Solve m_{g}h=mgh_{f}1/2m_v^2

$$m _ { g } h = m g h _ { f } \frac { 1 } { 2 } m _ { v } ^ { 2 }$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=\frac{2hm_{g}}{h_{f}mm_{v}^{2}}\text{, }&m_{v}\neq 0\text{ and }h_{f}\neq 0\text{ and }m\neq 0\\g\in \mathrm{C}\text{, }&\left(m_{g}=0\text{ and }m=0\right)\text{ or }\left(m_{g}=0\text{ and }h_{f}=0\right)\text{ or }\left(m_{g}=0\text{ and }m_{v}=0\right)\text{ or }\left(h=0\text{ and }m=0\right)\text{ or }\left(h=0\text{ and }h_{f}=0\right)\text{ or }\left(h=0\text{ and }m_{v}=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$mgh_{f}\times \frac{1}{2}m_{v}^{2}=m_{g}h$$

The equation is in standard form.

$$\frac{h_{f}mm_{v}^{2}}{2}g=hm_{g}$$

Divide both sides by \frac{1}{2}mh_{f}m_{v}^{2}.

$$\frac{2\times \frac{h_{f}mm_{v}^{2}}{2}g}{h_{f}mm_{v}^{2}}=\frac{2hm_{g}}{h_{f}mm_{v}^{2}}$$

Dividing by \frac{1}{2}mh_{f}m_{v}^{2} undoes the multiplication by \frac{1}{2}mh_{f}m_{v}^{2}.

$$g=\frac{2hm_{g}}{h_{f}mm_{v}^{2}}$$

Solve for h (complex solution)

$$\left\{\begin{matrix}h=\frac{gh_{f}mm_{v}^{2}}{2m_{g}}\text{, }&m_{g}\neq 0\\h\in \mathrm{C}\text{, }&\left(m=0\text{ or }g=0\text{ or }h_{f}=0\text{ or }m_{v}=0\right)\text{ and }m_{g}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$m_{g}h=\frac{gh_{f}mm_{v}^{2}}{2}$$

Divide both sides by m_{g}.

$$\frac{m_{g}h}{m_{g}}=\frac{gh_{f}mm_{v}^{2}}{2m_{g}}$$

Dividing by m_{g} undoes the multiplication by m_{g}.

$$h=\frac{gh_{f}mm_{v}^{2}}{2m_{g}}$$

Solve for g

$$\left\{\begin{matrix}g=\frac{2hm_{g}}{h_{f}mm_{v}^{2}}\text{, }&m_{v}\neq 0\text{ and }h_{f}\neq 0\text{ and }m\neq 0\\g\in \mathrm{R}\text{, }&\left(m_{g}=0\text{ and }m=0\right)\text{ or }\left(m_{g}=0\text{ and }h_{f}=0\right)\text{ or }\left(m_{g}=0\text{ and }m_{v}=0\right)\text{ or }\left(h=0\text{ and }m=0\right)\text{ or }\left(h=0\text{ and }h_{f}=0\right)\text{ or }\left(h=0\text{ and }m_{v}=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$mgh_{f}\times \frac{1}{2}m_{v}^{2}=m_{g}h$$

The equation is in standard form.

$$\frac{h_{f}mm_{v}^{2}}{2}g=hm_{g}$$

Divide both sides by \frac{1}{2}mh_{f}m_{v}^{2}.

$$\frac{2\times \frac{h_{f}mm_{v}^{2}}{2}g}{h_{f}mm_{v}^{2}}=\frac{2hm_{g}}{h_{f}mm_{v}^{2}}$$

Dividing by \frac{1}{2}mh_{f}m_{v}^{2} undoes the multiplication by \frac{1}{2}mh_{f}m_{v}^{2}.

$$g=\frac{2hm_{g}}{h_{f}mm_{v}^{2}}$$

Solve for h

$$\left\{\begin{matrix}h=\frac{gh_{f}mm_{v}^{2}}{2m_{g}}\text{, }&m_{g}\neq 0\\h\in \mathrm{R}\text{, }&\left(m=0\text{ or }g=0\text{ or }h_{f}=0\text{ or }m_{v}=0\right)\text{ and }m_{g}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$m_{g}h=\frac{gh_{f}mm_{v}^{2}}{2}$$

Divide both sides by m_{g}.

$$\frac{m_{g}h}{m_{g}}=\frac{gh_{f}mm_{v}^{2}}{2m_{g}}$$

Dividing by m_{g} undoes the multiplication by m_{g}.

$$h=\frac{gh_{f}mm_{v}^{2}}{2m_{g}}$$