Setting Up the Tableau

The Initial Tableau

The initial tableau records the slack basis, the right side, and the objective row in one exact grid. This lesson reads the starting basic feasible solution directly from that grid. The point is bookkeeping: every entry is a rational value simplex can pivot on.

highlighted = computed this step

Slack basis

The initial basis is the two slack columns. Setting x=0 and y=0 gives s1=4 and s2=4. Why: a basic feasible solution reads the right side from the basic rows.

x=0y=0x=0\quad y=0
Initial simplex tableauThe initial tableau displays the slack basis and the unreduced objective row.Initial tableau2110412014-1-1000xys1s2rhss1s2z

Objective row

The z-row starts with -1 under x and -1 under y. Why: the row encodes z minus the objective terms equals zero.

z−x−y=0z-x-y=0
Initial simplex tableauThe initial tableau displays the slack basis and the unreduced objective row.Initial tableau2110412014-1-1000xys1s2rhss1s2z

Origin value

At the starting basis, the decision point is (0, 0) and z=0. Why: simplex starts at a vertex where the slack variables carry all available capacity.

z=0z=0
Initial simplex tableauThe initial tableau displays the slack basis and the unreduced objective row.Initial tableau2110412014-1-1000xys1s2rhss1s2z

Diagram note

The tableau is a bookkeeping view of the same feasible origin used by the geometry book. Pixel positions are rounded for layout; every number shown is exact.

basis plus right side gives the starting point\text{basis plus right side gives the starting point}
Initial simplex tableauThe initial tableau displays the slack basis and the unreduced objective row.Initial tableau2110412014-1-1000xys1s2rhss1s2z