Editing Simplex algorithm (section)

===Example===
{{see also|Revised simplex algorithm#Numerical example}}
Consider the linear program
:Minimize
::<math>Z = -2 x - 3 y - 4 z\,</math>
:Subject to
::<math>\begin{align}
  3 x + 2 y + z &\le 10\\
  2 x + 5 y + 3 z &\le 15\\
  x,\,y,\,z &\ge 0
\end{align}</math>

With the addition of slack variables ''s'' and ''t'', this is represented by the canonical tableau
:<math>
  \begin{bmatrix}
    1 & 2 & 3 & 4 & 0 & 0 &  0 \\   
    0 & 3 & 2 & 1 & 1 & 0 & 10 \\
    0 & 2 & 5 & 3 & 0 & 1 & 15
  \end{bmatrix}
</math>

where columns 5 and 6 represent the basic variables ''s'' and ''t'' and the corresponding basic feasible solution is 
:<math>x=y=z=0,\,s=10,\,t=15.</math>

Columns 2, 3, and 4 can be selected as pivot columns, for this example column 4 is selected. The values of ''z'' resulting from the choice of rows 2 and 3 as pivot rows are 10/1&nbsp;=&nbsp;10 and 15/3&nbsp;=&nbsp;5 respectively. Of these the minimum is 5, so row 3 must be the pivot row. Performing the pivot produces
:<math>
  \begin{bmatrix}
    1 &            -\frac{2}{3} &             -\frac{11}{3} & 0 & 0 & -\frac{4}{3} & -20 \\   
    0 &  \frac{7}{3} &   \frac{1}{3} & 0 & 1 &  -\frac{1}{3} &  5 \\
    0 &  \frac{2}{3} &   \frac{5}{3} & 1 & 0 &  \frac{1}{3} & 5
  \end{bmatrix}
</math>

Now columns 4 and 5 represent the basic variables ''z'' and ''s'' and the corresponding basic feasible solution is 
:<math>x=y=t=0,\,z=5,\,s=5.</math>

For the next step, there are no positive entries in the objective row and in fact
:<math display="block">Z = -20 + \frac{2}{3}x+\frac{11}{3}y+\frac{4}{3}t</math>
so the minimum value of ''Z'' is&nbsp;&minus;20.