Y-intercept of a regression line

In $\hat{y} = a + bx$ (read y-hat for $\hat{y}$), the y-intercept $a$ is the height of the line at $x = 0$. It is not fitted on its own. Once the slope is settled, the intercept is forced by $a = \bar{y} - b\bar{x}$ (read y-bar and x-bar), because the least-squares line has to pass through the point of averages.

Four used trucks give odometer readings of 40, 60, 80 and 100 thousand miles against prices of 26, 22, 19 and 17 thousand dollars. Least squares returns $\hat{y} = 31.5 - 0.15x$. The formula agrees: $\bar{x} = 70$, $\bar{y} = 21$, so $a = 21 - (-0.15)(70) = 31.5$. That intercept predicts 31.5 thousand dollars for a truck with zero miles, 40 thousand miles below the least-driven truck in the data.

The wrong conclusion is the second half of this: "$x = 0$ makes no sense here, so the intercept is meaningless and we should drop it." The first half is fair. The second is not, because deleting $a$ forces the line through the origin and changes every prediction on the page. The best line of the form $\hat{y} = bx$ here is $\hat{y} = 0.2583x$, which predicts 10.3 thousand dollars for the truck that actually sold for 26. An intercept with no contextual reading still has a job: it positions the line over the range where the data live.

It is also worth seeing how little the intercept says about the relationship. Measure each odometer reading as a distance from the group mean, $x' = x - 70$, and refit. The slope is unchanged at -0.15, every residual is unchanged, and the intercept is now 21, which is exactly $\bar{y}$. Move where zero sits on the horizontal axis and the intercept moves with it.

Interpret it in context only when $x = 0$ is a value that occurs. Otherwise report it as part of the equation and say so.