Weierstraß normal form

General equation

Karl Theodor Wilhelm Weierstraß (31 Oct 1815 – 19 Feb 1897)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gz^3 + hz^2w + jzw^2 + kw^3}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {} + mz^2 + pzw + qw^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {} + rz + sw}

${}+t\qquad =0.$

Linear transformation

${\begin{bmatrix}z\\w\end{bmatrix}}={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = \alpha x + \beta y + \zeta}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w = \gamma x + \delta y + \eta}

Problem

Substitute $\alpha x+\beta y+\zeta$ and $\gamma x+\delta y+\eta$ for z and w in the general equation, simplify by collecting like terms in respective powers of x and y, and solve for α, β, γ, δ, ζ, η, a and b so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 = x^3 + ax + b.}