Calculate the slope of a line from two points. Determine line equations, y-intercepts, and angles.
Slope measures the steepness and direction of a line in a coordinate plane — the rate at which the y-coordinate changes relative to a change in x. Expressed as rise over run (Δy/Δx), slope is foundational to algebra, calculus, physics, engineering, economics, and data analysis. Every linear relationship, trend line, and rate of change calculation relies on slope computation.
This calculator accepts two coordinate points (x₁,y₁) and (x₂,y₂) and computes: slope m = (y₂-y₁)/(x₂-x₁), the angle of inclination in degrees (arctan(m)), the Euclidean distance between the points, the midpoint coordinates, and the complete line equation in slope-intercept form (y = mx + b), point-slope form (y-y₁ = m(x-x₁)), and standard form (Ax + By = C).
Slope sign and magnitude carry geometric meaning. A positive slope means the line rises left to right — y increases as x increases. Negative slope means the line falls left to right. Zero slope produces a horizontal line (constant y regardless of x). Undefined slope (division by zero when x₁=x₂) corresponds to a vertical line. Greater absolute slope values indicate steeper lines: slope of 10 is steeper than slope of 2, and slope of -5 is steeper than slope of -1.
Perpendicular lines have slopes that are negative reciprocals of each other: if line L₁ has slope m, any perpendicular line L₂ has slope -1/m. Parallel lines have identical slopes. These relationships are fundamental in geometry for constructing perpendicular bisectors, angle bisectors, and verifying right angles in coordinate proofs.
In calculus, slope is generalized by the derivative. The slope of a tangent line to a curve at a point x equals the derivative f'(x) at that point. Average rate of change over an interval [a,b] equals the slope of the secant line: [f(b)-f(a)]/(b-a). The fundamental theorem of calculus connects slope (differentiation) to area (integration).
In physics, slope appears ubiquitously in kinematic graphs. Position vs time graph slope gives velocity (Δx/Δt). Velocity vs time graph slope gives acceleration (Δv/Δt). Force vs displacement graph slope gives spring constant in Hooke's law. Voltage vs current graph slope gives resistance (Ohm's law). Understanding slope as a rate of change ratio is therefore prerequisite to physical science literacy.
Slope m = (y₂ - y₁) / (x₂ - x₁) — rise divided by run. For points (3,5) and (7,13): m = (13-5)/(7-3) = 8/4 = 2. The slope is 2, meaning for every 1 unit increase in x, y increases by 2 units. Order of points doesn't matter as long as you're consistent: (y₂-y₁)/(x₂-x₁) = (y₁-y₂)/(x₁-x₂).
A slope of zero means the line is perfectly horizontal — y does not change regardless of x. The line equation simplifies to y = c (a constant). In physics, a horizontal position-time graph means the object is stationary; a horizontal velocity-time graph means constant velocity (zero acceleration). Horizontal lines have no rise — only run.
The angle θ that a line makes with the positive x-axis relates to slope by: m = tan(θ). Therefore θ = arctan(m). A slope of 1 corresponds to a 45° angle. A slope of √3 corresponds to 60°. Negative slopes correspond to angles between 90° and 180°. The angle is always between 0° and 180° (exclusive) for non-vertical lines.
Use point-slope form: y - y₁ = m(x - x₁), where (x₁,y₁) is the known point and m is the slope. To convert to slope-intercept form (y = mx + b), distribute and solve for y. To find b directly: b = y₁ - m×x₁. Example: slope=3, point (2,7): y-7 = 3(x-2) → y = 3x - 6 + 7 → y = 3x + 1.
Negative slope indicates an inverse relationship — as one variable increases, the other decreases. Examples: supply/demand curves (as price rises, quantity demanded falls), depreciation curves (asset value decreases with age), cooling curves (temperature decreases over time). In economics, the slope of a demand curve is negative by the law of demand.