The equation of the plane containing the line 2x - y + z = 3; x + y + 4z = 5, and parallel to the plane x + 3y + 6z = 1 is x + 3y + 6z = 7, 2x + 6y + 12z = 13

One point on planes 2x - y + z = 3 and x + y + 4z = 5 is (4,1,0). Now, equation of plane parallel to x + 3y + 6z = 1 is given by x + 3y + 6z + k = 0. Also this plane is passing through (4,1,0) ⇒ 4 + 3.1 + 0 + k = 0 ⇒ k = -7. Hence, required plane is, x + 3y + 6z = 7