Solve frac{R_{1}}{R_{2}}=frac{R_{3}}{R4}+frac{R_{3}}{R_{5}}

$$\frac { R _ { 1 } } { R _ { 2 } } = \frac { R _ { 3 } } { R 4 } + \frac { R _ { 3 } } { R _ { 5 } }$$

Solve for R_1

$$R_{1}=\frac{R_{2}R_{3}\left(R_{4}+R_{5}\right)}{R_{4}R_{5}}$$

$$R_{2}\neq 0\text{ and }R_{4}\neq 0\text{ and }R_{5}\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by R_{2}R_{4}R_{5}, the least common multiple of R_{2},R_{4},R_{5}.

$$R_{4}R_{5}R_{1}=R_{2}R_{5}R_{3}+R_{2}R_{4}R_{3}$$

Reorder the terms.

$$R_{1}R_{4}R_{5}=R_{2}R_{3}R_{4}+R_{2}R_{3}R_{5}$$

The equation is in standard form.

$$R_{4}R_{5}R_{1}=R_{2}R_{3}R_{4}+R_{2}R_{3}R_{5}$$

Divide both sides by R_{4}R_{5}.

$$\frac{R_{4}R_{5}R_{1}}{R_{4}R_{5}}=\frac{R_{2}R_{3}\left(R_{4}+R_{5}\right)}{R_{4}R_{5}}$$

Dividing by R_{4}R_{5} undoes the multiplication by R_{4}R_{5}.

$$R_{1}=\frac{R_{2}R_{3}\left(R_{4}+R_{5}\right)}{R_{4}R_{5}}$$

Solve for R_2

$$\left\{\begin{matrix}R_{2}=\frac{R_{1}R_{4}R_{5}}{R_{3}\left(R_{4}+R_{5}\right)}\text{, }&R_{5}\neq 0\text{ and }R_{4}\neq 0\text{ and }R_{1}\neq 0\text{ and }R_{4}\neq -R_{5}\text{ and }R_{3}\neq 0\\R_{2}\neq 0\text{, }&\left(R_{4}=-R_{5}\text{ or }R_{4}\neq 0\right)\text{ and }\left(R_{4}=-R_{5}\text{ or }R_{3}=0\right)\text{ and }R_{1}=0\text{ and }R_{5}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable R_{2} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by R_{2}R_{4}R_{5}, the least common multiple of R_{2},R_{4},R_{5}.

$$R_{4}R_{5}R_{1}=R_{2}R_{5}R_{3}+R_{2}R_{4}R_{3}$$

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

$$R_{2}R_{5}R_{3}+R_{2}R_{4}R_{3}=R_{4}R_{5}R_{1}$$

Combine all terms containing R_{2}.

$$\left(R_{5}R_{3}+R_{4}R_{3}\right)R_{2}=R_{4}R_{5}R_{1}$$

The equation is in standard form.

$$\left(R_{3}R_{4}+R_{3}R_{5}\right)R_{2}=R_{1}R_{4}R_{5}$$

Divide both sides by R_{5}R_{3}+R_{4}R_{3}.

$$\frac{\left(R_{3}R_{4}+R_{3}R_{5}\right)R_{2}}{R_{3}R_{4}+R_{3}R_{5}}=\frac{R_{1}R_{4}R_{5}}{R_{3}R_{4}+R_{3}R_{5}}$$

Dividing by R_{5}R_{3}+R_{4}R_{3} undoes the multiplication by R_{5}R_{3}+R_{4}R_{3}.

$$R_{2}=\frac{R_{1}R_{4}R_{5}}{R_{3}R_{4}+R_{3}R_{5}}$$

Divide R_{4}R_{5}R_{1} by R_{5}R_{3}+R_{4}R_{3}.

$$R_{2}=\frac{R_{1}R_{4}R_{5}}{R_{3}\left(R_{4}+R_{5}\right)}$$

Variable R_{2} cannot be equal to 0.

$$R_{2}=\frac{R_{1}R_{4}R_{5}}{R_{3}\left(R_{4}+R_{5}\right)}\text{, }R_{2}\neq 0$$