Let's analyze the constraints and identify the feasible region’s vertices:
1. [tex]\(x + y \leq 7\)[/tex]
2. [tex]\(x - 2y \leq -2\)[/tex]
3. [tex]\(x \geq 0\)[/tex]
4. [tex]\(y \geq 0\)[/tex]
We are given the potential vertices:
- (0, 0)
- (0, 1)
- (4, 3)
- (7, 0)
To determine if each vertex is feasible, we need to check if it satisfies all the given constraints.
### Vertex (0, 0)
1. [tex]\(0 + 0 \leq 7\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
2. [tex]\(0 - 2(0) \leq -2\)[/tex] [tex]\(\rightarrow\)[/tex] Not satisfied
3. [tex]\(0 \geq 0\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
4. [tex]\(0 \geq 0\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
(0, 0) is not feasible because it does not satisfy the second constraint.
### Vertex (0, 1)
1. [tex]\(0 + 1 \leq 7\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
2. [tex]\(0 - 2(1) \leq -2\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
3. [tex]\(0 \geq 0\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
4. [tex]\(1 \geq 0\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
(0, 1) is feasible because it satisfies all constraints.
### Vertex (4, 3)
1. [tex]\(4 + 3 \leq 7\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
2. [tex]\(4 - 2(3) \leq -2\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
3. [tex]\(4 \geq 0\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
4. [tex]\(3 \geq 0\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
(4, 3) is feasible because it satisfies all constraints.
### Vertex (7, 0)
1. [tex]\(7 + 0 \leq 7\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
2. [tex]\(7 - 2(0) \leq -2\)[/tex] [tex]\(\rightarrow\)[/tex] Not satisfied
3. [tex]\(7 \geq 0\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
4. [tex]\(0 \geq 0\)[/tex] [tex]\(\rightarrow\)[/tex] Satisfied
(7, 0) is not feasible because it does not satisfy the second constraint.
The feasible vertices are those that satisfy all constraints:
[tex]\( \boxed{(0, 1), (4, 3)} \)[/tex].
Therefore, the vertices of the feasible region are:
[tex]\[ \boxed{(0, 1), (4, 3)} \][/tex]
