I am learning the simplex method and was wondering why we always search for solutions of the form basic $\cup$ nonbasic where nonbasic variables take on zero value.
Let's look at a concrete example:
let 8$x_1$+10$x_2$+7$x_3$= z
$x_1,x_2,x_3,s_1,s_2 \geq 0$
We start with $s_1$, $s_2$ as basic slack variables. They are nonnegative while $x_1,x_2,x_3$ are all zero nonbasic variables. This is a feasible solution to the linear programming problem.
We then find a "better solution" by finding a pivot that will tell us the incoming and outgoing variables. In this case the 5 is the pivot value with outgoing variable $s_2$ and incoming variable $x_2$.
After row reduction we read off from the tableau that $x_2$ and $s_1$ are basic variables that take on nonnegative values while $x_1$, $x_3$, $s_2$ take on zero values as nonbasic variables.
Why do we assume nonbasic variables are always zero?
Does it have something to do with searching for extreme points? Can anyone reference a proof for this?