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Intersection of Two Lines

Find where two lines ax+by+c=0 meet using Cramer's rule. Handles intersecting, parallel, and coincident lines. D = a₁b₂ − a₂b₁ determines the case.

Concept Fundamentals
D = a₁b₂ − a₂b₁
Determinant
x = (b₁c₂−b₂c₁)/D
Cramer

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D = 0 means direction vectors (a₁,b₁) and (a₂,b₂) are parallel. Quick parallel check: a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Vertical line x = k: use a=1, b=0, c=−k.

Key quantities
D = a₁b₂ − a₂b₁
Determinant
Key relation
x = (b₁c₂−b₂c₁)/D
Cramer
Key relation

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Why: Line intersection is fundamental in collision detection, ray tracing, and solving linear systems.

How: Compute D = a₁b₂ − a₂b₁. If D ≠ 0, lines intersect at one point. Use Cramer's rule for x and y. If D = 0, lines are parallel or coincident.

D = 0 means direction vectors (a₁,b₁) and (a₂,b₂) are parallel.Quick parallel check: a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
Sources:Wikipedia - Cramer's rule

Run the calculator when you are ready.

Start CalculatingEnter coefficients for both lines in standard form.

Line Equations (ax + by + c = 0)

Line 1

Line 2

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

Cramer's rule uses determinants to solve linear systems.

— Algebra

📊

D = 0 means direction vectors are parallel.

— Geometry

Key Takeaways

  • • Standard form: a1x+b1y+c1=0a_1 x + b_1 y + c_1 = 0 and a2x+b2y+c2=0a_2 x + b_2 y + c_2 = 0
  • • Determinant D=a1b2−a2b1D = a_1 b_2 - a_2 b_1: D ≠ 0 → unique intersection
  • • Cramer's rule: x=fracb1c2−b2c1D,;y=fracc1a2−c2a1Dx = \\frac{b_1 c_2 - b_2 c_1}{D}, \\; y = \\frac{c_1 a_2 - c_2 a_1}{D}
  • • D = 0 → parallel or coincident (check if coefficients are proportional)
  • • Coincident = same line; parallel = distinct lines, no intersection

Did You Know?

Cramer's Rule

Cramer's rule uses determinants to solve linear systems. For 2×2 systems it gives explicit formulas for x and y.

Geometric Meaning

D = 0 means the direction vectors (a₁,b₁) and (a₂,b₂) are parallel, so the lines are parallel or coincident.

Slope-Intercept

Convert to y = mx + b: slope m = −a/b. Parallel lines have the same slope.

Applications

Used in collision detection, ray tracing, and solving systems of linear equations.

Three Cases

Exactly one point (intersect), no point (parallel), or infinitely many (coincident).

Numerical Stability

When D is very small, the lines are nearly parallel and results can be sensitive to rounding.

Understanding the Formulas

For lines a1x+b1y+c1=0a_1 x + b_1 y + c_1 = 0 and a2x+b2y+c2=0a_2 x + b_2 y + c_2 = 0, the determinant is:

D=a1b2−a2b1D = a_1 b_2 - a_2 b_1

If D ≠ 0, the intersection point is:

x=fracb1c2−b2c1D,quady=fracc1a2−c2a1Dx = \\frac{b_1 c_2 - b_2 c_1}{D}, \\quad y = \\frac{c_1 a_2 - c_2 a_1}{D}

Expert Tips

Quick Parallel Check

If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, lines are parallel. If all three ratios equal, they're coincident.

Vertical Line

For x = k, use a=1, b=0, c=−k. For y = k, use a=0, b=1, c=−k.

Avoid Zero

Ensure at least one of a, b is non-zero for each line. Otherwise the equation is invalid.

Verification

Substitute (x,y) into both equations to verify. Both should equal 0.

Frequently Asked Questions

When do two lines intersect?

When the determinant D = a₁b₂ − a₂b₁ ≠ 0. Then they cross at exactly one point.

When are lines parallel?

When D = 0 and the coefficients are not proportional. The lines never meet.

What does coincident mean?

The lines are the same. Every point on the line is an intersection. D = 0 and coefficients are proportional.

How do I enter a vertical line?

Use a=1, b=0, c=−k for the line x = k.

What is Cramer's rule?

A method to solve linear systems using determinants. For 2×2 systems it gives explicit formulas.

Can the intersection be at the origin?

Yes, when c₁ = c₂ = 0 and D ≠ 0, the intersection is (0, 0).

What if both lines are the same?

Then they are coincident. Enter proportional coefficients (e.g. 1,2,3 and 2,4,6).

How to Use This Calculator

  1. Enter coefficients for Line 1 (a₁x + b₁y + c₁ = 0) and Line 2 (a₂x + b₂y + c₂ = 0).
  2. Click a sample example to auto-fill and calculate.
  3. Click Calculate to find the intersection (or determine parallel/coincident).
  4. Review the result, visualization, and step-by-step solution.
  5. Use Copy Results to share.

Disclaimer: This calculator is for educational use. Ensure at least one of a, b is non-zero for each line.

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