Exercise 2
Shown below are some traffic patterns in the downtown area of a large city. The figures give the number of cars per hour traveling along each road. Any car that drives into an intersection must also leave the intersection. This means that the number of cars entering an intersection in an hour is equal to the number of cars leaving the intersection.

Let's begin with the following traffic pattern.
How many cars per hour enter the upper left intersection? How many cars per hour leave this intersection? Use this to form a linear equation in the variables \(x\text{,}\) \(y\text{,}\) \(z\text{,}\) and \(w\text{.}\)
Form three more linear equations from the other three intersections to form a linear system having four equations in four unknowns. Then use Sage to find the solution space to this system.
Is there exactly one solution or infinitely many solutions? Explain why you would expect this given the information provided.

Another traffic pattern is shown below.
Once again, write a system of equations for the quantities \(x\text{,}\) \(y\text{,}\) \(z\text{,}\) and \(w\) and solve the system using the Sage cell below.
What can you say about the solution of this linear system? Is there exactly one solution or infinitely many solutions? Explain why you would expect this given the information provided.
What is the smallest amount of traffic flowing through \(x\text{?}\)