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

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

Solve for x

$$x=5$$

$$y\neq -1$$

Steps for Solving Linear Equation

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+1\right)\left(y+1\right), the least common multiple of xy+x+y+1,3.

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

Use the distributive property to multiply 3 by xy-y+x-1.

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

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

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

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

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

Subtract 2xy from both sides.

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

Combine 3xy and -2xy to get xy.

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

Subtract 2x from both sides.

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

Combine 3x and -2x to get x.

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

Add 3y to both sides.

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

Combine 2y and 3y to get 5y.

$$xy+x-3=5y+2$$

Add 3 to both sides.

$$xy+x=5y+2+3$$

Add 2 and 3 to get 5.

$$xy+x=5y+5$$

Combine all terms containing x.

$$\left(y+1\right)x=5y+5$$

Divide both sides by 1+y.

$$\frac{\left(y+1\right)x}{y+1}=\frac{5y+5}{y+1}$$

Dividing by 1+y undoes the multiplication by 1+y.

$$x=\frac{5y+5}{y+1}$$

Divide 5+5y by 1+y.

$$x=5$$

Solve for y

$$y\neq -1$$

$$x=5$$

Steps for Solving Linear Equation

Variable y cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+1\right)\left(y+1\right), the least common multiple of xy+x+y+1,3.

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

Use the distributive property to multiply 3 by xy-y+x-1.

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

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

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

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

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

Subtract 2xy from both sides.

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

Combine 3xy and -2xy to get xy.

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

Subtract 2y from both sides.

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

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

$$xy-5y+3x-3=2x+2$$

Subtract 3x from both sides.

$$xy-5y-3=2x+2-3x$$

Combine 2x and -3x to get -x.

$$xy-5y-3=-x+2$$

Add 3 to both sides.

$$xy-5y=-x+2+3$$

Add 2 and 3 to get 5.

$$xy-5y=-x+5$$

Combine all terms containing y.

$$\left(x-5\right)y=-x+5$$

The equation is in standard form.

$$\left(x-5\right)y=5-x$$

Divide both sides by x-5.

$$\frac{\left(x-5\right)y}{x-5}=\frac{5-x}{x-5}$$

Dividing by x-5 undoes the multiplication by x-5.

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

Divide -x+5 by x-5.

$$y=-1$$

Variable y cannot be equal to -1.

$$y\in \emptyset$$