Solve z^3+left(x-2right)z^2+left(y-2xright)z-2y=(z^3+2(sqrt{2}-1)z^2+4(1-sqrt{2})z-8)

$${ z }^{ 3 } + \left( x-2 \right) { z }^{ 2 } + \left( y-2x \right) z-2y=( { z }^{ 3 } +2( \sqrt{ 2 } -1) { z }^{ 2 } +4(1- \sqrt{ 2 } )z-8)$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{-2\sqrt{2}z+y-4}{z}\text{, }&z\neq 2\text{ and }z\neq 0\\x\in \mathrm{R}\text{, }&\left(z=-\frac{\sqrt{2}\left(-\sqrt{y^{2}-8\sqrt{2}y-8y+32\sqrt{2}+48}-y+4-4\sqrt{2}\right)}{8}\text{ and }y\leq 4\sqrt{2}+4\right)\text{ or }\left(z=0\text{ and }y=4\right)\text{ or }\left(z=-\frac{\sqrt{2}\left(\sqrt{y^{2}-8\sqrt{2}y-8y+32\sqrt{2}+48}-y+4-4\sqrt{2}\right)}{8}\text{ and }y\geq 4\sqrt{2}+4\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-2 by z^{2}.

$$z^{3}+xz^{2}-2z^{2}+\left(y-2x\right)z-2y=z^{3}+2\left(\sqrt{2}-1\right)z^{2}+4\left(1-\sqrt{2}\right)z-8$$

Use the distributive property to multiply y-2x by z.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\left(\sqrt{2}-1\right)z^{2}+4\left(1-\sqrt{2}\right)z-8$$

Use the distributive property to multiply 2 by \sqrt{2}-1.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+\left(2\sqrt{2}-2\right)z^{2}+4\left(1-\sqrt{2}\right)z-8$$

Use the distributive property to multiply 2\sqrt{2}-2 by z^{2}.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4\left(1-\sqrt{2}\right)z-8$$

Use the distributive property to multiply 4 by 1-\sqrt{2}.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+\left(4-4\sqrt{2}\right)z-8$$

Use the distributive property to multiply 4-4\sqrt{2} by z.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8$$

Subtract z^{3} from both sides.

$$xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-z^{3}$$

Combine z^{3} and -z^{3} to get 0.

$$xz^{2}-2z^{2}+yz-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8$$

Add 2z^{2} to both sides.

$$xz^{2}+yz-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8+2z^{2}$$

Combine -2z^{2} and 2z^{2} to get 0.

$$xz^{2}+yz-2xz-2y=2\sqrt{2}z^{2}+4z-4\sqrt{2}z-8$$

Subtract yz from both sides.

$$xz^{2}-2xz-2y=2\sqrt{2}z^{2}+4z-4\sqrt{2}z-8-yz$$

Add 2y to both sides.

$$xz^{2}-2xz=2\sqrt{2}z^{2}+4z-4\sqrt{2}z-8-yz+2y$$

Combine all terms containing x.

$$\left(z^{2}-2z\right)x=2\sqrt{2}z^{2}+4z-4\sqrt{2}z-8-yz+2y$$

The equation is in standard form.

$$\left(z^{2}-2z\right)x=-yz+2\sqrt{2}z^{2}-4\sqrt{2}z+2y+4z-8$$

Divide both sides by z^{2}-2z.

$$\frac{\left(z^{2}-2z\right)x}{z^{2}-2z}=\frac{-yz+2\sqrt{2}z^{2}-4\sqrt{2}z+2y+4z-8}{z^{2}-2z}$$

Dividing by z^{2}-2z undoes the multiplication by z^{2}-2z.

$$x=\frac{-yz+2\sqrt{2}z^{2}-4\sqrt{2}z+2y+4z-8}{z^{2}-2z}$$

Divide 2z^{2}\sqrt{2}+4z-4\sqrt{2}z-8-yz+2y by z^{2}-2z.

