### 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}=20x_{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!

x1 – x3 = 75 not 55

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

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