Network Flow Application

Linear Algebra (Matrices) · Axiom Academy

One rule at every intersection — flow in = flow out — turns a whole road network into a system of linear equations you can solve. Cars stream into a downtown grid: 60 , 40 , and 50 cars/min come in; 20 , 90 , and 40 leave. The hidden numbers are the flows on the inner streets — and one rule pins them down. At any intersection cars can't pile up or vanish — whatever drives in must drive out. Drag the two incoming streets and watch the outgoing flow balance the books. That balance is one linear equation. Four intersections give four equations, but there are five inner streets — so one flow is free to choose, and the other four follow. Drag x₁ (the A→B street) and watch the rest snap into place with every node still balanced. These streets are one-way, so no flow can run negative. Slide the free flow across its whole range and watch where a street goes red — the green band is every traffic plan that actually works. One rule — flow in = flow out — turned a road grid into 4 equations, 5 unknowns : solve it and you get a whole family of valid traffic plans, with one flow free to choose and non-negativity fencing in the rest. The same conservation move models water in pipes , current in circuits (Kirchhoff), and packets through a network .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.