Simplex method

[frctl]

Instructions:

Maximize
z = 2x₁ + 3x₂ + 2x₃,
subject to the system
2x₁ + x₂ + 2x₃ ≤ 13
x₁ + x₂ - 3x₃ ≤ 8
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

My work:

Introduce slack variables and write initial system

2x₁ + x₂ + 2x₃ + s₁ = 13
x₁ + x₂ - 3x₃ + s₂ = 8
-2x₁ - 3x₂ - 2x₃ + P = 0

Write the Simplex tableau



Perform pivot operations (my attempt)



I looked online for help with this and obtained the following:


But I don't understand which operations are being completed.

Could someone help me complete this question using the Simplex method?

 

[HallsofIvy]

In simplex table "#5 simplex table 2", the second number in the second row, 1, is in red. That indicates that they have chosen it as the "pivot". If you look at "#6 simplex table 3" you will see that the other numbers in column 2 are 0. That is because they have subtracted the second row from the first row, column by column: 2- 1= 1, 1- 1= 0, 2- (-3)= 5, 1- 0= 1, 0- 1= -1, 13- 8= 5.
For the third row, they added 3 times the second row to the third row:
-2+ 3(1)= 1, -3+ 3(1)= 0, -2+ 3(-3)= 11, 0+ 3(0)= 0, 0+ 3(1)= 3, 0+ 3(8)= 24.
Those are the rows shown in "#6 table 3".

And in that table, the third number in the first row, 5, is In red. That means they have chosen that number as the "pivot". In "#7 table 4" that column is "\(\displaystyle \begin{array}{ccc}1 \\ 0 \\ 0 \end{array}\)". They first divided each number in the first row by 5 to get that "1" and the rest of the first row, then added 3 times the new first row to the second row to get 1+ 3(1/5)= 8/5, 1+ 3(0)= 1, -3+ 3(1)= 0, 1+ 3(-1/5)= 2/5, and finally added 11 times the new first row to the second row to get 1+ 11(1/5)= 16/5, 0+ 11(0)= 0, 0+ 11(1/5)= 11/5, 3+ 11(-1/5)= 4/5, 24+ 11(1)= 35.

The key is that we want a "1" at the pivot position and "0"s in the rest of the column. And we get those by "row operations"- dividing the entire row by something or add a multiple of one row to another,