Adjacent extreme points differ in only one variable

The first observation indicates that we can identify the extreme points of the solution space algebraically by setting zero as many variables as the difference between the number of unknowns and the number of equations. This is a unique property of the extreme points.

The unique property of the extreme points yields the following general procedure for determining the extreme points algebraically. Assume that the standard form has m equations and n variables (m ≤ n) together with the non-negativity restrictions. All the feasible extreme points are determined by considering all the unique non-negative solutions of the m equations in which exactly n - m variables are set equal to zero.

Mathematically, the unique solutions resulting from setting n - m variables equal to zero are called basic solutions. If a basic solution satisfies the non-negativity restrictions, it is called a feasible basic solution. The variables set equal to zero are called non basic variables; the remaining ones are called basic variables.

The general conclusion is that the algebraic definition of basic solutions in the simplex method now takes the place of the extreme point in the graphical solution space.

The second of the two observations is very useful computationally because the simplex method moves from a current extreme point to an adjacent one.  Since adjacent extreme points differ only in one variable we can determine the next (adjacent) extreme point by interchanging a current non basic (zero) variable with a current basic valuable. This idea greatly simplifies the simplex method computations.

The basic-non basic interchange process gives rise to two suggestive names. The entering variable is a current non basic variable that will "enter" the set of basic variables at the next (adjacent extreme point) iteration. The leaving variable is a current basic variable that will "leave" the basic solution in the next iteration.