Solve x^2y+y=x+1

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

Solve for y (complex solution)

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

$$x\neq -i\text{ and }x\neq i$$

Steps for Solving Linear Equation

Combine all terms containing y.

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

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

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

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

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

Solve for y

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

Steps for Solving Linear Equation

Combine all terms containing y.

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

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

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

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{1+4y-4y^{2}}+1}{2y}\text{; }x=\frac{-\sqrt{1+4y-4y^{2}}+1}{2y}\text{, }&y\neq 0\\x=-1\text{, }&y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{1+4y-4y^{2}}+1}{2y}\text{; }x=\frac{-\sqrt{1+4y-4y^{2}}+1}{2y}\text{, }&y\neq 0\text{ and }y\geq \frac{1-\sqrt{2}}{2}\text{ and }y\leq \frac{\sqrt{2}+1}{2}\\x=-1\text{, }&y=0\end{matrix}\right.$$