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.