Science:Math Exam Resources/Courses/MATH221/December 2009/Question 03 (b)
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Question 03 (b) 

Problem 3. Consider the traffic flow diagram: <INSERT PICTURE HERE> The system is clearly inconsistent because the total in flow does not equal total outﬂow. Find all least squares solutions to the system. 
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Science:Math Exam Resources/Courses/MATH221/December 2009/Question 03 (b)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. See Q3 (a) for the flow diagram and system of equations. The least squares equation takes the form: In general, , has a diagonal of the number of edges connected to the respective node and has offdiagonals entries of 1 if the nodes are connected and 0 if not. It turns out that all of the nodes are connected to 2 other nodes by some edge. Simplifying the least squares equation we have: Solving this system of equations through row reduction yields the following least squares solution: 