Problem 16 - Algebra for College Students

Question

$$\text { Solve each formula for the indicated variable.}$$ $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \text { for } R_{2}$$

Step-by-Step Solution

Answer

R_2 = \frac{R R_1}{R_1 - R}

Step 1 - State the Given Formula

The given formula is: \ \[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\]

Step 2 - Subtract \(\frac{1}{R_1}\) from Both Sides

To isolate \(\frac{1}{R_2}\), subtract \(\frac{1}{R_1}\) from both sides: \ \[\frac{1}{R} - \frac{1}{R_1} = \frac{1}{R_2}\]

Step 3 - 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 \(R R_1\): \ \[\frac{R_1 - R}{R R_1} = \frac{1}{R_2}\]

Step 4 - Invert Both Sides

To solve for \(R_2\), take the reciprocal of both sides: \ \[R_2 = \frac{R R_1}{R_1 - R}\]

Key Concepts

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.

In the exercise, solving the given formula for \(R_2\) means performing specific steps to rearrange the equation. We used operations like subtraction and fraction manipulation to isolate \(\frac{1}{R_2}\) and eventually found the value of \(R_2\). This method is applicable to many types of problems, making it a crucial concept in algebra.

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.

In our problem, we subtract fractions with different denominators. The least common denominator, for both \(R\) and \(R_1\), is \(R R_1\). By adjusting the fractions to this common ground, we simplified the equation accurately, allowing for further manipulation. Remember, handling fractions with confidence takes practice!

Formula Isolation

Isolating a formula means rearranging it to make a specific variable the subject. It’s like solving a puzzle where you must move pieces around to reveal the hidden picture. Here’s 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.

In the given exercise, we isolated \(\frac{1}{R_2}\) by subtracting \(\frac{1}{R_1}\) from both sides of the original equation: \(\frac{1}{R} - \frac{1}{R_1} = \frac{1}{R_2}\). This step is crucial as it gets closer to isolating \(R_2\) once further manipulation, like taking the reciprocal, is applied. Isolating a formula is a key skill, massively useful in various algebra problems.

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.

In the exercise, after simplifying the equation to \(\frac{R_1 - R}{R R_1} = \frac{1}{R_2}\), we took the reciprocal of both sides to isolate \(R_2\): \ R_2 = \frac{R R_1}{R_1 - R}\. This final step provided the solution by flipping the fraction on the left. Understanding reciprocals is vital for equation-solving proficiency, especially when fractions are involved.