# Line Intersection Calculator

> Find the point where two lines cross from slope-intercept or ax + by = c form, the angle between them, and whether they are parallel or the same line.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/line-intersection-calculator

Two lines in a plane meet at exactly one point unless they are parallel. Solving the two equations together gives that point.

Also called: Intersection point calculator.

## Inputs
- Equation form (options: y = mx + b, ax + by = c; default slope)
- Line 1: slope m₁ (default 2)
- Line 1: intercept b₁ (default 1)
- Line 2: slope m₂ (default -1)
- Line 2: intercept b₂ (default 4)
- Line 1: a₁ (default 1)
- Line 1: b₁ (default 1)
- Line 1: c₁ (default 5)
- Line 2: a₂ (default 1)
- Line 2: b₂ (default -1)
- Line 2: c₂ (default 1)

## Results
- Intersection point (x, y)
- Result
- x
- y
- Acute angle between the lines

## Formula

```
For a₁x + b₁y = c₁ and a₂x + b₂y = c₂:  D = a₁b₂ − a₂b₁;  x = (c₁b₂ − c₂b₁) ÷ D;  y = (a₁c₂ − a₂c₁) ÷ D;  tan θ = |D| ÷ |a₁a₂ + b₁b₂|.  y = mx + b is −mx + y = b.
```

Where:
- a, b, c: coefficients of each line
- m, b: slope and y-intercept
- D: determinant; D = 0 means parallel or identical lines
- θ: acute angle between the lines

## Worked example
1. Set the lines equal: 2x + 1 = −x + 4, so 3x = 3 and x = 1.
2. Substitute: y = 2(1) + 1 = 3.
3. Angle: tan θ = |(m₁ − m₂) ÷ (1 + m₁m₂)| = |3 ÷ (−1)| = 3, so θ ≈ 71.5651°.

**Result:** The lines y = 2x + 1 and y = −x + 4 cross at (1, 3), at an angle of about 71.57°.

## What the result means

The intersection is the one (x, y) pair that satisfies both equations, so this is also a way to solve a pair of simultaneous linear equations.

## Assumptions and limits
- Both lines lie in the same plane (2D).
- A vertical line (x = constant) cannot be written as y = mx + b; use the ax + by = c form with b = 0.
- Lines in 3D usually do not meet at all (they are skew); this tool covers the plane only.

## Questions

### How do I enter a vertical line such as x = 3?

Choose ax + by = c and enter a = 1, b = 0, c = 3.

### When are two lines perpendicular?

When m₁ × m₂ = −1, or in standard form when a₁a₂ + b₁b₂ = 0.

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Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/line-intersection-calculator