Slope Formula

Slope measures how steep a line is: the change in y for each unit change in x.

Formulas

m = (y₂ − y₁) ÷ (x₂ − x₁)

slope-intercept form: y = mx + b

point-slope form: y − y₁ = m(x − x₁)

angle = arctan(m)

grade % = m × 100

Example

Through (2, 3) and (6, 11)

  1. Rise: 11 − 3 = 8
  2. Run: 6 − 2 = 4
  3. m = 8 ÷ 4 = 2
  4. b = 3 − 2 × 2 = −1

y = 2x − 1

Special cases

  • Horizontal line: slope 0.
  • Vertical line: slope undefined (the run is zero).
  • Parallel lines have equal slopes; perpendicular slopes multiply to −1.
  • A road “grade” of 8% rises 8 m for every 100 m of horizontal distance (about 4.6°).