✖️ Intersection of Two Lines Calculator
Find the intersection point of two lines
Line 1: a₁x + b₁y = c₁
Line 2: a₂x + b₂y = c₂
How to Use This Calculator
Enter Line 1 Equation
Input the coefficients a₁, b₁, and c₁ for the first line equation a₁x + b₁y = c₁.
Enter Line 2 Equation
Input the coefficients a₂, b₂, and c₂ for the second line equation a₂x + b₂y = c₂.
Click Find Intersection
Press "Find Intersection" to calculate the intersection point, or detect if lines are parallel or coincident.
Formula
System: a₁x + b₁y = c₁, a₂x + b₂y = c₂
Determinant: D = a₁b₂ - a₂b₁
If D ≠ 0: x = (c₁b₂ - c₂b₁)/D, y = (a₁c₂ - a₂c₁)/D
Cases:
- D ≠ 0: Unique intersection point
- D = 0 and lines are proportional: Coincident (infinite points)
- D = 0 and lines are not proportional: Parallel (no intersection)
Example: x + y = 5, x - y = 1
D = 1×(-1) - 1×1 = -2 ≠ 0
x = (5×(-1) - 1×1)/(-2) = (-6)/(-2) = 3
y = (1×1 - 1×5)/(-2) = (-4)/(-2) = 2
Intersection: (3, 2)
About Intersection of Two Lines Calculator
The Intersection of Two Lines Calculator finds the point where two lines intersect in 2D space, or detects if lines are parallel or coincident.
When to Use This Calculator
- Geometry: Find intersection points in coordinate geometry
- Mathematics: Solve systems of linear equations
- Engineering: Calculate crossing points in designs
- Computer Graphics: Detect line intersections for rendering
- Algebra: Solve linear equation systems
- Education: Learn about line intersections
Why Use Our Calculator?
- ✅ Multiple Cases: Handles intersection, parallel, and coincident lines
- ✅ Determinant Method: Uses Cramer's rule for solving
- ✅ Works with All Numbers: Handles decimals, fractions, and negatives
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Educational: Helps understand linear systems
- ✅ Completely Free: No registration required
Understanding Line Intersections
Two lines in 2D space can have three possible relationships: they intersect at a unique point, they are parallel (never meet), or they are coincident (same line, infinite intersection points).
- Unique intersection: Determinant D = a₁b₂ - a₂b₁ ≠ 0
- Parallel lines: D = 0 and lines are not proportional
- Coincident lines: D = 0 and lines are proportional
- Uses Cramer's rule for solving the system
- General form: ax + by = c
Real-World Applications
Navigation: Finding intersection points of routes, paths, or flight paths.
Computer Graphics: Detecting line intersections for collision detection, clipping, and rendering.
Engineering: Calculating crossing points in road networks, pipelines, or electrical circuits.
Frequently Asked Questions
How do I find the intersection of two lines?
Solve the system of linear equations: a₁x + b₁y = c₁ and a₂x + b₂y = c₂. Use Cramer's rule: x = (c₁b₂ - c₂b₁)/D, y = (a₁c₂ - a₂c₁)/D, where D = a₁b₂ - a₂b₁ is the determinant.
What if lines don't intersect?
If lines are parallel (D = 0 and not proportional), there's no intersection. If lines are coincident (D = 0 and proportional), they have infinite intersection points (every point on the line).
What is the determinant?
The determinant D = a₁b₂ - a₂b₁ determines whether lines intersect. D ≠ 0 means unique intersection. D = 0 means parallel or coincident lines.
Can I use slope-intercept form?
Yes! Convert y = mx + b to ax + by = c by rearranging: -mx + y = b, so a = -m, b = 1, c = b.
How do I convert from point-slope form?
Point-slope: y - y₁ = m(x - x₁). Rearrange to -mx + y = y₁ - mx₁, so a = -m, b = 1, c = y₁ - mx₁.
What about 3D line intersections?
This calculator is for 2D lines. 3D line intersections are more complex and typically require parametric equations or vector methods.
Is this the same as solving simultaneous equations?
Yes! Finding line intersections is exactly solving a system of two linear equations in two variables. This calculator uses Cramer's rule, which is a method for solving such systems.