A traffic signal system can be modeled using a system of linear equations when trying to determine the timings or operations of different signals at an intersection. For simplicity, let's say we have a traffic light at an intersection with two main directions (for example, North-South and East-West). The system of linear equations will represent the duration or sequencing of traffic light changes.

Let’s assume the following setup for the traffic signals:

• Red light for North-South (R_ns)
• Green light for North-South (G_ns)
• Red light for East-West (R_ew)
• Green light for East-West (G_ew)

The total time for the traffic signal cycle might be split into these variables. Let’s suppose the cycle has a total duration of $T$ seconds, and we are concerned with how the signals change.

Example System of Equations

Consider the following constraints:

1. The total cycle time $T$ is made up of the times spent in different states (Green and Red) for both directions.
2. We have:
   • $R_ns + G_ns = t_1$ (time for North-South direction signals)
   • $R_ew + G_ew = t_2$ (time for East-West direction signals)
3. The green lights for North-South and East-West must alternate in some way (no overlap), meaning:
   • $G_ns + G_ew = T$ (total green light time for both directions equals total cycle time)

Now, let’s set up the system of linear equations:

R_ns + G_ns &= t_1 \\ R_ew + G_ew &= t_2 \\ G_ns + G_ew &= T \end{aligned}$$ ### Solving the System Let's solve this system: 1. From the third equation, solve for $$G_ew$$: $$G_ew = T - G_ns$$ 2. Substitute $$G_ew = T - G_ns$$ into the second equation: $$R_ew + (T - G_ns) = t_2$$ Simplify this: $$R_ew + T - G_ns = t_2$$ $$R_ew = t_2 - T + G_ns$$ 3. Now, substitute $$R_ew$$ into the first equation: $$R_ns + G_ns = t_1$$ Since $$R_ns$$ will be $$t_1 - G_ns$$, we can now write the system completely in terms of $$G_ns$$ and $$T$$. This way, we solve for the green light durations based on $$T$$, the total time, and the given total red light durations. ### Example Problem Suppose we are given: - The total cycle time $$T = 120$$ seconds. - The North-South red light time $$t_1 = 50$$ seconds. - The East-West red light time $$t_2 = 40$$ seconds. Now we can apply the system to solve for the green light durations. --- Would you like more details on the setup, or is there a particular aspect you'd like to dive deeper into? Feel free to ask! Here are some follow-up questions for you: 1. How would you adapt this model if we had a left-turn signal involved? 2. What happens if the traffic signal cycles are not synchronized between different intersections? 3. How can you model variable cycle times depending on traffic demand? 4. How do you account for pedestrian crossing signals in this system of equations? 5. Can this system be used to model adaptive traffic signal systems? **Tip:** When modeling real-world systems with linear equations, it's crucial to consider all variables and constraints, including time limits, overlap, and sequence of operations.