Question

What are the vertices of the feasible region bounded by the constraints?

x + 2y =8
2x + y =10
x=0 and y=0

A. (0,5); (0,8); (2, 4)
B. (0,5); (8,0); (4,2)
C. (5,0); (8,0); (4,2)
D. (5,0); (0,8); (4,2)

Answer 1

The vertices of the feasible region formed by the given constraints are at the points where the constraint lines intersect. The correct vertices are (0,5), (8,0), and (4,2), corresponding to Option B from the given choices.

Step-by-step explanation:

The question is asking for the vertices of the feasible region defined by a set of linear inequalities:

• x + 2y ≤ 8
• 2x + y ≤ 10
• x ≥ 0
• y ≥ 0

To find the vertices, we need to determine the intersection points of these inequalities. The intersection points occur where two of these equations are both true, and they will form the corners of the feasible region. Let's find these:

1. Intersect x + 2y = 8 with y = 0, which gives us the point (8, 0).
2. Intersect 2x + y = 10 with x = 0, which gives us the point (0, 10).
3. Intersect x = 0 with y = 0, which gives us the origin (0, 0) (This is actually not a vertex since the region is unbounded in the negative direction for both x and y constraints).
4. Intersect the lines x + 2y = 8 with 2x + y = 10 via substitution or elimination which gives us the point (4, 2).

Thus, the vertices of the feasible region are (8,0), (0,10), and (4,2). Among the options provided, Option B has the correct vertices (0,5) which corresponds to the intersection of 2x + y = 10 with x = 0, (8,0) and (4,2).