Solve Q*f(x)=x/x-1

$$Q \cdot f ( x ) = \frac { x } { x - 1 }$$

Solve for Q (complex solution)

$$\left\{\begin{matrix}Q=\frac{1}{f\left(x-1\right)}\text{, }&x\neq 1\text{ and }f\neq 0\\Q\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x-1.

$$Qfx\left(x-1\right)=x$$

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

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

Combine all terms containing Q.

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

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

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

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

$$Q=\frac{x}{fx^{2}-fx}$$

Divide x by fx^{2}-fx.

$$Q=\frac{1}{f\left(x-1\right)}$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{1}{Q\left(x-1\right)}\text{, }&x\neq 1\text{ and }Q\neq 0\\f\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x-1.

$$Qfx\left(x-1\right)=x$$

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

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

Combine all terms containing f.

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

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

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

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

$$f=\frac{x}{Qx^{2}-Qx}$$

Divide x by Qx^{2}-Qx.

$$f=\frac{1}{Q\left(x-1\right)}$$

Solve for Q

$$\left\{\begin{matrix}Q=\frac{1}{f\left(x-1\right)}\text{, }&x\neq 1\text{ and }f\neq 0\\Q\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x-1.

$$Qfx\left(x-1\right)=x$$

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

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

Combine all terms containing Q.

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

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

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

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

$$Q=\frac{x}{fx^{2}-fx}$$

Divide x by fx^{2}-fx.

$$Q=\frac{1}{f\left(x-1\right)}$$

Solve for f

$$\left\{\begin{matrix}f=\frac{1}{Q\left(x-1\right)}\text{, }&x\neq 1\text{ and }Q\neq 0\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x-1.

$$Qfx\left(x-1\right)=x$$

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

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

Combine all terms containing f.

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

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

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

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

$$f=\frac{x}{Qx^{2}-Qx}$$

Divide x by Qx^{2}-Qx.

$$f=\frac{1}{Q\left(x-1\right)}$$