To solve the given minimization problem, we have the following linear programming formulation:

Objective function:

$\min Z = 5x + 7y$

Subject to the constraints:

1. $x + y = 10$
2. $x \geq 2$
3. $y \geq 3$
4. $x \geq 0$
5. $y \geq 0$

Let's solve this problem step by step.

Step 1: Express the constraints

From the first constraint: $x + y = 10 \quad \Rightarrow \quad y = 10 - x$

Step 2: Consider the boundaries of the other constraints:

• From $x \geq 2$, this implies that $x$ must be at least 2.
• From $y \geq 3$, and knowing $y = 10 - x$, we can substitute to get: $10 - x \geq 3 \quad \Rightarrow \quad x \leq 7$ Thus, $x$ must be between 2 and 7: $2 \leq x \leq 7$

Step 3: Minimize the objective function

Now, substitute $y = 10 - x$ into the objective function: $Z = 5x + 7y = 5x + 7(10 - x) = 5x + 70 - 7x = -2x + 70$

To minimize $Z$, we need to minimize $-2x + 70$, which is a linear function decreasing as $x$ increases. Therefore, the minimum value occurs when $x$ is at its maximum value, which is $x = 7$.

Step 4: Find the corresponding value of $y$

When $x = 7$, using the equation $x + y = 10$: $y = 10 - 7 = 3$

Step 5: Calculate the minimum value of $Z$

Substitute $x = 7$ and $y = 3$ into the objective function: $Z = 5(7) + 7(3) = 35 + 21 = 56$

Final Answer:

The minimum value of $Z$ is 56, occurring at $x = 7$ and $y = 3$.

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Would you like more details or have any questions?

Here are 5 questions to expand on the problem:

1. What happens if the objective function was maximized instead of minimized?
2. How would additional constraints affect the solution?
3. How can graphical methods be used to solve this problem?
4. What would change if the coefficients of $x$ and $y$ in the objective function were different?
5. Can this problem be solved using software tools like Excel or Python?

Tip: In linear programming, the optimal solution often lies at the boundaries of the feasible region.