Lahenda x+z/x-z-x-z/x+z=z(2x^2+zy)/x^2-z^2

$$\frac{ x+z }{ x-z } - \frac{ x-z }{ x+z } = \frac{ z(2 { x }^{ 2 } +zy) }{ { x }^{ 2 } - { z }^{ 2 } }$$

Lahendage ja leidke y (complex solution)

$$\left\{\begin{matrix}y=-\frac{2x\left(x-2\right)}{z}\text{, }&z\neq 0\text{ and }x\neq z\text{ and }x\neq -z\\y\in \mathrm{C}\text{, }&z=0\text{ and }x\neq 0\end{matrix}\right,$$

Juhised lineaarvõrrandi lahendamiseks

Korrutage võrrandi mõlemad pooled arvuga \left(x-z\right)\left(-x-z\right), mis on arvu x-z,x+z,x^{2}-z^{2} vähim ühiskordne.

$$\left(-x-z\right)\left(x+z\right)-\left(-x+z\right)\left(x-z\right)=-z\left(2x^{2}+zy\right)$$

Kasutage distributiivsusomadust, et korrutada -x-z ja x+z, ning koondage sarnased liikmed.

$$-x^{2}-2xz-z^{2}-\left(-x+z\right)\left(x-z\right)=-z\left(2x^{2}+zy\right)$$

Kasutage distributiivsusomadust, et korrutada -x+z ja x-z, ning koondage sarnased liikmed.

$$-x^{2}-2xz-z^{2}-\left(-x^{2}+2xz-z^{2}\right)=-z\left(2x^{2}+zy\right)$$

Avaldise "-x^{2}+2xz-z^{2}" vastandi leidmiseks tuleb leida iga liikme vastand.

$$-x^{2}-2xz-z^{2}+x^{2}-2xz+z^{2}=-z\left(2x^{2}+zy\right)$$

Kombineerige -x^{2} ja x^{2}, et leida 0.

$$-2xz-z^{2}-2xz+z^{2}=-z\left(2x^{2}+zy\right)$$

Kombineerige -2xz ja -2xz, et leida -4xz.

$$-4xz-z^{2}+z^{2}=-z\left(2x^{2}+zy\right)$$

Kombineerige -z^{2} ja z^{2}, et leida 0.

$$-4xz=-z\left(2x^{2}+zy\right)$$

Kasutage distributiivsusomadust, et korrutada -z ja 2x^{2}+zy.

$$-4xz=-2zx^{2}-yz^{2}$$

Vahetage pooled nii, et kõik muutuvad liikmed asuksid vasakul.

$$-2zx^{2}-yz^{2}=-4xz$$

Liitke 2zx^{2} mõlemale poolele.

$$-yz^{2}=-4xz+2zx^{2}$$

Võrrand on standardkujul.

$$\left(-z^{2}\right)y=2zx^{2}-4xz$$

Jagage mõlemad pooled -z^{2}-ga.

$$\frac{\left(-z^{2}\right)y}{-z^{2}}=\frac{2xz\left(x-2\right)}{-z^{2}}$$

-z^{2}-ga jagamine võtab -z^{2}-ga korrutamise tagasi.

$$y=\frac{2xz\left(x-2\right)}{-z^{2}}$$

Jagage 2xz\left(-2+x\right) väärtusega -z^{2}.

$$y=-\frac{2x\left(x-2\right)}{z}$$

Lahendage ja leidke y

$$\left\{\begin{matrix}y=-\frac{2x\left(x-2\right)}{z}\text{, }&z\neq 0\text{ and }|x|\neq |z|\\y\in \mathrm{R}\text{, }&z=0\text{ and }x\neq 0\end{matrix}\right,$$

Juhised lineaarvõrrandi lahendamiseks

Korrutage võrrandi mõlemad pooled arvuga \left(x-z\right)\left(-x-z\right), mis on arvu x-z,x+z,x^{2}-z^{2} vähim ühiskordne.

$$\left(-x-z\right)\left(x+z\right)-\left(-x+z\right)\left(x-z\right)=-z\left(2x^{2}+zy\right)$$

Kasutage distributiivsusomadust, et korrutada -x-z ja x+z, ning koondage sarnased liikmed.

$$-x^{2}-2xz-z^{2}-\left(-x+z\right)\left(x-z\right)=-z\left(2x^{2}+zy\right)$$

Kasutage distributiivsusomadust, et korrutada -x+z ja x-z, ning koondage sarnased liikmed.

$$-x^{2}-2xz-z^{2}-\left(-x^{2}+2xz-z^{2}\right)=-z\left(2x^{2}+zy\right)$$

Avaldise "-x^{2}+2xz-z^{2}" vastandi leidmiseks tuleb leida iga liikme vastand.

$$-x^{2}-2xz-z^{2}+x^{2}-2xz+z^{2}=-z\left(2x^{2}+zy\right)$$

Kombineerige -x^{2} ja x^{2}, et leida 0.

