Problem 76 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) for \(R_{2}\)

Step-by-Step Solution

Answer

\( R_2 = \frac{1}{\frac{1}{R} - \frac{1}{R_1} - \frac{1}{R_3}} \)

Step 1: Combine the fractions on the right hand side

The equation is \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \). To solve for \( R_2 \), start by combining the fractions on the right hand side: \[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \rightarrow \frac{1}{R} = \frac{R_2 R_3 + R_1 R_3 + R_1 R_2}{R_1 R_2 R_3} \].

Step 2: Isolate the term involving \( R_2 \)

Now, we want to isolate \( \frac{1}{R_2} \) from the equation. Subtract \( \frac{R_2 R_3 + R_1 R_3}{R_1 R_2 R_3} \) from both sides to get: \[ \frac{1}{R} - \frac{1}{R_1} - \frac{1}{R_3} = \frac{1}{R_2} \].

Step 3: Simplify to solve for \( R_2 \)

Finally, invert both sides of the equation from Step 2 to solve for \( R_2 \). This gives us: \[ R_2 = \frac{1}{\frac{1}{R} - \frac{1}{R_1} - \frac{1}{R_3}} \].

Key Concepts

Variable Isolation

Variable isolation is a fundamental part of solving equations, especially in algebra. It involves rearranging an equation to get the desired variable alone on one side of the equation. This process is crucial since it allows us to clearly see the relationship between different variables.
When isolating a variable, you perform operations that simplify the equation step by step. You might need to add, subtract, multiply, divide, or apply other algebraic manipulations to both sides until the variable you want is by itself.
For example, in the equation \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \), we want to solve for \( R_2 \). We isolate \( \frac{1}{R_2} \) by moving the other terms to the opposite side. We do this by subtracting the fractions for \( R_1 \) and \( R_3 \) from both sides, leading us to:

• Step: \[ \frac{1}{R} - \frac{1}{R_1} - \frac{1}{R_3} = \frac{1}{R_2} \]

This gives clarity on how \( R_2 \) is related to \( R, R_1, \) and \( R_3 \). Once isolated, you can proceed to find its direct value, simplifying your equation further.Variable isolation lays the foundation for clearer problem-solving strategies and helps to eliminate confusion when dealing with more complex formulae.

Fraction Operations

Fraction operations are crucial when dealing with equations involving fractions. Operations on fractions include addition, subtraction, multiplication, and division. Each has specific rules to ensure accuracy and consistency in problem-solving.
When combining fractions, a common denominator is essential. This allows for straightforward addition or subtraction. Multiply numerators and denominators across fractions when multiplying, whereas flipping the second fraction and multiplying should be done when dividing a fraction by another.
In the exercise, fractions on the right side of the equation were combined:

• From this: \[ \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]
• To this: \[ \frac{R_2 R_3 + R_1 R_3 + R_1 R_2}{R_1 R_2 R_3} \]

This combination utilized a common denominator \(R_1 R_2 R_3\). Understanding these basics helps solve and simplify equations smoothly.
Ultimately, having a strong grasp of fraction operations provides the tools needed to tackle equations like the ones commonly seen in mathematics focused on fractions.

Electrical Circuits

Electrical circuits often rely on equations involving resistance, current, and voltage to describe their behavior. The formula mentioned in the exercise - \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \) - is used to calculate the total resistance \( R \) of parallel resistors in a circuit.
In parallel circuits, the total resistance is always less than the smallest individual resistor. This is because the formula calculates how pathways are added, thus reducing the overall resistance of the circuit. Understanding formulas like these allow engineers to design circuits that will function correctly under specific loads.
Here’s a brief look at the concepts in electrical circuits:

• Resistance: Opposes the flow of electric current; measured in ohms.
• Series Circuits: Resistances add up directly: \( R_{total} = R_1 + R_2 + R_3 \).
• Parallel Circuits: The reciprocal sum of individual resistances: \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \).

With a strong understanding of these concepts, you can effectively solve more complex problems related to electrical circuits. By mastering these, you ensure that designs and computations in electronics are both accurate and efficient.