Line Intersection Calculator
Find where two 2D lines intersect from standard form, slope-intercept form, or two points on each line. The calculator also detects parallel and coincident lines.
How to Find the Intersection of Two Lines
The intersection point of two lines is the solution of a two-equation linear system. This calculator accepts standard-form coefficients, slope-intercept form, or two points on each line.
The tool works with infinite lines, not finite line segments. Segment intersection requires an extra boundary check after the line intersection is found.
Line Intersection Formula
A1, B1, A2, and B2 are line coefficients; Delta is the system determinant.
xP is the x-coordinate of the intersection point.
yP is the y-coordinate of the intersection point.
A nonzero determinant means the lines have one intersection point. A zero determinant means they are parallel or coincident.
Supported Line Inputs
| Input form | Best for | Limit |
|---|---|---|
| Standard form coefficients | problems already written as Ax + By + | A and B cannot both be zero |
| Slope and intercept | known slope and y-intercept | vertical lines cannot be entered this way |
| Two points | when no equation has been written yet | the two points must be distinct |
Parallel and Coincident Lines
When the determinant is zero, there is no single intersection point. The calculator checks coefficient proportionality to distinguish parallel lines from the same line written twice.
| Condition | Geometry | Result |
|---|---|---|
| nonzero determinant | lines cross | one point |
| zero determinant and different constants | parallel lines | no intersection point |
| zero determinant and proportional coefficients | coincident lines | infinitely many common points |
| one vertical line | use standard form or two points | slope-intercept form is not available |
Worked Example
Take two lines: + 1 and. After rewriting them in standard form, solve the system for the shared point.
x and y are the coordinates of the shared point.
The nonzero determinant confirms that the lines meet at one point.
xP and yP are the coordinates of point P.
Substitution checks that the point satisfies both original line equations.
Common Mistakes
- Expecting a free-form equation parser. This tool uses numeric coefficients, slope-intercept values, or point coordinates.
- Entering a vertical line in slope-intercept mode. Use standard form or two points with the same x-coordinate instead.
- Confusing infinite lines with finite segments. Segment boundaries are not checked here.
- Using zero for both A and B in standard form, which does not define a line.
- Using two identical points, which cannot define a unique line.
- Ignoring rounding for nearly parallel lines, where a tiny determinant can make coordinates unstable.
Frequently Asked Questions
Sources and References
Calculations are based on the listed reference sources. Links open in a new tab.
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