Solve 2xlambda=2x+y^2-1

$$2 x \lambda = 2 x + y ^ { 2 } - 1$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{y^{2}-1}{2\left(\lambda -1\right)}\text{, }&\lambda \neq 1\\x\in \mathrm{C}\text{, }&\left(y=-1\text{ or }y=1\right)\text{ and }\lambda =1\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 2x from both sides.

$$2x\lambda -2x=y^{2}-1$$

Combine all terms containing x.

$$\left(2\lambda -2\right)x=y^{2}-1$$

Divide both sides by 2\lambda -2.

$$\frac{\left(2\lambda -2\right)x}{2\lambda -2}=\frac{y^{2}-1}{2\lambda -2}$$

Dividing by 2\lambda -2 undoes the multiplication by 2\lambda -2.

$$x=\frac{y^{2}-1}{2\lambda -2}$$

Divide -1+y^{2} by 2\lambda -2.

$$x=\frac{y^{2}-1}{2\left(\lambda -1\right)}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{y^{2}-1}{2\left(\lambda -1\right)}\text{, }&\lambda \neq 1\\x\in \mathrm{R}\text{, }&\lambda =1\text{ and }|y|=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 2x from both sides.

$$2x\lambda -2x=y^{2}-1$$

Combine all terms containing x.

$$\left(2\lambda -2\right)x=y^{2}-1$$

Divide both sides by 2\lambda -2.

$$\frac{\left(2\lambda -2\right)x}{2\lambda -2}=\frac{y^{2}-1}{2\lambda -2}$$

Dividing by 2\lambda -2 undoes the multiplication by 2\lambda -2.

$$x=\frac{y^{2}-1}{2\lambda -2}$$

Divide -1+y^{2} by 2\lambda -2.

$$x=\frac{y^{2}-1}{2\left(\lambda -1\right)}$$

Solve for y (complex solution)

$$y=-\sqrt{2x\lambda -2x+1}$$

$$y=\sqrt{2x\lambda -2x+1}$$

Solve for y

$$y=\sqrt{2x\lambda -2x+1}$$

$$y=-\sqrt{2x\lambda -2x+1}\text{, }\left(\lambda \leq 1\text{ or }x\geq -\frac{1}{2\lambda -2}\right)\text{ and }\left(\lambda \geq 1\text{ or }x\leq -\frac{1}{2\lambda -2}\right)$$