Enter two line equations and compute their intersection point (x, y)
x +y += 0
x +y += 0
Result
Intersection Point
Step-by-Step Derivation
Line Intersection Formula
Given A₁x + B₁y + C₁ = 0 and A₂x + B₂y + C₂ = 0 D = A₁B₂ - A₂B₁, Dx = C₁B₂ - C₂B₁, Dy = A₁C₂ - A₂C₁ If D ≠ 0, the intersection is ( -Dx/D , -Dy/D )
The intersection of two lines is the common solution that satisfies both line equations. Using Cramer's rule (determinant method), we solve the system of two linear equations to obtain the x and y coordinates of the intersection.
⚠Note: Parallel lines (D = 0) have no intersection; coincident lines have infinitely many intersection points and require separate judgment.
What Is a Line Intersection?
The intersection of two lines is the unique point that satisfies both line equations simultaneously. Finding the intersection means solving the two line equations together as a system of linear equations.
Cramer's Rule
Uses the determinant D = A₁B₂ - A₂B₁ to compute the intersection coordinates. A classic method for solving two-variable linear systems.
Parallel → No Intersection
When two lines are parallel (D=0), they have the same slope and never intersect. The system has no solution.
Coincident → Infinite Intersections
When two lines are identical (D=0), they have infinitely many intersection points. The system has infinitely many solutions.
Geometric Meaning
The intersection point is the unique location where two lines cross each other when plotted on a coordinate plane.
💡 Teaching Example: Lines x - y - 1 = 0 and x + y - 3 = 0.
System: { x - y = 1; x + y = 3 } → Adding gives 2x = 4 → x = 2, y = 1 → Intersection at (2, 1).
Applications
Linear ProgrammingComputer GraphicsSurveyingRoad DesignRobot LocalizationData Analysis
Frequently Asked Questions
Do parallel lines have an intersection point?▼
Parallel lines have no intersection. When the determinant D = A₁B₂ - A₂B₁ = 0, the two lines have the same slope and are parallel or coincident, with no unique intersection.
What is Cramer's rule?▼
Cramer's rule uses determinants to solve systems of linear equations. For the system A₁x + B₁y = -C₁, A₂x + B₂y = -C₂, let D = A₁B₂ - A₂B₁, Dₓ = -C₁B₂ + C₂B₁, Dᵧ = -A₁C₂ + A₂C₁, then x = Dₓ/D, y = Dᵧ/D.
Do coincident lines count as having an intersection?▼
Coincident lines have infinitely many intersection points (all points lie on both lines). Cramer's rule cannot handle this case — you must check whether the two equations are scalar multiples of each other.
Can intersection coordinates be fractions?▼
Yes. The solutions from Cramer's rule may be rational numbers (fractions). This tool automatically displays them in decimal form.
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