Solve 2/x-3-x+y/x^2-9-1/x=0

$$\frac { 2 } { x - 3 } - \frac { x + y } { x ^ { 2 } - 9 } - \frac { 1 } { x } = 0$$

Solve for y (complex solution)

$$y=6+\frac{9}{x}$$

$$x\neq 3\text{ and }x\neq -3\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x.

$$x\left(x+3\right)\times 2-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$\left(x^{2}+3x\right)\times 2-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$2x^{2}+6x-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$2x^{2}+6x-\left(x^{2}+xy\right)-\left(x^{2}-9\right)=0$$

To find the opposite of x^{2}+xy, find the opposite of each term.

$$2x^{2}+6x-x^{2}-xy-\left(x^{2}-9\right)=0$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+6x-xy-\left(x^{2}-9\right)=0$$

To find the opposite of x^{2}-9, find the opposite of each term.

$$x^{2}+6x-xy-x^{2}+9=0$$

Combine x^{2} and -x^{2} to get 0.

$$6x-xy+9=0$$

Subtract 6x from both sides. Anything subtracted from zero gives its negation.

$$-xy+9=-6x$$

Subtract 9 from both sides.

$$-xy=-6x-9$$

The equation is in standard form.

$$\left(-x\right)y=-6x-9$$

Divide both sides by -x.

$$\frac{\left(-x\right)y}{-x}=\frac{-6x-9}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$y=\frac{-6x-9}{-x}$$

Divide -9-6x by -x.

$$y=6+\frac{9}{x}$$

Solve for x

$$x=-\frac{9}{6-y}$$

$$y\neq 9\text{ and }y\neq 6\text{ and }y\neq 3$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x.

$$x\left(x+3\right)\times 2-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$\left(x^{2}+3x\right)\times 2-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$2x^{2}+6x-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$2x^{2}+6x-\left(x^{2}+xy\right)-\left(x^{2}-9\right)=0$$

To find the opposite of x^{2}+xy, find the opposite of each term.

$$2x^{2}+6x-x^{2}-xy-\left(x^{2}-9\right)=0$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+6x-xy-\left(x^{2}-9\right)=0$$

To find the opposite of x^{2}-9, find the opposite of each term.

$$x^{2}+6x-xy-x^{2}+9=0$$

Combine x^{2} and -x^{2} to get 0.

$$6x-xy+9=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$6x-xy=-9$$

Combine all terms containing x.

$$\left(6-y\right)x=-9$$

Divide both sides by 6-y.

$$\frac{\left(6-y\right)x}{6-y}=-\frac{9}{6-y}$$

Dividing by 6-y undoes the multiplication by 6-y.

$$x=-\frac{9}{6-y}$$

Variable x cannot be equal to any of the values -3,3,0.

$$x=-\frac{9}{6-y}\text{, }x\neq -3\text{ and }x\neq 3\text{ and }x\neq 0$$

Solve for y

$$y=6+\frac{9}{x}$$

$$x\neq 0\text{ and }|x|\neq 3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}-9,x.

$$x\left(x+3\right)\times 2-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$\left(x^{2}+3x\right)\times 2-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$2x^{2}+6x-x\left(x+y\right)-\left(x^{2}-9\right)=0$$

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

$$2x^{2}+6x-\left(x^{2}+xy\right)-\left(x^{2}-9\right)=0$$

To find the opposite of x^{2}+xy, find the opposite of each term.

$$2x^{2}+6x-x^{2}-xy-\left(x^{2}-9\right)=0$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}+6x-xy-\left(x^{2}-9\right)=0$$

To find the opposite of x^{2}-9, find the opposite of each term.

$$x^{2}+6x-xy-x^{2}+9=0$$

Combine x^{2} and -x^{2} to get 0.

$$6x-xy+9=0$$

Subtract 6x from both sides. Anything subtracted from zero gives its negation.

$$-xy+9=-6x$$

Subtract 9 from both sides.

$$-xy=-6x-9$$

The equation is in standard form.

$$\left(-x\right)y=-6x-9$$

Divide both sides by -x.

$$\frac{\left(-x\right)y}{-x}=\frac{-6x-9}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$y=\frac{-6x-9}{-x}$$

Divide -9-6x by -x.

$$y=6+\frac{9}{x}$$