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Slope Calculator

Find the slope between two points, or convert a point and slope into a line equation. Results include slope, angle, distance between points, midpoint, and the full equation y = mx + b.

A — Two Points → Slope & Equation

Slope (m):1.333333 (rises left to right)
Angle:53.1301°
Distance:5
Midpoint:(1.5, 2)
Equation:y = 1.333333x + 0
Given: Point 1 = (0, 0), Point 2 = (3, 4)

Slope:
  m = (y₂ - y₁) / (x₂ - x₁)
  m = (4 - 0) / (3 - 0)
  m = 4 / 3
  m = 1.333333

Angle:
  θ = arctan(m) = arctan(1.333333)
  θ = 53.1301°

Distance:
  d = √((x₂-x₁)² + (y₂-y₁)²)
  d = √(3² + 4²)
  d = √(9 + 16)
  d = 5

Midpoint:
  M = ((x₁+x₂)/2, (y₁+y₂)/2)
  M = ((0+3)/2, (0+4)/2)
  M = (1.5, 2)

Line equation:
  y - y₁ = m(x - x₁)
  y - 0 = 1.333333(x - 0)
  y = 1.333333x + 0

The Slope Formula

Slope measures the steepness of a line. Given two points (x1, y1) and (x2, y2), the slope m is:

m = (y2 − y1) / (x2 − x1) = Δy / Δx

This is often described as "rise over run" — the vertical change divided by the horizontal change between any two points on the line.

Types of Slope

Slope value Type Visual
m > 0 Positive slope Line rises left to right
m < 0 Negative slope Line falls left to right
m = 0 Zero slope Horizontal line (y = constant)
m = undefined Undefined slope Vertical line (x = constant)
|m| > 1 Steep More vertical than 45°
|m| < 1 Shallow Less vertical than 45°
m = 1 Unit slope Exactly 45° angle

Line Equations

There are three common forms for writing the equation of a line:

Converting between forms: Start with point-slope, distribute m, then isolate y to get slope-intercept form. To get standard form, multiply through and move all terms to the left side.

Example: Line through (2, 5) with slope 3:

Parallel and Perpendicular Lines

Two lines are parallel if they have equal slopes and different y-intercepts. Parallel lines never intersect.

Two lines are perpendicular if their slopes are negative reciprocals: m1 × m2 = −1.

Given line Parallel slope Perpendicular slope
y = 2x + 3 (m = 2)m = 2m = −1/2
y = −3x + 1 (m = −3)m = −3m = 1/3
y = x (m = 1)m = 1m = −1
y = 0.5x (m = 0.5)m = 0.5m = −2

Distance and Midpoint Formulas

Given points (x1, y1) and (x2, y2):

Real-World Applications of Slope

Frequently Asked Questions

What is slope and how do you calculate it?

Slope (m) measures how steep a line is — the rise over run. Given two points (x₁, y₁) and (x₂, y₂), slope = (y₂ − y₁) / (x₂ − x₁). A positive slope rises left to right; a negative slope falls. A slope of 0 is horizontal; undefined slope is vertical (division by zero when x₁ = x₂).

What is the y-intercept and how is it found?

The y-intercept (b) is where the line crosses the y-axis (where x = 0). Once you know slope m and a point (x₁, y₁), use: b = y₁ − m × x₁. The full slope-intercept form is y = mx + b. For example, with slope 2 through point (3, 7): b = 7 − 2(3) = 1, giving y = 2x + 1.

What are parallel and perpendicular lines?

Parallel lines have equal slopes (m₁ = m₂) and never intersect. Perpendicular lines have slopes that are negative reciprocals: m₁ × m₂ = −1, or m₂ = −1/m₁. For example, if one line has slope 2, a perpendicular line has slope −1/2. Horizontal lines (slope 0) are perpendicular to vertical lines (undefined slope).

What is the point-slope form of a line?

Point-slope form is y − y₁ = m(x − x₁), where (x₁, y₁) is a known point and m is the slope. This is useful when you know the slope and one point but not the y-intercept. You can rearrange to slope-intercept form (y = mx + b) by distributing and solving for y.

How is slope used in real life?

Slope appears everywhere: road grade (a 6% grade means 6 units of rise per 100 units of run), roof pitch, ramp accessibility (ADA requires maximum 1:12 slope for wheelchair ramps), ski run difficulty, speed on a distance-time graph, and rate of change in economics (marginal cost = slope of cost curve). In physics, slope of a velocity-time graph gives acceleration.

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