To graph the system of constraints, we start by graphing each individual constraint on the xy-plane.
1. x >= 0:
This constraint indicates that x must be greater than or equal to 0, which means all points to the right of the y-axis are valid solutions. We shade the area to the right of the y-axis.
2. y >= 0:
This constraint indicates that y must be greater than or equal to 0, which means all points above the x-axis are valid solutions. We shade the area above the x-axis.
3. y <= 3:
This constraint indicates that y must be less than or equal to 3, which means all points below or on the line y = 3 are valid solutions. We draw a solid line at y = 3 and shade the area below it.
4. y <= -x + 5:
This constraint indicates that y must be less than or equal to -x + 5, which means all points below or on the line y = -x + 5 are valid solutions. We draw a solid line with a slope of -1 passing through the point (5, 0) and shade the area below it.
Now, we identify the feasible region, which is the shaded area that satisfies all the constraints. In this case, it is the triangle formed by the intersection of the shaded areas.
To find the value of x and y that maximize the objective function, we need to find the vertex of the feasible region that maximizes the objective function C = -5x + 3y. Since the objective function is linear, it will have the maximum value at one of the extreme points of the feasible region.
Upon analyzing the graph, we can see that the vertex located at (0,3) is the highest point within the feasible region. Plugging in these values into the objective function, we find:
C = -5(0) + 3(3) = 9
Therefore, the maximum value of the objective function is C = 9, and the values of x and y that maximize it are x = 0 and y = 3.