Let $\mathbb{P}^1$ be the projective line over a base field $k$, with homogeneous coordinates $[y_0 : y_1]$. Consider the sheaf of $\mathcal{O}_{\mathbb{P}^1}$-algebras $\mathcal{A} = \mathcal{O}_{\mathbb{P}^1}[y_0^2 t, y_0 y_1 t, y_1^2 t]$, where $t$ is just a variable to keep track of the grading.
I want to show that the relative spectrum $\operatorname{Spec}_{\mathcal{O}_{\mathbb{P}^1}}(\mathcal{A})$ is isomorphic to the line bundle $L_2$ given by the locally free sheaf $\mathcal{O}_{\mathbb{P}^1}(2)$, by checking that the line bundle transition functions are the same.
I want to do this by checking that the transition function of $\operatorname{Spec}_{\mathcal{O}_{\mathbb{P}^1}}(\mathcal{A})$ over $\{ y_1 \neq 0 \}$ to over $\{ y_0 \neq 0 \}$ is $(\frac{y_1}{y_0})^2$, the same as for $L_2$. Another way of saying this is that $L_2$ is isomorphic to the gluing of $\operatorname{Spec} k[\frac{y_0}{y_1},x_1]$ and $\operatorname{Spec} k[\frac{y_1}{y_0},x_0]$ via the isomorphism $\operatorname{Spec} k[\frac{y_0}{y_1},\frac{y_1}{y_0},x_1] \cong \operatorname{Spec} k[\frac{y_0}{y_1},\frac{y_1}{y_0},x_0]$ that sends $y_0$ to $y_0$, $y_1$ to $y_1$, and $x_0$ to $(\frac{y_1}{y_0})^2 x_1$.
We know $\mathcal{O}_{\mathbb{P}^1}(\{ y_0, y_1 \neq 0 \})[y_0^2 t, y_0 y_1 t, y_1^2 t]$ is isomorphic to $k[\frac{y_0}{y_1},\frac{y_1}{y_0}][y_0^2 t, y_0 y_1 t, y_1^2 t] = k[\frac{y_0}{y_1},\frac{y_1}{y_0}][y_1^2 t] = k[\frac{y_0}{y_1},\frac{y_1}{y_0}][y_0^2 t]$. However, I am strugling to use this to define isomorphisms to $k[\frac{y_0}{y_1},\frac{y_1}{y_0},x_1]$ and $k[\frac{y_0}{y_1},\frac{y_1}{y_0},x_0]$ that glue by $x_0 \mapsto (\frac{y_1}{y_0})^2 x_1$.
