= Solution
Cover $Y$ by
$$
U_0=Y\setminus D_Y,
\qquad U_y=D(y),
\qquad U_w=D(w).
$$
These sets cover because a point of $D_Y$ has $(y,w)\ne(0,0)$. On $U_y$, the relation $x=zw/y$ shows that $D_Y$ has local equation $z$; on $U_w$, the relation $z=xy/w$ gives local equation $x$; and on $U_0$ its local equation is $1$. Thus $D_Y$ is a <Cartier divisor>.
For the <line bundle associated to a divisor> $\mathcal O_Y(D_Y)$, choose local frames
$$
e_0=1,
\qquad e_y=\frac1z,
\qquad e_w=\frac1x.
$$
The transition functions are
$$
e_y=z^{-1}e_0,
\qquad
e_w=x^{-1}e_0,
\qquad
e_y=\frac{x}{z}e_w=\frac wy e_w
$$
on the corresponding overlaps. Every displayed ratio is a <unit in a ring> in the corresponding ring of <regular functions>, and the <Čech cocycle condition> follows immediately.
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