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Line-Line Intersection — Definition, Formula & Examples

A line-line intersection is the single point where two non-parallel lines cross each other on a plane. You find it by solving both line equations simultaneously for the coordinates that satisfy both.

Given two distinct lines ℓ1\ell_1 and ℓ2\ell_2 in R2\mathbb{R}^2 that are not parallel, their intersection is the unique point (x,y)(x, y) that lies on both lines — equivalently, the solution to the system formed by their two linear equations.

Key Formula

x=c1b2−c2b1a1b2−a2b1,y=a1c2−a2c1a1b2−a2b1x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}, \qquad y = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1}
Where:
  • a1,b1,c1a_1, b_1, c_1 = Coefficients of the first line: $a_1 x + b_1 y = c_1$
  • a2,b2,c2a_2, b_2, c_2 = Coefficients of the second line: $a_2 x + b_2 y = c_2$
  • a1b2−a2b1a_1 b_2 - a_2 b_1 = Determinant of the coefficient matrix; must be nonzero (lines are not parallel)

How It Works

Write both lines in the form ax+by=cax + by = c. Use substitution or elimination to solve the two-equation system. If the lines have different slopes, you will always get exactly one solution point. If the lines are parallel (same slope, different intercepts), no intersection exists. If the equations describe the same line, every point is an intersection.

Worked Example

Problem: Find the intersection of the lines y = 2x + 1 and y = -x + 7.
Set the equations equal: Since both expressions equal y, set them equal to each other.
2x+1=−x+72x + 1 = -x + 7
Solve for x: Add x to both sides and subtract 1 from both sides.
3x=6  ⟹  x=23x = 6 \implies x = 2
Solve for y: Substitute x = 2 into either original equation.
y=2(2)+1=5y = 2(2) + 1 = 5
Answer: The two lines intersect at the point (2, 5)(2,\, 5).

Why It Matters

Finding where lines meet is central to solving systems of linear equations in algebra and precalculus. In physics and engineering, line-line intersections determine collision points, equilibrium positions, and where supply-meets-demand curves cross in economics.

Common Mistakes

Mistake: Forgetting to check whether the lines are parallel before solving.
Correction: If both lines have the same slope (or the determinant a1b2−a2b1=0a_1 b_2 - a_2 b_1 = 0), no unique intersection exists. Always compare slopes first or verify the determinant is nonzero.