The optimal solution for the linear programming problem is as follows:

### Optimal Values of the Variables:
- \( x_1 = 0 \)
- \( x_2 = 100 \)
- \( x_3 = 230 \)

### Maximum Value of the Objective Function:
- \( w_{\text{max}} = 1350 \)

This means that by setting \( x_1 = 0 \), \( x_2 = 100 \), and \( x_3 = 230 \), you achieve the maximum possible value of \( w = 1350 \), while satisfying all the given constraints.

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**Relative Questions:**
1. How would the solution change if the coefficients in the objective function were modified?
2. What happens if one of the constraints is altered or removed?
3. How would you interpret the shadow prices (dual values) in this context?
4. What is the significance of the slack variables in the solution?
5. Can this problem be solved graphically, and how would the solution compare?

**Tip:** When solving linear programming problems, always ensure that your constraints are properly defined, as they can significantly impact the optimal solution.

Discover the optimal solution for a linear programming problem where x1 = 0, x2 = 100, and x3 = 230, achieving a maximum objective function value of 1350. Explore the implications of modifying coefficients and constraints, and understand the significance of shadow prices and slack variables. Suitable for advanced college-level students and professionals.

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