Solve z=x*f(y/x)+f(y/x)

$$z = x \cdot f ( \frac { y } { x } ) + f ( \frac { y } { x } )$$

Solve for f

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$zx=xfy+fy$$

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

$$xfy+fy=zx$$

Combine all terms containing f.

$$\left(xy+y\right)f=zx$$

The equation is in standard form.

$$\left(xy+y\right)f=xz$$

Divide both sides by xy+y.

$$\frac{\left(xy+y\right)f}{xy+y}=\frac{xz}{xy+y}$$

Dividing by xy+y undoes the multiplication by xy+y.

$$f=\frac{xz}{xy+y}$$

Divide zx by xy+y.

$$f=\frac{xz}{y\left(x+1\right)}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{fy}{z-fy}\text{, }&f\neq 0\text{ and }y\neq 0\text{ and }z\neq fy\\x\neq 0\text{, }&\left(y=0\text{ or }f=0\right)\text{ and }z=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$zx=xfy+fy$$

Subtract xfy from both sides.

$$zx-xfy=fy$$

Reorder the terms.

$$-fxy+xz=fy$$

Combine all terms containing x.

$$\left(-fy+z\right)x=fy$$

The equation is in standard form.

$$\left(z-fy\right)x=fy$$

Divide both sides by z-yf.

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

Dividing by z-yf undoes the multiplication by z-yf.

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

Variable x cannot be equal to 0.

$$x=\frac{fy}{z-fy}\text{, }x\neq 0$$