The Quebec$-$New Brunswick Railway Company $($QNBRC$)$ has a railway network that

covers the lower south shore of Quebec and New Brunswick. Rail lines connect the

system to Maine, New York, Nova Scotia, and Western Quebec. Freight $($measured

in railcar loads per day$)$ enters and exits the system. A lot of freight is also exported

through the port at L$$vis$,$ just south of Qu$$bec City.

A map of the QNBRC railway network is given below, showing the locations of the

track junctions, as well as the number of railway cars entering and exiting the

system through its various connections to other networks.

Like all rail companies, the QNBRC is looking to cut costs. In particular, they are

considering closing the rail sections between Qu$$bec City and Sherbrooke and

between St$-$Jean and Sherbrooke. One concern over this plan is that the rail line

between Qu$$bec City and Moncton is vulnerable to extended closures due to snow

storms.

The company president wants a thorough analysis of the situation. In particular:

a$)$ What is the traffic flow pattern before any rail line closures?

b$)$ If the planned rail line closures are carried out and the Qu$$bec City$-$Moncton

line is closed due to inclement weather, will the remaining network be able to

handle the rail traffic flow?

c$)$ If the planned rail line closures are carried out but the Qu$$bec City$-$Moncton

line is open, what will the traffic flow pattern be $($i$.$e$.,$ how many trains on each

line and in which direction$)?$

d$)$ What is the volume of freight along the Qu$$bec City$-$Moncton line before any

line closures, and will the planned closures increase or decrease the amount of

freight?

NOTE: To complete this question you may use the company records, which indicate

that an average of $20$ freight cars per day travelled from Sherbrooke to Saint John

and $40$ freight cars per day travelled from Montr$$al to Qu$$bec City.

Analyze the rail network using systems of linear equations and provide the

company president with the information he needs to know.