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To solve this linear programming problem, we aim to maximize \( Z = 2x_1 + x_2 \) under the given constraints.

### Problem Statement
Objective function:
\[
\text{Maximize } Z = 2x_1 + x_2
\]

Subject to the constraints:
\[
-x_1 + x_2 \leq 1
\]
\[
x_1 - 2x_2 \leq 2
\]
\[
x_1, x_2 \geq 0
\]

### Step-by-Step Solution
1. **Convert inequalities into boundary lines:**
   - For \( -x_1 + x_2 = 1 \):
     \[
     x_2 = x_1 + 1
     \]
   - For \( x_1 - 2x_2 = 2 \):
     \[
     x_2 = \frac{x_1 - 2}{2} = 0.5x_1 - 1
     \]

2. **Identify the feasible region:**
   - The inequalities define a feasible region in the first quadrant (\( x_1, x_2 \geq 0 \)) where the lines \( x_2 = x_1 + 1 \) and \( x_2 = 0.5x_1 - 1 \) intersect or bound the region.

3. **Find the points of intersection:**
   - Set \( x_1 + 1 = 0.5x_1 - 1 \) to find the intersection of the two lines:
     \[
     x_1 - 0.5x_1 = -2
     \]
     \[
     0.5x_1 = 2
     \]
     \[
     x_1 = 4
     \]
   - Substitute \( x_1 = 4 \) into \( x_2 = x_1 + 1 \):
     \[
     x_2 = 4 + 1 = 5
     \]
   - Thus, the point of intersection is \( (4, 5) \).

4. **Check the vertices of the feasible region:**
   - The vertices of the feasible region (where the constraints are satisfied) include:
     - Intersection with \( x_1 = 0 \):
       - \( x_2 = 0 + 1 = 1 \) from \( x_2 = x_1 + 1 \), so we have \( (0, 1) \).
     - Intersection with \( x_2 = 0 \):
       - \( x_1 = 2 \) from \( x_1 - 2x_2 = 2 \), so we have \( (2, 0) \).
     - The point \( (4, 5) \) where the two lines intersect.

5. **Calculate \( Z = 2x_1 + x_2 \) at each vertex:**
   - At \( (0, 1) \):
     \[
     Z = 2(0) + 1 = 1
     \]
   - At \( (2, 0) \):
     \[
     Z = 2(2) + 0 = 4
     \]
   - At \( (4, 5) \):
     \[
     Z = 2(4) + 5 = 8 + 5 = 13
     \]

6. **Conclusion:**
   The maximum value of \( Z = 2x_1 + x_2 \) is achieved at \( (4, 5) \) with \( Z = 13 \).

### Final Answer
\[
\text{Maximum } Z = 13 \text{ at } (x_1, x_2) = (4, 5)
\]

Would you like more details on any specific part, or do you have further questions?

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**Additional Questions:**
1. What would change if we were to minimize \( Z = 2x_1 + x_2 \) instead?
2. How would the solution be affected if the inequality \( -x_1 + x_2 \leq 1 \) was replaced by \( -x_1 + x_2 \geq 1 \)?
3. Can you visualize the feasible region on a graph to confirm the vertices?
4. How does the choice of objective function coefficients (e.g., \( 2x_1 + x_2 \)) influence the maximum point?
5. What happens if we add an additional constraint, like \( x_1 + x_2 \leq 6 \)?

**Tip:** Graphical visualization of constraints can greatly help in confirming the feasible region and solution in linear programming.

Maximize Z = 2x1 + x2 under the constraints -x1 + x2 ≤ 1, x1 - 2x2 ≤ 2, x1, x2 ≥ 0.

Solve a linear programming problem by maximizing Z = 2x1 + x2 with constraints -x1 + x2 ≤ 1, x1 - 2x2 ≤ 2, and non-negative variables x1, x2. Detailed solution includes finding intersection points and calculating Z at vertices. Suitable for Grade 11-12 students.

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