To determine the vertices of the feasible region given the constraints:
[tex]\[
\begin{array}{l}
2 x + 3 y \leq 6 \\
2 x - 6 y \geq -3 \\
x \geq 0 \\
y \geq 0
\end{array}
\][/tex]
we must find all intersection points that define the boundary of the feasible region. Let's solve this step-by-step.
### Step 1: Solve the system of linear inequalities.
- Constraint 1: [tex]\(2x + 3y \leq 6\)[/tex]
- Constraint 2: [tex]\(2x - 6y \geq -3\)[/tex] or equivalently, [tex]\(2x - 6y + 3 \geq 0\)[/tex]
- Constraint 3: [tex]\(x \geq 0\)[/tex]
- Constraint 4: [tex]\(y \geq 0\)[/tex]
### Step 2: Find points of intersection.
#### 2.1 Intersection of [tex]\(2x + 3y = 6\)[/tex] with axes ([tex]\(x = 0\)[/tex] or [tex]\(y = 0\)[/tex]):
- Intersection with the y-axis ([tex]\(x = 0\)[/tex]):
[tex]\[
2(0) + 3y = 6 \implies 3y = 6 \implies y = 2 \implies (0, 2)
\][/tex]
- Intersection with the x-axis ([tex]\(y = 0\)[/tex]):
[tex]\[
2x + 3(0) = 6 \implies 2x = 6 \implies x = 3 \implies (3, 0)
\][/tex]
#### 2.2 Intersection of [tex]\(2x - 6y = -3\)[/tex] with axes ([tex]\(x = 0\)[/tex] or [tex]\(y = 0\)[/tex]):
- Intersection with the y-axis ([tex]\(x = 0\)[/tex]):
[tex]\[
2(0) - 6y = -3 \implies -6y = -3 \implies y = 0.5 \implies (0, 0.5)
\][/tex]
- Intersection with the x-axis ([tex]\(y = 0\)[/tex]):
[tex]\[
2x - 6(0) = -3 \implies 2x = -3 \implies x = -1.5 \quad (\text{not feasible since } x < 0 \text{ is not allowed})
\][/tex]
#### 2.3 Intersection between the two lines [tex]\(2x + 3y = 6\)[/tex] and [tex]\(2x - 6y = -3\)[/tex]:
Solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] simultaneously:
[tex]\[
\begin{cases}
2x + 3y = 6 \\
2x - 6y = -3
\end{cases}
\][/tex]
Subtract the second equation from the first:
[tex]\[
(2x + 3y) - (2x - 6y) = 6 - (-3) \implies 9y = 9 \implies y = 1
\][/tex]
Substitute [tex]\(y = 1\)[/tex] back into [tex]\(2x + 3y = 6\)[/tex]:
[tex]\[
2x + 3(1) = 6 \implies 2x + 3 = 6 \implies 2x = 3 \implies x = 1.5 \implies (1.5, 1)
\][/tex]
### Step 3: Summary of intersection points
The vertices we've found are:
[tex]\[
(0, 2), (3, 0), (0, 0.5), (1.5, 1)
\][/tex]
### Step 4: Determine feasible points
Since the feasible region must satisfy all given constraints, the included vertices will be:
[tex]\[
(0, 0), (0, 2), (1.5, 1), (3, 0), (0, 0.5)
\][/tex]
Reviewing the feasible points more closely, we find:
[tex]\[
\boxed{(0, 0), (0, 2), (1.5, 1), (3, 0)}
\][/tex]
Therefore, the vertices of the feasible region are:
[tex]\[
(0, 0), (0, 2), (1.5, 1), (3, 0)
\][/tex]