$$-2xz-z^{2}-2xz+z^{2}=-z\left(2x^{2}+zy\right)$$

Kombineerige -2xz ja -2xz, et leida -4xz.

$$-4xz-z^{2}+z^{2}=-z\left(2x^{2}+zy\right)$$

Kombineerige -z^{2} ja z^{2}, et leida 0.

$$-4xz=-z\left(2x^{2}+zy\right)$$

Kasutage distributiivsusomadust, et korrutada -z ja 2x^{2}+zy.

$$-4xz=-2zx^{2}-yz^{2}$$

Vahetage pooled nii, et kõik muutuvad liikmed asuksid vasakul.

$$-2zx^{2}-yz^{2}=-4xz$$

Liitke 2zx^{2} mõlemale poolele.

$$-yz^{2}=-4xz+2zx^{2}$$

Võrrand on standardkujul.

$$\left(-z^{2}\right)y=2zx^{2}-4xz$$

Jagage mõlemad pooled -z^{2}-ga.

$$\frac{\left(-z^{2}\right)y}{-z^{2}}=\frac{2xz\left(x-2\right)}{-z^{2}}$$

-z^{2}-ga jagamine võtab -z^{2}-ga korrutamise tagasi.

$$y=\frac{2xz\left(x-2\right)}{-z^{2}}$$

Jagage 2xz\left(-2+x\right) väärtusega -z^{2}.

$$y=-\frac{2x\left(x-2\right)}{z}$$

Lahendage ja leidke x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{4-2yz}}{2}+1\text{, }&\left(z\neq -\frac{y}{2}+2\text{ and }z\neq -\frac{y}{2}-2\right)\text{ or }\left(z\neq -\frac{y}{2}+2\text{ and }y\neq -2\text{ and }arg(-\frac{y}{2}-1)<\pi \right)\text{ or }\left(arg(2-y)\geq \pi \text{ and }y\neq 2\text{ and }z\neq -\frac{y}{2}-2\right)\text{ or }\left(arg(2-y)\geq \pi \text{ and }y\neq 2\text{ and }arg(-\frac{y}{2}-1)<\pi \right)\\x=-\frac{\sqrt{4-2yz}}{2}+1\text{, }&\left(z\neq 0\text{ and }z\neq -\frac{y}{2}+2\text{ and }arg(-\frac{y}{2}-1)\geq \pi \text{ and }z\neq -1\right)\text{ or }\left(z\neq 0\text{ and }z\neq -\frac{y}{2}+2\text{ and }arg(-\frac{y}{2}-1)\geq \pi \text{ and }y\neq -2\right)\text{ or }\left(z\neq 0\text{ and }z\neq -\frac{y}{2}+2\text{ and }z\neq -\frac{y}{2}-2\text{ and }z\neq -1\right)\text{ or }\left(z\neq 0\text{ and }z\neq -\frac{y}{2}+2\text{ and }z\neq -\frac{y}{2}-2\text{ and }y\neq -2\right)\text{ or }\left(z\neq 0\text{ and }z\neq 1\text{ and }arg(y-2)\geq \pi \text{ and }arg(-\frac{y}{2}-1)\geq \pi \text{ and }z\neq -1\right)\text{ or }\left(z\neq 0\text{ and }z\neq 1\text{ and }arg(y-2)\geq \pi \text{ and }arg(-\frac{y}{2}-1)\geq \pi \text{ and }y\neq -2\right)\text{ or }\left(z\neq 0\text{ and }z\neq 1\text{ and }arg(y-2)\geq \pi \text{ and }z\neq -\frac{y}{2}-2\text{ and }z\neq -1\right)\text{ or }\left(z\neq 0\text{ and }z\neq 1\text{ and }arg(y-2)\geq \pi \text{ and }z\neq -\frac{y}{2}-2\text{ and }y\neq -2\right)\text{ or }\left(z\neq 0\text{ and }y\neq 2\text{ and }arg(y-2)\geq \pi \text{ and }arg(-\frac{y}{2}-1)\geq \pi \text{ and }z\neq -1\right)\text{ or }\left(z\neq 0\text{ and }y\neq 2\text{ and }arg(y-2)\geq \pi \text{ and }arg(-\frac{y}{2}-1)\geq \pi \text{ and }y\neq -2\right)\text{ or }\left(z\neq 0\text{ and }y\neq 2\text{ and }arg(y-2)\geq \pi \text{ and }z\neq -\frac{y}{2}-2\text{ and }z\neq -1\right)\text{ or }\left(z\neq 0\text{ and }y\neq 2\text{ and }arg(y-2)\geq \pi \text{ and }z\neq -\frac{y}{2}-2\text{ and }y\neq -2\right)\\x\neq 0\text{, }&z=0\end{matrix}\right,$$