I was stuck with finding lagrange multiplier for the following problem: $f(x,y,z)=xyz^2$ with constraints $g_1(x,y,z)=8x+y-9z^2-10=0$ and $g_2(x,y,z)=3x+4y^2-z-12=0$. I was able to get these equations with five unknowns where $\lambda$ and $\mu$ are the multipliers.
$yz^2-8\lambda-3\mu=0$
$xz^2-\lambda-2\mu y=0$
$2xyz+18\lambda z+\mu=0$
$8x+y-9z^2 =10$
$3x+4y^2-z=12$ I am not able to get past this in finding the variables.