How do we determine if two lines are parallel, perpendicular, or intersecting?

First of all, you know that two lines are parallel if they never touch, so if they are parallel, then they are neither perpendicular nor intersecting.

Also, you know that perpendicular lines do intersect, but in a special way because their intersection results in right (i.e., 90°) angles.

To see if two lines are parallel, perpendicular, or simply intersecting, you  can graph the two lines and answer the question by observation, or you
can look at the slopes of the two lines.

Parallel lines have the same slope.
Perpendicular lines have slopes that are opposite inverses of each other (that is, the product of the two lines equals -1). Intersecting lines have slopes that differ from each other.

For example, consider the two lines
y = 5x -1 and y = 5x +3 are they parallel, perpendicular, or intersecting?  how do you know?

Consider these two lines: y = 2x +3 and y+1/2x = 7. 
Parallel, perpendicular or intersecting? How do you know?