# Day 151: Traffic Flow Problems

As our final application of augmented matrices, we tackled the area of traffic flow solutions.  We kept it simple using one way streets and only 3-6 intersections.  We did discuss that these problems require computers to handle cities with 100s or 1000s of intersections, but we were using a simplified situation to understand the underlying constructs of the situation.  The simple diagram we used is:

Here was our introduction problem situation:

The figure shows the intersections of four one-way streets.  As you study the figure, notice that 300 cars per hour want to enter intersection I1 from the north on 27th Avenue.  Also, 200 cars per hour want to head east from intersection I2 on Palm Drive.  The letters w, x, y, and z stand for the number of cars passing between the intersections.

1. If the traffic is to keep moving, at each intersection the number of cars entering per hour must equal the number of cars leaving per hour. Use this idea to set up a system of linear equations involving w, x, y, and z.

2. Write the related augmented matrix. Use your calculator to solve the equation – but be sure to understand how to use row operations to solve as well.

3. If construction on 27th Avenue limits z to 50 cars per hour, how many cars per hour must pass between the other intersections to keep traffic flowing?

Once again, my students were fascinated and interested in another way matrices can be used in real situations.