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Geometry

Line Intersection Calculator

Find the intersection of two infinite lines written in standard form ax + by = c.

Use consistent dimensionsArea and volume units explained below
Geometry desk
Geometry desk

Match each dimension to the shape and keep units consistent.

See the method

Set the dimensions

Runs on your device

Your measurement

Your result

Let’s calculate.

Use the Calculate button to see your result.

Review the method below for assumptions and conventions.

How to use this tool

  1. Enter line 1 coefficient a, line 1 coefficient b, line 1 constant c, line 2 coefficient a, line 2 coefficient b, line 2 constant c.
  2. Select Calculate to view the result.
  3. Check the method and assumptions below before using the result.

The method, explained

The determinant Δ=a×e−b×d decides whether the intersection is unique. When nonzero, x=(c×e−b×f)/Δ and y=(a×f−c×d)/Δ.

A WORKED EXAMPLE

Using line 1 coefficient a = 1, line 1 coefficient b = 1, line 1 constant c = 3, line 2 coefficient a = 1, line 2 coefficient b = -1, line 2 constant c = 1, the result is (2, 1). Change these example inputs to match your task; use the method above to check each step.

Understanding your result

Yes. Standard form represents a vertical line with b=0. Unlike slope-intercept form, it does not need an infinite slope.

What to keep in mind

Infinite lines, not finite segments. Nearly parallel lines are numerically sensitive. Use the indicated units consistently. Ideal shapes are assumed; dimensions must describe the same object. Display rounding does not increase measurement precision.

Common questions

Can this handle vertical lines?

Yes. Standard form represents a vertical line with b=0. Unlike slope-intercept form, it does not need an infinite slope.

How can I check the result?

The determinant Δ=a×e−b×d decides whether the intersection is unique. When nonzero, x=(c×e−b×f)/Δ and y=(a×f−c×d)/Δ. The worked example uses editable inputs. Calculations run on your device; no external API or account is required.

Methodology maintained by ClarityKit. How these tools are built and checked.