# ¶ General form of equation

\begin{array}{ccc}g{z}^{3}& +h{z}^{2}w+jz{w}^{2}+k{w}^{3}& \\ & +m{z}^{2}+pzw+q{w}^{2}& \\ & +rz+sw& \\ & +t& =0\end{array}\begin\left\{aligned\right\} gz^3 & + hz^2w + jzw^2 + kw^3 & \\ & + mz^2 + pzw + qw^2 & \\ & + rz + sw & \\ & + t & = 0 \end\left\{aligned\right\}

# ¶ Linear transformation

The plane is skewed, rotated, and translated by a linear transformation.

\begin{array}{cc}\left[\begin{array}{c}z\\ w\end{array}\right]& =\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}\zeta \\ \eta \end{array}\right]\\ z& =\alpha x+\beta y+\zeta \\ w& =\gamma x+\delta y+\eta \end{array}\begin\left\{aligned\right\} \begin\left\{bmatrix\right\} z \\ w \end\left\{bmatrix\right\} & = \begin\left\{bmatrix\right\} \alpha & \beta \\ \gamma & \delta \end\left\{bmatrix\right\} \begin\left\{bmatrix\right\} x \\ y \end\left\{bmatrix\right\} +\begin\left\{bmatrix\right\} \zeta \\ \eta \end\left\{bmatrix\right\} \\ z & = \alpha x + \beta y + \zeta \\ w & = \gamma x + \delta y + \eta \end\left\{aligned\right\}

# ¶ Problem

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

${y}^{2}={x}^{3}+ax+b\mathrm{.}y^2 = x^3 + ax + b .$