Minimize z = 3x + 6y subject to:

4x + y ≥ 20
x + y ≤ 20
x + y ≥ 10
x ≥ 0
y ≥ 0
Graph the feasibility region.
Identify all applicable corner points of the feasibility region.
Find the point(s) (x,y) that minimizes the objective function z = 3x + 6y.

I am having trouble with this one as well. please help!

To graph the feasibility region, we need to plot the boundary lines of the inequalities and shade the region that satisfies all the constraints.

First, let's graph the line 4x + y = 20:
1. Find two points on this line. Let's assume x = 0, then y = 20. Another point could be x = 5, then y = 0.
2. Plot the two points on a graph and draw a straight line passing through them.

Next, let's graph the line x + y = 20:
1. Find two points on this line. Let's assume x = 0, then y = 20. Another point could be x = 20, then y = 0.
2. Plot the two points on a graph and draw a straight line passing through them.

Now, let's graph the line x + y = 10:
1. Find two points on this line. Let's assume x = 0, then y = 10. Another point could be x = 10, then y = 0.
2. Plot the two points on a graph and draw a straight line passing through them.

Finally, let's graph the horizontal and vertical axes by drawing two perpendicular lines intersecting at the origin (0, 0).

The feasibility region is the shaded region that satisfies all the constraints. It is the area where all the lines intersect or fall within.

To find the corner points of the feasibility region, find the points of intersection between the lines. These points will be the corners of the shaded region. In this case, there are three corner points: (0, 10), (10, 0), and (5, 15).

To find the point(s) (x, y) that minimize the objective function z = 3x + 6y, evaluate the objective function at each corner point and compare the values. The point that gives the smallest value of z will be the solution.

Evaluate z = 3x + 6y at each corner point:
1. For (0, 10), z = 3 * 0 + 6 * 10 = 60.
2. For (10, 0), z = 3 * 10 + 6 * 0 = 30.
3. For (5, 15), z = 3 * 5 + 6 * 15 = 105.

From the evaluations, the point that minimizes z is (10, 0) with a value of 30. Therefore, the point (10, 0) minimizes the objective function z = 3x + 6y in the given feasibility region.