Graph the system of constraints and find the value of x and y that maximize the objective function
x ≥ 0
y ≥ 0
y ≤ 3
y ≤ -x+5
objective function: C=-6x+2y
First, let's graph the constraints:
1) x ≥ 0: This constraint means that x must be greater than or equal to 0. It represents all the points to the right of the y-axis.
2) y ≥ 0: This constraint means that y must be greater than or equal to 0. It represents all the points above the x-axis.
3) y ≤ 3: This constraint means that y must be less than or equal to 3. It represents a horizontal line at y = 3.
4) y ≤ -x + 5: This constraint represents a line with a slope of -1 and a y-intercept of 5. It intersects the x-axis at x = 5 and the y-axis at y = 5.
Now we can graph the systems of constraints:
To find the maximum value of the objective function: C = -6x + 2y, we need to find the point (x, y) that falls within the feasible region (the shaded area) and maximizes the objective function. The feasible region is the overlapping area of the constraints.
To find this point, we need to evaluate the objective function at the corner points of the feasible region:
A: (0, 0)
C = -6(0) + 2(0)
C = 0
B: (0, 3)
C = -6(0) + 2(3)
C = 6
C: (5, 3)
C = -6(5) + 2(3)
C = -30 + 6
C = -24
D: (5, 0)
C = -6(5) + 2(0)
C = -30
From these calculations, we can see that the maximum value of the objective function occurs at point (0, 3) with C = 6.