Question

Graph the system of constraints and find the values of x and y that maximize P for the objective function P = 3x + 2y

x+y ≤ 6

2x+y ≤ 8

x ≥ 0

Y ≥ 0

What are the values of x and y that maximize P?

(1 point)
Responses

x=0, y=6
x=0, y=6

x=4, y=2
x=4, y=2

x=2, y=4
x=2, y=4

x=3, y=3
x=3, y=3
Question 2
What is the maximum profit?(1 point)
Responses

$18
$18

$6
$6

$12
$12

$14

Answer 1

To solve the problem, we need to graph the system of constraints and find the feasible region to maximize the objective function \( P = 3x + 2y \).

**Step 1: Graph the Constraints**

1. \( x + y \leq 6 \)
This can be rearranged to \( y = 6 - x \). The line intersects the axes at (6, 0) and (0, 6).

2. \( 2x + y \leq 8 \)
This can be rearranged to \( y = 8 - 2x \). The line intersects the axes at (4, 0) and (0, 8).

3. \( x \geq 0 \)
This is the right half of the graph (all points where \( x \) is non-negative).

4. \( y \geq 0 \)
This is the upper half of the graph (all points where \( y \) is non-negative).

Now we will plot these equations on a graph.

**Step 2: Find the Feasible Region**

The feasible region is the area that satisfies all of the constraints. It will be bounded by the lines from the constraints mentioned above.

**Step 3: Identify Vertices of the Feasible Region**

1. **Intersection of \( x + y = 6 \) and \( 2x + y = 8 \)**:
- Substitute \( y = 6 - x \) into \( 2x + y = 8 \):
- \( 2x + (6 - x) = 8 \) ⟹ \( 2x - x + 6 = 8 \) ⟹ \( x = 2 \)
- Then \( y = 6 - 2 = 4 \) ⟹ Vertex: (2, 4)

2. **Intersection of \( x + y = 6 \) and \( y = 0 \)**:
- Set \( y = 0 \) in \( x + y = 6 \) ⟹ \( x = 6 \) ⟹ Vertex: (6, 0)

3. **Intersection of \( 2x + y = 8 \) and \( y = 0 \)**:
- Set \( y = 0 \) in \( 2x + y = 8 \) ⟹ \( 2x = 8 \) ⟹ \( x = 4 \) ⟹ Vertex: (4, 0)

4. **Intersection of \( x = 0 \) and \( y = 6 \)**:
- This gives the vertex (0, 6).

5. **Intersection of \( x = 0 \) and \( 2x + y = 8 \)**:
- Set \( x = 0 \) ⟹ \( y = 8 \), but since \( y \) must be ≤ 6 from the first constraint, we disregard this point as it's outside the feasible region.

**Step 4: Calculate the Objective Function at Each Vertex**

Now we check the value of \( P = 3x + 2y \) at the feasible vertices:

- At (2, 4):
\( P = 3(2) + 2(4) = 6 + 8 = 14 \)

- At (6, 0):
\( P = 3(6) + 2(0) = 18 + 0 = 18 \)

- At (4, 0):
\( P = 3(4) + 2(0) = 12 + 0 = 12 \)

- At (0, 6):
\( P = 3(0) + 2(6) = 0 + 12 = 12 \)

**Step 5: Determine the Maximum Value of P**

The maximum value of \( P = 18 \) occurs at the vertex (6, 0).

**Final Answers:**

1. The values of \( x \) and \( y \) that maximize \( P \) are:
- \( x = 6, y = 0 \)

2. The maximum profit is:
- \( \$18 \)