Find an equation of the tangent plane to the surface at the given point. $$ z=8-\frac{8}{3} x-y, \quad(3,-5,5) $$

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We are given the surface

$$
z = 8 - \frac{8}{3}x - y,
$$

and the point $$(3, -5, 5)$$. The first step is to check that the point lies on the surface.

Substitute $$ x = 3 $$ and $$ y = -5 $$ into the surface equation:

$$
z = 8 - \frac{8}{3}(3) - (-5) = 8 - 8 + 5 = 5.
$$

Since $$ z = 5 $$, the point $$(3, -5, 5)$$ is indeed on the surface.

Next, note that the given surface is a plane. Therefore, the tangent plane at any point on a plane is the plane itself.

However, we can also derive the equation of the tangent plane using the formula for the tangent plane to a surface $$ z = f(x, y) $$. For a point $$(x_0, y_0, z_0)$$, the tangent plane is given by

$$
z = f(x_0, y_0) + f_x(x_0, y_0) (x - x_0) + f_y(x_0, y_0) (y - y_0),
$$

where $$ f_x $$ and $$ f_y $$ are the partial derivatives of $$ f $$ with respect to $$ x $$ and $$ y $$, respectively.

For our $$ f(x,y) = 8 - \frac{8}{3}x - y $$, compute the partial derivatives:

$$
f_x(x, y) = -\frac{8}{3}, \quad f_y(x, y) = -1.
$$

At the point $$(3, -5)$$, these derivatives remain the same (since they are constant).

Now, substitute into the tangent plane formula:

$$
z = f(3, -5) + \left(-\frac{8}{3}\right)(x - 3) + (-1)(y - (-5)).
$$

We already found $$ f(3, -5) = 5 $$, and note that $$ y - (-5) = y + 5 $$. Thus,

$$
z = 5 - \frac{8}{3}(x - 3) - (y + 5).
$$

Simplify step-by-step:

1. Expand the terms:

$$
   z = 5 - \frac{8}{3}x + \frac{8}{3}\cdot 3 - y - 5.
   $$

2. Since $$\frac{8}{3} \cdot 3 = 8$$, we have:

$$
   z = 5 - \frac{8}{3}x + 8 - y - 5.
   $$

3. Combine like terms:

$$
   z = (5 + 8 - 5) - \frac{8}{3}x - y = 8 - \frac{8}{3}x - y.
   $$

Thus, the equation of the tangent plane is

$$
z = 8 - \frac{8}{3}x - y.
$$

This is the same as the given surface, which confirms that the tangent plane to a plane is the plane itself.
The equation of the tangent plane at the point $$(3, -5, 5)$$ is: $$ z = 8 - \frac{8}{3}x - y $$

Find an equation of the tangent plane to the surface at the given point.

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