How to Write Equations of Perpendicular & Parallel Lines

Parallel lines extend to infinity without crossing.
••• train line 1 image by Christopher Hall from

Parallel lines are straight lines that extend to infinity without touching at any point. Perpendicular lines cross each other at a 90-degree 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 straight-line 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 y-axis.

Parallel Lines

    Write the equation for the first line and identify the slope and y-intercept.

    Example: y = 4x + 3 m = slope = 4 b = y-intercept = 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 y-intercept different from the original line. Regardless of the magnitude of the new y-intercept, 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 y-intercept, as with the parallel lines.

    Example: y = 4x + 3 m = slope = 4 b = y-intercept = 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' = x_cos(90) - y_sin(90) y' = x_sin(90) + y_cos(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 - 3/4

    The 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.


    • For three-dimensional lines, the process is the same but the calculations are much more complex. A study of Euler angles will help understand three-dimensional transformations.

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