Problem 32 - Precalculus Mathematics for Calculus

Question

Solve the equation for the indicated variable. $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} ; \quad \text { for } R_{1}$$

Step-by-Step Solution

Answer

The solution is \( R_{1} = \frac{-RR_{2}}{R - R_{2}} \).

Step 1: Understand the equation

The given equation is \( \frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} \) and we need to solve for \( R_{1} \). This equation is typically used in electronics to calculate the equivalent resistance of two resistors in parallel.

Step 2: Make a common denominator

Since the equation involves fractions, we need to manipulate it to isolate \( \frac{1}{R_{1}} \). The right-hand side is a sum of two fractions. We can write it as \( \frac{R_{2} + R_{1}}{R_{1}R_{2}} \), finding a common denominator \( R_{1}R_{2} \).

Step 3: Set the common denominators equal

Equate the left-hand side fraction with a denominator \( R \) to the right-hand side fraction \( \frac{R_{2} + R_{1}}{R_{1}R_{2}} \). This gives us: \( \frac{1}{R} = \frac{R_{1} + R_{2}}{R_{1}R_{2}} \).

Step 4: Cross-multiply to eliminate fractions

Cross-multiply to eliminate fractions, obtaining \( R(R_{1} + R_{2}) = R_{1}R_{2} \).

Step 5: Rearrange the equation

Expand the equation to \( RR_{1} + RR_{2} = R_{1}R_{2} \). We need to solve for \( R_{1} \), so rearrange it to isolate terms involving \( R_{1} \): \( RR_{1} - R_{1}R_{2} = -RR_{2} \).

Step 6: Factor out \( R_{1} \)

Factor \( R_{1} \) from the terms: \( R_{1}(R - R_{2}) = -RR_{2} \).

Step 7: Solve for \( R_{1} \)

Divide both sides by \( (R - R_{2}) \) to get \( R_{1} = \frac{-RR_{2}}{R - R_{2}} \).

Key Concepts

Equation Solving

Equation solving is about finding the value of a variable that makes an equation true. It's a fundamental concept in algebra and precalculus. In the given problem, we have an equation in which we need to solve for the variable \( R_{1} \). The original equation represents the relationship between resistors in parallel:

\[\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}\]To solve for \( R_{1} \), we must perform various steps to isolate it on one side of the equation. This involves algebraic manipulation techniques such as finding a common denominator, cross-multiplying, and rearranging terms. Understanding these steps and performing them correctly helps in arriving at the solution efficiently.

Here are tips for effective equation solving:

• Understand the initial equation, identifying the terms involved and their relationships.
• Clear any fractions by finding a common denominator or using cross-multiplication.
• Rearrange terms systematically to isolate the variable of interest.
• Perform algebraic manipulations carefully to avoid errors.

Fractions

Fractions often appear in equations, especially when dealing with problems involving parts or ratios. Solving equations with fractions requires understanding their properties and how to manipulate them. In the problem, we manage fractions by finding a common denominator:

The given fractions are:

• \( \frac{1}{R} \)
• \( \frac{1}{R_{1}} + \frac{1}{R_{2}} \)

To combine these into a single fraction, both parts on the right-hand side are combined over a common denominator \( R_{1}R_{2} \). This allows us to work with the equation more easily and prepare for further algebraic manipulation:

\[ \frac{1}{R} = \frac{R_{1} + R_{2}}{R_{1}R_{2}} \]Properly handling fractions ensures we accurately represent the relationships in the equation and simplify the solving process.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to solve for a variable or simplify expressions. It is crucial when faced with complex equations that are difficult to solve directly. In this exercise, manipulation starts with cross-multiplying to remove fractions. This results in:

\[ R(R_{1} + R_{2}) = R_{1} R_{2} \]Next, expand and rearrange terms to isolate \( R_{1} \):

1. Expand the equation: \( R R_{1} + R R_{2} = R_{1} R_{2} \).2. Move terms: \( R R_{1} - R_{1} R_{2} = -R R_{2} \).3. Factor out \( R_{1} \): \( R_{1}(R - R_{2}) = -R R_{2} \).

Finally, solve for \( R_{1} \) by dividing through:

\[ R_{1} = \frac{-R R_{2}}{R - R_{2}} \]Each step of manipulation brings us closer to the solution, carefully transforming the equation while maintaining its equality. Learning these techniques is vital for solving a broad range of algebraic problems.