Maximize P = 4x + 20y subject to these constraints:

2x + 15y ≤ 700
5x + 10y ≤ 1150
5x + 5y ≤ 1000
4x + 15y ≤ 980
x ≥ 0, y ≥ 0

Maximum value for P = ?.

This value of P occurs when x = ? and y = ?

If u1, u2, u3, and u4 represent the slack variables in the four counstraints (u1 the slack variable in the top constraint listed above, u2 for the second, etc.), then give the values for each of the four slack variables when the optimal solution is reached:

u1 = ?
u2 = ?
u3 = ?
u4 = ?

To maximize the objective function P = 4x + 20y, subject to the given constraints, we can use the method of linear programming.

Step 1: Convert the inequalities to equations and introduce slack variables:

2x + 15y + u1 = 700
5x + 10y + u2 = 1150
5x + 5y + u3 = 1000
4x + 15y + u4 = 980

Step 2: Write the objective function in terms of the variables and slack variables:

P = 4x + 20y

Step 3: Create a table to solve the system of equations using the Simplex method:

| x | y | u1 | u2 | u3 | u4 | RHS |
-----------------------------------------------------------
Eq1 | 2 | 15 | 1 | 0 | 0 | 0 | 700 |
Eq2 | 5 | 10 | 0 | 1 | 0 | 0 | 1150 |
Eq3 | 5 | 5 | 0 | 0 | 1 | 0 | 1000 |
Eq4 | 4 | 15 | 0 | 0 | 0 | 1 | 980 |
Obj | -4 | -20 | 0 | 0 | 0 | 0 | 0 |

Step 4: Apply the Simplex method to find the optimal solution:

Perform row operations to make the objective function row the pivot row. Divide the pivot row by the pivot element (in this case, -4).

| x | y | u1 | u2 | u3 | u4 | RHS |
-----------------------------------------------------------
Obj | 1 | 5 | 0 | 0 | 0 | 0 | 0 |
Eq2 | 5/4 | 9/4 | 0 | 1 | 0 | 0 | 1150 |
Eq3 | 5/4 | 1/4 | 0 | 0 | 1 | 0 | 1000 |
Eq4 | 1 | 3 | 0 | 0 | 0 | 1 | 980 |
Eq1 | 1/2 | 15/4| 1 | 0 | 0 | 0 | 700 |

Perform row operations to make all other elements in the pivot column zero:

| x | y | u1 | u2 | u3 | u4 | RHS |
-----------------------------------------------------------
Obj | 0 | 17/4 | 0 | 0 | 0 | 0 | 2125 |
Eq2 | 0 | 1 | 0 | 3/4 | 0 | 0 | 230 |
Eq3 | 0 |-3/4 | 0 | -1/4| 1 | 0 | 800 |
Eq4 | 0 | 18 | 0 | -3 | 0 | 1 | 180 |
Eq1 | 1 | 15/8 | 2 | 15/8 | 0 | 0 | 650 |

Step 5: Repeat Steps 4 until the objective function coefficients are all nonnegative:

| x | y | u1 | u2 | u3 | u4 | RHS |
--------------------------------------------------------------
Obj | 0 | 0 | 3/2 | 0 | 0 | 875/3 | 6375/2 |
Eq2 | 0 | 0 | -1 | 0 | 0 | 230 | 2740 |
Eq3 | 0 | 0 | -1/2| 0 | 1 |-375/2 | 1250 |
Eq4 | 1 | 0 | 1/6 | 0 | 0 | 60 | 280 |
Eq1 | 0 | 1 |-1/16 | 1 | 0 |-325/8 | 75 |

Step 6: The maximum value for P = 2125 and occurs when x = 280 and y = 75.

Step 7: The values for the slack variables when the optimal solution is reached are:

u1 = 325/8
u2 = 0
u3 = 0
u4 = 0