Slope Between Two Points Calculator with Steps
Calculate slope between two points with complete step-by-step explanations. Learn rise over run, slope formula, and line equations. Free calculator.
Quick Answer
Slope (m) = (y₂ - y₁) / (x₂ - x₁), representing rise over run between two points. Positive slope rises left to right, negative slope falls, zero slope is horizontal, undefined slope is vertical. Essential for understanding line behavior and rates of change.
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Calculate With Full ToolWhat is Slope?
Slope measures the steepness and direction of a line between two points. It represents the rate of change in the vertical direction (rise) for each unit of change in the horizontal direction (run). Slope is fundamental in coordinate geometry.
How Slope Works
The slope is calculated as the difference in y-coordinates divided by the difference in x-coordinates between two points. This ratio tells us how much the line rises or falls for each unit of horizontal distance, providing insight into the line's behavior.
Types of Slopes
Positive slope rises from left to right, negative slope falls, zero slope is horizontal, and undefined slope is vertical. Each type represents different line behaviors and has specific applications in mathematics and real-world scenarios.
Slope Formula
m = (y₂ - y₁) / (x₂ - x₁)
Slope Formula: m = (y₂ - y₁) / (x₂ - x₁)
Where: (x₁, y₁) and (x₂, y₂) are two points on the line
Rise: y₂ - y₁ (vertical change)
Run: x₂ - x₁ (horizontal change)
Point-Slope Form: y - y₁ = m(x - x₁)
Step-by-Step Example
Example: Find slope between points (2, 3) and (6, 11)
Step 1: Identify points: (x₁, y₁) = (2, 3), (x₂, y₂) = (6, 11)
Step 2: Calculate rise: y₂ - y₁ = 11 - 3 = 8
Step 3: Calculate run: x₂ - x₁ = 6 - 2 = 4
Step 4: Divide rise by run: m = 8 ÷ 4 = 2
Step 5: Interpret: For every 1 unit right, the line rises 2 units
The slope of 2 means the line rises 2 units vertically for every 1 unit of horizontal movement. This positive slope indicates the line rises from left to right with moderate steepness.
Who Should Use This Calculator?
Students
Learn coordinate geometry and algebra concepts
Engineers
Calculate gradients and rates of change
Economists
Analyze trends and marginal changes
Architects
Design ramps and inclined structures
Frequently Asked Questions
What does a slope of 0 mean?
A slope of 0 means the line is horizontal. There is no vertical change as x increases, so y remains constant. This represents a constant function with no rate of change.
When is slope undefined?
Slope is undefined when the run (x₂ - x₁) equals 0, which occurs for vertical lines. This represents an infinite rate of change, as x doesn't change while y varies.
How does slope relate to real-world applications?
Slope represents rates of change in real-world scenarios like speed (distance/time), economic growth (GDP/time), or incline angles (height/distance). It helps quantify relationships between variables and predict behavior.
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