Explain the process ---STEP 6. Perform pivoting to make all other entries in this column zero.
Pivoting is a process of obtaining a 1 in the location of the pivot element, and then making all other entries zeros in that column. So now our job is to make our pivot element a 1 by dividing the entire second row by 2. i.e., new **R₂ = R₂ / 2**.
The result follows.
Tableau 1:
x1 x2 y1 y2 Z
1 1 1 0 0 | 12
1 1/2 0 1/2 0 | 8
-40 -30 0 0 1 | 0
To obtain a zero in the entry first above the pivot element, we multiply the second row by -1 and add it to row 1. i.e., new **R₁ = -1*R₂ + R₁**
We get
Tableau 2:
x1 x2 y1 y2 Z
0 1/2 1 -1/2 0 | 4 y1
1 1/2 0 1/2 0 | 8 x1
-40 -30 0 0 1 | 0
To obtain a zero in the element below the pivot, we multiply the second row by 40 and add it to the last row. i.e., new **R₃ = 40*R₂ + R₃**
Tableau 3:
x1 x2 y1 y2 Z
0 1/2 1 -1/2 0 | 4 y1
1 1/2 0 1/2 0 | 8 x1
0 -10 0 20 1 | 320 Z
We now determine the basic solution associated with this tableau. By arbitrarily choosing x₂ = 0 and y₂ = 0, we obtain x₁=8, y₁=4, and z=320. If we write the augmented matrix, whose left side is a matrix with columns that have one 1 and all other entries zeros, we get the following matrix stating the same thing.
Matrix:
[ x1 y1 Z | C ]
[ 0 1 0 | 4 ]
[ 1 0 0 | 8 ]
[ 0 0 1 | 320 ]
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Step 6 of the simplex method involves pivoting to make all other entries in the pivot column zero. We start with our tableau where we've identified the pivot element as one half in the x2 column. The pivoting process transforms this element to 1 and eliminates all other entries in this column through systematic row operations.
The first step in pivoting is to make the pivot element equal to 1. We do this by dividing the entire second row by 2, since our pivot element is currently one half. This operation, written as new R2 equals R2 divided by 2, transforms every element in the second row. The pivot element becomes 1, and all other elements in that row are also divided by 2.
Now we eliminate the entry above the pivot element. We need to make the 1 in the first row, second column become zero. We do this by multiplying the second row by negative 1 and adding it to the first row. This operation is written as new R1 equals negative 1 times R2 plus R1. After this calculation, the entry above the pivot becomes zero, and we get Tableau 2.
The final step is to eliminate the entry below the pivot element. We multiply the second row by 40 and add it to the third row. This operation is written as new R3 equals 40 times R2 plus R3. However, note that the result in the x2 column is negative 10, not zero as the pivoting goal would suggest. This gives us our final Tableau 3 with the pivot column partially processed.