Question 1. Network Flow

An island's main network contains bridges and main roads as in the following Figure. Arrows indicate the traffic's direction.

The number of vehicles crossing each bridge at peak hour (called flow) is indicated on the figure. We are interested in the ow on the main roads (denoted with variables a to g). Our model consists of the following two laws:

1. Flow is constant along each road.

2. The total flow coming in any intersection is equal to the total flow coming out of the intersection.

Please answer the following questions.

(a) Find a system of linear equations that describe the flow according to those laws. Enter the augmented matrix of the system in SageMath.

(b) Use SageMath to find the reduced row echelon form of the matrix. Write the solutions to the system in parametric form.

(c) What is the maximal flow on road b? Justify your answer.

(d) What is the minimal flow on road a? Justify your answer.

(e) As a truck convoy blocks road b, an anti-war protest blocks road c. What is then the solution to the system? What do you suggest in order to solve the problem?