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Chapter 6: Problem 80

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

Short Answer

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

Step by step solution

01

Understand the Equation

The equation given is the formula for current, which is expressed as \( I = \frac{k E}{R} \) where \( I \) is the current, \( k \) is a constant, \( E \) is the electromotive force, and \( R \) is the resistance. The goal is to solve for \( E \).
02

Multiply Both Sides by \( R \)

To isolate \( E \), start by eliminating the fraction. Multiply both sides of the equation by \( R \): \( I R = \frac{k E}{R} \times R \)
03

Simplify the Equation

After multiplying, the equation simplifies to: \( I R = k E \) The \( R \) in the denominator cancels out with the \( R \) on the numerator.
04

Isolate \( E \)

To solve for \( E \), divide both sides of the equation by \( k \): \( E = \frac{I R}{k} \)

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Key Concepts

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

solving for a variable
When solving for a variable in an equation, the objective is to isolate that variable on one side.
This means you perform a series of operations to get the variable by itself. For example, in the formula for current given by \( I = \frac{k E}{R} \), we needed to solve for \( E \).

Steps to solve for a variable:
  • Identify the variable you need to solve for. In this case, it's \( E \).
  • Use algebraic manipulation to move other terms to the opposite side of the equation.
  • Perform inverse operations to isolate the variable completely.
Let's take a closer look at these steps in the context of our problem.
isolating a variable
Isolating a variable involves performing operations that will get the variable you are solving for alone on one side of the equation.
In our exercise, we need to isolate \( E \).

Here's how we do it:
  • Start with the equation: \( I = \frac{k E}{R} \)
  • Multiply both sides by \( R \) to eliminate the fraction: \( I R = k E \)
  • Now, \( E \) is almost isolated, as it is only multiplied by \( k \).
  • Finally, divide both sides by \( k \) to get \( E \) alone: \( E = \frac{I R}{k} \).

By following these steps, we successfully isolated \( E \).
algebraic manipulation
Algebraic manipulation is a set of techniques used to rearrange equations and expressions to solve for a specific variable.
These techniques include adding, subtracting, multiplying, dividing, and various other operations.
In the example given:
  • We started with \( I = \frac{k E}{R} \).
  • We performed multiplication on both sides to eliminate the denominator \( R \): \( I R = k E \).

The key is to apply the same operation on both sides of the equation to maintain the equality.
This ensures we progressively simplify and rearrange the equation in a logical and step-by-step manner.
equation simplification
Simplifying an equation is an essential step in solving for a variable.
It involves reducing the equation to its simplest form while maintaining the equality.
Consider our example:
  • We began with \( I = \frac{k E}{R} \).
  • By multiplying both sides by \( R \), we simplified to \( I R = k E \).
  • This operation removed the fraction and thus simplified the equation.
  • Finally, by dividing both sides by \( k \), we isolated \( E \) as \( E = \frac{I R}{k} \).

Simplifying equations helps to make the process of solving for a variable more straightforward and easy to follow.

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Most popular questions from this chapter

Joshua can do a job in \(r\) hours. What portion of the job is done in 1 hr?

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