Solved: Derive Eqs. 3.443.49 from Eqs. 3.413.43. The following two hints should help you

Chapter 3, Problem 3.65

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Derive Eqs. 3.44–3.49 from Eqs. 3.41–3.43. The following two hints should help you get started in the right direction:

1) To find \(R_{1}\) as a function of \(R_{a}, R_{b} \text {, and } R_{c}\), first subtract Eq. 3.42 from Eq. 3.43 and then add this result to Eq. 3.41. Use similar manipulations to find \(R_{2}\) and \(R_{3}\) as functions of \(R_{a}, R_{b}\), and \(R_{c}\).

2) To find \(R_{b}\) as a function of \(R_{1}, R_{2}\), and \(R_{3}\), take advantage of the derivations obtained by hint (1), namely, Eqs. 3.44–3.46. Note that these equations can be divided to obtain

\(\frac{R_{2}}{R_{3}}=\frac{R_{c}}{R_{b}}\), or \(R_{c}=\frac{R_{2}}{R_{3}} R_{b}\),

and

\(\frac{R_{1}}{R_{2}}=\frac{R_{b}}{R_{a}}\), or \(R_{a}=\frac{R_{2}}{R_{1}} R_{b}\).

Now use these ratios in Eq. 3.43 to eliminate \(R_{a}\) and \(R_{c}\). Use similar manipulations to find \(R_{a}\) and \(R_{c}\) as functions of \(R_{1}, R_{2}\), and \(R_{3}\).

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