Find the values of x and y that maximize or minimize the objective function.

x+y < or equal to 8
2x+y < or equal to 10
x> or equal to 0, y > or equal to 0

A. (0,5) Maximum value is 100
B.(1,7) Maximum value is 220
C.(2,6) Maximum value is 280
D. (5,0) Maximum value is 400

I'm so confused. Please help.

Sorry, I forgot to add the objective function.

C=80x+20y

Probably a little late, but this is how you solve it:

You are given two formulas:
x + y = 8
2x + y = 10

First manipulate them so that you have a common variable alone. I did:
y = 8 - x
y = 10 - 2x

Since y = y you can combine these formulas to create:
8 - x = 10 - 2x

Solve this to find x.
x = 2

Plug x into one of the original formulas:
(x) + y = 8
(2) + y = 8

Solve for y.
y = 6

You now have x and y.
(2,6)

Plug x and y into the objective function:
C = 80(x) + 20(y)
C = 80(2) + 20(6)

Solve for C.
C = 160 + 120
C = 280

You have your answer:
C.(2,6) Maximum value is 280

To determine the values of x and y that maximize or minimize the objective function, we need to solve the given system of inequalities and check the objective function values for each solution.

1. We start by graphing the inequalities on a coordinate plane.

The first inequality x+y ≤ 8 represents the area below the line x+y = 8.

The second inequality 2x+y ≤ 10 represents the area below the line 2x+y = 10.

The feasible region is the area where the shaded regions from both inequalities overlap.

2. We need to check each of the provided answer choices to see if the objective function value is maximized.

A. (0,5): The objective function value for this point is (0*50) + (5*10) = 50. This is not the maximum value.

B. (1,7): The objective function value for this point is (1*50) + (7*10) = 90. This is not the maximum value.

C. (2,6): The objective function value for this point is (2*50) + (6*10) = 140. This is not the maximum value.

D. (5,0): The objective function value for this point is (5*50) + (0*10) = 250.

Since the feasible region is bounded, we need to check the objective function value for every point in the feasible region. As we can see, the point (5,0) gives the maximum value.

Therefore, the correct answer is D. (5,0) with a maximum value of 400.

To find the values of x and y that maximize or minimize the objective function, you need to use the concept of linear programming and solve a system of inequalities. Here's how you can approach this problem:

1. Graph the inequalities: Start by graphing the inequalities on a coordinate plane. Since it's a two-variable problem, you can plot the points and shade the areas that satisfy each inequality. The feasible region is the overlapping shaded area.

2. Identify the corners of the feasible region: The corners of the feasible region are the points where the lines intersect. These points are also known as vertices. You can find the coordinates of each vertex by solving the equations of the intersecting lines.

3. Evaluate the objective function at each vertex: Once you have the coordinates of each vertex, substitute them into the objective function and calculate the corresponding value. The objective function represents the quantity you want to maximize or minimize.

4. Find the maximum or minimum value: Compare the values of the objective function at each vertex and determine the maximum or minimum value. Based on the given options, choose the option that matches the coordinates of the vertex with the corresponding value.

Let's go through the steps using the given problem:

1. Graph the inequalities:
Start by graphing the lines x+y = 8 and 2x+y = 10. To graph each inequality, rewrite them as equations by replacing the inequality sign with an equal sign. Then, shade the area that satisfies the inequality.

2. Identify the corners of the feasible region:
The intersection points of the lines x+y = 8 and 2x+y = 10 are (0,8), (5,0), and (2,6).

3. Evaluate the objective function at each vertex:
Substitute the coordinates of each vertex into the objective function and calculate the value:
For (0,8): f(0,8) = 0*100 + 8*20 = 160
For (5,0): f(5,0) = 5*100 + 0*20 = 500
For (2,6): f(2,6) = 2*100 + 6*20 = 280

4. Find the maximum value:
Based on the given options, you can see that the vertex (5,0) yields the maximum value of 500.

Therefore, the correct answer is option D: (5,0) with a maximum value of 400.

Note: The given options in the question may have some incorrect values or errors, so always double-check your calculations and make sure they match the options provided.