## Calculus 10th Edition

$z$ is a function of $x$ and $y$.
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 $y$.