Find the implicit functions

Question

Consider the system of equations $$xyz+\sin{z}=0$$ $$\log{xy}+e^{xz}=1$$

I need to prove that the system define implicit functions $y=y(x)$ and $z=z(x)$, $C^{\infty}$, in an open neighbourhood $U$ of the point $x=1$, with $y(1)=1$ and $z(1)=0$

Answer 1

$(1,1,0)$ is a solution of the system and if $F_1$ and $F_2$ are the left sides of the two equations, the Jacobian matrix of $[F_1, F_2]$ with respect to $(x,y,z)$ at $(1,1,0)$ is

$$ J(1,1,0) = \pmatrix{0 & 0 & 2\cr 1 & 1 & 1\cr} $$

Since the second and third columns form an invertible matrix, the Implicit Function Theorem says in a neighbourhood of $1$ there are smooth functions $Y(x)$ and $Z(x)$ with $Y(1) = 1$, $Z(1)=0$, and $(x,y,z) = (x, Y(x), Z(x))$ satisfies the system.

But, contrary to the title of your question, you're not going to "find" the implicit function, i.e. don't expect closed-form formulas.