$$x=\frac{2\sqrt{2}z-y+4}{z}$$

Solve for y

$$\left\{\begin{matrix}y=-xz+2\sqrt{2}z+4\text{, }&z\neq 2\\y\in \mathrm{R}\text{, }&\left(z=\frac{-\sqrt{x^{2}-4\sqrt{2}x-4x+8\sqrt{2}+12}+x+2-2\sqrt{2}}{x-2\sqrt{2}}\text{ and }x\leq 2\sqrt{2}+2\text{ and }x>2\sqrt{2}\right)\text{ or }\left(z=\frac{-\sqrt{x^{2}-4\sqrt{2}x-4x+8\sqrt{2}+12}+x+2-2\sqrt{2}}{x-2\sqrt{2}}\text{ and }x<2\sqrt{2}\right)\text{ or }\left(x\geq 2\sqrt{2}+2\text{ and }z=\frac{\sqrt{x^{2}-4\sqrt{2}x-4x+8\sqrt{2}+12}+x+2-2\sqrt{2}}{x-2\sqrt{2}}\right)\text{ or }\left(z=2\text{ and }x=2\sqrt{2}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-2 by z^{2}.

$$z^{3}+xz^{2}-2z^{2}+\left(y-2x\right)z-2y=z^{3}+2\left(\sqrt{2}-1\right)z^{2}+4\left(1-\sqrt{2}\right)z-8$$

Use the distributive property to multiply y-2x by z.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\left(\sqrt{2}-1\right)z^{2}+4\left(1-\sqrt{2}\right)z-8$$

Use the distributive property to multiply 2 by \sqrt{2}-1.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+\left(2\sqrt{2}-2\right)z^{2}+4\left(1-\sqrt{2}\right)z-8$$

Use the distributive property to multiply 2\sqrt{2}-2 by z^{2}.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4\left(1-\sqrt{2}\right)z-8$$

Use the distributive property to multiply 4 by 1-\sqrt{2}.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+\left(4-4\sqrt{2}\right)z-8$$

Use the distributive property to multiply 4-4\sqrt{2} by z.

$$z^{3}+xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8$$

Subtract z^{3} from both sides.

$$xz^{2}-2z^{2}+yz-2xz-2y=z^{3}+2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-z^{3}$$

Combine z^{3} and -z^{3} to get 0.

$$xz^{2}-2z^{2}+yz-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8$$

Subtract xz^{2} from both sides.

$$-2z^{2}+yz-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-xz^{2}$$

Add 2z^{2} to both sides.

$$yz-2xz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-xz^{2}+2z^{2}$$

Add 2xz to both sides.

$$yz-2y=2\sqrt{2}z^{2}-2z^{2}+4z-4\sqrt{2}z-8-xz^{2}+2z^{2}+2xz$$

Combine -2z^{2} and 2z^{2} to get 0.

$$yz-2y=2\sqrt{2}z^{2}+4z-4\sqrt{2}z-8-xz^{2}+2xz$$

Combine all terms containing y.

$$\left(z-2\right)y=2\sqrt{2}z^{2}+4z-4\sqrt{2}z-8-xz^{2}+2xz$$

The equation is in standard form.

$$\left(z-2\right)y=-xz^{2}+2\sqrt{2}z^{2}+2xz-4\sqrt{2}z+4z-8$$

Divide both sides by -2+z.

$$\frac{\left(z-2\right)y}{z-2}=\frac{-xz^{2}+2\sqrt{2}z^{2}+2xz-4\sqrt{2}z+4z-8}{z-2}$$

Dividing by -2+z undoes the multiplication by -2+z.

$$y=\frac{-xz^{2}+2\sqrt{2}z^{2}+2xz-4\sqrt{2}z+4z-8}{z-2}$$

Divide 2z^{2}\sqrt{2}+4z-4\sqrt{2}z-8-xz^{2}+2xz by -2+z.

$$y=-xz+2\sqrt{2}z+4$$