Solve l:x-y=y+x

$$l : x - y = y + x$$

Solve for l

$$l=x\left(x+2y\right)$$

$$x\neq 0$$

Solution Steps

Multiply both sides of the equation by x.

$$l-yx=xy+xx$$

Multiply x and x to get x^{2}.

$$l-yx=xy+x^{2}$$

Add yx to both sides.

$$l=xy+x^{2}+yx$$

Combine xy and yx to get 2xy.

$$l=2xy+x^{2}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\sqrt{y^{2}+l}-y\text{, }&\left(arg(y)\geq \pi \text{ and }y\neq 0\right)\text{ or }l\neq 0\\x=-\sqrt{y^{2}+l}-y\text{, }&\left(arg(y)<\pi \text{ and }y\neq 0\right)\text{ or }l\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{y^{2}+l}-y\text{, }&\left(l\neq 0\text{ or }y<0\right)\text{ and }l\geq -y^{2}\\x=-\sqrt{y^{2}+l}-y\text{, }&\left(l\neq 0\text{ or }y>0\right)\text{ and }l\geq -y^{2}\end{matrix}\right.$$