How to find where two lines meet
Use the calculator
Write each line as Ax + By = C and enter its A, B and C coefficients. Zero is allowed for one direction coefficient: x = 3 is 1x + 0y = 3, while y = 2 is 0x + 1y = 2.
Select Calculate intersection. The result distinguishes a unique crossing, a perpendicular crossing, parallel lines and the same coincident line.
Determinant and formulas
For A₁x + B₁y = C₁ and A₂x + B₂y = C₂, Δ = A₁B₂ − A₂B₁. If Δ is not zero, x = (C₁B₂ − C₂B₁) / Δ and y = (A₁C₂ − A₂C₁) / Δ.
When Δ is zero, proportional checks including C distinguish parallel equations from two versions of the same line. Perpendicular lines satisfy A₁A₂ + B₁B₂ = 0.
Scope and limits
This page handles two infinite 2D lines only. It does not parse typed equations, fit a regression, test finite segments or edit a graph.
Displayed coordinates use at most two ordinary decimal places; the calculation keeps the browser number until display. Extremely large, ill-conditioned coefficients may require specialist exact arithmetic.
Worked examples
One perpendicular intersection
x + y = 5 and x − y = 1 have Δ = −2 and meet at (3, 2). Their normals have dot product 1×1 + 1×(−1) = 0, so the lines are perpendicular.
Parallel and coincident lines
2x + 2y = 4 and x + y = 3 are parallel because only A and B are proportional. With x + y = 2 instead, every coefficient is proportional and the lines are coincident.
Frequently asked questions
Can this handle vertical and horizontal lines?
Yes. Use B = 0 for a vertical line and A = 0 for a horizontal line.
Why is there no intersection point for coincident lines?
Coincident lines share infinitely many points, so there is no single ordered pair to report.
Does rounding change the relationship test?
The relationship is tested before display rounding with a scale-aware floating-point tolerance.