Line Intersection Calculator — Solve for Intersection of Two Linear Equations

Find where two lines meet (or learn if they don’t). Private by design—everything runs locally in your browser.

Equations & Actions

Intersection = —

For Ax+By=C: if Δ = A₁B₂ − A₂B₁ ≠ 0, then x = (C₁B₂ − C₂B₁)/Δ, y = (A₁C₂ − A₂C₁)/Δ. If Δ = 0 and constants align proportionally ⇒ coincident; otherwise ⇒ parallel. Tip: Press Enter in any field to calculate.

Preview

Shows the two infinite lines (not segments). The red dot is the unique intersection, when it exists.

Understanding Line Intersections

Two distinct lines in the plane intersect in at most one point. Using standard form A₁x + B₁y = C₁ and A₂x + B₂y = C₂, define the determinant Δ = A₁B₂ − A₂B₁.

  • Unique intersection if Δ ≠ 0.
  • Parallel (no solution) if Δ = 0 and the constants don’t match proportionally.
  • Coincident (infinite solutions) if all coefficients are proportional.

How This Calculator Works

  • Accepts inputs in Ax+By=C or y=mx+b and converts to standard form internally.
  • Solves the 2×2 system using determinants (Cramer’s Rule).
  • Displays a shareable URL and an instant visual.
  • Everything runs locally for full privacy.

5 Fun Facts about Line Intersections

Determinant decides fate

The little number Δ = A₁B₂ − A₂B₁ tells all: nonzero means a unique intersection; zero means parallel or the same line.

One test

Parallel isn’t always “no point”

If ratios A₁:A₂, B₁:B₂, and C₁:C₂ all match, the lines are identical—infinitely many intersections.

Coincident catch

Lines vs. segments

Two infinite lines almost always meet; two segments can miss even if the lines cross. Always check the bounding boxes for segment tests.

Geometry gotcha

Skew is a 3D twist

In 3D, lines can be skew: not parallel, never intersecting, and not coplanar. The 2D determinant trick no longer settles it.

Out of plane

Cramer’s Rule cameo

The intersection formulas are Cramer’s Rule for a 2×2 system. Linear algebra hiding in plain sight in every crossing of two lines.

Algebra under hood

Line Intersection: Frequently Asked Questions

What forms can I enter?

Use standard form Ax+By=C for maximum robustness (handles vertical lines). Use slope–intercept y=mx+b if both lines are non-vertical.

What if I get “parallel” or “coincident”?

Parallel means the lines never meet. Coincident means they’re the exact same infinite line (infinitely many intersection points).

How precise is the result?

Results are computed with full floating-point precision and displayed rounded; copy the LaTeX for exact symbolic form of the computed values.

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