X_2 +...+f_{20}X_{20} \ge q_5$$ $$g_1X_1 + g_2X_2 +...+g_{20}X_{20} \ge q_6$$ $$h_1X_1 + h_2X_2 +...+h_{20}X_{20} \ge q_7$$ $$X_1, X_2, ..., X_{20} \ge 0$$
To my understanding, per the dual method, I put these equations into a matrix, take its transpose, and rewrite my equations then. That would give me 20 equations with 8 variables. So then when I introduce my slack variables (which will be my original X1 X2 etc), I should only have 19, which is one less than the amount of variables I had. Am I to choose one to ignore somehow? Or have I made a mistake in my logic somewhere.
Any help would be deeply appreciated, thank you.