How to Tell If Lines Are Parallel, Perpendicular or Neither

By Kelvin O'Donahue
Any data can be used to form a line's equation, including its slope, x-intercept, y-intercept or points on the line.
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Every straight line has a specific linear equation, which can be reduced to the standard form of y = mx + b. In that equation, the value of m is equal to the line's slope when plotted on a graph. The value of the constant, b, equals the y intercept, the point at which the line crosses the Y-axis (vertical line) of its graph. The slopes of lines that are perpendicular or parallel have very specific relationships, so if you reduce two lines' equations to their standard form, the geometry of their relationship becomes clear.

Reduce the two linear equations to their standard form, with the y variable alone on one side, the x variable and constant (if any) on the other, and the coefficient of y equal to 1. For instance, given a line with the equation 8x -- 2y + 4 = 0, first add 2y to both sides to get 8x + 4 = 2y, then divide both sides by 2 to yield 4x+2 = y. In this case, the slope of the line is 4 (it rises 4 units for every 1 unit sideways) and the intercept is 2 (it crosses the Y intercept at 2).

Compare the slopes of the two lines for parallelism. If the slopes are identical, as long as the intercepts are not equal, the lines are parallel. For example, the line with the equation 4x -- y + 7 = 0 is parallel to 8x -- 2y +4 = 0, while 2x - 3y -- 3 = 0 is not parallel, because its slope equals 2/3 instead of 4.

Compare the two slopes for perpendicularity. Perpendicular lines slope in opposite directions, so one line has a positive slope, and the other has a negative slope. The slope of one line must be the negative reciprocal of the other for the two to be perpendicular: the second line's slope must equal -1 divided by the slope of the first line. For example, lines with slopes of -2 and 1/2 are perpendicular, because -2 is the negative reciprocal of 1/2.