Pivoting

One Exact Pivot

A simplex pivot is exact row reduction on the tableau. This lesson shows the first post-pivot tableau and the adjacent vertex it represents. The fractions are the computation, not decoration.

highlighted = computed this step

Exact row reduction

The pivot makes x basic and produces entries such as 1/2, 3/2, and -1/2. Why: pivoting is Gaussian elimination on one column.

pivoting clears the entering column\text{pivoting clears the entering column}
After one exact pivotThe post-pivot tableau is recomputed by exact row reduction.After first pivot11/21/20203/2-1/2120-1/21/202xys1s2rhsxs2z

New vertex

The new basic point is (2, 0) with z=2. Why: a pivot moves from one feasible vertex to an adjacent one.

z=2 at (2,0)z=2\text{ at }(2, 0)
After one exact pivotThe post-pivot tableau is recomputed by exact row reduction.After first pivot11/21/20203/2-1/2120-1/21/202xys1s2rhsxs2z

Diagram note

Every fraction in the post-pivot tableau comes from exact row operations. Pixel positions are rounded for layout; every number shown is exact.

one pivot gives the next vertex\text{one pivot gives the next vertex}
After one exact pivotThe post-pivot tableau is recomputed by exact row reduction.After first pivot11/21/20203/2-1/2120-1/21/202xys1s2rhsxs2z