Parallel lines are straight lines that extend to infinity without touching at any point. Perpendicular lines cross each other at a 90degree angle. Both sets of lines are important for many geometric proofs, so it is important to recognize them graphically and algebraically. You must know the structure of a straightline equation before you can write equations for parallel or perpendicular lines. The standard form of the equation is "y = mx + b," in which "m" is the slope of the line and "b" is the point where the line crosses the yaxis.
Parallel Lines

Write the equation for the first line and identify the slope and yintercept.
Example:
y = 4x + 3
m = slope = 4
b = yintercept = 3 
Copy the first half of the equation for the parallel line. A line is parallel to another if their slopes are identical.
Example:
Original line: y = 4x + 3
Parallel line: y = 4x 
Choose a yintercept different from the original line. Regardless of the magnitude of the new yintercept, as long as the slope is identical, the two lines will be parallel.
Example:
Original line: y = 4x + 3
Parallel line 1: y = 4x + 7
Parallel line 2: y = 4x  6
Parallel line 3: y = 4x + 15,328.35
Perpendicular Lines

Write the equation for the first line and identify the slope and yintercept, as with the parallel lines.
Example:
y = 4x + 3
m = slope = 4
b = yintercept = 3 
Transform for the "x" and "y" variable. The angle of rotation is 90 degrees because a perpendicular line intersects the original line at 90 degrees.
Example:
x' = xcos(90)  ysin(90)
y' = xsin(90) + ycos(90)x' = y
y' = x 
Substitute "y'" and "x'" for "x" and "y" and then write the equation in standard form.
Example:
Original line: y = 4x + 3
Substitute: x' = 4y' + 3
Standard form: y' = (1/4)*x  3/4The original line, y = 4x + b, is perpendicular to new line, y' = (1/4)x  3/4, and any line parallel to the new line, such as y' = (1/4)x  10.
Tips & Warnings
 For threedimensional lines, the process is the same but the calculations are much more complex. A study of Euler angles will help understand threedimensional transformations.
References
 "Linear Algebra and its Applications"; Gilbert Strang; 1988
 Photo Credit train line 1 image by Christopher Hall from Fotolia.com