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Chapter 6: Problem 83

Solve formula for the specified variable. \(I=\frac{E}{R+r}\) for \(R\)

Short Answer

Expert verified
R = \(\frac{E - Ir}{I}\)

Step by step solution

01

Understand the Formula

The given formula is: \[ I = \frac{E}{R + r} \]We are to solve this formula for the variable \(R\).
02

Isolate the Denominator

Multiply both sides of the equation by \(R + r\) to get rid of the denominator:\[ I(R + r) = E \]
03

Distribute \(I\)

Distribute \(I\) on the left-hand side:\[ IR + Ir = E \]
04

Isolate the term with \(R\)

Subtract \(Ir\) from both sides of the equation to isolate the term with \(R\):\[ IR = E - Ir \]
05

Solve for \(R\)

Divide both sides of the equation by \(I\) to solve for \(R\):\[ R = \frac{E - Ir}{I} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Manipulation
Algebraic manipulation involves reworking an equation to express it in a different form. This can involve a few different processes, but the goal is to transform the given formula while maintaining equality. In our example, we start with the formula:
\( I = \frac{E}{R + r} \)
To solve for the variable \( R \), we apply specific operations such as multiplication, distribution, addition, and subtraction. These steps allow us to rearrange the terms to isolate the target variable.
  • Multiplication: We multiply both sides of the formula by \( R + r \) to eliminate the denominator.
  • Distribution: We distribute the multiplication over addition inside the parentheses.
  • Addition/Subtraction: We rearrange the terms by adding or subtracting quantities to both sides of the equation.
  • Division: Finally, we divide to solve for the desired variable.
Whenever you manipulate an equation, always perform the same operation on both sides to keep the equation balanced.
Isolating Variables
Isolating variables is a key part of solving equations. To isolate a specific variable means to get that variable alone on one side of the equation.
In our example, we want to isolate \( R \) in the equation \( I = \frac{E}{R + r} \).
To achieve isolation:
  • Step 1: Multiply both sides of the equation by \( R + r \) to remove the fraction: \( I(R + r) = E \).
  • Step 2: Distribute \( I \) across the terms inside the parentheses: \( IR + Ir = E \).
  • Step 3: Remove the term that does not contain the target variable by subtracting \( Ir \) from both sides: \( IR = E - Ir \).
  • Step 4: Finally, divide both sides by \( I \) to isolate \( R \): \( R = \frac{E - Ir}{I} \).
These steps ensure you isolate the variable correctly, transforming the equation into a form where the variable of interest is by itself.
Linear Equations
A linear equation is an equation that represents a straight line when plotted on a graph. It usually has the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.
In our example, the underlying principle mirrors solving a linear equation, even though the initial formula is in fraction form.
Once we apply algebraic manipulation and isolate the variable, it becomes more straightforward:
  • We start with \( I = \frac{E}{R + r} \), a form that can initially seem complex due to the fraction.
  • After isolating \( R \), we get \( IR + Ir = E \), which aligns more closely with a linear format.
  • Ultimately, our final equation, \( R = \frac{E - Ir}{I} \), presents \( R \) in a clear, linear manner.
Understanding how to convert complex formulas into simple linear equations enables us to solve for specific variables with ease.

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Most popular questions from this chapter

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{1+p^{3}}{1+p} $$

Simplify by starting at "the bottom" and working upward. $$ 1+\frac{1}{1+\frac{1}{1+1}} $$

Multiply or divide. Write each answer in lowest terms. $$ \frac{y^{2}+y-2}{y^{2}+3 y-4} \div \frac{y+2}{y+3} $$

Multiply. Write each answer in lowest terms. $$ \frac{(x-y)^{2}}{2} \cdot \frac{24}{3(x-y)} $$

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{2}{3}+\frac{8}{27} $$
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