Solve x(y-z)+y(x-z)+z(x-y)=0

$$x(y-z)+y(x-z)+z(x-y)=0$$

Solve for x

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

Steps for Solving Linear Equation

Use the distributive property to multiply x by y-z.

$$xy-xz+y\left(x-z\right)+z\left(x-y\right)=0$$

Use the distributive property to multiply y by x-z.

$$xy-xz+yx-yz+z\left(x-y\right)=0$$

Combine xy and yx to get 2xy.

$$2xy-xz-yz+z\left(x-y\right)=0$$

Use the distributive property to multiply z by x-y.

$$2xy-xz-yz+zx-zy=0$$

Combine -xz and zx to get 0.

$$2xy-yz-zy=0$$

Combine -yz and -zy to get -2yz.

$$2xy-2yz=0$$

Add 2yz to both sides. Anything plus zero gives itself.

$$2xy=2yz$$

Cancel out 2 on both sides.

$$xy=yz$$

The equation is in standard form.

$$yx=yz$$

Divide both sides by y.

$$\frac{yx}{y}=\frac{yz}{y}$$

Dividing by y undoes the multiplication by y.

$$x=\frac{yz}{y}$$

Divide yz by y.

$$x=z$$

Solve for y

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

Steps for Solving Linear Equation

Use the distributive property to multiply x by y-z.

$$xy-xz+y\left(x-z\right)+z\left(x-y\right)=0$$

Use the distributive property to multiply y by x-z.

$$xy-xz+yx-yz+z\left(x-y\right)=0$$

Combine xy and yx to get 2xy.

$$2xy-xz-yz+z\left(x-y\right)=0$$

Use the distributive property to multiply z by x-y.

$$2xy-xz-yz+zx-zy=0$$

Combine -xz and zx to get 0.

$$2xy-yz-zy=0$$

Combine -yz and -zy to get -2yz.

$$2xy-2yz=0$$

Combine all terms containing y.

$$\left(2x-2z\right)y=0$$

Divide 0 by 2x-2z.

$$y=0$$