Solve y+1/y-x+3/x-2=1/xy-2y

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

Solve for x

$$x=5y+3$$

$$y\neq -\frac{1}{5}\text{ and }y\neq 0$$

Solution Steps

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by y\left(x-2\right), the least common multiple of y,x-2,xy-2y.

$$\left(x-2\right)\left(y+1\right)-y\left(x+3\right)=1$$

Use the distributive property to multiply x-2 by y+1.

$$xy+x-2y-2-y\left(x+3\right)=1$$

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

$$xy+x-2y-2-\left(yx+3y\right)=1$$

To find the opposite of yx+3y, find the opposite of each term.

$$xy+x-2y-2-yx-3y=1$$

Combine xy and -yx to get 0.

$$x-2y-2-3y=1$$

Combine -2y and -3y to get -5y.

$$x-5y-2=1$$

Add 5y to both sides.

$$x-2=1+5y$$

Add 2 to both sides.

$$x=1+5y+2$$

Add 1 and 2 to get 3.

$$x=3+5y$$

Variable x cannot be equal to 2.

$$x=3+5y\text{, }x\neq 2$$

Solve for y

$$y=\frac{x-3}{5}$$

$$x\neq 3\text{ and }x\neq 2$$

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 y\left(x-2\right), the least common multiple of y,x-2,xy-2y.

$$\left(x-2\right)\left(y+1\right)-y\left(x+3\right)=1$$

Use the distributive property to multiply x-2 by y+1.

$$xy+x-2y-2-y\left(x+3\right)=1$$

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

$$xy+x-2y-2-\left(yx+3y\right)=1$$

To find the opposite of yx+3y, find the opposite of each term.

$$xy+x-2y-2-yx-3y=1$$

Combine xy and -yx to get 0.

$$x-2y-2-3y=1$$

Combine -2y and -3y to get -5y.

$$x-5y-2=1$$

Subtract x from both sides.

$$-5y-2=1-x$$

Add 2 to both sides.

$$-5y=1-x+2$$

Add 1 and 2 to get 3.

$$-5y=3-x$$

Divide both sides by -5.

$$\frac{-5y}{-5}=\frac{3-x}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$y=\frac{3-x}{-5}$$

Divide 3-x by -5.

$$y=\frac{x-3}{5}$$

Variable y cannot be equal to 0.

$$y=\frac{x-3}{5}\text{, }y\neq 0$$