How to Solve Traffic Flow Problems Using Linear Algebra?

[vs55]

Homework Statement

the following statement shows the traffic flow(in cars/minute) along 3 one way streets through the intersections A, B and C
Set up and solve a system of linear equations(using the augmented matrix for the system) to help you find the flows x,y and z. Be sure to generate the RREF of the system
a link of the drawing:
http://img18.imageshack.us/img18/2276/trafficflow.jpg

Homework Equations

traffic flow in must equal flow out

The Attempt at a Solution

so to start things off, equations i put:
3 + z = 3
6 + x = 7
5 + y = 4
is this at all right?

 

[Mark44]

Homework Statement

the following statement shows the traffic flow along 3 one way streets through the intersections A, B and C
Set up and solve a system of linear equations(using the augmented matrix for the system) to help you find the flows x,y and z. Be sure to generate the RREF of the system
a link of the drawing:
http://img216.imagevenue.com/img.php?image=27718_trafficflow_122_18lo.jpg

Homework Equations

traffic flow in must equal flow out

The Attempt at a Solution

so to start things off, equations i put:
3 + z = 3
6 + x = 7
5 + y = 4
is this at all right?

I don't think so. For your first equation, 3 + z = 3 ==> z = 0, which doesn't make any sense. How about giving us the problem exactly as it was worded? Useful information would be what the numbers in your drawing represent.

I believe that what you have as "relevant equations" means that the traffic flow into and out of an intersection must be equal.

 

[vs55]

this is the problem exactly as it was worded, but if u insist...i added "cars/minute" :|
i also added the label on the intersections in the image A,B and C
and sorry i thought the numbers were obvious :S, they represent the flow out that that arrow

i'm guessing i have to go take away some of the flow as well? I'm just not sure how to start the problem

 

[Mark44]

If my previous interpretation of what traffic flow in equals traffic flow out is on track, here's what you have:
Node A (upper left intersection) x + 5 = y + 7
Node B (upper right int.) z + 6 = x + 3
Node C (lower int.) y + 3 = z + 4


Solve this system.

 

[vs55]

i got:
-x + z =-3
-z +y = 1
-y + x = z
to help set up the matrix:
-x 0 z
0 y -z
x -y 0
augmented matrix:
-1 0 1 |-3 ---->(1)
0 1 -1 |1------>(2)
1 -1 0 |2------>(3)

after some math:
(1)/-1
(3)-(1)
(3)+(2)

1 0 -1 | 3
0 1 -1 | 1
0 0 0 | 0

1x -1z = 3
1y -1z =1
z=free

z=1,y=2,x=4
is this right?

 

[Mark44]

i got:
-x + z =-3
-z +y = 1
-y + x = z

The last equation should be -y + x = 2, but otherwise I got the same solution that you showed.

 

[vs55]

thx for ur help!