Solve the linear programming problem by the method of corners.

Minimize C = 5x + y

subject to
x + 3y ≥ 43
x + y ≥ 27
x ≥ 0, y ≥ 0

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To solve the linear programming problem using the method of corners, we need to find the coordinates of the corners of the feasible region formed by the given constraints and then evaluate the objective function at each corner to find the minimum value.

Let's start by graphing the feasible region formed by the constraints.

First, we plot the line x + 3y = 43. To do this, we need to find two points that satisfy this equation. Let's set y = 0 and y = 10 to find the x-intercept and y-intercept, respectively.

When y = 0, x = 43. So, one point is (43, 0).

When x = 0, 3y = 43, so y ≈ 14.33. So, the other point is (0, 14.33).

Plotting these two points and drawing a line through them gives us the line x + 3y = 43.

Next, we plot the line x + y = 27 using a similar process.

When y = 0, x = 27. So, one point is (27, 0).

When x = 0, y = 27. So, the other point is (0, 27).

Plotting these two points and drawing a line through them gives us the line x + y = 27.

Now, we need to shade the region that satisfies the inequalities x + 3y ≥ 43 and x + y ≥ 27. To do this, we can choose a test point within the region and check if it satisfies both inequalities. If it does, then we shade the region that contains the test point; otherwise, we shade the region that does not contain the test point.

We can choose the point (0, 0) as our test point. Plugging it into the inequalities:

For x + 3y ≥ 43: 0 + 3(0) = 0 ≥ 43 (False)
For x + y ≥ 27: 0 + 0 = 0 ≥ 27 (False)

Since (0, 0) does not satisfy both inequalities, we shade the region opposite from the origin.

Now, we can find the corners of the feasible region.

The corners are the points where the lines intersect. In this case, the corners are the points of intersection between the two lines x + 3y = 43 and x + y = 27. By solving the system of equations, we can find these points.

Subtracting the second equation from the first equation, we get:

(x + 3y) - (x + y) = 43 - 27
2y = 16
y = 8

Substituting this value for y into the second equation, we get:

x + 8 = 27
x = 19

Thus, one corner point is (19, 8).

Similarly, by substituting this value for y into the first equation, we get:

x + 3(8) = 43
x = 19

Thus, the other corner point is (19, 8) as well.

Now, we can evaluate the objective function C = 5x + y at the corner points to find the minimum value.

For the corner point (19, 8):
C = 5(19) + 8 = 95 + 8 = 103

Therefore, the minimum value of C is 103 at the point (19, 8).

In conclusion, the linear programming problem is minimized at the point (19, 8) with a value of C = 103.