Maximize $$ \left.\begin{array}{r}p=x-7 y ext { subject to } \ 2 x+y \geq 6 \ y \leq 5 \ x \geq 0, y \geq 0 \ p=-34.5\end{array} ight) $$

Question

Maximize $$ \left.\begin{array}{r}p=x-7 y 	ext { subject to } \ 2 x+y \geq 6 \ y \leq 5 \ x \geq 0, y \geq 0 \ p=-34.5\end{array}ight) $$

UpStudy AI · Accepted Answer

To maximize the objective function $$ p = x - 7y $$ subject to the constraints:

1. $$ 2x + y \geq 6 $$
2. $$ y \leq 5 $$
3. $$ x \geq 0 $$
4. $$ y \geq 0 $$

We will first rewrite the constraints in a more manageable form and then find the feasible region. After that, we will evaluate the objective function at the vertices of the feasible region to find the maximum value.

### Step 1: Rewrite the Constraints

1. From $$ 2x + y \geq 6 $$, we can express it as $$ y \geq 6 - 2x $$.
2. The second constraint $$ y \leq 5 $$ remains as is.
3. The non-negativity constraints $$ x \geq 0 $$ and $$ y \geq 0 $$ also remain.

### Step 2: Find the Feasible Region

Now we will find the intersection points of the lines defined by the constraints:

1. **Intersection of $$ y = 6 - 2x $$ and $$ y = 5 $$**:
   $$
   6 - 2x = 5 \implies 2x = 1 \implies x = 0.5
   $$
   So, the point is $$ (0.5, 5) $$.

2. **Intersection of $$ y = 6 - 2x $$ and $$ y = 0 $$**:
   $$
   6 - 2x = 0 \implies 2x = 6 \implies x = 3
   $$
   So, the point is $$ (3, 0) $$.

3. **Intersection of $$ y = 5 $$ and $$ y = 0 $$**:
   This does not provide a new point since $$ y = 0 $$ is the x-axis.

### Step 3: Evaluate the Objective Function at the Vertices

The vertices of the feasible region are:
1. $$ (0.5, 5) $$
2. $$ (3, 0) $$
3. $$ (0, 0) $$

Now we will evaluate $$ p = x - 7y $$ at these points:

1. At $$ (0.5, 5) $$:
   $$
   p = 0.5 - 7(5) = 0.5 - 35 = -34.5
   $$

2. At $$ (3, 0) $$:
   $$
   p = 3 - 7(0) = 3
   $$

3. At $$ (0, 0) $$:
   $$
   p = 0 - 7(0) = 0
   $$

### Step 4: Determine the Maximum Value

The maximum value of $$ p $$ occurs at the point $$ (3, 0) $$ where $$ p = 3 $$.

### Conclusion

The maximum value of $$ p = x - 7y $$ subject to the given constraints is $$ 3 $$ at the point $$ (3, 0) $$.
The maximum value of $$ p = x - 7y $$ is 3 when $$ x = 3 $$ and $$ y = 0 $$.

Maximize

Upstudy AI Solution

Answer

Solution

Step 1: Rewrite the Constraints

Step 2: Find the Feasible Region

Step 3: Evaluate the Objective Function at the Vertices

Step 4: Determine the Maximum Value

Conclusion

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