Chapter 7, Problem 18CT

To determine

To solve: the system for the traffic flow represented by $x_i$ .

Answer to Problem 18CT

The system for the traffic flow has the solutions:

  $\begin{array}{l}{x}_1=700-s-t,x_2=300-s-t\\ x_3=s,x_4=100-t,x_5=t\end{array}$

Explanation of Solution

Given information:

The flow of traffic through a network of streets is shown at the right.

Where $i=1,2,3,4,$ and 5

Given figure

  

Calculation:

For the given figure in the question, the conservation of flow leads to a system of linear equations:

For the top left intersection:

Traffic in $=x_2+400$

Traffic out $=x_1$

Thus,

  $\begin{array}{l}{x}_1=x_2+400\\ x_1-x_2=400\end{array}$

For the top right intersection:

Traffic in $=x_1+x_3$

Traffic out $=x_4+600$

Thus,

  $\begin{array}{l}{x}_1+x_3=x_4+600\\ x_1+x_3-x_4=600\end{array}$

For the bottom left intersection:

Traffic in $=300$

Traffic out $=x_2+x_3+x_5$

Thus,

  $\begin{array}{l}300=x_2+x_3+x_5\\ x_2+x_3+x_5=300\end{array}$

For the bottom right intersection:

Traffic in $=x_4+x_5$

Traffic out $=100$

Thus,

  $x_4+x_5=100$

Thus, we have a system of equations:

  $\begin{array}{l}{x}_1-x_2=\mathrm{400...........}(1)\\ x_1+x_3-x_4=\mathrm{600.......}(2)\\ x_2+x_3+x_5=\mathrm{300.......}(3)\\ x_4+x_5=\mathrm{100..........}(4)\end{array}$

The above system has 4 equations and 5 unknowns.

Therefore, the system has infinite number of solutions.

Suppose, $x_5=t$ ; therefore, (4) implies:

  $\begin{array}{c}{x}_4+t=100\\ x_4=100-t\end{array}$

Suppose, $x_3=s$ ; therefore (3) implies:

  $\begin{array}{c}{x}_2+s+t=300\\ x_2=300-s-t\end{array}$

And (1) implies:

  $\begin{array}{c}{x}_1-\left(300-s-t\right)=400\left[\text{As}{x}_2=300-s-t\right]\\ x_1=700-s-t\end{array}$

Thus, the system for the traffic flow has the solutions:

  $\begin{array}{l}{x}_1=700-s-t,x_2=300-s-t\\ x_3=s,x_4=100-t,x_5=t\end{array}$