Solve 3xy+3x+3y+2=3(x+1)(y+1)-1

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

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

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

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

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

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

Subtract 1 from 3 to get 2.

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

Subtract 3xy from both sides.

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

Combine 3xy and -3xy to get 0.

$$3x+3y+2=3x+3y+2$$

Subtract 3x from both sides.

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

Combine 3x and -3x to get 0.

$$3y+2=3y+2$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for y (complex solution)

$$y\in \mathrm{C}$$

Solution Steps

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

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

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

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

Subtract 1 from 3 to get 2.

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

Subtract 3xy from both sides.

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

Combine 3xy and -3xy to get 0.

$$3x+3y+2=3x+3y+2$$

Subtract 3y from both sides.

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

Combine 3y and -3y to get 0.

$$3x+2=3x+2$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

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

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

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

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

Subtract 1 from 3 to get 2.

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

Subtract 3xy from both sides.

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

Combine 3xy and -3xy to get 0.

$$3x+3y+2=3x+3y+2$$

Subtract 3x from both sides.

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

Combine 3x and -3x to get 0.

$$3y+2=3y+2$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Solve for y

$$y\in \mathrm{R}$$

Solution Steps

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

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

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

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

Subtract 1 from 3 to get 2.

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

Subtract 3xy from both sides.

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

Combine 3xy and -3xy to get 0.

$$3x+3y+2=3x+3y+2$$

Subtract 3y from both sides.

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

Combine 3y and -3y to get 0.

$$3x+2=3x+2$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{R}$$