An Application Involving Traffic Modeling

According to Blitzer (2009), checkpoint four "shows a system of four one-way streets" (p. 573). The numbers in the figure denote the number of cars per minute that travel in the direction shown.
a. Use the requirement that the number of cars entering each of the intersections per minute must equal the number of cars leaving per minute to set up a system of equations in w, x, y, and z.

For Intersection I1: Because 5+10=15 cars enter and w+z cars leave the intersection I1, then w+z=15.
For Intersection I2: Because w+x cars enter the intersection and 10+20=30 cars leave, then w+x=30.
For Intersection I3: Because 15+30=45 cars enter the intersection and x+y cars leave, then x+y=45.
For Intersection I4: Because y+z cars enter the intersection and 10+20=30 cars leave. traffic will keep flowing if y+z=30.
Arrange the equations where the variables are on one side and the constants are on the other side of the equal sign. w+z=15 w+x=30 x+y=45 y+z=30

System of equations: w+z=15 w+x=30 x+y=45 y+z=30

b. Use Gaussian elimination to solve the system.

System of equations: w+z=15 w+x=30 x+y=45 y+z=30

Put system of equations in matrix form.

w x y z
1 0 0 1 15
1 1 0 0 30
0 1 1 0 45
0 0 1 1 30

Solve the system of equations.
-1R1+R2
-1R2+R3
-1R3+R4

w x y z
1 0 0 1 15
0 1 0 -1 15
0 0 1 1 30
0 0 0 0 0

w+z=15 x-z=15 y+z=30

The last row of the matrix shows that the system in the four intersections has no unique solutions but has dependent equations and infinitely many solutions. To write the solution set containing these infinitely many solutions, let z equal any real number.

Express w, x, and y in terms of z. w=15-z x=15+z y=30-z
The solution set is {(15-z, 15+z, 30-z, z)}

c. If construction limits to 10 cars per minute, how many cars per minute must pass between the other intersections to keep traffic flowing?
The solution set is {(15-z, 15+z, 30-z, z)}
When z=10, then w, x, and y are the following: w=15-z =15-10 =5 x=15+z =15+10 =25 y=30-z =30-10 =20
The solution set is {(5, 25, 20, 10)}
Since the construction has limited z to 10 cars per minute, then to keep traffic flowing w=5, x=25, and y=20.

References

Blitzer, R. F. (2009). College Algebra. (5th ed.). Boston, MA: Pearson.