The Ultimate Crash Course for STEM Majors algebra equations and inequalities free preview

Solving Literal Equations and Formulas: Resistance and Geometric Series Examples

This page is nothing more than a free preview of The Ultimate Crash Course for STEM Majors. This algebra lesson works through equations and inequalities by solving formulas for an indicated variable, including a resistance formula involving \(R\), \(R_1\), and \(R_2\), followed by solving \(S=\frac{a}{1-r}\) for \(r\).

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1.1.1 Equations and Inequalities

Solve each formula for the indicated variable

Solve for \(R\), [1]

\[ \frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}. \]

Using the formula

\[ \frac{a}{b}\pm\frac{c}{d} = \frac{ad+cb}{bd}, \qquad \frac{a}{b}=\frac{c}{d} \iff \frac{b}{a}=\frac{d}{c} \]

\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \Rightarrow \frac{1}{R} = \frac{R_2+R_1}{R_1R_2} \Rightarrow R = \frac{R_1R_2}{R_2+R_1}. \quad \therefore \quad R = \frac{R_1R_2}{R_2+R_1}. \]

NOTE*

\[ R\neq0, \qquad R_1\neq0, \qquad R_2\neq0 \]

\[ \color{red}{\frac{1}{R}} = \frac{1}{R_1} + \frac{1}{R_2} \iff \color{red}{f(R)} = g(R_1,R_2). \]

Solve for \(r\), [1]

\[ S=\frac{a}{1-r}. \]

\[ S = \frac{a}{1-r} \Rightarrow (1-r) \left[ S=\frac{a}{1-r} \right] \Rightarrow (1-r)S = (1-r)\frac{a}{1-r} \]

\[ \Rightarrow (1-r)S = \cancel{(1-r)} \frac{a}{\cancel{(1-r)}} \Rightarrow (1-r)S=a \Rightarrow S-rS=a \]

\[ \Rightarrow S-rS=a \Rightarrow -rS=a-S \Rightarrow (-1)[-rS=a-S] \]

\[ \Rightarrow rS=-a+S=S-a \Rightarrow \frac{1}{S}[rS] = \frac{1}{S}[S-a] \]

\[ \Rightarrow r = \frac{S-a}{S} = \frac{S}{S} – \frac{a}{S} = 1-\frac{a}{S}. \]

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