How will computers be programmed to control traffic flow and avoid congestion?
They will be required to solve systems continually based on the following premise: If traffic is to keep moving, during any period of time the number of cars entering an intersection must equal the number of cars leaving that intersection. Let’s see what this means by looking at the intersections of four one-way city streets.
Traffic Control
Figure 6.3shows the intersections of four one-way streets.As you study the figure,notice that 300 cars per hour want to enter intersection from the north on 27th Avenue.Also,
200 cars per hour want to head east from intersection on Palm Drive. The letters and stand for the number of cars passing between the intersections.
a. 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 linear system of equations involving and
b. Use Gaussian elimination to solve the system.
c. If construction on 27th Avenue limits to 50 cars per hour, how many cars per hour must pass between the other intersections to keep traffic flowing?
Solution
a. Set up the system by considering one intersection at a time, referring to Figure 6.3.
For Intersection Because cars enter and cars leave the intersection, then
For Intersection Because cars enter the intersection and cars leave then
For Intersection Figure 6.3 indicates that cars enter and leave, so
For Intersection With cars entering and cars exiting, traffic will keep flowing if y + z = 600.
I4: y + z 200 + 400 = 600 x + y x + y = 700.
I3: 300 + 400 = 700
I w + x = 1100. 2 200 + 900 = 1100 ,
I2: w + x w + z = 1000.
I1: 300 + 700 = 1000 I1 w + z z w, x, y, z. w, x, y, z
I2
I1
EXAMPLE 4 b x + 2y + 3z = 70 x + y + z = 60.
515z + 4, -3z + 2, z26. z, 15z + 4, -3z + 2, z2
 Solve problems involving systems without unique solutions. I1 I2
I4 I3 z x y w
300
Cars/hr
900
Cars/hr
400
Cars/hr
300
Cars/hr
200
Cars/hr
400
Cars/hr
700
Cars/hr
200
Cars/hr
Sunset Drive
Palm Drive
27th Ave
S 37th Ave
N
W E
Figure 6.3 The intersections of four one-way streets
ISBN 0-558-97848-7
College Algebra, Fifth Edition, by Robert Blitzer. Published by Prentice Hall. Copyright © 2010 by Pearson Education, Inc.Section 6.2 Inconsistent and Dependent Systems and Their Applications 573
The system of equations that models this situation is given by
b. To solve this system using Gaussian elimination, we begin with the augmented matrix. d w + z = 1000 w + x = 1100 x + y = 700 y + z = 600.
System of Linear Equations
(showing missing variables with 0 coefficients) Augmented Matrix d 1w + 0x + 0y + 1z = 1000
1w + 1x + 0y + 0z = 1100
0w + 1x + 1y + 0z = 700
0w + 0x + 1y + 1z = 600 D
1001
1100
0110
0011 4
1000
1100
700
600T
We can now use row operations to obtain the following matrix:
The last row of the matrix shows that the system in the voice balloons has dependent equations and infinitely many solutions. To write the solution set containing these infinitely many solutions, let equal any real number. Use the three equations in the voice balloons to express and in terms of
With arbitrary, the alphabetically ordered solution enables us to express the system’s solution set as
c. We are given that construction limits to 50 cars per hour. Because we substitute 50 for in the system’s ordered solution:
Use the system’s solution.
Thus, and (See Figure 6.4.) With construction on
27th Avenue, this means that to keep traffic flowing, 950 cars per hour must be routed between and 150 per hour between and and 550 per hour between and
Check Point 4 Figure 6.5 shows a system of four one-way streets.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 and
b. Use Gaussian elimination to solve the system.
c. If construction limits to 10 cars per minute, how many cars per minute must pass between the other intersections to keep traffic flowing?