Solve frac{sqrt{2}-2}{2}x^2=(frac{sqrt{2}-2}{2}x+x)y

$$\frac{ \sqrt{ 2 } -2 }{ 2 } { x }^{ 2 } =( \frac{ \sqrt{ 2 } -2 }{ 2 } x+x)y$$

Solve for y (complex solution)

$$\left\{\begin{matrix}\\y=\left(1-\sqrt{2}\right)x\text{, }&\text{unconditionally}\\y\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

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

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

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

Express \frac{\sqrt{2}-2}{2}x as a single fraction.

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

Use the distributive property to multiply 2 by \frac{\left(\sqrt{2}-2\right)x}{2}+x.

$$\sqrt{2}x^{2}-2x^{2}=\left(2\times \frac{\left(\sqrt{2}-2\right)x}{2}+2x\right)y$$

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

$$\sqrt{2}x^{2}-2x^{2}=\left(2\times \frac{\sqrt{2}x-2x}{2}+2x\right)y$$

Express 2\times \frac{\sqrt{2}x-2x}{2} as a single fraction.

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

Cancel out 2 and 2.

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

Combine -2x and 2x to get 0.

$$\sqrt{2}x^{2}-2x^{2}=\sqrt{2}xy$$

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{2}xy=\sqrt{2}x^{2}-2x^{2}$$

Divide both sides by \sqrt{2}x.

$$\frac{\sqrt{2}xy}{\sqrt{2}x}=\frac{\left(\sqrt{2}-2\right)x^{2}}{\sqrt{2}x}$$

Dividing by \sqrt{2}x undoes the multiplication by \sqrt{2}x.

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

Divide \left(\sqrt{2}-2\right)x^{2} by \sqrt{2}x.

$$y=-\sqrt{2}x+x$$

Solve for y

$$\left\{\begin{matrix}\\y=\left(1-\sqrt{2}\right)x\text{, }&\text{unconditionally}\\y\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

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

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

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

Express \frac{\sqrt{2}-2}{2}x as a single fraction.

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

Use the distributive property to multiply 2 by \frac{\left(\sqrt{2}-2\right)x}{2}+x.

$$\sqrt{2}x^{2}-2x^{2}=\left(2\times \frac{\left(\sqrt{2}-2\right)x}{2}+2x\right)y$$

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

$$\sqrt{2}x^{2}-2x^{2}=\left(2\times \frac{\sqrt{2}x-2x}{2}+2x\right)y$$

Express 2\times \frac{\sqrt{2}x-2x}{2} as a single fraction.

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

Cancel out 2 and 2.

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

Combine -2x and 2x to get 0.

$$\sqrt{2}x^{2}-2x^{2}=\sqrt{2}xy$$

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{2}xy=\sqrt{2}x^{2}-2x^{2}$$

Divide both sides by \sqrt{2}x.

$$\frac{\sqrt{2}xy}{\sqrt{2}x}=\frac{\left(\sqrt{2}-2\right)x^{2}}{\sqrt{2}x}$$

Dividing by \sqrt{2}x undoes the multiplication by \sqrt{2}x.

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

Divide \left(\sqrt{2}-2\right)x^{2} by \sqrt{2}x.

$$y=-\sqrt{2}x+x$$

Solve for x

$$x=-\left(\sqrt{2}+1\right)y$$

$$x=0$$