Intersection of Two Lines Calculator

The intersection of two lines is the point where they cross each other. In 2D space, two lines intersect at exactly one point — unless they are parallel (no intersection) or identical (infinite intersections). The intersection point satisfies both line equations simultaneously.

Given y = a₁x + b₁ and y = a₂x + b₂, set the right-hand sides equal: a₁x + b₁ = a₂x + b₂. Solve for x: x = (b₂ − b₁) / (a₁ − a₂). Then substitute that x back into either equation to find y.

Two lines in 2D will intersect if and only if they have different slopes (a₁ ≠ a₂). If the slopes are equal but the y-intercepts differ, the lines are parallel and never meet. If both the slopes and y-intercepts are equal, the lines are identical and overlap entirely.

Set x + 3 = 2x + 1, which gives x = 2. Substituting back: y = 2 + 3 = 5. So the lines intersect at the point (2, 5).

No — in 3D space, two lines can be non-parallel yet still not intersect. These are called skew lines. True intersection in 3D requires both lines to share a common point, which means their parametric equations must yield the same (x, y, z) for some parameter values.

If both lines have the same slope (a₁ = a₂) but different y-intercepts, they run in the same direction and never meet. There is no point of intersection, and the calculator will indicate this as parallel lines.

Only if they are the same line. Two distinct lines in a plane can intersect at most at one single point. If the lines are identical (same slope and same y-intercept), every point on the line is an intersection point.

This calculator uses the slope-intercept form y = ax + b, where 'a' is the slope and 'b' is the y-intercept. You enter a and b for each of the two lines, and the tool solves the system of equations to find the exact intersection coordinates.