Solve z=f(yx/z+z)

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

Solve for f (complex solution)

$$f=\frac{z^{2}}{xy+z^{2}}$$

$$z\neq -i\sqrt{x}\sqrt{y}\text{ and }z\neq i\sqrt{x}\sqrt{y}\text{ and }z\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by z.

$$zz=f\left(\frac{yx}{z}+z\right)z$$

Multiply z and z to get z^{2}.

$$z^{2}=f\left(\frac{yx}{z}+z\right)z$$

To add or subtract expressions, expand them to make their denominators the same. Multiply z times \frac{z}{z}.

$$z^{2}=f\left(\frac{yx}{z}+\frac{zz}{z}\right)z$$

Since \frac{yx}{z} and \frac{zz}{z} have the same denominator, add them by adding their numerators.

$$z^{2}=f\times \frac{yx+zz}{z}z$$

Do the multiplications in yx+zz.

$$z^{2}=f\times \frac{yx+z^{2}}{z}z$$

Express f\times \frac{yx+z^{2}}{z} as a single fraction.

$$z^{2}=\frac{f\left(yx+z^{2}\right)}{z}z$$

Express \frac{f\left(yx+z^{2}\right)}{z}z as a single fraction.

$$z^{2}=\frac{f\left(yx+z^{2}\right)z}{z}$$

Cancel out z in both numerator and denominator.

$$z^{2}=f\left(xy+z^{2}\right)$$

Use the distributive property to multiply f by xy+z^{2}.

$$z^{2}=fxy+fz^{2}$$

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

$$fxy+fz^{2}=z^{2}$$

Combine all terms containing f.

$$\left(xy+z^{2}\right)f=z^{2}$$

Divide both sides by yx+z^{2}.

$$\frac{\left(xy+z^{2}\right)f}{xy+z^{2}}=\frac{z^{2}}{xy+z^{2}}$$

Dividing by yx+z^{2} undoes the multiplication by yx+z^{2}.

$$f=\frac{z^{2}}{xy+z^{2}}$$

Solve for x (complex solution)

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

Steps for Solving Linear Equation

Multiply both sides of the equation by z.

$$zz=f\left(\frac{yx}{z}+z\right)z$$

Multiply z and z to get z^{2}.

$$z^{2}=f\left(\frac{yx}{z}+z\right)z$$

To add or subtract expressions, expand them to make their denominators the same. Multiply z times \frac{z}{z}.

$$z^{2}=f\left(\frac{yx}{z}+\frac{zz}{z}\right)z$$

Since \frac{yx}{z} and \frac{zz}{z} have the same denominator, add them by adding their numerators.

$$z^{2}=f\times \frac{yx+zz}{z}z$$

Do the multiplications in yx+zz.

$$z^{2}=f\times \frac{yx+z^{2}}{z}z$$

Express f\times \frac{yx+z^{2}}{z} as a single fraction.

$$z^{2}=\frac{f\left(yx+z^{2}\right)}{z}z$$

Express \frac{f\left(yx+z^{2}\right)}{z}z as a single fraction.

$$z^{2}=\frac{f\left(yx+z^{2}\right)z}{z}$$

Cancel out z in both numerator and denominator.

$$z^{2}=f\left(xy+z^{2}\right)$$

Use the distributive property to multiply f by xy+z^{2}.

$$z^{2}=fxy+fz^{2}$$

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

$$fxy+fz^{2}=z^{2}$$

Subtract fz^{2} from both sides.

$$fxy=z^{2}-fz^{2}$$

Reorder the terms.

$$fxy=-fz^{2}+z^{2}$$

The equation is in standard form.

$$fyx=z^{2}-fz^{2}$$

Divide both sides by fy.

$$\frac{fyx}{fy}=\frac{\left(1-f\right)z^{2}}{fy}$$

Dividing by fy undoes the multiplication by fy.

