Solve f(x)-Zf(x)=3x+2

$$f ( x ) - Z f ( x ) = 3 x + 2$$

Solve for Z (complex solution)

$$\left\{\begin{matrix}Z=\frac{fx-3x-2}{fx}\text{, }&x\neq 0\text{ and }f\neq 0\\Z\in \mathrm{C}\text{, }&x=-\frac{2}{3}\text{ and }f=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract fx from both sides.

$$-Zfx=3x+2-fx$$

The equation is in standard form.

$$\left(-fx\right)Z=2+3x-fx$$

Divide both sides by -fx.

$$\frac{\left(-fx\right)Z}{-fx}=\frac{2+3x-fx}{-fx}$$

Dividing by -fx undoes the multiplication by -fx.

$$Z=\frac{2+3x-fx}{-fx}$$

Divide 2+3x-xf by -fx.

$$Z=1-\frac{3x+2}{fx}$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{3x+2}{x\left(1-Z\right)}\text{, }&Z\neq 1\text{ and }x\neq 0\\f\in \mathrm{C}\text{, }&x=-\frac{2}{3}\text{ and }Z=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Reorder the terms.

$$-Zfx+fx=3x+2$$

Combine all terms containing f.

$$\left(-Zx+x\right)f=3x+2$$

The equation is in standard form.

$$\left(x-Zx\right)f=3x+2$$

Divide both sides by x-xZ.

$$\frac{\left(x-Zx\right)f}{x-Zx}=\frac{3x+2}{x-Zx}$$

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

$$f=\frac{3x+2}{x-Zx}$$

Divide 3x+2 by x-xZ.

$$f=\frac{3x+2}{x\left(1-Z\right)}$$

Solve for Z

$$\left\{\begin{matrix}Z=\frac{fx-3x-2}{fx}\text{, }&x\neq 0\text{ and }f\neq 0\\Z\in \mathrm{R}\text{, }&x=-\frac{2}{3}\text{ and }f=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract fx from both sides.

$$-Zfx=3x+2-fx$$

The equation is in standard form.

$$\left(-fx\right)Z=2+3x-fx$$

Divide both sides by -fx.

$$\frac{\left(-fx\right)Z}{-fx}=\frac{2+3x-fx}{-fx}$$

Dividing by -fx undoes the multiplication by -fx.

$$Z=\frac{2+3x-fx}{-fx}$$

Divide 3x+2-fx by -fx.

$$Z=1-\frac{3x+2}{fx}$$

Solve for f

$$\left\{\begin{matrix}f=\frac{3x+2}{x\left(1-Z\right)}\text{, }&Z\neq 1\text{ and }x\neq 0\\f\in \mathrm{R}\text{, }&x=-\frac{2}{3}\text{ and }Z=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Reorder the terms.

$$-Zfx+fx=3x+2$$

Combine all terms containing f.

$$\left(-Zx+x\right)f=3x+2$$

The equation is in standard form.

$$\left(x-Zx\right)f=3x+2$$

Divide both sides by x-xZ.

$$\frac{\left(x-Zx\right)f}{x-Zx}=\frac{3x+2}{x-Zx}$$

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

$$f=\frac{3x+2}{x-Zx}$$

Divide 3x+2 by x-xZ.

$$f=\frac{3x+2}{x\left(1-Z\right)}$$