Solve x^2y-2xy-12x-3y=-16/x+3

$${ x }^{ 2 } y-2xy-12x-3y= \frac{ -16 }{ x+3 }$$

Solve for y

$$y=-\frac{4\left(4-9x-3x^{2}\right)}{x^{3}+x^{2}-9x-9}$$

$$x\neq -1\text{ and }|x|\neq 3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+3.

$$x^{2}y\left(x+3\right)-2xy\left(x+3\right)-12x\left(x+3\right)-3y\left(x+3\right)=-16$$

Use the distributive property to multiply x^{2}y by x+3.

$$yx^{3}+3x^{2}y-2xy\left(x+3\right)-12x\left(x+3\right)-3y\left(x+3\right)=-16$$

Use the distributive property to multiply -2xy by x+3.

$$yx^{3}+3x^{2}y-2yx^{2}-6xy-12x\left(x+3\right)-3y\left(x+3\right)=-16$$

Combine 3x^{2}y and -2yx^{2} to get x^{2}y.

$$yx^{3}+x^{2}y-6xy-12x\left(x+3\right)-3y\left(x+3\right)=-16$$

Use the distributive property to multiply -12x by x+3.

$$yx^{3}+x^{2}y-6xy-12x^{2}-36x-3y\left(x+3\right)=-16$$

Use the distributive property to multiply -3y by x+3.

$$yx^{3}+x^{2}y-6xy-12x^{2}-36x-3yx-9y=-16$$

Combine -6xy and -3yx to get -9xy.

$$yx^{3}+x^{2}y-9xy-12x^{2}-36x-9y=-16$$

Add 12x^{2} to both sides.

$$yx^{3}+x^{2}y-9xy-36x-9y=-16+12x^{2}$$

Add 36x to both sides.

$$yx^{3}+x^{2}y-9xy-9y=-16+12x^{2}+36x$$

Combine all terms containing y.

$$\left(x^{3}+x^{2}-9x-9\right)y=-16+12x^{2}+36x$$

The equation is in standard form.

$$\left(x^{3}+x^{2}-9x-9\right)y=12x^{2}+36x-16$$

Divide both sides by x^{3}-9+x^{2}-9x.

$$\frac{\left(x^{3}+x^{2}-9x-9\right)y}{x^{3}+x^{2}-9x-9}=\frac{12x^{2}+36x-16}{x^{3}+x^{2}-9x-9}$$

Dividing by x^{3}-9+x^{2}-9x undoes the multiplication by x^{3}-9+x^{2}-9x.

$$y=\frac{12x^{2}+36x-16}{x^{3}+x^{2}-9x-9}$$

Divide -16+12x^{2}+36x by x^{3}-9+x^{2}-9x.

$$y=\frac{4\left(3x^{2}+9x-4\right)}{x^{3}+x^{2}-9x-9}$$