Even though we know how many vehicles are coming in and out of the town at each intersection, we don't know exactly how much traffic is on our four roads surrounding the town square, which is why we need your help. Please respond to the questions listed below. Use linear algebra terminology in your responses, because Professor Brown has promised to translate your answers for us$.$ Write a professional report and submit it to Professor Brown by class time of February $19.$

Thank you for all your help,

Ernie Chelon,

Mayor of Popperville

Questions

Set up a linear equation for each of the four intersections of Popperville: $A,B,C,$ and $D.$ At each intersection, the traffic exiting the intersection must equal the traffic entering. For example, at intersection $A,x_1+300$ must equal $x_4+400.$ Now write the system of linear equations as a matrix equation Avec$(x)=$vec$(b),$ and solve the system. Are there infinite solutions, a unique solution, or no solutions?

Using your solution set from above, what are the bounds on the traffic for all four roads $,$ and $($:$x_4\}?$ Keep in mind, these are one$-$way roads, so negative traffic is not allowed.

We are confident that there are no more than $720$ vehicles per hour on

any one of the roads, give a solution that would correspond to one of the road's traffic reaching this maximum value.

Road $x_4$ is desperate for repair. Returning to your bounds from question $2,$ what is the minimum number of cars we can allow on $x_4,$ and how will this affect the traffic on the other $3$ roads?

Once we have the repair work done as mentioned above, we have a bold new plan for our town square. We feel that we can accomodate our guests much better by building a $400$ car underground parking garage under our town square. The proposed new street layout is as follows:

Since the parking garage holds $400$ vehicles, we anticipate $400$ cars entering $G$ every hour, and $400$ cars exiting $G$ every hour. $($People only spend about an hour at our celebration.$)$ Repeat questions $1$ thru $3$ for this new layout $($including equation$($s$)$ for $G)$; and include your final analysis as to the affect this parking garage will have on the flow of traffic in Popperville.