The slope of a straight line represents the rate of change of one variable with respect to another — specifically, how much the vertical value (y) changes for every unit change in the horizontal value (x).
Mathematically, slope is defined as:
slope (m) = rise / run = (change in y) / (change in x) = (y₂ − y₁) / (x₂ − x₁)
for any two points (x₁, y₁) and (x₂, y₂) on the line.
Because a straight line has a constant slope everywhere along it, that single number fully describes the line's steepness and direction:
- Positive slope: the line rises from left to right (as x increases, y increases).
- Negative slope: the line falls from left to right (as x increases, y decreases).
- Zero slope: the line is horizontal (y stays constant regardless of x).
- Undefined slope: the line is vertical (x stays constant; there's no "run," so division by zero occurs).
The magnitude of the slope indicates steepness: a larger absolute value means a steeper line, while a value close to zero means a nearly flat line.
Beyond pure geometry, slope has real-world interpretations depending on context. In physics, the slope of a position-vs-time graph represents velocity, and the slope of a velocity-vs-time graph represents acceleration. In economics, the slope of a cost function might represent marginal cost — the additional cost per unit produced. In everyday terms, like a road's incline, slope reflects how much elevation changes per unit of horizontal distance.
In algebra, slope also appears in the equation of a line, most commonly the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis). Slope is also used to determine whether two lines are parallel (equal slopes) or perpendicular (slopes that are negative reciprocals of each other).