Weierstrass form of elliptic curves over $\mathbb Q$

Question

I am confused about elliptic curves. How can we replace this $$E: ay^2=bx^3+c$$ such that $a,b,c\in\mathbb Z, a,b\not=0$ to be its Weierstrass form? Can we just transform them like this? Substitute $y$ by $y/a^2b$ and $x$ by $x/ab$ yields $$E: y^2=x^3+a^3b^2c.$$ Is easy way like this justified so that the above is its Weierstrass form? Actually, I got the first elliptic curve and I want to minimize its discriminant. There is a theorem stating that starting from Weierstrass equation then there is a relation (in figure) so that I can relate it to a minimal form (where here I want to use $R=\mathbb Z$)