Let $E$ be an elliptic curve over field $K$ defined by affine curve
$E:y^2 +a_1 xy+ a_3y =x^3 + a_2x^2 + a_4x + a_6$（all coefficients are in $K$）
By changing variables, $z＝-x/y$,$w＝-1/y$
We get new affine curve
$E':w＝z^3＋a_1 zw+ a_2z^2w＋a_3w^2 + a_4zw^2 + a_6w^3$
My question is, why $E'$（strictly speaking, projective closure of $E'$）is still elliptic curve?
How can I formally prove that?
My approach: If $E'$ is proved to be smooth, then genus formula deduces $E'$ has genus 1, and it has also base point, so is elliptic curve.
So,My question is that : How can I show $E'$ is smooth curve?（Is there no choice but to use Jacobi criteria?）
Thank you in advance.