Slope measures the steepness and direction of a line. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line: m = rise / run = (y₂ − y₁) / (x₂ − x₁). A slope of 2 means the line rises 2 units for every 1 unit it moves to the right. Slope can be positive (line rises left to right), negative (line falls left to right), zero (horizontal line), or undefined (vertical line).

How do you find the slope between two points?

To find the slope between (x₁, y₁) and (x₂, y₂): subtract the y-coordinates (y₂ − y₁) to get the rise, subtract the x-coordinates (x₂ − x₁) to get the run, then divide rise by run: m = (y₂ − y₁) / (x₂ − x₁). For points (1, 3) and (4, 9): rise = 9 − 3 = 6, run = 4 − 1 = 3, m = 6/3 = 2. If x₁ = x₂ (both points have the same x-coordinate), the run is 0 and the slope is undefined — the line is vertical.

What is rise over run?

"Rise over run" is the plain-English description of slope. "Rise" is how much the line goes up (or down, if negative) between two points. "Run" is how much it goes sideways (always measured left to right). Slope = rise ÷ run. A slope of 3/4 means for every 4 units you move to the right, the line rises 3 units. This interpretation makes slope intuitive: a slope of 1 is a 45-degree line; a slope of 0.1 is nearly flat; a slope of 10 is nearly vertical.

What does a negative slope mean?

A negative slope means the line falls as you move from left to right — it has a downward trend. For example, a slope of −2 means the line drops 2 units for every 1 unit you move to the right. Negative slopes are common in economics (inverse relationships, like price and demand), physics (deceleration), and everyday situations like a downhill road grade. The absolute value of the slope tells you how steep the line is; the sign tells you whether it rises or falls.

What is the y-intercept?

The y-intercept is the point where the line crosses the y-axis — where x = 0. In the slope-intercept form y = mx + b, the y-intercept is b. To find it: b = y₁ − m·x₁ (use any point on the line with the slope). For a line through (2, 5) with slope 3: b = 5 − 3(2) = 5 − 6 = −1. The line equation is y = 3x − 1. The y-intercept represents the starting value of y when the independent variable (x) is zero — meaningful in many applied contexts (initial balance, starting temperature, etc.).

How do you write the equation of a line?

The most common form is slope-intercept: y = mx + b, where m is the slope and b is the y-intercept. To write the equation from two points: (1) Find the slope m = (y₂−y₁)/(x₂−x₁). (2) Find the y-intercept b = y₁ − m·x₁. (3) Substitute into y = mx + b. For points (0, 3) and (2, 7): m = (7−3)/(2−0) = 2, b = 3, equation: y = 2x + 3. For vertical lines (undefined slope), the equation is x = constant (e.g., x = 4). For horizontal lines (slope = 0), the equation is y = constant.

What is the difference between slope and grade?

Slope and grade measure the same thing — rise divided by run — but expressed differently. Slope is a pure ratio or fraction (e.g., 0.08 or 1/12), while grade (also called gradient) is expressed as a percentage: grade % = slope × 100. A slope of 0.08 is an 8% grade. Road grades are always stated as percentages: U.S. interstate highways are limited to about 6% grade; mountain roads can reach 10–12%; the steepest streets in San Francisco exceed 30%. ADA-compliant wheelchair ramps must stay at or below an 8.33% grade (1:12 slope).

How steep is a slope of 1, 2, or 10?

The angle a slope makes with horizontal (the inclination angle) gives an intuitive feel for steepness. A slope of 1 (m = 1) corresponds to a 45° angle — a "staircase" diagonal. A slope of 2 is about 63°, noticeably steeper than 45°. A slope of 10 is about 84°, nearly vertical. In practical terms: gentle walking paths have slopes below 0.1 (< 6°); comfortable stairs are around 0.7–0.8 (35–40°); ladders and climbing walls approach slopes of 1–2 (45–65°). A slope greater than about 5.67 (80°) is too steep to walk up safely without handholds.

How does slope connect to calculus?

In calculus, slope becomes the derivative. For a curved function y = f(x), the slope at any specific point is the instantaneous rate of change — found by taking the derivative f′(x) and evaluating it at that point. The slope formula from algebra, (y₂ − y₁)/(x₂ − x₁), computes the average rate of change (the slope of a secant line) between two points. As the two points get infinitely close together, that average slope approaches the instantaneous slope (tangent line). This limiting process is the definition of the derivative, making slope the conceptual foundation of differential calculus.

