Final answer:
To find the equation of the tangent plane to the surface defined by z = ln(x - 6y) at the point (7, 1, 0), we need to find the partial derivatives of z with respect to x and y and use them to determine the equation of the tangent plane.
Step-by-step explanation:
To find the equation of the tangent plane to the surface defined by the equation z = ln(x - 6y) at the point (7, 1, 0), we need to find the partial derivatives of z with respect to x and y. The partial derivative of z with respect to x is 1/(x - 6y), and the partial derivative of z with respect to y is (-6)/(x - 6y). We can use these partial derivatives along with the point (7, 1, 0) to find the equation of the tangent plane.
Using the point-normal form of a plane equation (Ax + By + Cz = D), we can substitute the values of x, y, and z from the given point into the equation. The values of A, B, and C can be determined by the partial derivatives of z with respect to x and y. Finally, we can simplify the equation to get the equation of the tangent plane.
The equation of the tangent plane to the given surface at the point (7, 1, 0) is 7(x - 6y) - 6 = 0.