Question

find the equation for the tangent plane to the surface -10x+9y-2z=-23 at the poin (1,-1,2)

Answer 1

It is given by:
\[\nabla f \cdot <(x-x_0),(y-y_0),(z-z_0)>=0\]
\[\nabla f=<-10,9,-2>\]
\[<-10,9,-2> \cdot <(x-1),(y+1),(z-2)>=-10x+10+9y+9-2z+4=0\]
I was going to say this but I wanted to show you how to work it out. Since your problem gives you a plane, the only tangent plane would be the plane itself, as you saw.