Solve for r i=e/(R+r)

$i = \frac{e}{R + r}$

Step 1

Rewrite the equation as $\frac{e}{R + r} = i$ .

$\frac{e}{R + r} = i$

Step 2

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$R + r, 1$

Remove parentheses.

$R + r, 1$

The LCM of one and any expression is the expression.

$R + r$

Step 3

Multiply each term in $\frac{e}{R + r} = i$ by $R + r$ to eliminate the fractions.

Multiply each term in $\frac{e}{R + r} = i$ by $R + r$ .

$\frac{e}{R + r} (R + r) = i (R + r)$

Simplify the left side.

Cancel the common factor of $R + r$ .

Cancel the common factor.

$\frac{e}{R + r} (R + r) = i (R + r)$

Rewrite the expression.

$e = i (R + r)$

Simplify the right side.

Apply the distributive property.

$e = i R + i r$

Step 4

Solve the equation.

Rewrite the equation as $i R + i r = e$ .

$i R + i r = e$

Subtract $i R$ from both sides of the equation.

$i r = e - i R$

Divide each term in $i r = e - i R$ by $i$ and simplify.

Divide each term in $i r = e - i R$ by $i$ .

$\frac{i r}{i} = \frac{e}{i} + \frac{- i R}{i}$

Simplify the left side.

Cancel the common factor of $i$ .

Cancel the common factor.

$\frac{i r}{i} = \frac{e}{i} + \frac{- i R}{i}$

Divide $r$ by $1$ .

$r = \frac{e}{i} + \frac{- i R}{i}$

Simplify the right side.

Simplify each term.

Multiply the numerator and denominator of $\frac{e}{i}$ by the conjugate of $i$ to make the denominator real.

$r = \frac{e}{i} \cdot \frac{i}{i} + \frac{- i R}{i}$

Multiply.

Combine.

$r = \frac{e i}{i i} + \frac{- i R}{i}$

Simplify the denominator.

Raise $i$ to the power of $1$ .

$r = \frac{e i}{i^{1} i} + \frac{- i R}{i}$

Raise $i$ to the power of $1$ .

$r = \frac{e i}{i^{1} i^{1}} + \frac{- i R}{i}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \frac{e i}{i^{1 + 1}} + \frac{- i R}{i}$

Add $1$ and $1$ .

$r = \frac{e i}{i^{2}} + \frac{- i R}{i}$

Rewrite $i^{2}$ as $- 1$ .

$r = \frac{e i}{- 1} + \frac{- i R}{i}$

Move the negative one from the denominator of $\frac{e i}{- 1}$ .

$r = - 1 \cdot (e i) + \frac{- i R}{i}$

Rewrite $- 1 \cdot (e i)$ as $- (e i)$ .

$r = - (e i) + \frac{- i R}{i}$

Cancel the common factor of $i$ .

Cancel the common factor.

$r = - e i + \frac{- i R}{i}$

Divide $- 1 R$ by $1$ .

$r = - e i - 1 R$

Rewrite $- 1 R$ as $- R$ .

$r = - e i - R$

Solve for r i=e/(R+r)

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