Question

The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (Assume a 300 and b 50.ג x1 X2 14 (a) Solve this system for xi, 1, 2, 3, 4. (If the system has an infinite number of solutions, express x1, x2, x3, and x4 in terms of the param eter t.) (x1, X2, X3, X4)-+100,t- 300,t400,t x (b) Find the traffic flow when x40 (xi, x2, x3, xa) - (c) Find the traffic flow when x4300 xx2,x3, x4)- (d) Find the traffic flow when x13x2 (X1, X2, X3, x4)- Need Help?Read It Talk to a Tutor

Answer 1

The incoming traffic at each junction has to be equal to the outgoing traffic . Therefore,

b+x₂= x₁ or, x₁-x₂ = b…(1)

x₁+a = x₃ or, x₁- x₃ = -a …(2)

x₃ = b+x₄ or, x₃- x₄ = b…(3) and

x₄ = a+x₂ or, x₂- x₄ = -a…(4)

The augmented matrix of this linear system of equations is A (say) =

1	-1	0	0	b
1	0	-1	0	-a
0	0	1	-1	b
0	1	0	-1	-a

To solve this linear system of equations, we have to reduce A to its RREF which is

1	0	0	-1	-a+b
0	1	0	-1	-a
0	0	1	-1	b
0	0	0	0	0

Thus, the above linear system of equations is equivalent to x₁-x₄ =-a+b or, x₁= x₄-a+b,x₂-x₄=-a or, x₂=x₄ -a x₃-x₄= b or, x₃=x₄+b. Now, let x = t. Then, (x₁,x₂,x₃,x₄) = (t-a +b, t-a, t+ b, t).

(b). When x₄ = 0, we have (x₁,x₂,x₃,x₄) = (-a +b, -a, b, 0) ( on substituting t = x₄ = 0).

(c ). When x₄ = 300, we have (x₁,x₂,x₃,x₄) = (300-a +b, 300-a, 300+b, 300) ( on substituting t = x₄ = 300).

(d). When x₁= 3x₂, we have x₄-a+b = 3(x₄-a) or, 2x₄ = 2a+b or, t = x₄ = a+b/2. Then, (x₁,x₂,x₃,x₄) = (3b/2, b/2, a+ 3b/2, a+b/2).