How to Write Equations of Perpendicular & Parallel Lines

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.


  1. Parallel Lines

    • 1

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

      y = 4x + 3
      m = slope = 4
      b = y-intercept = 3

    • 2

      Copy the first half of the equation for the parallel line. A line is parallel to another if their slopes are identical.

      Original line: y = 4x + 3
      Parallel line: y = 4x

    • 3

      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.

      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

    • 4

      Write the equation for the first line and identify the slope and y-intercept, as with the parallel lines.

      y = 4x + 3
      m = slope = 4
      b = y-intercept = 3

    • 5

      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.

      x' = xcos(90) - ysin(90)
      y' = xsin(90) + ycos(90)

      x' = -y
      y' = x

    • 6

      Substitute "y'" and "x'" for "x" and "y" and then write the equation in standard form.

      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.

Tips & Warnings

  • 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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  • "Linear Algebra and its Applications"; Gilbert Strang; 1988
  • Photo Credit train line 1 image by Christopher Hall from

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