Question

using the geometric approach, we want to minimize the objective function (z = 4x + 5y) subject to the following constraints: (\begin{cases}x + yleq8\\y - xgeq2\xgeq0\ygeq0end{cases}). the graph below shows the boundary - lines for the above constraint inequalities but is not shaded. select the one answer that contains the correct coordinates of the corner points of what must be shaded as the solution region, and identify the associated minimum value given the above objective function, (z).

Explanation:

Step1: Find intersection of \( y - x = 2 \) and \( x = 0 \)

Solve \( y - 0 = 2 \Rightarrow y=2 \). Point: (0,2).

Step2: Find intersection of \( x + y = 8 \) and \( x = 0 \)

Solve \( 0 + y = 8 \Rightarrow y=8 \). Point: (0,8).

Step3: Find intersection of \( y - x = 2 \) and \( x + y = 8 \)

Substitute \( y = x + 2 \) into \( x + y = 8 \): \( x + x + 2 = 8 \Rightarrow 2x=6 \Rightarrow x=3 \), \( y=5 \). Point: (3,5).

Step4: Calculate \( Z = 4x + 5y \) at corners

At (0,2): \( Z=4(0)+5(2)=10 \); (0,8): \( Z=40 \); (3,5): \( Z=37 \). Minimum Z=10.

Answer:

Corner points: (0,2), (0,8), (3,5); Minimum Z=10