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To solve this problem, we need to find the region that satisfies the system of inequalities and identify the unknown corner point.

### Given System of Inequalities:
1. \( x + 3y \leq 15 \)
2. \( 2x + y \leq 10 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Steps to Solve:

1. **Graph the inequalities:**
   - Plot the lines \( x + 3y = 15 \) and \( 2x + y = 10 \). Then identify the regions that satisfy the inequalities.
   - The conditions \( x \geq 0 \) and \( y \geq 0 \) indicate the solution must lie in the first quadrant.

2. **Identify the region where all conditions are satisfied:**
   - The feasible region is the overlapping area that satisfies all inequalities.

3. **Determine the corner points:**
   - The intersections of the boundary lines will give us the corner points.
   - The given graph shows corner points at \( (0,0) \), \( (0,5) \), \( (5,0) \), and an unknown point.

4. **Find the unknown corner point:**
   - This point will be the intersection of \( x + 3y = 15 \) and \( 2x + y = 10 \).

### Solving for the Unknown Corner Point:

To find the intersection of \( x + 3y = 15 \) and \( 2x + y = 10 \), solve these equations simultaneously:

1. Solve \( 2x + y = 10 \) for \( y \):
   \[
   y = 10 - 2x
   \]

2. Substitute \( y = 10 - 2x \) into \( x + 3y = 15 \):
   \[
   x + 3(10 - 2x) = 15
   \]
   \[
   x + 30 - 6x = 15
   \]
   \[
   -5x = -15
   \]
   \[
   x = 3
   \]

3. Substitute \( x = 3 \) back into \( y = 10 - 2x \):
   \[
   y = 10 - 2(3) = 4
   \]

Therefore, the unknown corner point is \( (3,4) \).

### Matching the Region:
The feasible region that satisfies all the inequalities is Region II in the graph.

### Final Answers:
- The solution region is **Region II**.
- The coordinates of the unknown corner point are \( (3,4) \).

Would you like a more detailed explanation or have any further questions?

Here are five related questions you could explore:
1. How would the solution change if one of the inequalities were reversed?
2. What is the importance of the conditions \( x \geq 0 \) and \( y \geq 0 \) in this problem?
3. How can we verify the solution region using a test point?
4. What methods can be used to solve systems of inequalities graphically?
5. How can this system of inequalities be solved using matrix operations?

**Tip:** When graphing inequalities, always shade the region that satisfies the inequality, and clearly mark any intersection points for clarity.

Match the solution region of the following system of linear inequalities with one of the four regions shown in the figure. Identify the unknown corner point of the solution region.

Learn how to solve a system of linear inequalities by graphing and finding the feasible region. This problem covers graphing lines, identifying regions, and finding the intersection points of lines. Suitable for students in Grades 9-12.

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