This problem involves finding the minimum and maximum values of a linear objective function subject to a set of constraints using linear programming.

Given Problem:

• Objective Function: $f(x, y) = 7x + 9y$
• Constraints:
  • $x \geq 0$
  • $y \geq 0$
  • $x + y \leq 5$

Solution Approach:

1. Plot the Constraints: The feasible region is determined by the intersection of the constraints:

   • $x \geq 0$ and $y \geq 0$ indicate that the solution is in the first quadrant.
   • $x + y \leq 5$ is a line with intercepts $(5, 0)$ and $(0, 5)$.

2. Find Feasible Region: The feasible region is a triangle bounded by:

   • Points $(0, 0)$, $(5, 0)$, and $(0, 5)$.

3. Evaluate the Objective Function at Vertices: Evaluate $f(x, y) = 7x + 9y$ at each vertex:

   • At $(0, 0)$: $f(0, 0) = 7(0) + 9(0) = 0$
   • At $(5, 0)$: $f(5, 0) = 7(5) + 9(0) = 35$
   • At $(0, 5)$: $f(0, 5) = 7(0) + 9(5) = 45$

4. Determine the Minimum and Maximum Values:

   • Minimum value: $f(0, 0) = 0$ at $(0, 0)$.
   • Maximum value: $f(0, 5) = 45$ at $(0, 5)$.

Final Answer:

• Minimum value: 0 at $(0, 0)$.
• Maximum value: 45 at $(0, 5)$.

Would you like more details or have any questions about this problem?

Related Questions:

1. How do you graph linear constraints to find a feasible region?
2. What is the role of vertices in linear programming problems?
3. How can we determine if a linear programming problem is unbounded?
4. How do you solve a linear programming problem with more complex constraints?
5. What are the real-world applications of linear programming in optimization problems?

Tip:

In linear programming, the optimal solution for a bounded feasible region always occurs at one of the vertices (corner points) of the feasible region.