The equation of a plane through the intersection of π₁ = 0 and π₂ = 0 has the form

1. Let π₁ and π₂ be linear expressions in x, y, z each equal to zero on its plane. 2. Their linear combination π₁ + λπ₂ vanishes at any common point of the two planes. 3. Varying λ traces the pencil of planes through the intersection line. 4. So the family is π₁ + λπ₂ = 0. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.6.5_