Question

Use the simplex method to maximize the given function. Assume all variables are nonnegative. Maximize f = 3x + 22y subject to 14x + 7y S 35 5x + 5y S 50. (x, y) = f= Need Help? Watch It Master It

Answer 1

The simplex method maximizes a linear objective function subject to linear inequalities by introducing slack variables, setting up a simplex tableau, performing pivot operations, and iterating until no negative numbers remain in the objective function row.

Step-by-step explanation:

To solve the linear programming problem with the simplex method, you start by converting the inequalities into equations by introducing slack variables. For the constraints given:

• 
  
• 14x + 7y ≤ 35
• 
  
• 5x + 5y ≤ 50

We add slack variables s1 and s2, respectively:

• 
  
• 14x + 7y + s1 = 35
• 
  
• 5x + 5y + s2 = 50

Now, we set up the initial simplex tableau. Our objective function is:

Maximize f = 3x + 22y

We express it as:

-3x - 22y (because we seek to maximize, we take the negative for the simplex method)

The initial tableau will represent the constraints and the objective function. The columns correspond to the variables x, y, s1, s2, and the solution column:

x
y
s1
s2
Solution


14
7
1
0
35


5
5
0
1
50


-3
-22
0
0
0
Following the steps of the simplex method, you would identify the pivot element to bring into the basis (the most negative element in the objective function row), perform the row operations to change this pivot to 1 and other elements in this pivot column to 0, and iterate the process until no negative numbers remain in the objective function row, which signifies that the optimum solution has been reached.

Answer 2

The maximum value is f = 110 at (x,y) = (0,5)

Using the simplex method

The simplex method can be used to solve the minimization or maximization of a given linear objective function related to its linear equality and inequality constraints.

In the question, we are told to maximize the function f(x) = 3x + 22y subject to;

14x + 7y ≤ 35

5x + 5y ≤ 50

x ≥ 0

y ≥ 0

Using the graph to maximize the function the feasible region that is shaded in black is shown in the image attached below.

Evaluating the value of f= 3x + 22y at each corner point can be represented as:

Corner point f = 3x + 22y

(0,0) 3(0) + 22(0) = 0

(2.5, 0) 3(2.5) + 22(0) = 7.5

(0,5) 3(0) + 22(15) = 110

Therefore, we can conclude that the maximum value is f = 110 at (x,y) = (0,5)