Solve the given LP problem. If no optimal solution exists, indicate whether the feas
(Enter EMPTY if the region is empty. Enter UNBOUNDED if the function is unbounde
Maximize subject to

Ask by Paul Flynn. in the United States

Mar 20,2025

Question

Solve the given LP problem. If no optimal solution exists, indicate whether the feas
(Enter EMPTY if the region is empty. Enter UNBOUNDED if the function is unbounde
Maximize subject to

Ask by Paul Flynn. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

To solve the given linear programming (LP) problem, we need to follow these steps:

1. **Identify the objective function and constraints:**
   - Objective function: Maximize $$ p = x - 7y $$
   - Constraints:
     1. $$ 2x + y \geq 6 $$
     2. $$ x \geq 0 $$
     3. $$ y \geq 0 $$

2. **Graph the constraints:**
   - We will convert the inequality $$ 2x + y \geq 6 $$ into an equation to find the boundary line: $$ 2x + y = 6 $$.
   - The intercepts can be found by setting $$ x = 0 $$ and $$ y = 0 $$:
     - If $$ x = 0 $$: $$ y = 6 $$ (point (0, 6))
     - If $$ y = 0 $$: $$ 2x = 6 $$ → $$ x = 3 $$ (point (3, 0))

3. **Determine the feasible region:**
   - The feasible region is where the constraints overlap, which is above the line $$ 2x + y = 6 $$ in the first quadrant (since $$ x \geq 0 $$ and $$ y \geq 0 $$).

4. **Evaluate the objective function at the vertices of the feasible region:**
   - The vertices of the feasible region are (0, 6) and (3, 0).
   - We will evaluate $$ p = x - 7y $$ at these points:
     - At (0, 6): $$ p = 0 - 7(6) = -42 $$
     - At (3, 0): $$ p = 3 - 7(0) = 3 $$

5. **Determine the optimal solution:**
   - The maximum value of $$ p $$ occurs at the point (3, 0) with $$ p = 3 $$.

6. **Check for boundedness:**
   - Since the feasible region is bounded by the constraints and the objective function has a maximum value at a vertex, we conclude that there is an optimal solution.

Thus, the optimal solution exists, and the maximum value of $$ p $$ is $$ 3 $$ at the point $$ (3, 0) $$.
The optimal solution is $$ p = 3 $$ at the point $$ (3, 0) $$.

Solve the given LP problem. If no optimal solution exists, indicate whether the feas
(Enter EMPTY if the region is empty. Enter UNBOUNDED if the function is unbounde
Maximize subject to

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Solve the given LP problem. If no optimal solution exists, indicate whether the feas (Enter EMPTY if the region is empty. Enter UNBOUNDED if the function is unbounde Maximize subject to

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Solve the given LP problem. If no optimal solution exists, indicate whether the feas
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