To solve this modified linear programming problem, we incorporate the condition that $x$ and $y$ can only take integer values (since values between 1 and 2 are disallowed, and fractional values of $x$ and $y$ are not allowed). This transforms the problem into an integer linear programming (ILP) problem.

The problem can be written as:

Objective function: Maximize $3x + 5y$

Constraints:

1. $x + 2y \leq 5$
2. $2x + 2y \leq 7$
3. $4x + 3y \leq 12$
4. $x, y \geq 0$
5. $x$ and $y$ must be integers.

We will now calculate the optimal values of $x$ and $y$ using integer programming methods.

Step 1: Determine feasible integer solutions

We systematically evaluate feasible integer values of $x$ and $y$ by satisfying the constraints:

• From $x + 2y \leq 5$, $x \leq 5 - 2y$.
• From $2x + 2y \leq 7$, $x + y \leq 3.5$, or $x + y \leq 3$ for integer solutions.
• From $4x + 3y \leq 12$, $x \leq 3 - 0.75y$.

Feasible pairs for $x$ and $y$:

We test values systematically:

1. $y = 0$:
   • $x \leq 5$, $x \leq 3.5$, $x \leq 3$ → $x = 0, 1, 2, 3$.
2. $y = 1$:
   • $x \leq 3$, $x \leq 2.5$, $x \leq 2.25$ → $x = 0, 1, 2$.
3. $y = 2$:
   • $x \leq 1$, $x \leq 1.5$, $x \leq 1.75$ → $x = 0, 1$.
4. $y = 3$: Not feasible, as it violates constraints.

Step 2: Evaluate the objective function

We evaluate $3x + 5y$ for all feasible integer pairs:

• $(x, y) = (0, 0)$: $3(0) + 5(0) = 0$
• $(x, y) = (1, 0)$: $3(1) + 5(0) = 3$
• $(x, y) = (2, 0)$: $3(2) + 5(0) = 6$
• $(x, y) = (3, 0)$: $3(3) + 5(0) = 9$
• $(x, y) = (0, 1)$: $3(0) + 5(1) = 5$
• $(x, y) = (1, 1)$: $3(1) + 5(1) = 8$
• $(x, y) = (2, 1)$: $3(2) + 5(1) = 11$
• $(x, y) = (0, 2)$: $3(0) + 5(2) = 10$
• $(x, y) = (1, 2)$: $3(1) + 5(2) = 13$.

Step 3: Identify the optimal solution

The maximum value of the objective function is 13, achieved at $(x, y) = (1, 2)$.

Final Answer:

• Objective function value: 13
• $x$: 1
• $y$: 2

Would you like further clarification or a step-by-step breakdown of any part of this process?

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Expanding Questions:

1. What are the differences between linear programming and integer programming?
2. How would the solution change if $x$ and $y$ were allowed to take fractional values?
3. What are practical examples of problems that use integer programming?
4. Can you explain how the feasible region is constructed graphically for this problem?
5. What optimization methods can be used for more complex integer programming problems?

Tip: Always ensure the feasibility of solutions by checking all constraints when solving optimization problems.