By graphing the system of constraints, find the values of x and y that minimize the objective function.

Question

x+2y>=8
x>=2
y>=0
minimum for C = x + 3y
A. (8, 0)
B. (0, 10)
C. (10, 0)

Answer 1

To find the values of x and y that minimize the objective function, we need to graph the system of constraints and find the point that satisfies all the constraints and is closest to the origin.

First, let's graph the line x + 2y = 8. To do this, we can rewrite the equation in slope-intercept form as y = (-1/2)x + 4. Let's plot this line:

Next, let's graph the line x = 2. This is a vertical line passing through x = 2. Let's plot this line:

Finally, let's graph the line y = 0. This is a horizontal line passing through y = 0. Let's plot this line:

Now, let's find the region that satisfies all the constraints. This is the shaded region in the graph below:

To minimize the objective function C = x + 3y, we need to find the point in the shaded region that is closest to the origin. In other words, we need to find the point that minimizes the distance from the origin.

From the graph, it is clear that the point (8, 0) is the closest point to the origin that satisfies all the constraints. Therefore, the values of x and y that minimize the objective function are x = 8 and y = 0.

So, the answer is A. (8, 0).