Linear Algebra System of Equations

[MDTeen]

Hello Everyone,

I am new to the forum, but I am kind of at a loss and I could use a little guidance. I currently have an assignment where we are given a traffic circle, with 4 on ramps and 4 off ramps (labeled a through h) which are constants. We are then given the traffic flow in between each of the on and off ramps as variables x1 through x8. None of the constants or variables are given an actual value (we are supposed to treat these as real numbers).


We are supposed to analyze the traffic circle, create a system of equations in order to solve for the values of x1 through x8 (the traffic flow) and analyze the constraints of the constants (a through h - on/off ramps) and the values of x1 through x8.


I have a diagram of the traffic circle, with the on/off ramps and sections of the circle labeled with the appropriate variables and constants (including work I have done so far) in the attached PDF (sorry for the horrible handwriting).


Normally when given a problem like this, we are least given some data to work with, in order to find the values of the variables, but since we are given no data (just that the constants are a through h and that they mark the different on / off ramps) I'm just a little confused as to how to solve this. I started by creating a system of equations regarding the constants and the variables we were given, then tried to solve the system of equations in order to get values for x1 through x8, now I am just stuck.

I am unsure if I even created the correct system of equations for this problem, and if I did, did I solve the system correctly?

I am not asking for someone to give me the direct answer to the problem, but any assistance in helping me with this problem could really help.


Please give me your opinions on what I have done so far, and if I did anything wrong, please let me know what.

Thank You

 

[flyingpig]

These problems are similar to Kirchoff's circuit problems. Flow in = Flow out

I quickly glanced through your constraints, they look good. I'll also assume you are correct in your final answer.

All you have to do is sub in x_1, x_3, x_5, x_7 into the other variables to get your final answer

 

[MDTeen]

In regards to your response:

"These problems are similar to Kirchoff's circuit problems. Flow in = Flow out

I quickly glanced through your constraints, they look good. I'll also assume you are correct in your final answer.

All you have to do is sub in x_1, x_3, x_5, x_7 into the other variables to get your final answer"

I reworked the problem a little and substituted x1, x3, x5, and x7 into the remaining variables of x2, x4, x6, x8 as shown in the attached PDF.

Is this what you were meaning?

 

[flyingpig]

In regards to your response:

"These problems are similar to Kirchoff's circuit problems. Flow in = Flow out

I quickly glanced through your constraints, they look good. I'll also assume you are correct in your final answer.

All you have to do is sub in x_1, x_3, x_5, x_7 into the other variables to get your final answer"

I reworked the problem a little and substituted x1, x3, x5, and x7 into the remaining variables of x2, x4, x6, x8 as shown in the attached PDF.

Is this what you were meaning?

I am not following, how did you get RSTU?

 

[MDTeen]

I was about to give the variables a letter to represent it but then I realized it really was not needed in this case and finished the work at the bottom of the page without without erasing my previous notes for RTSU.

Please see my work at the bottom of the page and let me know what you think (sorry for the mishap).

 

[flyingpig]

I was about to give the variables a letter to represent it but then I realized it really was not needed in this case and finished the work at the bottom of the page without without erasing my previous notes for RTSU.

Please see my work at the bottom of the page and let me know what you think (sorry for the mishap).

I still not sure what the real problem is. In the beginning you already have written the constraints to solve for x_8 and the others. You solved what x_1, x_7 etc... are from your row operations. Just put those values back into the constraints you had in the very beginning

 

[MDTeen]

Sorry it might be because I'm tired, but I didn't even know I created the constraints... Which items I wrote were the constraints?

As for the answers I got:

x8 = x1 + d
x2 = x3 + b
x4 = x5 + h
x6 = x7 + f

Where are these answers supposed to be inserted into? Or was I completely off and that latest page I uploaded was wrong and I already had the answer?

I know I must be confusing right now, maybe I should get back online tomorrow after I had some rest and look this over again and see if I can make sense of any of this..

 

[flyingpig]

You have

$$x_1 = (g - f) + (a-h) + (c-b) + (e - d)^2$$

And

$$x_8 = x_1 + d$$

You know what x_1 is, what is the confusion?