Final answer:
To find the equation of the tangent plane to the given surface at the specified point, we find the partial derivatives of the surface equation and substitute them into the equation for a plane.
Step-by-step explanation:
To find the equation of the tangent plane to the given surface, we need to find the partial derivatives of the surface equation with respect to x and y. The given surface equation is z = 232 + y - 5y. Taking the partial derivatives, we get ∂z/∂x = 0 and ∂z/∂y = 1 - 5 = -4.
At the specified point (1, 2, -4), we can use the equation for a plane, which is z - z0 = (∂z/∂x)(x - x0) + (∂z/∂y)(y - y0), where (x0, y0, z0) is the specified point. Substituting the values, we get z - (-4) = 0(x - 1) + (-4)(y - 2).
Simplifying, we have z + 4 = -4(y - 2), or z = -4y + 12.