Final answer:
To find the equation of the tangent plane to the surface z = ln(x - 7y) at the point (8, 1, 0), we need to find the partial derivatives and evaluate them at the given point. The equation of the tangent plane is z = x - 7y - 7.
Step-by-step explanation:
To find the equation of the tangent plane to the surface z = ln(x - 7y) at the point (8, 1, 0), we need to find the partial derivatives and evaluate them at the given point. First, find the partial derivative with respect to x by differentiating z = ln(x - 7y) with respect to x, which gives us dz/dx = 1/(x - 7y). Next, find the partial derivative with respect to y by differentiating z = ln(x - 7y) with respect to y, which gives us dz/dy = -7/(x - 7y).
Now, evaluate these partial derivatives at the point (8, 1, 0): dz/dx = 1/(8 - 7(1)) = 1/1 = 1, and dz/dy = -7/(8 - 7(1)) = -7/1 = -7.
Thus, the equation of the tangent plane to the surface z = ln(x - 7y) at the point (8, 1, 0) is given by z - 0 = 1(x - 8) - 7(y - 1), which simplifies to z = x - 7y -7.