The answer to 'why is my supply line constantly running' is that the most common cause is a faulty fill valve or flapper in the toilet tank. This allows water to continuously flow from the tank into the bowl, causing the supply line to run constantly to refill the tank.
Other possible causes include a leak in the supply line itself, a problem with the shut-off valve, or an issue with the toilet's internal components. If the supply line is running constantly, it's important to address the issue promptly to prevent water waste and potential damage to your home.