Question

The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (Assume a = 100 and 400.) b XI a 33 X4 (a) Solve this system for Xi 1, 2, 3, 4. (If the system has an infinite number of solutions, express X1, X2, Xy, and x4 in terms of the parameter t.) (X1, X2, X3, X4) (b) Find the traffic flow when X4 = 0. (X1, X2, X3, X4) = (c) Find the traffic flow when X4 = 100. (X1, X2, X3, X4) = (d) Find the traffic flow when X1 = 2x2. (X1, X2, X3, 4) =

Answer 1

b = 400 6 xa B a A ra D ly Xz vb and a = 100 la At node Rode - 4t a=Xz 0 x-Xz =-100 At node - B X & b. = x X - X 400 At node -( xq = fg ta 100 3) Xg -xa At nodeD xq tb xz G - x4 x4 = 400

Augmented matrin [Alb] = T OT 0 0 OOIT 100 400 ~100 400 0 Ra Ra - R, o - 0 ļu 0 1 O -100 Soo -100 400 -I - RzR3 + Rg 0 1 -10 ㅔ 100 -Soo 400 400 o pool o O Pa R4-R ī01 01-100 - 500 400 0 0 11 o 0 - o 0 0 3 Rank (A) = rank (Alb rank(Alb] =3 Infinity Hence there are many Solutions Reduced system of equchong

- -100 ५ - २ - 3 23-4 = -500 - 400 Led xq = * Then - 400+t en er - -500 +400+t - t-100 ४१ - -100 + ५ = -100 + 400++ म - 300-+t 300+t x - t-100 yo - 400+ X4 है when x4 co (60 ie t=0

300 -100 Kg = 400 xq = 0 when xq 100 Ha 100 ie x = 400 & goo O x₂ =500 XA = 100 @ when 19 = 2 x 300 +4 = gt 200 t = soo X = 800 Il ll My = 400 = 900 Sq = 500 х3