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Magazine Chapter 6: Problem 16
$$\text { Solve each formula for the indicated variable.}$$ $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \text { for } R_{2}$$
Short Answer
Expert verified
R_2 = \frac{R R_1}{R_1 - R}
Step by step solution
01
- State the Given Formula
The given formula is: \ \[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\]
02
- Subtract \(\frac{1}{R_1}\) from Both Sides
To isolate \(\frac{1}{R_2}\), subtract \(\frac{1}{R_1}\) from both sides: \ \[\frac{1}{R} - \frac{1}{R_1} = \frac{1}{R_2}\]
03
- Combine the Fractions on the Left Side
Combine the terms on the left side by getting a common denominator. The common denominator for \(R\) and \(R_1\) is \(R R_1\): \ \[\frac{R_1 - R}{R R_1} = \frac{1}{R_2}\]
04
- Invert Both Sides
To solve for \(R_2\), take the reciprocal of both sides: \ \[R_2 = \frac{R R_1}{R_1 - R}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Equations
When we solve equations, we look for the value of a variable that makes the equation true. It's like finding the key to a lock. You must perform the same operation on both sides of the equation to keep it balanced, like a perfectly balanced seesaw. Here's what you need to keep in mind:
- Identify what you need to find (the variable to solve for).
- Make the variable you are solving for the subject of the formula.
- Use arithmetic operations (addition, subtraction, multiplication, and division) to isolate the variable.
Fractions
Fractions can seem tricky, but they are just numbers that represent parts of a whole. Understanding how to work with fractions is essential in algebra. There are a few key operations you need to know:
- Adding/Subtracting Fractions: To add or subtract fractions, they must have the same denominator.
- Multiplying Fractions: Multiply the numerators together and denominators together.
- Dividing Fractions: Multiply by the reciprocal of the divisor.
Formula Isolation
Isolating a formula means rearranging it to make a specific variable the subject. It鈥檚 like solving a puzzle where you must move pieces around to reveal the hidden picture. Here鈥檚 the general approach:
- Identify the variable that needs to be isolated.
- Use inverse operations to move other terms away from the target variable.
- Perform consistent operations on both sides of the equation.
Reciprocal
The reciprocal of a number is one divided by that number. In fractions, flipping the numerator and denominator gives the reciprocal. This is especially useful in solving equations involving fractions. Here's what to remember:
- Finding Reciprocals: For a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).
- Using Reciprocals: To solve equations, especially when dividing fractions, because multiplying by a reciprocal is equivalent to division.
- Reciprocal in Equations: Using reciprocals allows us to simplify and solve equations for variables.
