# The volume of traffic for a collection of intersections 2

### The volume of traffic for a collection of intersections is shown in the figure below. Find all possible values for $x_{1}$, $x_{2}$, and $x_{3}$. What is the minimum volume of traffic from C to A? # Solution:

Write down the equations for each intersection. (On the left side, we write down the traffic coming in, on the right side, we write down the traffic going out)

Intersection A : $x_{2}=20+x_{3}$

Intersection B : $x_{3}+35+50=10+x_{1}$

Intersection C : $x_{1}+40=95+x_{2}$

Rearrange the equations so that the variables are on the left side.

$x_{2}-x_{3}=20$

$x_{1}-x_{2}=55$

$x_{1}-x_{3}=55$

Solve the equations, easiest way is to use a matrix.

$\begin{bmatrix}1 & 0 &-1 &|75\\1 & -1 & 0 & |55\\0 & 1 & -1 & |20\end{bmatrix}$

Put the matrix in reduced row echelon form.

$\begin{bmatrix}1 & 0 &-1 &|75\\0 & 1 & -1 & |20\\0 & 0 & 0 & |0\end{bmatrix}$

Let $s_{1}$ represent a free parameter.

$x_{3} = s_{1}$

$x_{2} = 20+ s_{1}$

$x_{1} = 75+ s_{1}$

As $x_{2}$ represents traffic going between C and A (labeled on diagram), we can see that at the very least, 20 cars must go through.

See a mistake? Comment below so we can fix it!

## 2 thoughts on “The volume of traffic for a collection of intersections”

• Jarod Koopman

x1 – x3 = 75 not 55

Reduced Echelon last column should read (75, 20, 0)
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• Question Solutions

Thank you very much, appreciate the help a lot. We fixed it!