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From the provided image depicting traffic flows in downtown Raleigh, NC, we can derive constraints and formulate a system of linear equations to model the traffic situation. Here’s the breakdown:

### Constraints and Explanations:
1. **Total Flow Conservation**: At each intersection, the total number of cars entering must equal the total number leaving (assuming no cars are permanently parked at the intersection).

2. **Specific Intersection Flows**:
   - At the intersection of Jones St. and Wilmington St., traffic flow from west to east (blue arrows, incoming) plus the northward flow must equal the outgoing eastward and southward flows.
   - Similar principles apply to the other intersections shown, adapted to their respective traffic directions and counts.

### System of Linear Equations:
- Let \( x \), \( u \), \( v \), and \( z \) represent the traffic flow in cars per day between intersections along Jones St., Wilmington St., Edenton St., and Salisbury St., respectively.
- \( y \) represents the total cars per day exiting the intersection of Jones St. and Wilmington St. to the east along Jones St.

#### Formulating the Equations:
1. **Intersection at Wilmington St. and Jones St.**:
   - Incoming: \( 1200 \) (from west) + \( x \) (from south)
   - Outgoing: \( y \) (eastward) + \( u \) (northward)
   - Equation: \( 1200 + x = y + u \)

2. **Intersection at Edenton St. and Jones St.**:
   - Incoming: \( y \) (from west) + \( 2300 \) (from south)
   - Outgoing: \( 5100 \) (eastward) + \( v \) (northward)
   - Equation: \( y + 2300 = 5100 + v \)

3. **Intersection at Edenton St. and Salisbury St.**:
   - Incoming: \( 5100 \) (from west) + \( 3800 \) (from south)
   - Outgoing: \( z \) (eastward) + \( v \) (northward)
   - Equation: \( 5100 + 3800 = z + v \)

4. **Intersection at Wilmington St. and Salisbury St.**:
   - Incoming: \( z \) (from west) + \( 4100 \) (from south)
   - Outgoing: \( 6600 \) (eastward) + \( u \) (northward)
   - Equation: \( z + 4100 = 6600 + u \)

This system of equations helps model the traffic flows between intersections, which can be used to optimize traffic control measures. If you have any questions or need further explanations on this, feel free to ask! Here are some follow-up points you might consider:

1. How can we solve this system of equations for unknowns \( x, y, u, v, z \)?
2. What are the implications of having more cars entering an intersection than exiting?
3. Could additional data on traffic light timings or pedestrian crossings refine this model?
4. How might these traffic patterns change during peak and off-peak hours?
5. What strategies could be implemented to reduce congestion based on this model?

**Tip:** When dealing with systems of linear equations in traffic flow analysis, consider using matrix methods or computer software for more efficient computation, especially with larger networks.

Solve part a of the problem involving traffic flow at four intersections in downtown Raleigh, NC. Define x, u, v, and z as the traffic flow in cars per day between intersections along specific streets. Use the given traffic flow data to create a system of linear equations.

Learn how to model traffic flow using a system of linear equations based on real-world data from four intersections in downtown Raleigh, NC. Understand key concepts like flow conservation and solve for unknowns in traffic patterns. Ideal for Grades 9-12 students studying linear algebra.

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