Building a Linear Model

Building a linear model means finding the constant rate and the initial value in the situation, writing f(x) = mx + b from them, and testing the model on a known moment.

A tank holds 50 gallons and fills at 10 gallons per minute. Rate: m = 10. Initial value: b = 50. The model: V(t) = 10t + 50. Test it on a moment that can be reasoned by hand: after 3 minutes, the tank should hold the start plus three fills — 50 + 30 = 80 — and the model answers V(3) = 30 + 50 = 80. Agreement is the receipt. The two questions that build every linear model are the same two every time: what is the steady rate, and where did things stand at zero? A situation that cannot answer both is not linear.