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

$$x ^ { 3 } + y ^ { 3 } = ( x + y ) ( x ^ { 2 } - x y + y )$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-y\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&y=1\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+y by x^{2}-xy+y and combine like terms.

$$x^{3}+y^{3}=x^{3}+xy-xy^{2}+y^{2}$$

Subtract x^{3} from both sides.

$$x^{3}+y^{3}-x^{3}=xy-xy^{2}+y^{2}$$

Combine x^{3} and -x^{3} to get 0.

$$y^{3}=xy-xy^{2}+y^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$xy-xy^{2}+y^{2}=y^{3}$$

Subtract y^{2} from both sides.

$$xy-xy^{2}=y^{3}-y^{2}$$

Combine all terms containing x.

$$\left(y-y^{2}\right)x=y^{3}-y^{2}$$

Divide both sides by y-y^{2}.

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

Dividing by y-y^{2} undoes the multiplication by y-y^{2}.

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

Divide \left(-1+y\right)y^{2} by y-y^{2}.

$$x=-y$$

Solve for x

$$\left\{\begin{matrix}\\x=-y\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&y=1\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+y by x^{2}-xy+y and combine like terms.

$$x^{3}+y^{3}=x^{3}+xy-xy^{2}+y^{2}$$

Subtract x^{3} from both sides.

$$x^{3}+y^{3}-x^{3}=xy-xy^{2}+y^{2}$$

Combine x^{3} and -x^{3} to get 0.

$$y^{3}=xy-xy^{2}+y^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$xy-xy^{2}+y^{2}=y^{3}$$

Subtract y^{2} from both sides.

$$xy-xy^{2}=y^{3}-y^{2}$$

Combine all terms containing x.

$$\left(y-y^{2}\right)x=y^{3}-y^{2}$$

Divide both sides by y-y^{2}.

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

Dividing by y-y^{2} undoes the multiplication by y-y^{2}.

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

Divide \left(-1+y\right)y^{2} by y-y^{2}.

$$x=-y$$

Solve for y

$$y=1$$

$$y=0$$

$$y=-x$$