I am reading a review on Toric Geometry for string theorists by Harald Skarke (arXiv:hep-th/9806059). In section 3, the author says
A standard way of describing an elliptic curve is by embedding it into $\mathbb{P}^2 = (\mathbb{C}^3\setminus\{0\})/(\mathbb{C}\setminus\{0\})$.....The elliptic curve is embedded in this space via the Weierstrass equation $$ \label{eq:w1}y^2 z = x^3 + a x z^2 + b z^3$$
An alternative description can be given in terms of the weighted projective space $\mathbb{P}^{(2,3,1)}$ defined just like $\mathbb{P}^2$, but with the equivalence relation changed to
$$(x, y, z) \sim (\lambda^2 x, \lambda^3 y, \lambda z) \quad \text{ for any } \lambda \in \mathbb{C}\setminus\{0\}$$
and the Weierstrass equation changed to
$$\label{eq:w2}y^2 = x^3 + a x z^4 + b z^6$$
Questions
Given the description of the two projective spaces as embeddings in $\mathbb{C}^3$, how does one obtain the two Weierstrass equations?
Is there a way to transform the first Weierstrass equation into the other?
Thanks in advance!