Slope Calculator
Calculate the slope (gradient) of a line passing through two points. Find the equation of the line in slope-intercept and point-slope forms, plus angle of inclination and perpendicular slope.
What is slope and how is it calculated?
Slope (m) measures the steepness and direction of a line. Formula: m = (y₂ - y₁) / (x₂ - x₁) or "rise over run". Positive slope = line goes up left to right. Negative slope = line goes down left to right. Zero slope = horizontal line. Undefined slope = vertical line (division by zero). Example: Points (2,3) and (5,9): m = (9-3)/(5-2) = 6/3 = 2. For every 1 unit right, line rises 2 units. Used in: Road grades (6% = 0.06 slope), roof pitch, stairs, ramps, graphing linear equations.
How do I find the equation of a line from slope and a point?
Use point-slope form: y - y₁ = m(x - x₁), where m = slope, (x₁, y₁) = known point. Example: Slope = 3, point (2,5). Equation: y - 5 = 3(x - 2). Simplify: y - 5 = 3x - 6. Standard form: y = 3x - 1. Or slope-intercept form: y = mx + b. Find b by plugging in point: 5 = 3(2) + b, so b = -1. Thus y = 3x - 1. Use this to find any point on the line: at x = 4, y = 3(4) - 1 = 11.
What does the slope tell you about parallel and perpendicular lines?
Parallel lines: Same slope. Lines y = 2x + 3 and y = 2x - 5 are parallel (both m = 2). Never intersect. Perpendicular lines: Slopes are negative reciprocals (multiply to -1). If line 1 has m = 2, perpendicular line has m = -1/2. Example: y = 3x + 1 and y = (-1/3)x + 4 are perpendicular (3 × -1/3 = -1). They intersect at 90°. Special cases: Horizontal (m=0) is perpendicular to vertical (m=undefined). Used in: Construction (perpendicular walls), coordinate geometry, engineering.
How do you interpret slope in real-world applications?
Slope represents rate of change. Examples: Speed/velocity: m = distance/time (slope = 60 means 60 mph). Cost: m = total cost/quantity (slope = 5 means $5 per item). Temperature change: m = temp change/time (slope = -3°F/hour means cooling 3° per hour). Road grade: 6% slope = 0.06 = 6 ft rise per 100 ft horizontal (6:100). Roof pitch: 6:12 pitch = 6" rise per 12" run = slope of 0.5. Economics: marginal cost, rate of inflation. Slope magnitude = how fast change occurs. Sign = direction of change (positive = increasing, negative = decreasing).
What are the different forms of linear equations?
Slope-intercept form: y = mx + b. m = slope, b = y-intercept. Easiest for graphing. Example: y = 2x + 3 (slope 2, y-int 3). Point-slope form: y - y₁ = m(x - x₁). Best when you know slope and one point. Standard form: Ax + By = C. No fractions, A positive. Useful for finding intercepts quickly. Example: 2x + 3y = 6. Two-point form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁). When you only know two points. Convert between forms as needed. Slope-intercept is most common in algebra and calculus.