Solve left(1+xright)^2+left(1+yright)^2=left(x+yright)^2

$${ \left(1+x \right) }^{ 2 } + { \left(1+y \right) }^{ 2 } = { \left(x+y \right) }^{ 2 }$$

Solve for x

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

$$y\neq 1$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$1+2x+x^{2}+\left(1+y\right)^{2}=\left(x+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+y\right)^{2}.

$$1+2x+x^{2}+1+2y+y^{2}=\left(x+y\right)^{2}$$

Add 1 and 1 to get 2.

$$2+2x+x^{2}+2y+y^{2}=\left(x+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.

$$2+2x+x^{2}+2y+y^{2}=x^{2}+2xy+y^{2}$$

Subtract x^{2} from both sides.

$$2+2x+x^{2}+2y+y^{2}-x^{2}=2xy+y^{2}$$

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

$$2+2x+2y+y^{2}=2xy+y^{2}$$

Subtract 2xy from both sides.

$$2+2x+2y+y^{2}-2xy=y^{2}$$

Subtract 2 from both sides.

$$2x+2y+y^{2}-2xy=y^{2}-2$$

Subtract 2y from both sides.

$$2x+y^{2}-2xy=y^{2}-2-2y$$

Subtract y^{2} from both sides.

$$2x-2xy=y^{2}-2-2y-y^{2}$$

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

$$2x-2xy=-2-2y$$

Combine all terms containing x.

$$\left(2-2y\right)x=-2-2y$$

The equation is in standard form.

$$\left(2-2y\right)x=-2y-2$$

Divide both sides by -2y+2.

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

Dividing by -2y+2 undoes the multiplication by -2y+2.

$$x=\frac{-2y-2}{2-2y}$$

Divide -2-2y by -2y+2.

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

Solve for y

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

$$x\neq 1$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$1+2x+x^{2}+\left(1+y\right)^{2}=\left(x+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+y\right)^{2}.

$$1+2x+x^{2}+1+2y+y^{2}=\left(x+y\right)^{2}$$

Add 1 and 1 to get 2.

$$2+2x+x^{2}+2y+y^{2}=\left(x+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.

$$2+2x+x^{2}+2y+y^{2}=x^{2}+2xy+y^{2}$$

Subtract 2xy from both sides.

$$2+2x+x^{2}+2y+y^{2}-2xy=x^{2}+y^{2}$$

Subtract y^{2} from both sides.

$$2+2x+x^{2}+2y+y^{2}-2xy-y^{2}=x^{2}$$

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

$$2+2x+x^{2}+2y-2xy=x^{2}$$

Subtract 2 from both sides.

$$2x+x^{2}+2y-2xy=x^{2}-2$$

Subtract 2x from both sides.

$$x^{2}+2y-2xy=x^{2}-2-2x$$

Subtract x^{2} from both sides.

$$2y-2xy=x^{2}-2-2x-x^{2}$$

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

$$2y-2xy=-2-2x$$

Combine all terms containing y.

$$\left(2-2x\right)y=-2-2x$$

The equation is in standard form.

$$\left(2-2x\right)y=-2x-2$$

Divide both sides by -2x+2.

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

Dividing by -2x+2 undoes the multiplication by -2x+2.

$$y=\frac{-2x-2}{2-2x}$$

Divide -2-2x by -2x+2.

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