Traffic Control At rush hours, substantial traffic...

Traffic Control At rush hours, substantial traffic congestion is encountered at the traffic intersections shown in the figure on the next page. (The streets are one-way, as shown by the arrows.) GRAPH CANNOT COPY The city wishes to improve the signals at these corners so as to speed the flow of traffic. The traffic engineers first gather data. As the figure shows, 700 cars per hour come down M Street to intersection A, and 300 cars per hour come down 10th Street to intersection $A .$ A total of $x_{1}$ of these cars leave $A$ on $\mathrm{M}$ Street, and $x_{4}$ cars leave $A$ on 10 th Street. The number of cars entering $A$ must equal the number leaving, so that $$x_{1}+x_{4}=700+300$$ Or $$x_{1}+x_{4}=1000$$ For intersection $B, x_{1}$ cars enter on $\mathrm{M}$ Street and $x_{2}$ on 11 th Street. The figure shows that 900 cars leave $B$ on 11 th and 200 on $\mathrm{M}$ . Thus, $$\begin{array}{l}{x_{1}+x_{2}=900+200} \\ {x_{1}+x_{2}=1100}\end{array}$$ a. Write two equations representing the traffic entering and leaving intersections C and D. b. Use the four equations to set up an augmented matrix, and solve the system by the Gauss-Jordan method, using as the parameter. c. Based on your solution to part b, what are the largest and smallest possible values for the number of cars leaving intersection A on 10th Street? d. Answer the question in part c for the other three variables. e. Verify that you could have discarded any one of the four original equations without changing the solution. What does this tell you about the original problem?

Traffic Control At rush hours, substantial traffic congestion is encountered at the traffic intersections shown in the figure on the next page. (The streets are one-way, as shown by the arrows.)
GRAPH CANNOT COPY
The city wishes to improve the signals at these corners so as to speed the flow of traffic. The traffic engineers first gather data. As the figure shows, 700 cars per hour come down M Street to intersection A, and 300 cars per hour come down 10th Street to intersection $A .$ A total of $x_{1}$ of these cars leave $A$ on $\mathrm{M}$ Street, and $x_{4}$ cars leave $A$ on 10 th Street. The number of cars entering $A$ must equal the number leaving, so that
$$x_{1}+x_{4}=700+300$$
Or
$$x_{1}+x_{4}=1000$$
For intersection $B, x_{1}$ cars enter on $\mathrm{M}$ Street and $x_{2}$ on 11 th Street. The figure shows that 900 cars leave $B$ on 11 th and 200 on $\mathrm{M}$ . Thus,
$$\begin{array}{l}{x_{1}+x_{2}=900+200} \\ {x_{1}+x_{2}=1100}\end{array}$$
a. Write two equations representing the traffic entering and leaving intersections C and D.
b. Use the four equations to set up an augmented matrix, and solve the system by the Gauss-Jordan method, using as the parameter.
c. Based on your solution to part b, what are the largest and smallest possible values for the number of cars leaving intersection A on 10th Street?
d. Answer the question in part c for the other three variables.
e. Verify that you could have discarded any one of the four original equations without changing the solution. What does this tell you about the original problem?

Key Concepts

Redundancy in Linear Systems

A system of linear equations may have redundancy if one or more equations can be derived from the others. This redundancy indicates that not all equations provide independent information. Recognizing redundant equations helps simplify the analysis of the system and shows that the overall problem has inherent consistency and interdependency among the constraints.

Gauss-Jordan Elimination

Gauss-Jordan elimination is a method for solving systems of linear equations by performing row operations on an augmented matrix. The goal is to reduce the matrix to reduced row-echelon form, which directly reveals the solution to the system. This step-by-step process simplifies solving the equations and helps identify free parameters in underdetermined systems.

Parameterization in Linear Systems

When a system of equations has infinitely many solutions, one or more variables can be free parameters. Parameterization involves expressing the solution set in terms of these free parameters, providing a complete description of all possible solutions. This concept is key when dealing with systems that do not have a unique solution.

Systems of Linear Equations

Systems of linear equations arise when multiple relationships (such as traffic flows at various intersections) must be satisfied simultaneously. Each equation represents a balance condition at an intersection, and together they describe the overall behavior of the traffic system. Solving such systems allows for understanding how different flows interact and determining unknown traffic quantities.

Conservation of Flow

This principle, often referred to as the conservation law, states that for any given intersection or junction, the total number of vehicles entering must equal the total number leaving. In modeling traffic intersections, this concept is used to balance the flow of cars and to ensure that the equations accurately reflect physical traffic movement.

Augmented Matrix

An augmented matrix is a compact way of representing a system of linear equations, combining the coefficients of the variables and the constants from the equations into a single matrix. This form is particularly useful for applying systematic methods, such as Gauss-Jordan elimination, to solve the system efficiently.

Transcript

00:01 Okay, so for this problem, we're talking about traffic control.

00:05 So based off the diagram, i can go ahead and construct four equations.

00:09 So at the intersection of x2 and x3, so x2 and x3 would end up being 700, x3 and x4 will be 600, x1 and x4 will be 1 ,000, x1 and x2 will be 1 ,000, 1100.

00:38 So what i want to do is i want to set these up in a matrix and so that way i can go ahead and and solve them.

01:01 And so what i want to do is this is going to be a thousand and then 0 -1 -1 -0 -700 and then 0 -1 -6.

01:17 So now what i want to do is i want to go through and i want to cancel out this first row.

01:27 So i want to make that one zero zero zero.

01:29 So what i'm going to do is i'm going to subtract those first two rows.

01:33 And what i'm going to get is so i'm going to have my first row.

01:40 Then i'm going to have when i subtract them, i'm going to get 0 -1 -0 -0 -0 -9 -1, and then 100, 0 -1 -0, and then 700.

02:00 So now what i would like to do is i would like to actually clear out this row here.

02:08 Primarily, i'm sorry, not that row, this second column.

02:12 And i primarily want to get rid of the first and the third of these.

02:17 So when i do that, by subtracting them from that second row, i get 0 -0 -1 -1, and then 1 -1, and then 1 ,100 ,000, and then 1 ,600.

02:38 600, 600, and 600.

02:44 And then last but not least, i want to take care of this third column and see what happens.

02:51 So when i subtract the third and fourth rows, i am left with about as augmented of a matrix, a semi -augmented matrix.

03:03 So i am left with the bottom row being zeros, which is really what i want in this scenario, because what this tells me is that i have x1 plus x4 equals 1 ,000.

03:25 I have x2 minus x4 equals 100, x3 plus x4 equals 600...

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