Slope Calculator

Slope is one of the most fundamental concepts in mathematics — it quantifies the steepness and direction of a straight line between any two points on a coordinate plane. Slope appears everywhere: in physics (speed is slope on a distance-time graph), economics (marginal cost), engineering (road grades), and data science (linear regression).

Core formulas:

Slope (m) = (y₂ − y₁) / (x₂ − x₁) = Rise / Run

Slope-Intercept Form: y = mx + b

Point-Slope Form: y − y₁ = m(x − x₁)

Y-Intercept: b = y₁ − m × x₁

Distance Between Two Points: d = √((x₂ − x₁)² + (y₂ − y₁)²)

Midpoint: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Variable definitions:

• m: slope; the ratio of vertical change (rise) to horizontal change (run)
• b: y-intercept; where the line crosses the y-axis (x = 0)
• Rise: vertical change between two points (y₂ − y₁)
• Run: horizontal change between two points (x₂ − x₁)
• (x₁, y₁) and (x₂, y₂): any two distinct points on the line

Worked example: Two points: (2, 3) and (8, 15) Slope = (15 − 3) / (8 − 2) = 12 / 6 = 2 Y-intercept: b = 3 − 2 × 2 = −1 Line equation: y = 2x − 1 Distance = √((8−2)² + (15−3)²) = √(36 + 144) = √180 = 13.42 Midpoint = ((2+8)/2, (3+15)/2) = (5, 9)

Interpreting slope values:

Perpendicular and parallel slopes:

• Parallel lines share the same slope: m₁ = m₂
• Perpendicular lines: m₁ × m₂ = −1 (slopes are negative reciprocals)

Angle of inclination: θ = arctan(m) — at slope = 1, the angle is exactly 45°; at slope = 10, the angle is ~84.3°.