Intersection of Two Lines Calculator

Solve a1x+b1y=c1 and a2x+b2y=c2 with Cramer's rule to find the intersection point. Detects parallel and coincident lines, and shows each slope, the angle between the lines, and a coordinate graph.

Input

Enter the coefficients of two lines a1x+b1y=c1 and a2x+b2y=c2 to compute their intersection point.

Line 1: a1x + b1y = c1

Line 2: a2x + b2y = c2

Result

Intersection point

(2, 2)

Slope of line 1

Slope of line 2

-1

Angle between lines

90 deg

The intersection is found with Cramer''s rule. A zero determinant means the lines are parallel or coincident.

How it works

The intersection point is found with Cramer's rule. When the determinant det = a1·b2 − a2·b1 is nonzero, x = (c1·b2 − c2·b1) / det and y = (a1·c2 − a2·c1) / det give a unique intersection.
When the determinant det is zero, the two lines are parallel or coincident. If the coefficients and constant terms are proportional the lines coincide (infinitely many shared points); otherwise they are parallel with no shared point.
The slope of a line a x + b y = c is −a / b when b is nonzero. When b is zero the line is vertical and its slope is undefined.
The angle between the two lines is computed from the angle of their normal vectors (a, b) and normalized to the acute side, from 0 to 90 degrees.

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