solve the given linear programming problem using the simplex

solve the given linear programming problem using the simplex method. Also, set up and solve the dual of the given problem.

6. Minimize z=4x+6y

subject to x+3y>5

2x+ y>3

x>0, y>0.

please be clear and neat. thanks. and provide a step-by-step- process.

Step 1: First we need to form the augmented matrix for the problem

[1 3 ....5]

[2 1 .... 3]

[4 6 ..... 0]

now we need to transpose the augmented matrix we get

[1 2 .... 4]

[3 1 .... 6]

[5 3 .... 0]

Dual maximization problem

Maximize z\' = 5x + 3y

with constraints

x + 2y <= 4

3x + y <= 6

where x>0 and y>0

we can now apply the simplex method to the dual problem

y1 y2 s1 s2 b

1 2 1 0 4

3 1 0 1 6

-5 -3 0 0 0

hence on solving we get

1/2 1 1/2 0 2

3 1 0 1 6

-5 -3 0 0 0

now for departing we need to multiply 2*first row + 1*second row, we get

Final minimum value answer as 10