Final answer:
The equation of the plane that passes through the point (4, -7, -2) and is parallel to the plane 2x - y - z = 8 is 2x - y - z = 17.
Step-by-step explanation:
To find the equation of a plane that is parallel to the given plane and passes through a specific point, we need to first understand that parallel planes have the same normal vector. The normal vector of the given plane 2x - y - z = 8 is (2, -1, -1). So, the equation of the plane that is parallel to this plane can be written as 2x - y - z = k, where k is the constant we need to find.
Since the parallel plane also passes through the point (4, -7, -2), we can substitute these coordinates into the equation. 2(4) - (-7) - (-2) = k. Simplifying, we get 8 + 7 + 2 = k, which gives us k = 17.
Therefore, the equation of the plane that passes through the point (4, -7, -2) and is parallel to the plane 2x - y - z = 8 is 2x - y - z = 17.