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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.

Find the minimum and maximum values of the objective function f(x, y) = 7x + 9y subject to the constraints: x ≥ 0, y ≥ 0, and x + y ≤ 5.

Explore how to find the minimum and maximum values of the linear function f(x, y) = 7x + 9y subject to constraints using linear programming. This problem involves identifying the feasible region, evaluating corner points, and understanding optimization concepts. Suitable for high school students in Grades 10-12.

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