The family of planes through the line of intersection of P1: a1x+b1y+c1z=d1 and P2: a2x+b2y+c2z=d2 is P1 + lambdaP2 = 0, i.e., (a1+lambdaa2)x+(b1+lambdab2)y+(c1+lambdac2)z = d1+lambda*d2.

This one-parameter family generates all planes through the intersection line except P2 itself (obtained in the limit lambda -> infinity; to include it, write lambda*P1+P2=0 with lambda=0).

Typical problem types: (1) Find the plane through the intersection of two given planes and satisfying an additional condition (passing through a point, perpendicular to a plane, parallel to a line, etc.). (2) Set up the family, apply the condition to determine lambda, then write the specific plane.

Example: Plane through intersection of x+y+z=1 and 2x-y+z=2, passing through (0,0,0): (1+2*lambda)*0 + (1-lambda)0 + (1+lambda)0 = 1+2lambda. So 1+2lambda=0, lambda=-1/2. Plane: (1-1)x + (1+1/2)y + (1-1/2)z = 0, i.e., $\frac{3}{2}$y + $\frac{1}{2}$z = 0, or 3y+z=0.

This technique is cleaner than solving the line of intersection explicitly and then finding a plane through that line. It reduces to a single equation in one unknown (lambda).