$$x=\frac{\left(1-f\right)z^{2}}{fy}$$

Solve for f

$$f=\frac{z^{2}}{xy+z^{2}}$$

$$\left(x>0\text{ and }y>0\text{ and }z\neq 0\right)\text{ or }\left(x<0\text{ and }y<0\text{ and }z\neq 0\right)\text{ or }\left(|z|\neq \sqrt{-xy}\text{ and }z\neq 0\right)$$

Steps for Solving Linear Equation

Multiply both sides of the equation by z.

$$zz=f\left(\frac{yx}{z}+z\right)z$$

Multiply z and z to get z^{2}.

$$z^{2}=f\left(\frac{yx}{z}+z\right)z$$

To add or subtract expressions, expand them to make their denominators the same. Multiply z times \frac{z}{z}.

$$z^{2}=f\left(\frac{yx}{z}+\frac{zz}{z}\right)z$$

Since \frac{yx}{z} and \frac{zz}{z} have the same denominator, add them by adding their numerators.

$$z^{2}=f\times \frac{yx+zz}{z}z$$

Do the multiplications in yx+zz.

$$z^{2}=f\times \frac{yx+z^{2}}{z}z$$

Express f\times \frac{yx+z^{2}}{z} as a single fraction.

$$z^{2}=\frac{f\left(yx+z^{2}\right)}{z}z$$

Express \frac{f\left(yx+z^{2}\right)}{z}z as a single fraction.

$$z^{2}=\frac{f\left(yx+z^{2}\right)z}{z}$$

Cancel out z in both numerator and denominator.

$$z^{2}=f\left(xy+z^{2}\right)$$

Use the distributive property to multiply f by xy+z^{2}.

$$z^{2}=fxy+fz^{2}$$

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

$$fxy+fz^{2}=z^{2}$$

Combine all terms containing f.

$$\left(xy+z^{2}\right)f=z^{2}$$

Divide both sides by yx+z^{2}.

$$\frac{\left(xy+z^{2}\right)f}{xy+z^{2}}=\frac{z^{2}}{xy+z^{2}}$$

Dividing by yx+z^{2} undoes the multiplication by yx+z^{2}.

$$f=\frac{z^{2}}{xy+z^{2}}$$

Solve for x

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

Steps for Solving Linear Equation

Multiply both sides of the equation by z.

$$zz=f\left(\frac{yx}{z}+z\right)z$$

Multiply z and z to get z^{2}.

$$z^{2}=f\left(\frac{yx}{z}+z\right)z$$

To add or subtract expressions, expand them to make their denominators the same. Multiply z times \frac{z}{z}.

$$z^{2}=f\left(\frac{yx}{z}+\frac{zz}{z}\right)z$$

Since \frac{yx}{z} and \frac{zz}{z} have the same denominator, add them by adding their numerators.

$$z^{2}=f\times \frac{yx+zz}{z}z$$

Do the multiplications in yx+zz.

$$z^{2}=f\times \frac{yx+z^{2}}{z}z$$

Express f\times \frac{yx+z^{2}}{z} as a single fraction.

$$z^{2}=\frac{f\left(yx+z^{2}\right)}{z}z$$

Express \frac{f\left(yx+z^{2}\right)}{z}z as a single fraction.

$$z^{2}=\frac{f\left(yx+z^{2}\right)z}{z}$$

Cancel out z in both numerator and denominator.

$$z^{2}=f\left(xy+z^{2}\right)$$

Use the distributive property to multiply f by xy+z^{2}.

$$z^{2}=fxy+fz^{2}$$

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

$$fxy+fz^{2}=z^{2}$$

Subtract fz^{2} from both sides.

$$fxy=z^{2}-fz^{2}$$

Reorder the terms.

$$fxy=-fz^{2}+z^{2}$$

The equation is in standard form.

$$fyx=z^{2}-fz^{2}$$

Divide both sides by fy.

$$\frac{fyx}{fy}=\frac{\left(1-f\right)z^{2}}{fy}$$

Dividing by fy undoes the multiplication by fy.

$$x=\frac{\left(1-f\right)z^{2}}{fy}$$