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This is a problem involving a directed and weighted graph. The goal is to determine the value of \( y_2 \).

### Key observations from the graph:
- The graph has four vertices: \( P \), \( Q \), \( R \), and \( S \).
- The edge weights and directions are indicated:
  - \( P \to S \) has a weight of 15.
  - \( Q \to R \) has a weight of 25.
  - There are unknown weights \( y_1 \) and \( y_2 \) associated with the other edges:
    - \( S \to Q \) has a weight \( y_2 \).
    - \( S \to R \) has a weight \( y_1 \).

### Step-by-step solution:
We likely need to apply some graph theory principles such as balance of incoming and outgoing weights at certain nodes (vertices) in the directed graph.

1. **Identify the vertex balances**:
   - Consider the balance at vertex \( S \). The total incoming weight to vertex \( S \) is 15 (from vertex \( P \)).
   - The outgoing weights from vertex \( S \) are \( y_1 \) and \( y_2 \).
   
2. **Assume an equation for vertex \( S \)**:
   Since all weights are balanced, the incoming weight should equal the sum of the outgoing weights:
   \[
   15 = y_1 + y_2
   \]
   
3. **Solving for \( y_2 \)**:
   To solve for \( y_2 \), we would need additional information to determine the value of \( y_1 \). However, if no more information is provided, the equation will be:
   \[
   y_2 = 15 - y_1
   \]

Without the value of \( y_1 \), the best we can say is that \( y_2 = 15 - y_1 \). 

Would you like to provide additional information for \( y_1 \), or should I proceed assuming any typical value or constraint for \( y_1 \)? 

Let me know if you have any further questions! Here are five related questions to expand on the topic:

1. How can we calculate the value of \( y_1 \) based on the graph?
2. What happens to the graph’s balance if additional weights are introduced?
3. How can this method be applied to more complex directed graphs?
4. What are other ways of solving for unknown weights in a directed graph?
5. How would the problem change if the graph was undirected?

**Tip**: In directed graphs, the sum of incoming and outgoing edge weights at any vertex can help balance and determine unknown values.

Determine the value of y2 from the directed and weighted graph shown in Diagram 10.

Learn how to solve for the unknown weight y2 in a directed and weighted graph. The problem involves balancing incoming and outgoing weights at a vertex and applying graph theory principles. Suitable for students in Grades 10-12.

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