Solve x(x-3)+y(3-x)=0

$$x(x-3)+y(3-x)=0$$

Solve for y (complex solution)

$$\left\{\begin{matrix}\\y=x\text{, }&\text{unconditionally}\\y\in \mathrm{C}\text{, }&x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by x-3.

$$x^{2}-3x+y\left(3-x\right)=0$$

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

$$x^{2}-3x+3y-yx=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

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

Add 3x to both sides.

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

Combine all terms containing y.

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

The equation is in standard form.

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

Divide both sides by -x+3.

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

Dividing by -x+3 undoes the multiplication by -x+3.

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

Divide x\left(3-x\right) by -x+3.

$$y=x$$

Solve for y

$$\left\{\begin{matrix}\\y=x\text{, }&\text{unconditionally}\\y\in \mathrm{R}\text{, }&x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x by x-3.

$$x^{2}-3x+y\left(3-x\right)=0$$

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

$$x^{2}-3x+3y-yx=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

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

Add 3x to both sides.

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

Combine all terms containing y.

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

The equation is in standard form.

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

Divide both sides by 3-x.

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

Dividing by 3-x undoes the multiplication by 3-x.

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

Divide x\left(3-x\right) by 3-x.

$$y=x$$

Solve for x

$$x=y$$

$$x=3$$