Solve yx^2-y=x^3-1

$$y { x }^{ 2 } -y= { x }^{ 3 } -1$$

Solve for y (complex solution)

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

Steps for Solving Linear Equation

Combine all terms containing y.

$$\left(x^{2}-1\right)y=x^{3}-1$$

Divide both sides by x^{2}-1.

$$\frac{\left(x^{2}-1\right)y}{x^{2}-1}=\frac{x^{3}-1}{x^{2}-1}$$

Dividing by x^{2}-1 undoes the multiplication by x^{2}-1.

$$y=\frac{x^{3}-1}{x^{2}-1}$$

Divide x^{3}-1 by x^{2}-1.

$$y=\frac{x^{2}}{x+1}+1$$

Solve for y

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

Steps for Solving Linear Equation

Combine all terms containing y.

$$\left(x^{2}-1\right)y=x^{3}-1$$

Divide both sides by x^{2}-1.

$$\frac{\left(x^{2}-1\right)y}{x^{2}-1}=\frac{x^{3}-1}{x^{2}-1}$$

Dividing by x^{2}-1 undoes the multiplication by x^{2}-1.

$$y=\frac{x^{3}-1}{x^{2}-1}$$

Divide x^{3}-1 by x^{2}-1.

$$y=\frac{x^{2}}{x+1}+1$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{\left(y-1\right)\left(y+3\right)}+y-1}{2}$$

$$x=\frac{\sqrt{\left(y-1\right)\left(y+3\right)}+y-1}{2}$$

$$x=1$$

Solve for x

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x=\frac{-\sqrt{\left(y-1\right)\left(y+3\right)}+y-1}{2}\text{; }x=\frac{\sqrt{\left(y-1\right)\left(y+3\right)}+y-1}{2}\text{, }&y\leq -3\text{ or }y\geq 1\end{matrix}\right.$$