Solve for f (complex solution)
\left\{\begin{matrix}f=-\frac{-xy+y^{2}-7}{x^{2}}\text{, }&x\neq 0\\f\in \mathrm{C}\text{, }&\left(y=\sqrt{7}\text{ or }y=-\sqrt{7}\right)\text{ and }x=0\end{matrix}\right.
Solve for f
\left\{\begin{matrix}f=-\frac{-xy+y^{2}-7}{x^{2}}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&x=0\text{ and }|y|=\sqrt{7}\end{matrix}\right.
Solve for x (complex solution)
\left\{\begin{matrix}x=\frac{\sqrt{28f+y^{2}-4fy^{2}}+y}{2f}\text{; }x=\frac{-\sqrt{28f+y^{2}-4fy^{2}}+y}{2f}\text{, }&f\neq 0\\x=\frac{y^{2}-7}{y}\text{, }&f=0\text{ and }y\neq 0\end{matrix}\right.
Solve for x
\left\{\begin{matrix}x=\frac{\sqrt{28f+y^{2}-4fy^{2}}+y}{2f}\text{; }x=\frac{-\sqrt{28f+y^{2}-4fy^{2}}+y}{2f}\text{, }&\left(|y|\leq \sqrt{7}\text{ and }f\neq 0\text{ and }f\geq -\frac{y^{2}}{4\left(7-y^{2}\right)}\right)\text{ or }\left(y\neq 0\text{ and }f=-\frac{y^{2}}{4\left(7-y^{2}\right)}\text{ and }|y|\neq \sqrt{7}\right)\text{ or }\left(f=-\frac{y^{2}}{4\left(7-y^{2}\right)}\text{ and }|y|>\sqrt{7}\right)\text{ or }\left(f\neq 0\text{ and }f\leq -\frac{y^{2}}{4\left(7-y^{2}\right)}\text{ and }|y|\geq \sqrt{7}\right)\text{ or }\left(f\neq 0\text{ and }|y|=\sqrt{7}\right)\\x=\frac{y^{2}-7}{y}\text{, }&f=0\text{ and }y\neq 0\end{matrix}\right.
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fx^{2}+y^{2}=7+xy
Add xy to both sides.
fx^{2}=7+xy-y^{2}
Subtract y^{2} from both sides.
x^{2}f=xy-y^{2}+7
The equation is in standard form.
\frac{x^{2}f}{x^{2}}=\frac{xy-y^{2}+7}{x^{2}}
Divide both sides by x^{2}.
f=\frac{xy-y^{2}+7}{x^{2}}
Dividing by x^{2} undoes the multiplication by x^{2}.
fx^{2}+y^{2}=7+xy
Add xy to both sides.
fx^{2}=7+xy-y^{2}
Subtract y^{2} from both sides.
x^{2}f=xy-y^{2}+7
The equation is in standard form.
\frac{x^{2}f}{x^{2}}=\frac{xy-y^{2}+7}{x^{2}}
Divide both sides by x^{2}.
f=\frac{xy-y^{2}+7}{x^{2}}
Dividing by x^{2} undoes the multiplication by x^{2}.
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