Final answer:
The equation of the plane passing through the point (9, -9, -6) and parallel to the plane 8x - y - z = 8 is 8x - y - z = 69.
Step-by-step explanation:
The student is asking for the equation of a plane that passes through a specific point and is parallel to another plane. The given plane's equation is 8x − y − z = 8, which tells us that the normal vector to the plane is (8, -1, -1) because the coefficients of x, y, and z correspond to the components of the normal vector. Since parallel planes have the same normal vector, we can use the same normal vector for our unknown plane.
To find the equation of the plane through the point (9, −9, −6) and parallel to the given plane, we start with the general equation of a plane, which is A(x-x0) + B(y-y0) + C(z-z0) = 0, where (A, B, C) is the normal vector and (x0, y0, z0) is a point on the plane.
Substituting the point and normal vector into this general equation, we get 8(x-9) - (y+9) - (z+6) = 0. Simplifying, the equation of our plane is 8x − y − z = 69.