Let's break down the problem presented in the image.

Given:

• There are four intersections $A$, $B$, $C$, and $D$.
• The traffic flow (vehicles per hour) entering and leaving each intersection is represented by arrows, with the incoming and outgoing traffic flows labeled with the given values.

The variables $x_1, x_2, x_3, x_4$ represent unknown traffic flows at each intersection.

Task Breakdown:

a) Formulate the equations for each intersection:

At each intersection, the sum of incoming traffic should equal the sum of outgoing traffic (since no vehicles are being created or lost at intersections).

• Intersection A: $450 + x_2 = x_1 + 610$

• Intersection B: $480 + x_3 = x_2 + 520$

• Intersection C: $390 + 600 = x_3 + x_4$

• Intersection D: $310 + x_1 = 640 + x_4$

b) Write these equations as a matrix:

The system of equations can be written as a matrix equation $A \mathbf{x} = \mathbf{b}$, which we'll perform after simplifying the equations from part a.

c) Determine the traffic flow knowing that the difference between traffic at intersections C and D is 150 vehicles:

We are told that the difference in traffic flow between intersections $C$ and $D$ is 150 vehicles, i.e., $C - D = 150.$

Now, let's work through these steps and solve the system of equations. I'll calculate the equations for you first.The solution for the traffic flows at each intersection is as follows:

• $x_1 = 560$
• $x_2 = 720$
• $x_3 = 760$
• $x_4 = 230$

This gives the traffic values in vehicles per hour at each intersection.

For part (b), let's express these equations in matrix form and reduce it to row echelon form. I'll work on this next.The reduced row echelon form (RREF) of the system's augmented matrix is: