Solve x^2y+4xy-x^2+x+4y-6=0

$$x ^ { 2 } y + 4 x y - x ^ { 2 } + x + 4 y - 6 = 0$$

Solve for y

$$y=-\frac{-x^{2}+x-6}{\left(x+2\right)^{2}}$$

$$x\neq -2$$

Steps for Solving Linear Equation

Add x^{2} to both sides. Anything plus zero gives itself.

$$x^{2}y+4xy+x+4y-6=x^{2}$$

Subtract x from both sides.

$$x^{2}y+4xy+4y-6=x^{2}-x$$

Add 6 to both sides.

$$x^{2}y+4xy+4y=x^{2}-x+6$$

Combine all terms containing y.

$$\left(x^{2}+4x+4\right)y=x^{2}-x+6$$

Divide both sides by x^{2}+4x+4.

$$\frac{\left(x^{2}+4x+4\right)y}{x^{2}+4x+4}=\frac{x^{2}-x+6}{x^{2}+4x+4}$$

Dividing by x^{2}+4x+4 undoes the multiplication by x^{2}+4x+4.

$$y=\frac{x^{2}-x+6}{x^{2}+4x+4}$$

Divide x^{2}-x+6 by x^{2}+4x+4.

$$y=\frac{x^{2}-x+6}{\left(x+2\right)^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-4y+\sqrt{48y-23}-1}{2\left(y-1\right)}\text{; }x=-\frac{4y+\sqrt{48y-23}+1}{2\left(y-1\right)}\text{, }&y\neq 1\\x=\frac{2}{5}=0.4\text{, }&y=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-4y+\sqrt{48y-23}-1}{2\left(y-1\right)}\text{; }x=-\frac{4y+\sqrt{48y-23}+1}{2\left(y-1\right)}\text{, }&y\neq 1\text{ and }y\geq \frac{23}{48}\\x=\frac{2}{5}=0.4\text{, }&y=1\end{matrix}\right.$$