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Slope Formulas
Understanding Slope Values
出典と方法論
m = (y₂ - y₁) / (x₂ - x₁)Slope of a line from two points
How to Use the Slope Calculator
The slope calculator finds the slope of a line from two points, calculates the y-intercept, and provides the equation of the line in multiple forms. Visualize your line on an interactive graph that updates as you type.
The Slope Formula
Given two points (x₁, y₁) and (x₂, y₂), the slope is calculated as:
m = (y₂ - y₁) / (x₂ - x₁) = rise / run
The slope represents how much y changes for each unit change in x.
Understanding Slope Values
- Positive slope: Line goes up from left to right (m > 0)
- Negative slope: Line goes down from left to right (m < 0)
- Zero slope: Horizontal line (m = 0)
- Undefined slope: Vertical line (division by zero)
Forms of Linear Equations
- Slope-Intercept Form: y = mx + b (where m is slope, b is y-intercept)
- Point-Slope Form: y - y₁ = m(x - x₁)
- Standard Form: Ax + By = C
Parallel and Perpendicular Lines
Two lines are related if their slopes have special relationships:
- Parallel lines: Have the same slope (m₁ = m₂)
- Perpendicular lines: Have negative reciprocal slopes (m₁ × m₂ = -1)
Finding the Y-Intercept
Once you have the slope, substitute one point into y = mx + b and solve for b:
b = y₁ - m × x₁
Related Calculators
For finding the distance between two points, use our Pythagorean theorem calculator. For quadratic equations, try our scientific calculator.
よくある質問
Use the formula m = (y₂ - y₁) / (x₂ - x₁). Subtract the y-coordinates (rise) and divide by the difference in x-coordinates (run). Example: for points (2, 3) and (6, 11), slope = (11 - 3) / (6 - 2) = 8 / 4 = 2.
A negative slope means the line goes downward from left to right. As x increases, y decreases. For example, a slope of -2 means y decreases by 2 for every 1 unit increase in x. Negative slopes are common in decay, depreciation, and inverse relationships.
A horizontal line has a slope of zero. Since all points have the same y-value, the rise is 0, and 0 divided by any run equals 0. The equation is y = b (constant). Example: y = 5 is a horizontal line at height 5.
A vertical line has an undefined slope (not zero). Since all points have the same x-value, the run is 0, and division by zero is undefined. Vertical lines cannot be expressed as y = mx + b; they use the form x = a.
Parallel lines have equal slopes. If two lines never intersect, they have the same steepness and direction. For example, y = 2x + 1 and y = 2x + 5 are parallel (both have slope 2) but have different y-intercepts.
Perpendicular lines have slopes that are negative reciprocals of each other. If one line has slope m, a perpendicular line has slope -1/m. Their product is -1. Example: a line with slope 2 is perpendicular to a line with slope -1/2.
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