Slope Calculator — Find Slope Between Two Points

Calculates slope, y-intercept, line equation, distance, angle of inclination, rise, and run from two points.

How to Use the Slope Calculator

This slope calculator finds the slope, y-intercept, and full line equation between any two points — enter the coordinates (x₁, y₁) and (x₂, y₂) and it instantly returns the slope, y-intercept, full line equation, distance between the points, angle of inclination, rise, and run. Vertical lines (where x₁ = x₂) are handled gracefully — the calculator shows "undefined slope" and the equation x = constant. For finding the center of a line segment, see our midpoint calculator.

The result updates as you type. Use the Share button to copy a link with your inputs pre-filled for sharing or saving.

The Slope Formula

Slope is defined as the ratio of vertical change to horizontal change between any two points on a line:

m = (y₂ − y₁) / (x₂ − x₁) = rise / run

Once you have the slope, the y-intercept b follows from the point-slope form:

b = y₁ − m · x₁

And the complete line equation in slope-intercept form is:

y = mx + b

The angle of inclination θ (the angle the line makes with the positive x-axis) is:

θ = arctan(m), expressed in degrees

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Interpreting Slope

The slope tells you two things: the direction of the line and how steep it is.

Positive slope (m > 0) — line rises from left to right. Example: a savings account balance growing over time.
Negative slope (m < 0) — line falls from left to right. Example: a car depreciating in value over time.
Zero slope (m = 0) — horizontal line. The y-value does not change as x increases. Example: a constant temperature reading.
Undefined slope — vertical line (x₁ = x₂). The run is 0, so the ratio is undefined. Vertical lines cannot be written in slope-intercept form; their equation is x = constant.

The magnitude of the slope determines steepness: m = 0.1 is nearly flat; m = 1 is 45°; m = 10 is nearly vertical. For very steep lines, the angle of inclination (arctan of the slope) is more intuitive than the slope itself.

Slope in Real Life

Slope has direct, tangible meanings in many applied contexts:

Road grades — a 6% grade means a rise of 6 feet per 100 feet of horizontal run (m = 0.06). Steep for a road; typical for mountain highways.
Ramps and accessibility — ADA wheelchair ramps must not exceed a 1:12 slope (rise:run = 1/12 ≈ 8.3%), meaning 1 inch of rise per 12 inches of run.
Roofing — roof pitch is expressed as rise-over-run (e.g., 6/12 means 6 inches of rise per 12 inches of run, or m = 0.5).
Economics — in a linear supply or demand curve, slope tells you how sensitive price is to quantity (or vice versa).
Physics — velocity is the slope of a position-time graph; acceleration is the slope of a velocity-time graph.

Parallel and Perpendicular Lines

The relationship between slopes of special line pairs:

Parallel lines — same slope (m₁ = m₂), different y-intercepts. Example: y = 2x + 1 and y = 2x − 3 are parallel.
Perpendicular lines — slopes are negative reciprocals: m₁ × m₂ = −1, so m₂ = −1/m₁. Example: if m₁ = 3, then m₂ = −1/3. The angle between perpendicular lines is exactly 90°.
Collinear points — three points are collinear (on the same line) if and only if the slope between any two pairs of them is equal.

For distance-related geometry problems, our distance formula calculator computes the length of the segment between any two points.

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Step-by-Step Example

Problem: Find the slope, y-intercept, and equation of the line through (−2, 1) and (4, 13).

Rise = 13 − 1 = 12
Run = 4 − (−2) = 6
Slope m = 12 / 6 = 2
y-intercept b = 1 − 2·(−2) = 1 + 4 = 5
Line equation: y = 2x + 5
Distance: √(6² + 12²) = √(36 + 144) = √180 ≈ 13.42
Angle: arctan(2) ≈ 63.43°

Verification: plug x = −2 into y = 2x + 5: y = 2(−2) + 5 = 1 ✓. Plug x = 4: y = 2(4) + 5 = 13 ✓.

Sources & References

Slope of a Line — Khan Academy
Slope-Intercept Form — Math Is Fun

Slope Calculator