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Math calculator

Line Equation Calculator

Equation of the line through two plane points. Same engine as Coordinate Geometry. Vertical is x = x₁. Coincident points are invalid. Not Pythagorean. Not a second engine. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 11/11 tests · Production surface contract 3/3 · v1.4.0
Input interpretation
Enter values to calculate.
Result
—
Model
The unique line through two distinct plane points, as y = mx + b or x = x₁.
Scope
Euclidean plane
Verification
Engine tested · 11/11 tests · Production surface contract 3/3 · Specification checked · v1.4.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
5 golden · 1 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.4.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Two-pointy − y₁ = m(x − x₁)
Slopem = (y₂ − y₁)/(x₂ − x₁)
StandardAx + By + C = 0
iSame engine as /coordinate-geometry. Distance and midpoint stay on that page. Vertical is x = x₁. Not CAS.

How to use

1

Enter two distinct points

(0, 1) and (2, 5) is y = 2x + 1. The distance page stays at /coordinate-geometry.

2

Read slope-intercept or x = x₁

y = 2x + 1, also 2x − y + 1 = 0. A vertical pair is x = x₁.

Example calculations

Common configurations with formula and result.

ϟ

Slope-intercept

(0, 1) to (2, 5)

m = 2
y = 2x + 1
ϟ

Vertical

(2, 0) to (2, 5)

x₂ = x₁
x = 2
ϟ

Horizontal

(0, 3) to (4, 3)

m = 0
y = 3

Line Equation calculator specification

Version 1.4.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Two distinct points determine a unique line. This page is mode=line on math.geometry.coordinate. Non-vertical lines are y = mx + b; vertical lines are x = x₁. Coincident points are invalid. Not a second engine. Not Pythagorean. Not CAS.
What it calculates
The unique line through two distinct plane points, as y = mx + b or x = x₁.
Inputs
  • x1
  • y1
  • x2
  • y2
Outputs
  • equation
  • standard
  • slope
  • intercept
  • A
  • B
  • C
  • vertical
Formula
y − y₁ = m(x − x₁)
Assumptions
  • Euclidean plane
  • Two distinct points
  • Not 3-D / spherical
  • Not Pythagorean
  • Not CAS
Units
  • length (consistent units)
Boundary conditions
  • missing coordinates → MISSING_REQUIRED_INPUT
  • coincident points → INVALID_INPUT
  • unknown mode (including cas, pythagorean) → INVALID_MODE
Example
x1=0 y1=1 x2=2 y2=5 → y = 2x + 1
Validation cases

2 published on this page · 11/11 tests · Production surface contract 3/3 · View evidence

  • x1=0 y1=1 x2=2 y2=5 → y = 2x + 1
  • mode=cas x1=0 y1=1 x2=2 y2=5 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.4.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dot Product Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → Cone Calculator Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. › → Dot Product Calculator Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. Not a matrix. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second coordinate engine?

No. It is mode=line on math.geometry.coordinate, the same engine as /calc/math/coordinate-geometry.

What if the points coincide?

There is no unique line. The engine returns INVALID_INPUT.

Where are distance and midpoint?

/calc/math/coordinate-geometry is the same engine: mode=distance and mode=midpoint.