Find an equation for the plane consisting of all points that are equidistant from the points (−6, 1, 3) and (4, 3, 7).

The direction vector between the two points is perpendicular to the plane. That direction is
  (4, 3, 7) - (-6, 1, 3) = (10, 2, 4) or (5, 1, 2)

The plane goes through the midpoint of that line segment,
  ((4, 3, 7) + (-6, 1, 3))/2 = (-1, 2, 5)

So, the equation of the plane is
  5x +y +2z = 5(-1) +2 +2(5)
  5x +y +2z = 7