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a. Find the general traffic pattern in the freeway network shown in the figure. (Flow rates are in cars/minute.)b. Describe the general traffic pattern when the road whose flow is $x_{4}$ is closed.c. When $x_{4}=0,$ what is the minimum value of $x_{1} ?$figure can't copy

a. $\left\{\begin{aligned} x_{1}=& 100+x_{3}-x_{5} \\ x_{2}=& 100-x_{3}+x_{5} \\ & x_{3} \text { is free } \\ x_{4}=& 60-x_{5} \\ & x_{5} \text { is free } \end{aligned}\right.$b. See solutionc. See solution

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 6

Applications of Linear Systems

Introduction to Matrices

Missouri State University

University of Michigan - Ann Arbor

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Okay, So in this problem, we need to describe the general traffic pattern or to give them figure. Now, we just need to make a table out of these four intersections and with the flow kinda blowing and the flow out So here we have a B C D for intersections and flowing and blow out. Okay, So for the intersection, eh? We have X one closing and 40 last x three less x four blowout. And this should be equal or intersection B. We have 200. No wind and X one A sex, too. Blowout intersections. See, we have x two. And next three go in and 100. Uh, X five go out or into section D. We have X four in next five. Blow in and 60 comes out. Okay, So based on this table, we can make a matrix. That is one. Uh, we this denote the first column as ah X one. Next to x three. It's four. Ex fight. Okay, So the first question we have, uh, one of x one and put the x three from the right hand side. You left inside. So that is negative. One for detective for X three and it's the same four x four we have next one and forties Constant out there. So this should be 40 and one of x one and neck three connected. One, Rex three and export X three x four. So arrested will be zero. Okay. For the second or the second row, we have 11 and all zeros 60. Awake should be 200. Sorry. 200. And for the third row at 011 because we have one of ex to one of x three. And next one off expire next while backs five. And the council terms is 100. Excuse me. Okay, So for for the last roll at all zeros, we only have one of exploring the wild. Well, enough, expert. So that's 11 and constant term is 60. All right, so we can perform the cashing illumination to this matrix. Um, to do that, we first used a second row. Subs tracked first roll. So that is 100 Sorry. 10 connective 110 and working. So one minus one zero one minus zero. One on this 3110 and 1 60 110 1 60 okay. And keep the third row. And fourth rule 100 worker will be three zeros. Scuse me. Three zeros and too once and constant term is 60. Okay, the next we used a third row. Subtract a second role. So we first set down the first roll next to one negative one zero or tea. 0111 zero 1 60 Okay, um so third row, subtract second role. That is 00 Next one, a negative one and 60 connected 1 to 1. And like their 60. All right on we keep the nice roll. 11 16. All right, so right now, right now, we can First officer of the third row on the fourth, who will be the same. So fourth row will be cancelled. So that is okay, we have 10 notice. 1 91 zero 40 and 10111 Do? Yeah. 1 60 and 00 You know, 11 and 16. The last room is all heroes. Okay. Finally, we used a second rule. Subtract the third row. So this gives us first one. Do active 110 and 40 on dhe. Second role will be your 211 So one minus one, which is zoo on zero minus one Negative. One 1 16 minus 60 will be 100 and the third rule will keep the same. Oh, that is very little. 011 and 60. Yeah, 60. All right, So you're Here's our reduce station of form. Now we plug in our variables. That is x one minus x three minus x four is 20 and x two plus x three minus x five two plus X three minus X fly. It's 100 all right? And the last one have X floor, plus x five is 16 so x five will be our will be the free bearable here. Oh, right down x one turns off 40 plus x three US explore and next to will be negative x three. Plus it's five and x three is a free bearable. Well, that's four is 60 financed X five. So here we have 23 variables X three, x five and this is our traffic pattern. Okay, Um so for the second problem, we need to describe the General General Patrick of Pattern. When the road who's flow is exports closed. So that means we need to said x four as zero, because that's four is closed. So that means for according to our the fourth equation here, so enemies, our X five is 50. All right, so Corey's found early. We have Thanks. One is 40 plus x three. Thanks for. So this will be 40 those x three and next 480 And secondly, X two will be connective X three, class 50 because x five here is 50 and ex threes. Steel of free variable. This won't be changed, but X for his ex warrior zero and the last. The NASA question turns out to be x five People's 50. I'm sorry. 60 60 here. Yeah, X files should be 60 because x four here, we're assuming x 40 So x five should be 60. So I have to revised the value of X fire here. So that is 60. So it is 60 and next to is equal to negative extra plus x five so that this should be 60. All right, so here's the traffic pattern for our 2nd 2nd question. So for the last question, we have you export zero, then what is the minimum? Better off X one. So, for the quick first question here we have, we're we have X one equals 40 plus x three. So this is under the condition affects four equals zero. So we the thing we need to consider here is to minimize x three so that our RX one here will be the minimum. So in order to meet him, minimize x three. The best we can do is to set x 30 So x one will be the month, which is the meeting will be 40. And here's our solution, while for the second question and we're done.

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