The accompanying figure shows a network of one-way streets with traffic flowing

Question

The accompanying figure shows a network of one-way streets with traffic flowing in the direction indicated. The flow rates along the streets are measured as the average number of vehicles per hour.

Set up a linear system whose solution provides the unknown flow rates.

Solve the system for the unknown flow rates.

A construction company would like to close a street for one day to make it easier to get some work done on a building site adjacent to the street, while the city wants to keep the road open to avoid congestion in the downtown area. Is it possible to close the road from A to B for construction and keep traffic flowing on the other streets? Explain.

What would be your recommendation to your supervisor on the request to close the road from A to B? Are there any streets that could be closed and still allow traffic to flow in and out of the city center with no congestion?

This is just part of a project that is due tomorrow and my partners have abandoned me so I am working by myself. I know a similar question has been posted, but this has extra questions added that I need help with in addition to the other ones. I have the equations for flow in and out figured out. Thank you very much!

Explanation / Answer

we can write the equation like this

1) x1+x3=800

2) x3+x6=750

3) x1+x4=200+x2

4) x7+600=x6+x4

5) x5+450=x7+400

6) x2+100=600+x5

we solve the equation 1 & 2 for path a to b

if we romove path a to b then

x1=0 means x3=800

if we put value of x3 in equation 2 we get value x6 negative thats not posible so it can not we remove it also we observe that if coming traffic 800 and outgoing 750 so obious might be a conjection.

x2 path also not removed

------------------------------------path of x3 can be removed ------------------------------

if we remove x3 means x3=0

then we get from equation 1

x1=800

from equation 2 x6=750

other four equations

a) x4-x2=-600

b) x7-x4=150

c) x2-x5=500

d) x5-x7=-50

add equation a and c

we get equation 7) x4-x5=-100

add equation b & d

we get equation 8) x5-x4=150

by oberserving equation 7 & 8 we coclude for any value of x4 and x5 no conjection in system

so x3 can be removed and it also conclude that x4 and x5 also can be removed becouse equation satify for any value of x4 and x5 so they can be zero means can be removed

any query can be answerd by solving above equation through 1 to 6

x3 can be removed with no conjection

x4 can be removed with no conjection

x5 can be removed with no conjection

x6 can be removed with no conjection

x7 path also not removed

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