## Augmented form (slack form)

Linear programming problems must be converted into *augmented form* before being solved by the simplex algorithm. This form introduces non-negative *slack variables* to replace inequalities with equalities in the constraints. The problem can then be written in the following block matrix form:

- Maximize
*Z*in:

where are the newly introduced slack variables, and *Z* is the variable to be maximized.

### Example

The example above is converted into the following augmented form:

maximize | (objective function) | |

subject to | (augmented constraint) | |

(augmented constraint) | ||

(augmented constraint) | ||

where are (non-negative) slack variables, representing in this example the unused area, the amount of unused fertilizer, and the amount of unused insecticide.

In matrix form this becomes:

- Maximize
*Z*in: