Line Equation from Two Points Calculator
Find the equation of a line passing through two points in 2D or 3D space
Line Equation Calculator
First Point
Second Point
Invalid Input
Example: Line Through (1,1) and (3,5)
Given Points
Point 1: (1, 1)
Point 2: (3, 5)
Step-by-Step Solution
Step 1: Calculate slope: m = (y₂ - y₁) / (x₂ - x₁) = (5 - 1) / (3 - 1) = 4/2 = 2
Step 2: Calculate y-intercept: b = y₁ - m × x₁ = 1 - 2 × 1 = -1
Step 3: Write equation: y = 2x - 1
Line Equation Forms
Slope-Intercept
y = mx + b
Most common form
Standard Form
Ax + By + C = 0
Integer coefficients
Two-Point Form
(y - y₁) = m(x - x₁)
Direct from points
Key Formulas
Tips
Vertical lines have undefined slope
Horizontal lines have zero slope
Parallel lines have equal slopes
Perpendicular lines have negative reciprocal slopes
Understanding Line Equations from Two Points
2D Line Equations
Given two points (x₁, y₁) and (x₂, y₂), there exists a unique line passing through both points. This line can be expressed in several mathematical forms.
Slope-Intercept Form
The slope-intercept form y = mx + b is the most commonly used form where:
- •m is the slope: (y₂ - y₁) / (x₂ - x₁)
- •b is the y-intercept: y₁ - m × x₁
Standard Form
The standard form Ax + By + C = 0 where:
- •A = y₂ - y₁
- •B = x₁ - x₂
- •C = y₁(x₂ - x₁) - (y₂ - y₁)x₁
3D Line Equations
In 3D space, a line is defined using parametric equations or vector form since traditional y = mx + b doesn't extend to three dimensions.
Vector Form
The vector form expresses the line as: (x, y, z) = t⃗v + P₁
- •⃗v is the direction vector [x₂-x₁, y₂-y₁, z₂-z₁]
- •P₁ is one of the given points
- •t is a parameter that varies
Applications
- ✓Computer graphics and 3D modeling
- ✓Engineering design and CAD systems
- ✓Physics simulations and trajectories
- ✓Navigation and GPS systems