Simplex (in-depth):

Setup:
* For each regular constraint - add an artificial variable to make it =
* Convert the objective function to an = 0 equation (this is now equation 0)
* Choose nonbasic variables (#of variables - #of equations) and set them to 0 - 
typically these are our original variables.
* Set all other variables based on equations.

Steps:
0.) Test for optimality - is there a positive coefficient in eq0?

1.) Determine direction of movement: Looking at equation 0, the coefficients 
indicate the rate of change for variables in eq 0. Choose the largest. This is
our chosen variable to increase.

2.) Determine where to stop: Figure out how the basic variables change by 
looking at every constraint equation. If the equation has the variable, then
determine the minimum ratio for those basic variables. This is done by
dividing the current value for the equation's basic variable by the coefficient
in that equation. Choose the smallest basic variable.

3a.) Solve for a new feasible solution: increase the nonbasic variable by the
value determined in #2, it now becomes a basic variable. Decrease the chosen 
basic variable to 0. It now becomes a nonbasic variable.

3b.) Adjust the system of equations: since x2 is now the basic variable for
eq 2, we need to drop its coefficient to 1, and eliminate it from all other
equations. This is basic algebra. Start with eq2, then use eq2 to adjust the
others.

Repeat. At end, the values of x1 and x2, plus the Z value is our solution.