In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My question is at the end; I have put a red line across the point I am interested in)
Could someone explain to me how to "work out" the algebra part?
This seems pretty important derivation actually. I am afraid I don't even know how to start the algebra. I would appreciate any suggestions/hints. Thank you :)
Related. There is a close question I found here on MSE. However, that particular question asks more about geometrical insight. My question is how does one explicitly go from the general cubic to the equation $xy^2+(ax+b)y=cx^2+dx+e$. By the way, the equation for the general cubic is (just so the coefficients are consistent with the above formula): $ax^3+bx^2y+cxy^2+dy^3+ex^2+fxy+gy^2+hx+iy+j=0$