Slope-Intercept Form of a Line

$y = mx + b$ is the workhorse line equation, and it is worth reading as two separate claims. The $b$ is where you start: set $x = 0$ and everything else vanishes, so $b$ is the value of $y$ before anything has happened. The $m$ is the rate at which $y$ changes per unit of $x$. Between them those two numbers pin down every non-vertical straight line there is, which is why two parameters is all a linear model ever needs.

The reason this outlives school algebra is that it is the shape of any process with a fixed part and a per-unit part. A service call billed at $95 to show up plus $85 an hour is $y = 85x + 95$; four hours on site invoices at $435, and running it backwards, a $520 invoice accounts for 5 hours. Utility bills, freight rates, and equipment rentals are all this equation with different names on the constants, and the useful habit is asking, of any quoted price, which number is the $b$ and which is the $m$.

Two related forms cover what this one cannot. Point-slope, $y - y_1 = m(x - x_1)$, is the natural choice when you know a point on the line that is not the intercept — a calibration reading, say — and want the line through it. Standard form, $Ax + By = C$, is more general still, and unlike slope-intercept it can describe a vertical line, which $y = mx + b$ is structurally incapable of representing because a vertical line has no slope to put in $m$.

Four things go wrong. The first is confusing $b$ with the x-intercept: $b$ is where the line crosses the vertical axis, while the crossing of the horizontal axis is at $x = -b/m$, which is a different number and often the one actually wanted — a break-even point, for instance. The second is that solving for $m$ divides by $x$, so a point with $x = 0$ tells you nothing about the slope; every line through the intercept fits it equally. The third is sign carelessness with a negative intercept: $y = 3x - 4$ has $b = -4$, and typing 4 shifts the whole line eight units. The fourth is the conceptual one: a straight line is a model, not a law. Fitting one to two measurements and extrapolating far past them assumes a constancy that most real processes do not have, and the intercept in particular frequently describes a condition that never occurs — the value at zero hours, zero flow, or zero temperature may be pure arithmetic with no physical meaning at all.