Problem 86 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( R_{1} = \frac{R \cdot R_{2}}{R_{2} - R} \)

Step 1: Set Equation

Start with the equation given: \( \frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} \). We need to solve this equation for \( R_{1} \).

Step 2: Subtract Formula

Subtract \( \frac{1}{R_{2}} \) from both sides to isolate the \( \frac{1}{R_{1}} \) term: \( \frac{1}{R} - \frac{1}{R_{2}} = \frac{1}{R_{1}} \).

Step 3: Find Common Denominator

Calculate a common denominator for the expression: \( \frac{1}{R} - \frac{1}{R_{2}} = \frac{R_{2} - R}{R \cdot R_{2}} \).

Step 4: Invert the Fraction

Take the reciprocal of the right side to solve for \( R_{1} \): \( R_{1} = \frac{R \cdot R_{2}}{R_{2} - R} \).

Key Concepts

Equation Solving

Equation solving is one of the foundational skills in algebra. It involves finding the value of a variable that makes the equation true. When we solve equations, we perform a series of operations to isolate the variable of interest. In the case of the given equation \( \frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} \), the goal is to express \( R_{1} \) in terms of \( R \) and \( R_{2} \).

The process usually starts by simplifying both sides of the equation. This includes performing operations like addition, subtraction, multiplication, or division to balance both sides. The main trick is to keep whatever you do to one side, you do to the other side. Here, you begin by re-arranging the terms to isolate the term \( \frac{1}{R_{1}} \).

Key steps in solving equations typically involve:

• Isolating the variable you are solving for.
• Simplifying both sides as much as possible.
• Maintaining the equilibrium of the equation by performing identical operations on both sides.

Formula Manipulation

Formula manipulation involves rearranging equations to solve for a particular variable. This technique is especially handy when dealing with equations that relate several variables. In mathematics, it's common to start with a formula involving multiple variables and change it to solve for one specific variable of interest.

To manipulate a formula, follow these steps:

• Identify the variable you need to isolate.
• Use algebraic operations to move other terms away from the variable.
• Ensure you keep the equation balanced by performing equal operations on both sides.

In the problem at hand, we used subtraction to move \( \frac{1}{R_{2}} \) to the other side and then took the reciprocal to solve for \( R_{1} \). These manipulations converted the initial equation into a form that expressed \( R_{1} \) in terms of other known quantities. Understanding how to manipulate formulas is crucial for working with complex equations in algebra and beyond.

Resistor Equations

Resistor equations are critical in understanding electrical circuits where resistors are often connected in series or parallel. In parallel circuits, the equation \( \frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} \) denotes the total resistance \( R \) being affected by individual resistances \( R_{1} \) and \( R_{2} \).

When resistors are in parallel:

• The total resistance is always less than the smallest individual resistance.
• Adding more resistors in parallel paths increases the total conductance, leading to reduced overall resistance.

Solving for one resistor in terms of others allows engineers and scientists to design circuits according to desired specifications. In this case, understanding how to re-arrange the given equation is essential for determining the contribution of each resistor and for calculating what might be needed to achieve a certain total resistance. Resistor equations like these are part of the fundamental tools used in electrical engineering.