Find the equation for the tangent plane to the surface x^2+3xyz+y^2=-5z^2 at the point (1,1,1).
I found the answer to be x+y+3z=5 but my solution manual says -5x-5y+7z=-3.
It should be $\displaystyle x^2+3xyz+y^2=5z^2$ (no negative sign before $\displaystyle 5z^2$). Then
$\displaystyle f(x,y,z)=x^2+3xyz+y^2-5z^2=0$
$\displaystyle \nabla f = \langle 2x+3yz,3xz+2y,3xy-10z\rangle$
Plug in $\displaystyle (1,1,1)$ to get $\displaystyle \langle 5,5,-7\rangle$
So it's $\displaystyle 5(x-1)+5(y-1)-7(z-1)=0\implies 5x+5y-7z=3$ (or $\displaystyle -5x-5y+7z=-3$ if you prefer).