The statement is true
Your teacher worded things a bit strange, but basically if two lines have the same slope, but different y intercepts, then the lines are parallel. I recommend graphing out something like y = 3x+7 and y = 3x+5 to see two parallel lines forming. These lines will never intersect because each line goes up 3 each time you move to the right 1. So effectively the motion is done in lockstep.