The problem provides the formula $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$ and asks to solve for $R_1$ in part (a) and for $R_2$ in part (b).

1. Problem Description

The problem provides the formula $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$ and asks to solve for $R_1$ in part (a) and for $R_2$ in part (b).

2. Solution Steps

(a) Solving for $R_1$:

First, isolate the term with $R_1$ by subtracting $\frac{1}{R_2}$ from both sides of the equation:

$\frac{1}{R_T} - \frac{1}{R_2} = \frac{1}{R_1}$

Now, find a common denominator for the left side:

$\frac{R_2 - R_T}{R_T R_2} = \frac{1}{R_1}$

Next, take the reciprocal of both sides to solve for $R_1$:

$R_1 = \frac{R_T R_2}{R_2 - R_T}$

(b) Solving for $R_2$:

First, isolate the term with $R_2$ by subtracting $\frac{1}{R_1}$ from both sides of the equation:

$\frac{1}{R_T} - \frac{1}{R_1} = \frac{1}{R_2}$

Now, find a common denominator for the left side:

$\frac{R_1 - R_T}{R_T R_1} = \frac{1}{R_2}$

Next, take the reciprocal of both sides to solve for $R_2$:

$R_2 = \frac{R_T R_1}{R_1 - R_T}$

3. Final Answer

(a) $R_1 = \frac{R_T R_2}{R_2 - R_T}$

(b) $R_2 = \frac{R_T R_1}{R_1 - R_T}$