Answer
$z$ is a function of $x$ and $y$.
Work Step by Step
For $z$ to be the function of $x$ and $y$ then for each $x$ and $y$ we need to have only one value of $z$.
Here we have
$$x^2 z+3y^2-xy=10.$$
Solving for $z$:
$$x^2 z+3y^2-xy=10\Rightarrow x^2z=10+xy-3y^2\Rightarrow z=\frac{10+xy-3y^2}{x^2}$$
The last equality says that for each $x$ and $y$ we have ONLY ONE value for $z$ which is calculated from that equality so $z$ is a function of $x$ and $y$.
