To maximize \( z = x_1 - x_2 \), we need to understand the context of the problem and the constraints that might apply to \( x_1 \) and \( x_2 \). Without constraints, the function \( z = x_1 - x_2 \) can be unbounded, meaning we can make \( z \) as large as we want by choosing appropriate values for \( x_1 \) and \( x_2 \).

However, assuming there are no specific constraints given, let's consider two general approaches: 

1. **If there are no constraints**:
    - \( x_1 \) should be made as large as possible.
    - \( x_2 \) should be made as small as possible.

In this scenario, \( z \) would tend towards infinity as \( x_1 \) increases and \( x_2 \) decreases. 

2. **If there are constraints**:
    - Constraints typically limit the values \( x_1 \) and \( x_2 \) can take, such as \( 0 \leq x_1 \leq 10 \) and \( 0 \leq x_2 \leq 10 \).

### Example with constraints:
Suppose the constraints are:
\[ 0 \leq x_1 \leq 10 \]
\[ 0 \leq x_2 \leq 10 \]

In this case, to maximize \( z = x_1 - x_2 \):
- Set \( x_1 \) to its maximum value, which is 10.
- Set \( x_2 \) to its minimum value, which is 0.

Therefore,
\[ z = 10 - 0 = 10 \]

This is the maximum value of \( z \) under these constraints.

### General Solution Approach:
1. Identify any constraints on \( x_1 \) and \( x_2 \).
2. Maximize \( x_1 \) within its allowed range.
3. Minimize \( x_2 \) within its allowed range.

If specific constraints are provided, we can apply them to find the exact values for \( x_1 \) and \( x_2 \) that maximize \( z \).

Would you like to provide any constraints or additional information? Or do you need further details on any particular aspect?

Here are some related questions that might help expand your understanding:

1. What are the typical constraints in a linear programming problem?
2. How does changing the constraints affect the maximum value of \( z \)?
3. Can \( z = x_1 - x_2 \) have a maximum value if \( x_1 \) and \( x_2 \) are unrestricted?
4. What role do objective functions play in optimization problems?
5. How would adding a constraint like \( x_1 + x_2 \leq 15 \) affect the solution?
6. What methods can be used to solve optimization problems with constraints?
7. How does the graphical method work for solving two-variable linear programming problems?
8. Can \( x_1 \) and \( x_2 \) be negative in optimization problems?

**Tip:** In optimization problems, always clearly define the objective function and constraints before attempting to solve the problem.

Explore how to maximize z = x1 - x2 in linear programming by understanding constraints and applying optimization principles. Learn about objective functions and solving techniques for grades 11-12 students.

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