(x, y)

To solve this linear programming (LP) problem, let's analyze each constraint and determine the feasible region. Here’s a breakdown:

Problem Setup

Objective Function: $\text{Maximize } p = 3x + 2y$

Subject to:

1. $-5x + y \geq 5$
2. $x + 4y \leq 7$
3. $x \geq 0$
4. $y \geq 0$

Let's rewrite and interpret the inequalities for simplicity.

Step 1: Rewrite the Constraints

1. $-5x + y \geq 5$ is equivalent to $y \geq 5x + 5$.
2. $x + 4y \leq 7$ can be rearranged to $y \leq \frac{7 - x}{4}$.

Step 2: Graphical Analysis of Constraints

1. First Constraint: The line $y = 5x + 5$ has a slope of 5 and an intercept of 5. The region satisfying $y \geq 5x + 5$ is above this line.
2. Second Constraint: The line $y = \frac{7 - x}{4}$ has a slope of -1/4 and an intercept of $\frac{7}{4}$. The region satisfying $y \leq \frac{7 - x}{4}$ is below this line.
3. Non-Negativity Constraints: $x \geq 0$ and $y \geq 0$ restrict us to the first quadrant.

Step 3: Determine Feasibility

The line $y = 5x + 5$ starts at $y = 5$ when $x = 0$ and increases steeply. The line $y = \frac{7 - x}{4}$ only reaches a maximum $y$-value of $\frac{7}{4}$ when $x = 0$. Thus, the feasible region for these constraints is empty because the inequalities do not overlap in the first quadrant.

Conclusion

Since there is no feasible region that satisfies all constraints simultaneously:

$\text{Solution: } \text{EMPTY}$

Would you like more details or have questions on any part of this solution?

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Related Questions to Explore:

1. How do we find feasible regions in other linear programming problems?
2. What steps can we take to determine if a solution is unbounded?
3. Can changing coefficients of constraints affect feasibility?
4. How does graphical analysis help in understanding linear programming?
5. What are alternative methods to solve linear programming problems without graphs?

Tip: In LP problems, checking constraints graphically can provide quick insights into feasibility and boundedness.