Exoplanet detection with precise radial velocity (RV) observations is currently limited by spurious RV signals introduced by stellar activity. We show that machine-learning techniques such as linear regression and neural networks can effectively remove the activity signals (due to starspots/faculae) from RV observations. Previous efforts focused on carefully filtering out activity signals in time using modeling techniques like Gaussian process regression. Instead, we systematically remove activity signals using only changes to the average shape of spectral lines, and use no timing information. We trained our machine-learning models on both simulated data (generated with the SOAP 2.0 software) and observations of the Sun from the HARPS-N Solar Telescope. We find that these techniques can predict and remove stellar activity both from simulated data (improving RV scatter from 82 to 3cm/s) and from more than 600 real observations taken nearly daily over 3yr with the HARPS-N Solar Telescope (improving the RV scatter from 1.753 to 1.039m/s, a factor of ~1.7 improvement). In the future, these or similar techniques could remove activity signals from observations of stars outside our solar system and eventually help detect habitable-zone Earth-mass exoplanets around Sun-like stars.