Solve S=frac{I_{g}}{(I-I_{g})}G

$$S = \frac { I _ { g } } { ( I - I _ { g } ) } G$$

Solve for G (complex solution)

$$\left\{\begin{matrix}G=\frac{S\left(I-I_{g}\right)}{I_{g}}\text{, }&I_{g}\neq 0\text{ and }I_{g}\neq I\\G\in \mathrm{C}\text{, }&S=0\text{ and }I_{g}=0\text{ and }I\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by I-I_{g}.

$$S\left(I-I_{g}\right)=I_{g}G$$

Use the distributive property to multiply S by I-I_{g}.

$$SI-SI_{g}=I_{g}G$$

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

$$I_{g}G=SI-SI_{g}$$

The equation is in standard form.

$$I_{g}G=IS-I_{g}S$$

Divide both sides by I_{g}.

$$\frac{I_{g}G}{I_{g}}=\frac{S\left(I-I_{g}\right)}{I_{g}}$$

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

$$G=\frac{S\left(I-I_{g}\right)}{I_{g}}$$

Solve for G

$$\left\{\begin{matrix}G=\frac{S\left(I-I_{g}\right)}{I_{g}}\text{, }&I_{g}\neq 0\text{ and }I_{g}\neq I\\G\in \mathrm{R}\text{, }&S=0\text{ and }I_{g}=0\text{ and }I\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by I-I_{g}.

$$S\left(I-I_{g}\right)=I_{g}G$$

Use the distributive property to multiply S by I-I_{g}.

$$SI-SI_{g}=I_{g}G$$

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

$$I_{g}G=SI-SI_{g}$$

The equation is in standard form.

$$I_{g}G=IS-I_{g}S$$

Divide both sides by I_{g}.

$$\frac{I_{g}G}{I_{g}}=\frac{S\left(I-I_{g}\right)}{I_{g}}$$

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

$$G=\frac{S\left(I-I_{g}\right)}{I_{g}}$$

Solve for I

$$\left\{\begin{matrix}I=\frac{I_{g}\left(G+S\right)}{S}\text{, }&G\neq 0\text{ and }I_{g}\neq 0\text{ and }S\neq 0\\I\neq 0\text{, }&G\neq 0\text{ and }S=0\text{ and }I_{g}=0\\I\neq I_{g}\text{, }&S=0\text{ and }G=0\end{matrix}\right.$$