Problem 43 - Prealgebra and Introductory Algebra

Question

Solve the physics formula \(\frac{1}{R_{1}}+\frac{1}{R_{2}}=\frac{1}{R}\) for \(R_{2}\)

Step-by-Step Solution

Answer

The physics formula solved for \(R_{2}\) is \(R_{2} = \frac{1}{R - R_{1}}\)

Step 1: Rewrite the Equation

First we start by rewriting the equation to remove the fraction: \(R = \frac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}\)

Step 2: Separate the Denominator

Next, separate the denominator so that \(R_2\) is isolated on one side: \(R = \frac{1}{\frac{1}{R_{1}}} + \frac{1}{R_{2}} = R_{1} + \frac{1}{R_{2}}\)

Step 3: Isolate \(R_{2}\)

Move \(R_{1}\) to the other side of the equation: \(\frac{1}{R_{2}} = R - R_{1}\)

Step 4: Solve for \(R_{2}\)

Finally, isolate \(R_{2}\) by taking the reciprocal of both sides: \(R_{2} = \frac{1}{R - R_{1}}\)

Key Concepts

Understanding Reciprocals

A reciprocal is essentially the inverse of a number or a fraction. When you take the reciprocal of a number, you are flipping its numerator and denominator. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). If you have a whole number, like 5, its reciprocal is \( \frac{1}{5} \).

Reciprocals are useful in mathematics because multiplying a number by its reciprocal always results in 1. That is, \( a \times \frac{1}{a} = 1 \). In the context of our formula, taking the reciprocal is used to rearrange terms and solve equations more easily.

In the physics formula \( \frac{1}{R_{1}} + \frac{1}{R_{2}} = \frac{1}{R} \), each term on the left-hand side is already in reciprocal form. To solve for \( R_{2} \), taking the reciprocal of both sides of an equation helps in isolating \( R_{2} \), making it easier to find its value.

Solving for a Variable

Solving for a variable means rearranging an equation to make the variable the subject or main focus. For example, if you need to find \( R_{2} \) from the equation \( \frac{1}{R_{1}} + \frac{1}{R_{2}} = \frac{1}{R} \), our goal is to rearrange this equation so \( R_{2} \) is isolated on one side.

Steps to solve for \( R_{2} \):

• First, rewrite the original equation in a more manageable form. In this case, rewrite it considering the reciprocal operation.
• Isolate \( R_{2} \) by moving all other terms to the opposite side of the equation.
• As a result, you derive \( \frac{1}{R_{2}} \) as a function of \( R \) and \( R_{1} \).
• Finally, take the reciprocal of the isolated term to find \( R_{2} \).

These steps help in systematically breaking down complex formulas into simpler, easily solvable parts.

Fraction Manipulation Made Easy

Fraction manipulation involves changing the form of a fraction without altering its value. This includes operations like addition, subtraction, multiplication, division, and simplification of fractions.

In the given physics problem, you have fractions like \( \frac{1}{R_{1}} \) and \( \frac{1}{R_{2}} \). Understanding how to manipulate these fractions is crucial for solving the equation.

Useful tips for fraction manipulation:

• To add fractions, they must have a common denominator. If they don't, find the least common multiple and convert each to equivalent fractions with the same denominator.
• When simplifying fractions, like moving \( R_{1} \) to the other side, subtraction is key. Subtract \( \frac{1}{R_{1}} \) from both sides to begin simplifying the equation.
• Always remember that variables in the denominator can be isolated by adopting reciprocal operations.

Mastering fraction manipulation facilitates solving a broad range of mathematical and scientific problems, making equations simpler to deal with.