The line tension of a three-phase contact line is implicated in a wide variety of interfacial phenomena, but there is ongoing controversy, with existing measurements spanning six orders of magnitude in both signs. Here, we show that computationally obtained magnitudes, sign changes, and nontrivial variations of apparent line tension can be faithfully reproduced in a parsimonious model that incorporates only liquid-substrate interactions. Our results suggest that the origin for the remarkable variation lies in the failure of a widely used estimation method to eliminate body forces, leading measured line tensions to behave like an extensive quantity.