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Consider the line 6x+5y=8.
What is the slope of a line parallel to this line?
What is the slope of a line perpendicular to this line?
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Answer :

Let's isolate the variable y.

6x+5y = 8

5y = -6x+8

y = (-6x+8)/5

y = (-6/5)x + 8/5

The result is in slope-intercept form y = mx+b

  • m = -6/5 = slope
  • b = 8/5 = y intercept

Recall these key facts:

  1. Parallel lines have equal slopes but different y intercepts.
  2. Perpendicular lines have negative reciprocal slopes (the y intercepts may be the same or may be different).

Fact 1 will mean that the slope of any parallel line to 6x+5y = 8 is m = -6/5

Meanwhile, fact 2 will mean we flip the fraction (aka reciprocal) and flip the sign from negative to positive. The -6/5 becomes 5/6 which is the perpendicular slope. Notice the original slope and perpendicular slope multiply to -1. This applies to any pair of perpendicular lines as long as neither line is vertical nor horizontal.

I recommend that you stick to fraction form since the numerator and denominator of the slope fraction describe rise and run respectively.

slope = rise/run

rise = how far you move up or down

run = how far you move to the right

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Answers:

  • Parallel slope = -6/5
  • Perpendicular slope = 5/6

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