Solve frac{T_{b}}{I_{g}}=frac{R_{a}}{R_{a}+R_{b}}

$$\frac { T _ { b } } { I _ { g } } = \frac { R _ { a } } { R _ { a } + R _ { b } }$$

Solve for I_g

$$\left\{\begin{matrix}I_{g}=\frac{T_{b}\left(R_{a}+R_{b}\right)}{R_{a}}\text{, }&R_{a}\neq -R_{b}\text{ and }T_{b}\neq 0\text{ and }R_{a}\neq 0\\I_{g}\neq 0\text{, }&R_{a}=0\text{ and }T_{b}=0\text{ and }R_{b}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable I_{g} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by I_{g}\left(R_{a}+R_{b}\right), the least common multiple of I_{g},R_{a}+R_{b}.

$$\left(R_{a}+R_{b}\right)T_{b}=I_{g}R_{a}$$

Use the distributive property to multiply R_{a}+R_{b} by T_{b}.

$$R_{a}T_{b}+R_{b}T_{b}=I_{g}R_{a}$$

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

$$I_{g}R_{a}=R_{a}T_{b}+R_{b}T_{b}$$

The equation is in standard form.

$$R_{a}I_{g}=R_{a}T_{b}+R_{b}T_{b}$$

Divide both sides by R_{a}.

$$\frac{R_{a}I_{g}}{R_{a}}=\frac{T_{b}\left(R_{a}+R_{b}\right)}{R_{a}}$$

Dividing by R_{a} undoes the multiplication by R_{a}.

$$I_{g}=\frac{T_{b}\left(R_{a}+R_{b}\right)}{R_{a}}$$

Variable I_{g} cannot be equal to 0.

$$I_{g}=\frac{T_{b}\left(R_{a}+R_{b}\right)}{R_{a}}\text{, }I_{g}\neq 0$$

Solve for R_a

$$\left\{\begin{matrix}R_{a}=-\frac{R_{b}T_{b}}{T_{b}-I_{g}}\text{, }&R_{b}\neq 0\text{ and }I_{g}\neq 0\text{ and }T_{b}\neq I_{g}\\R_{a}\neq 0\text{, }&R_{b}=0\text{ and }T_{b}=I_{g}\text{ and }I_{g}\neq 0\end{matrix}\right.$$