Solve x=zx(1-x)

$$x=zx(1-x)$$

Solve for z

$$\left\{\begin{matrix}z=-\frac{1}{x-1}\text{, }&x\neq 1\\z\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply zx by 1-x.

$$x=zx-zx^{2}$$

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

$$zx-zx^{2}=x$$

Combine all terms containing z.

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

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

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

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

$$z=\frac{x}{x-x^{2}}$$

Divide x by x-x^{2}.

$$z=\frac{1}{1-x}$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{z-1}{z}\text{, }&z\neq 0\end{matrix}\right.$$