To graph the system of constraints, we need to plot the equations and inequalities.
First, let's plot the equation -2x + 2y = 20.
To do this, we can find two points on the line and draw a straight line passing through them.
When x = 0, we have -2(0) + 2y = 20, which gives us y = 10. So one point on the line is (0, 10).
When y = 0, we have -2x + 2(0) = 20, which gives us x = -10. So another point on the line is (-10, 0).
Plotting these points and drawing a line passing through them, we get:
Next, let's plot the inequality 5x + 2y <= 20.
To do this, we find two points on the line and draw a dashed line passing through them (since it's an inequality).
When x = 0, we have 5(0) + 2y <= 20, which gives us 2y <= 20. Dividing both sides by 2, we get y <= 10. So one point on the line is (0, 10).
When y = 0, we have 5x + 2(0) <= 20, which gives us 5x <= 20. Dividing both sides by 5, we get x <= 4. So another point on the line is (4, 0).
Plotting these points and drawing a dashed line passing through them, we get:
The shaded region represents the area where both constraints are satisfied.
Finally, let's find the value of x and y that maximize the objective function C = 7x - 3y within this feasible region.
To do this, we need to find the intersection of the lines/lines and inequalities. Looking at the graph, we see that the feasible region is a triangle with vertices at (0, 0), (0, 10), and (4, 0).
We evaluate the objective function C = 7x - 3y at each corner of the triangle:
C(0, 0) = 7(0) - 3(0) = 0
C(0, 10) = 7(0) - 3(10) = -30
C(4, 0) = 7(4) - 3(0) = 28
Therefore, the value of x and y that maximize the objective function is (4, 0).