Problem 90 - Elementary Algebra

Question

For the following problems, solve the rational equations. Solve \(I=\frac{E}{R+r}\) for \(r\)

Step-by-Step Solution

Answer

Question: Solve the rational equation, 𝐼=𝐸/(𝑅+𝑟), for r. Answer: 𝑟 = (𝐸 - 𝐼𝑅)/𝐼

Step 1: 1. Multiply both sides by (R+r)

To eliminate the denominator, we need to multiply both sides of the equation by (R+r): \(I(R+r) = \frac{E}{R+r} \cdot (R+r)\)

Step 2: 2. Simplify the equation

Next, we simplify the equation by canceling out (R+r) from the denominator and expanding the left side: \(IR + Ir = E\)

Step 3: 3. Solve for r

Now, we need to isolate r in the equation. In order to do that, we'll first move IR to the right side of the equation: \(Ir = E - IR\) We can now factor r out of the left side: \(r(I) = E - IR\) Lastly, we'll divide both sides by I to solve for r: \(r = \frac{E - IR}{I}\)

Key Concepts

Solving Equations

When you approach solving equations, especially rational equations, it's essential to organize your steps and think logically. The goal is to isolate the variable we're interested in, which involves manipulating the equation effectively.
In the provided exercise, the equation given was \( I = \frac{E}{R+r} \). To solve for the variable \( r \), we needed to first clear the fraction by multiplying both sides by the denominator \( (R+r) \). This step is crucial because it allows us to work with a simpler equation without fractions.
Once the fraction is eliminated, the next step is simplifying and re-arranging terms to isolate the variable \( r \). By methodically moving terms and using basic algebraic operations, we ensure clarity and precision in reaching the solution.
Maintaining a clear goal for each step ensures logical progression, leading to the accurate isolation of \( r \). A regimented approach makes complex equations more manageable.

Algebraic Manipulation

Algebraic manipulation involves using operations to simplify or rearrange equations, which is fundamental in solving rational equations. In our given problem, we utilized these techniques to handle the equation \( I(R+r) = E \). Here’s how it unfolded.
Initially, our goal was to get rid of the denominator. By multiplying everything by \( (R+r) \), it effectively "disappeared" from the fraction, streamlining the equation into \( IR + Ir = E \). This step shows how multiplication can simplify expressions and prepare them for further manipulation.
Following this, we rearranged the terms to isolate \( r \). We moved \( IR \) by subtracting it from both sides, leading to \( Ir = E - IR \). Such horizontal movements of terms across the equation utilize addition and subtraction principles that are core to algebraic manipulation.
Finally, recognizing \( Ir = E - IR \), we divided both sides by \( I \) to solve explicitly for \( r \), yielding \( r = \frac{E - IR}{I} \). This division step is an essential aspect of focusing the equation on the variable of interest. Practice simplifies complex equations by breaking them into smaller, manageable steps.

Variables in Denominators

Variables in denominators can pose a unique challenge when solving rational equations. These types of problems require careful handling to avoid mistakes and ensure accurate solutions.
Consider the equation \( I = \frac{E}{R+r} \). To deal with \( r \) effectively, our first step was to eliminate \( (R+r) \) from the denominator by multiplying both sides by it. This transforms the rational equation into a simpler form without fractions.
It's crucial to perform these operations accurately, as variables in the denominator can lead to restrictions. Remember, dividing by zero is undefined, so we must assume \( R+r eq 0 \) throughout the solution.
This variable arrangement affects how we manipulate the equation. Properly managing the transition from rational to simple equations relies on effectively using multiplication and aiming to set one side of the equation to the variable of interest.
Once the fraction is cleared, we can focus on simple algebraic steps to isolate and solve for the variable \( r \). Mastering these skills makes handling equations with variables in the denominator less daunting.