 Solution

Using the standard maximum form of a linear program, writewhere $x\in {\mathbb{R}}^n$, $A\in {\mathbb{R}}^{m\times n}$, $b\in {\mathbb{R}}^m$, and $c\in {\mathbb{R}}^n$. The inequalities are coordinatewise, and $c^{T}x$ is the objective function. General linear programs can be put in this form: reverse the sign of a minimization objective function, reverse lower-bound linear inequalities, replace an equality by two opposite linear inequalities, and replace each unrestricted variable by the difference of two nonnegative variables. Some conventions instead call the equality-constraint version standard form; the convention here is the maximum-with-upper-bounds convention.

Introduce nonnegative slack variables $x_{n+i}=b_i-{\sum }_{j=1}^n{a}_{ij}{x}_j$. The equivalent slack form of a linear program isThe original variables are initially nonbasic and the slack variables basic. More generally, a slack form of a linear program specifies disjoint index sets $N,B$ and writesThus each basic variable and the objective function are expressed using only nonbasic variables. Setting all nonbasic variables to zero gives the associated basic solution, which is a feasible solution exactly when the resulting basic values are nonnegative. The slack form of a linear program does not itself require that initial basic solution to be feasible.