Solve U_{G}=-frac{Gm_{1}m_{2}}{r}

$$U _ { G } = - \frac { G m _ { 1 } m _ { 2 } } { r }$$

Solve for G (complex solution)

$$\left\{\begin{matrix}G=-\frac{U_{G}r}{m_{1}m_{2}}\text{, }&m_{2}\neq 0\text{ and }m_{1}\neq 0\text{ and }r\neq 0\\G\in \mathrm{C}\text{, }&\left(m_{2}=0\text{ or }m_{1}=0\right)\text{ and }U_{G}=0\text{ and }r\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by r.

$$U_{G}r=-Gm_{1}m_{2}$$

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

$$-Gm_{1}m_{2}=U_{G}r$$

The equation is in standard form.

$$\left(-m_{1}m_{2}\right)G=U_{G}r$$

Divide both sides by -m_{1}m_{2}.

$$\frac{\left(-m_{1}m_{2}\right)G}{-m_{1}m_{2}}=\frac{U_{G}r}{-m_{1}m_{2}}$$

Dividing by -m_{1}m_{2} undoes the multiplication by -m_{1}m_{2}.

$$G=\frac{U_{G}r}{-m_{1}m_{2}}$$

Divide U_{G}r by -m_{1}m_{2}.

$$G=-\frac{U_{G}r}{m_{1}m_{2}}$$

Solve for G

$$\left\{\begin{matrix}G=-\frac{U_{G}r}{m_{1}m_{2}}\text{, }&m_{2}\neq 0\text{ and }m_{1}\neq 0\text{ and }r\neq 0\\G\in \mathrm{R}\text{, }&\left(m_{2}=0\text{ or }m_{1}=0\right)\text{ and }U_{G}=0\text{ and }r\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by r.

$$U_{G}r=-Gm_{1}m_{2}$$

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

$$-Gm_{1}m_{2}=U_{G}r$$

The equation is in standard form.

$$\left(-m_{1}m_{2}\right)G=U_{G}r$$

Divide both sides by -m_{1}m_{2}.

$$\frac{\left(-m_{1}m_{2}\right)G}{-m_{1}m_{2}}=\frac{U_{G}r}{-m_{1}m_{2}}$$

Dividing by -m_{1}m_{2} undoes the multiplication by -m_{1}m_{2}.

$$G=\frac{U_{G}r}{-m_{1}m_{2}}$$

Divide U_{G}r by -m_{1}m_{2}.

$$G=-\frac{U_{G}r}{m_{1}m_{2}}$$

Solve for U_G

$$U_{G}=-\frac{Gm_{1}m_{2}}{r}$$

$$r\neq 0$$