Map of downtown grid with one-way streets, Court House Square, and flows x, y, z, w
Figure 1
Diagram (a): a map of a downtown street grid forming a square block labeled Court House Square. Two vertical streets, North S. on the left and South S. on the right, and two horizontal streets, East S. on the top and West S. on the bottom, bound the square. An arrow at the top of North S. points upward with the number 30 leaving; an arrow at the top of South S. is labeled 20 entering downward. On the left, East S. has an arrow pointing left labeled 20 leaving. The top side of the square between the two vertical streets is labeled x. The left side of the square is labeled y and the right side is labeled w. The bottom side of the square is labeled z. At the lower left, 50 enters along West S. flowing right, and at the lower right an arrow points right labeled 20 leaving. At the bottom, North S. is labeled 30 and South S. has a downward arrow labeled 30. The panel is captioned (a).
Monument Circle traffic diagram with flows x, y, z, w, v and entering and leaving car counts
Figure 2
Diagram (b): a circle labeled Monument Circle with five roads meeting it. On the left, an arrow labeled 90 points into the circle horizontally. Upper left, an arrow labeled 40 points outward away from the circle along a road whose arc segment near it is labeled x. At the top right, an arrow labeled 60 points outward, with the arc segment labeled v at the top of the circle. On the right, the arc segment is labeled w and an arrow from the lower right labeled 30 points into the circle. At the lower left, an arrow labeled 20 points outward, and the arc segment on the lower left of the circle is labeled y, while the arc at the bottom is labeled z. The panel is captioned (b).

FIGURE 1.24 Two traffic patterns.

The final output to you should be a simple Yes or No. You do not want to see matrices!

  1. One of your assistant engineers comments that it would be easier for the technician in part (b) to use MATLAB's rank command rather than rref. What does your assistant have in mind?

1.3.2 Applications to Traffic Flow

An interesting context in which linearly dependent systems of equations can arise is in the study of traffic flow. Figure 1.24a is a map of the downtown area of a city. Each street is one-way in its respective direction. The numbers represent the average number of cars per minute that enter or leave a given street at 3:30 p.m. The variables also represent average numbers of cars per minute. Of course, barring accidents, the total number of cars entering any intersection must equal the total number leaving. Thus, from the intersection of East and North, we see that \( x + y = 50 \). Continuing counterclockwise around the square, we get the system

\[ \begin{array}{rcrcrcrcl} x &+& y & & & & &=& 50 \\ & & y &+& z & & &=& 80 \\ & & & & z &+& w &=& 50 \\ x & & & & &+& w &=& 20 \end{array} \qquad (1.37) \]

Notice also that in Figure 1.24, the total number of cars entering the street system per minute is \( 20 + 30 + 50 = 100 \) while the total number leaving is \( 20 + 30 + 20 + 30 = 100 \). All of our examples share the property that the total number of cars per minute that enter the street system per minute equals the total number that leave the street system per minute. We assume that the total number of cars per minute on the street system at any given time equals the total number of cars entering the street system per minute which equals the total number of cars leaving the street system per minute. Thus we augment system 1.37 with the additional equation

\[ x + y + z + w = 100 \]
Street grid map with one-way streets, flow variables s, t, x, y, z, u, v and entering and leaving car counts
Figure 3
A schematic map of a rectangular street grid used for a traffic-flow exercise. Three vertical streets run from left to right across the figure: First St., Second St., and Third St. Two horizontal streets cross them: West St. along the top and East St. along the bottom. Arrows indicate one-way directions and numbers give cars per minute entering or leaving the system. At the top, 200 enters downward on First St., 50 enters upward on Second St. (arrow pointing up, leaving), and 150 enters downward on Third St. On the top horizontal street (West St.), 300 leaves at the left (arrow pointing left) and 200 leaves at the right. The top horizontal segment between First and Second Streets is labeled s, and between Second and Third Streets is labeled t. The vertical segments between the two horizontal streets are labeled x on First St., y on Second St., and z on Third St. On the bottom horizontal street (East St.), 400 enters at the left and 100 leaves at the right (arrow pointing right). The bottom horizontal segment between First and Second Streets is labeled u, and between Second and Third Streets is labeled v. Below the bottom street, 450 leaves downward on First St., 300 leaves downward on Second St., and 350 leaves downward on Third St.

FIGURE 1.25 Exercise 3.

In this case, this equation is dependent on the other equations; it equals half their sum. In other cases, however, it can add additional information.

In the exercises that follow, you will analyze this system as well as the system that describes figure (b). The type of analysis done in these exercises can be applied in any context where some quantity is flowing through prescribed channels. In circuit theory, the statement that the amount of current entering a node equals the amount leaving is called Kirchhoff's current law.

Self-Study Questions

  1. 1.7 ✓In Figure 1.25 write the linear equation that describes the traffic flow at the intersection of West and Third Streets.
  2. 1.8 ✓In Figure 1.25, write the linear equation that describes the traffic flow at the intersection of East and Second Streets.
  3. 1.9 ✓In Figure 1.24b, write the linear equation that describes the traffic flow at the intersection of Monument Circle with the street labeled 90.

EXERCISES

  1. 1.90 Find all solutions to the system that describes Figure 1.24a. Use \( w \) as your free variable. Then answer the following:
    1. ✓Suppose that over the month of December the traffic on South Street in front of the courthouse at 3:30 p.m. ranged from six to eight cars per minute. Determine which of the streets on Courthouse Square had the greatest volume of traffic. What were the maximum and minimum levels of traffic flow on this street?
    2. Suppose that it is observed that in June the traffic flow past the courthouse is heaviest on West Street. Prove that \( 0 \le w < 10 \).
  1. Sleezie's Construction wants to close down West Street for six months. The City Council refused to grant the permit. Why? Explain on the basis of the solution to the system of Figure 1.24a.
  1. 1.91 Find all solutions to the system that describes Figure 1.24b. Use \( v \) as your free variable.
  2. 1.92 Figure 1.25 shows the traffic flow in a town at 3:30 p.m.
    1. Find the system of equations that describes the traffic flow?
    2. The section of Third Street between East and West is under construction. Assuming that as few cars as possible use this block, what are the possible ranges of traffic flow on each of the other blocks? [Hint: Note that none of the variables can take on negative values.]