Solve 6-x/2x^2y-2x/3xy^2=x/x^2y^2

$$\frac { 6 - x } { 2 x ^ { 2 } y } - \frac { 2 x } { 3 x y ^ { 2 } } = \frac { x } { x ^ { 2 } y ^ { 2 } }$$

Solve for y

$$y=\frac{2x\left(2x+3\right)}{3\left(6-x\right)}$$

$$x\neq -\frac{3}{2}\text{ and }x\neq 0\text{ and }x\neq 6$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6x^{2}y^{2}, the least common multiple of 2x^{2}y,3xy^{2},x^{2}y^{2}.

$$3y\left(6-x\right)-2x\times 2x=6x$$

Multiply 2x and 2x to get \left(2x\right)^{2}.

$$3y\left(6-x\right)-\left(2x\right)^{2}=6x$$

Use the distributive property to multiply 3y by 6-x.

$$18y-3yx-\left(2x\right)^{2}=6x$$

Expand \left(2x\right)^{2}.

$$18y-3yx-2^{2}x^{2}=6x$$

Calculate 2 to the power of 2 and get 4.

$$18y-3yx-4x^{2}=6x$$

Add 4x^{2} to both sides.

$$18y-3yx=6x+4x^{2}$$

Combine all terms containing y.

$$\left(18-3x\right)y=6x+4x^{2}$$

The equation is in standard form.

$$\left(18-3x\right)y=4x^{2}+6x$$

Divide both sides by 18-3x.

$$\frac{\left(18-3x\right)y}{18-3x}=\frac{2x\left(2x+3\right)}{18-3x}$$

Dividing by 18-3x undoes the multiplication by 18-3x.

$$y=\frac{2x\left(2x+3\right)}{18-3x}$$

Divide 2x\left(3+2x\right) by 18-3x.

$$y=\frac{2x\left(2x+3\right)}{3\left(6-x\right)}$$

Variable y cannot be equal to 0.

$$y=\frac{2x\left(2x+3\right)}{3\left(6-x\right)}\text{, }y\neq 0$$

Solve for x (complex solution)

$$x=-\frac{3\sqrt{y^{2}+36y+4}}{8}-\frac{3y}{8}-\frac{3}{4}$$

$$x=\frac{3\sqrt{y^{2}+36y+4}}{8}-\frac{3y}{8}-\frac{3}{4}\text{, }y\neq 0$$

Solve for x

$$x=-\frac{3\sqrt{y^{2}+36y+4}}{8}-\frac{3y}{8}-\frac{3}{4}$$

$$x=\frac{3\sqrt{y^{2}+36y+4}}{8}-\frac{3y}{8}-\frac{3}{4}\text{, }y\leq -8\sqrt{5}-18\text{ or }\left(y\neq 0\text{ and }y\geq 8\sqrt{5}-18\right)$$