Solve z/k=1/sqrt{x+y}+1/sqrt{x-y}

$$\frac { z } { k } = \frac { 1 } { \sqrt { x + y } } + \frac { 1 } { \sqrt { x - y } }$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=\frac{z\sqrt{x^{2}-y^{2}}}{\sqrt{x+y}+\sqrt{x-y}}\text{, }&x\neq y\text{ and }x\neq -y\text{ and }z\neq 0\text{ and }\left(|-\frac{arg(x)}{2}+arg(-\sqrt{x})|\geq \pi \text{ or }y\neq 0\right)\text{ and }\left(x\neq 0\text{ or }y\neq 0\right)\\k\neq 0\text{, }&y=0\text{ and }|-\frac{arg(x)}{2}+arg(-\sqrt{x})|<\pi \text{ and }z=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$z=k\left(x+y\right)^{-\frac{1}{2}}+k\left(x-y\right)^{-\frac{1}{2}}$$

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

$$k\left(x+y\right)^{-\frac{1}{2}}+k\left(x-y\right)^{-\frac{1}{2}}=z$$

Combine all terms containing k.

$$\left(\left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}\right)k=z$$

Divide both sides by \left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}.

$$\frac{\left(\left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}\right)k}{\left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}}=\frac{z}{\left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}}$$

Dividing by \left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}} undoes the multiplication by \left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}.

$$k=\frac{z}{\left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}}$$

Divide z by \left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}.

$$k=\frac{z\sqrt{x^{2}-y^{2}}}{\sqrt{x+y}+\sqrt{x-y}}$$

Variable k cannot be equal to 0.

$$k=\frac{z\sqrt{x^{2}-y^{2}}}{\sqrt{x+y}+\sqrt{x-y}}\text{, }k\neq 0$$

Solve for k

$$k=\frac{z\sqrt{x^{2}-y^{2}}}{\sqrt{x+y}+\sqrt{x-y}}$$

$$z\neq 0\text{ and }x>|y|$$

Steps for Solving Linear Equation

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

$$z=k\left(x+y\right)^{-\frac{1}{2}}+k\left(x-y\right)^{-\frac{1}{2}}$$

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

$$k\left(x+y\right)^{-\frac{1}{2}}+k\left(x-y\right)^{-\frac{1}{2}}=z$$

Combine all terms containing k.

$$\left(\left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}\right)k=z$$

The equation is in standard form.

$$\left(\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{x-y}}\right)k=z$$

Divide both sides by \left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}.

$$\frac{\left(\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{x-y}}\right)k}{\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{x-y}}}=\frac{z}{\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{x-y}}}$$

Dividing by \left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}} undoes the multiplication by \left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}.

$$k=\frac{z}{\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{x-y}}}$$

Divide z by \left(x+y\right)^{-\frac{1}{2}}+\left(x-y\right)^{-\frac{1}{2}}.

$$k=\frac{z\sqrt{x^{2}-y^{2}}}{\sqrt{x+y}+\sqrt{x-y}}$$

Variable k cannot be equal to 0.

$$k=\frac{z\sqrt{x^{2}-y^{2}}}{\sqrt{x+y}+\sqrt{x-y}}\text{, }k\neq 0$$