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.

Line Intersection Calculator: find where two lines meet

This line intersection calculator helps you find the exact point where two lines cross, or tells you if they are parallel or the same line. It is useful for geometry homework, graphing, engineering sketches, and any situation where you need the intersection point of two linear equations. You can enter lines in standard form or slope-intercept form and get a clear answer plus a visual preview.

Line intersection basics in plain language

In a 2D coordinate plane, two distinct lines can intersect at one point, never intersect (parallel lines), or overlap completely (coincident lines). The calculator converts your inputs into a standard form and checks whether the lines have a unique solution. If the slopes are the same, the lines are either parallel or identical; if the slopes differ, they meet exactly once.

How to use the calculator

  1. Choose a format for each line: standard form Ax + By = C or slope-intercept y = mx + b.
  2. Enter the coefficients or slope and intercept for Line 1 and Line 2.
  3. Press Calculate (or hit Enter) to compute the intersection.
  4. Read the result: a single point (x, y), or a message saying the lines are parallel or coincident.
  5. Check the preview chart to see the two infinite lines and the intersection point in red.

Why it works

The intersection point solves a system of two linear equations. In standard form, the lines are written as A1x + B1y = C1 and A2x + B2y = C2. The calculator uses the determinant Delta = A1B2 - A2B1 to decide the outcome: if Delta is not zero, there is a unique intersection; if Delta is zero, the lines are parallel or coincident depending on whether the constants match proportionally.

Real-world examples

  • Mapping and navigation: Find where two roads, paths, or bearings intersect.
  • Design and drafting: Locate crossing points when laying out walls or components.
  • Computer graphics: Solve where guidelines or rays cross in a coordinate system.
  • Algebra practice: Verify solutions to systems of linear equations quickly.

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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