Atrisiniet fx^2-xy+y^2=7

$$f x ^ { 2 } - x y + y ^ { 2 } = 7$$

Atrast f (complex solution)

$$\left\{\begin{matrix}f=-\frac{-xy+y^{2}-7}{x^{2}}\text{, }&x\neq 0\\f\in \mathrm{C}\text{, }&\left(y=\sqrt{7}\text{ or }y=-\sqrt{7}\right)\text{ and }x=0\end{matrix}\right,$$

Lineāra vienādojuma risināšanas darbības

Pievienot xy abās pusēs.

$$fx^{2}+y^{2}=7+xy$$

Atņemiet y^{2} no abām pusēm.

$$fx^{2}=7+xy-y^{2}$$

Vienādojums ir standarta formā.

$$x^{2}f=xy-y^{2}+7$$

Daliet abas puses ar x^{2}.

$$\frac{x^{2}f}{x^{2}}=\frac{xy-y^{2}+7}{x^{2}}$$

Dalīšana ar x^{2} atsauc reizināšanu ar x^{2}.

$$f=\frac{xy-y^{2}+7}{x^{2}}$$

Atrast f

$$\left\{\begin{matrix}f=-\frac{-xy+y^{2}-7}{x^{2}}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&x=0\text{ and }|y|=\sqrt{7}\end{matrix}\right,$$

Lineāra vienādojuma risināšanas darbības

Pievienot xy abās pusēs.

$$fx^{2}+y^{2}=7+xy$$

Atņemiet y^{2} no abām pusēm.

$$fx^{2}=7+xy-y^{2}$$

Vienādojums ir standarta formā.

$$x^{2}f=xy-y^{2}+7$$

Daliet abas puses ar x^{2}.

$$\frac{x^{2}f}{x^{2}}=\frac{xy-y^{2}+7}{x^{2}}$$

Dalīšana ar x^{2} atsauc reizināšanu ar x^{2}.

$$f=\frac{xy-y^{2}+7}{x^{2}}$$

Atrast x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{28f+y^{2}-4fy^{2}}+y}{2f}\text{; }x=\frac{-\sqrt{28f+y^{2}-4fy^{2}}+y}{2f}\text{, }&f\neq 0\\x=\frac{y^{2}-7}{y}\text{, }&f=0\text{ and }y\neq 0\end{matrix}\right,$$

Atrast x

$$\left\{\begin{matrix}x=\frac{\sqrt{28f+y^{2}-4fy^{2}}+y}{2f}\text{; }x=\frac{-\sqrt{28f+y^{2}-4fy^{2}}+y}{2f}\text{, }&\left(|y|\leq \sqrt{7}\text{ and }f\neq 0\text{ and }f\geq -\frac{y^{2}}{4\left(7-y^{2}\right)}\right)\text{ or }\left(y\neq 0\text{ and }f=-\frac{y^{2}}{4\left(7-y^{2}\right)}\text{ and }|y|\neq \sqrt{7}\right)\text{ or }\left(f=-\frac{y^{2}}{4\left(7-y^{2}\right)}\text{ and }|y|>\sqrt{7}\right)\text{ or }\left(f\neq 0\text{ and }f\leq -\frac{y^{2}}{4\left(7-y^{2}\right)}\text{ and }|y|\geq \sqrt{7}\right)\text{ or }\left(f\neq 0\text{ and }|y|=\sqrt{7}\right)\\x=\frac{y^{2}-7}{y}\text{, }&f=0\text{ and }y\neq 0\end{matrix}\right,$$