The figure shows the flow of traffic in vehicles per hour th

The figure shows the flow of traffic (in vehicles per hour) through a network of streets. Solve this system for x_i, i = 1, 2, ..., 5. Find the traffic flow when x_2 = 200 and x_3 = 50. Find the traffic flow when x_2 = 150 and x_3 = 0.

Solution

For the given network, the system of equations can be obtained by considering source and sink at each node. Sum of all the source will be equal to sum of all the sink.

Now considering top-left node; the equation is given by:

x₁ + x₂ = 300

considering bottom-left node; the equation is given by:

x₃ + x₅ = 200 + x₂

considering top-right node; the equation is given by:

x₁ + x₃ = 150 + x₄

considering bottom-right node; the equation is given by:

x₄ + x₅ = 350

a)

from above equations,

x₁ = 300 - x₂

x₂ = 300 - x₁

x₃ = 200 + x₂ - x₅ = 150 + x₄ - x₁

x₄ = 350 - x₅ = x₃ + x₁ -150

x₅ = 350 - x₄ = 200 + x₂ - x₃

b) When x₂ = 200 & x₃ = 50

x₁ = 300 - x₂ = 300 - 200 = 100

x₄ = = x₃ + x₁ -150 = 50 + 100 - 150 = 0

x₅ = 200 + x₂ - x₃ = 200 + 200 - 50 = 350

c) When x₂ = 150 & x₃ = 0

x₁ = 300 - x₂ = 300 - 150 = 150

x₄ = = x₃ + x₁ -150 = 0 + 150 - 150 = 0

x₅ = 200 + x₂ - x₃ = 200 + 150 - 0 = 350