When the slopes of different lines are equal, it means the lines must be parallel to each other. They have different y-intercepts, but are orientated in the same direction.They could also be the same line (same slope and same y-intercept), but I doubt this is what your teacher has in mind.
The two lines have equal slopes implies they are equally inclined to any given line of reference. In other words the two lines are parallel or under extreme case they may coincide into one single line.
Let y=m1x+c1 and
y= m2x+c2 be two lines with inclinations of slope m1 and m2 to the X axis and making intercepts c1 and c2 respectively with X axis.
So if m1 = m2 , then the two lines are parallel as long as c1 and c2 are different.
Further, if m1=m2 and c1=c2, the lines coincide to be a single line.