Importance of basic solution in simplex algorithm?
If all variables (structural and logical) are non-negative (i.e. x>=0 and slacks s>=0) then all non-basic variables are equal to zero. As they are fixed to zero we only have to solve for the m basic variables.
Essentially we have to solve
A x = b
Unfortunately this is a non-square system of equations (after adding slacks we always have more columns than rows). In LPs we can form a basic solution and partition this into
B x_B + N x_N = b
After setting x_N = 0 we have just a square system of linear equations with solution:
x_B = inv(B) b
There is a fundamental theorem that says we can restrict the search to only basic solutions i.e. solutions that can be partitioned in basic and non-basic variables
x = [ x_B ]
[ x_N ]
with x_B >= 0 and x_N = 0.
For more info open a book about Linear Programming; a very good one is Vanderbei.