How to use this calculator
- Enter or choose equation form.
- The intersection point (x, y) updates as you type. Use Copy result or Share to keep it.
Method
For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: D = a₁b₂ − a₂b₁; x = (c₁b₂ − c₂b₁) ÷ D; y = (a₁c₂ − a₂c₁) ÷ D; tan θ = |D| ÷ |a₁a₂ + b₁b₂|. y = mx + b is −mx + y = b.
- a, b, c
- coefficients of each line
- m, b
- slope and y-intercept
- D
- determinant; D = 0 means parallel or identical lines
- θ
- acute angle between the lines
Worked example
- Set the lines equal: 2x + 1 = −x + 4, so 3x = 3 and x = 1.
- Substitute: y = 2(1) + 1 = 3.
- Angle: tan θ = |(m₁ − m₂) ÷ (1 + m₁m₂)| = |3 ÷ (−1)| = 3, so θ ≈ 71.5651°.
The lines y = 2x + 1 and y = −x + 4 cross at (1, 3), at an angle of about 71.57°.
What the result means
The intersection is the one (x, y) pair that satisfies both equations, so this is also a way to solve a pair of simultaneous linear equations.
Assumptions and limits
- Both lines lie in the same plane (2D).
- A vertical line (x = constant) cannot be written as y = mx + b; use the ax + by = c form with b = 0.
- Lines in 3D usually do not meet at all (they are skew); this tool covers the plane only.
Questions
How do I enter a vertical line such as x = 3?
Choose ax + by = c and enter a = 1, b = 0, c = 3.
When are two lines perpendicular?
When m₁ × m₂ = −1, or in standard form when a₁a₂ + b₁b₂ = 0.