Solved: Derive Eqs. 3.443.49 from Eqs. 3.413.43. The following two hints should help you
Chapter 3, Problem 3.65(choose chapter or problem)
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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