Simplex Algorithm - Convert the LP problem to a system of linear equations.

Example1)
Maximize p = x + 2y + 3z subject to the constraints

7x + z <=  6
x + 2y <=  20
3y + 4z <=  30

The initial tableau  is:

 x y z s t u p Ans 7 0 1 1 0 0 0 6 1 2 0 0 1 0 0 20 0 3 4 0 0 1 0 30 -1 -2 -3 0 0 0 1 0

The final tableau is

 x y z s t u p Ans 7 0 1 1 0 0 0 6 59 0 0 8 3 -2 0 48 -28 3 0 -4 0 1 0 6 4 0 0 1 0 2 3 66

The x-column is not cleared, so x = 0.
Since the y-column is cleared with pivot 3, the value of y is 6/3 = 2.
Since the z-column is cleared with pivot 1, the value of z is 6/1 = 6.
Since the t-column is cleared with pivot 3, the value of t is 48/3 = 16.
Since the s and u-columns are not cleared, their value is 0.
Since the p-column is cleared with pivot 3, the value of y is 66/3 = 22.

Example2)
The constraints

4x - 3y + z <= 3
x + y + z <= 10
2x + y - z<= 10
in the above LP problem are written as equations by adding a new "slack" variable to the left-hand side of each to "take up the slack." In addition, the objective function
p = 2x - 3y + 4z
is rewritten with all the unknowns on the left-hand side.

-2x + 3y - 4z + p = 0

This gives the following system of equations.

 4x - 3y + z + s = 3 x + y + z + t = 10 2x + y - z + u = 10 -2x + 3y - 4z + p = 0
the initial tableau is as follows (notice how we separate the last row using a line).
 x y z s t u p Ans 4 -3 1 1 0 0 0 3 1 1 1 0 1 0 0 10 2 1 -1 0 0 1 0 10 -2 3 -4 0 0 0 1 0

Thus, the matrix containing all information has 4 rows and 8 columns.
Now enter edit matrix A under F7 using n=#rows=4 and m=#columns=8 .

MATRIX A will look as follows:

 4 -3 1 1 0 0 0 3 1 1 1 0 1 0 0 10 2 1 -1 0 0 1 0 10 -2 3 -4 0 0 0 1 0

The Solution is

x=0  ( as x column is not cleared)
y=7/4 = 1.75
z=33/2 =  16.5
p=111/4 = 27.75