1. **State the problem:** Find the equation of the tangent plane to the surface given by $$-7z - 3y - 8x = -19$$ at the point $$P(2,1)$$. 2. **Rewrite the surface equation to solve for $$z$$:** $$-7z = -19 + 3y + 8x$$ $$z = \frac{19 - 3y - 8x}{7}$$ 3. **Define the function:** $$f(x,y) = \frac{19 - 3y - 8x}{7}$$ 4. **Calculate partial derivatives:** $$f_x(x,y) = \frac{\partial}{\partial x} \left( \frac{19 - 3y - 8x}{7} \right) = \frac{-8}{7}$$ $$f_y(x,y) = \frac{\partial}{\partial y} \left( \frac{19 - 3y - 8x}{7} \right) = \frac{-3}{7}$$ 5. **Evaluate partial derivatives at point $$(2,1)$$:** $$f_x(2,1) = \frac{-8}{7}$$ $$f_y(2,1) = \frac{-3}{7}$$ 6. **Evaluate the function at point $$(2,1)$$:** $$f(2,1) = \frac{19 - 3(1) - 8(2)}{7} = \frac{19 - 3 - 16}{7} = \frac{0}{7} = 0$$ 7. **Write the equation of the tangent plane:** The tangent plane at $$(x_0,y_0) = (2,1)$$ is given by: $$z = f(2,1) + f_x(2,1)(x - 2) + f_y(2,1)(y - 1)$$ Substitute values: $$z = 0 + \frac{-8}{7}(x - 2) + \frac{-3}{7}(y - 1)$$ 8. **Simplify the tangent plane equation:** $$z = -\frac{8}{7}x + \frac{16}{7} - \frac{3}{7}y + \frac{3}{7}$$ $$z = -\frac{8}{7}x - \frac{3}{7}y + \frac{19}{7}$$ **Final answer:** $$\boxed{z = -\frac{8}{7}x - \frac{3}{7}y + \frac{19}{7}}$$