Problem 85 - Beginning and Intermediate Algebra

Question

Solve. \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}},\) for \(R_{2}\) (A formula for resistance)

Step-by-Step Solution

Answer

The short answer is: To solve for \(R_{2}\) in the equation \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\), follow these steps: 1. Subtract \(\frac{1}{R_1}\) from both sides: \(\frac{1}{R_2} = \frac{1}{R} - \frac{1}{R_1}\) 2. Find a common denominator: \(\frac{1}{R_2} = \frac{R_1 - R}{R \cdot R_1}\) 3. Take the reciprocal of both sides: \(R_2 = \frac{R \cdot R_1}{R_1 - R}\)

Step 1: We are given the equation: \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\). #Step 2: Isolate the term with R2#

To solve for R2, first subtract \(\frac{1}{R_1}\) from both sides of the equation: \[ \frac{1}{R_2} = \frac{1}{R} - \frac{1}{R_1} \] #Step 3: Find a common denominator for the right side of the equation#

Step 2: To simplify the right side of the equation, find a common denominator for the two fractions, which is R1 * R: \[ \frac{1}{R_2} = \frac{R_1 - R}{R \cdot R_1} \] #Step 4: Take the reciprocal of both sides of the equation#

Now, take the reciprocal of both sides of the equation to solve for R2: \[ R_2 = \frac{R \cdot R_1}{R_1 - R} \] Now we have isolated R2 and found its expression in terms of R and R1. The final formula for R2 is: \[ R_2 = \frac{R \cdot R_1}{R_1 - R} \]

Key Concepts

Formula for Resistance

In many electrical engineering problems, understanding resistance is crucial. Resistance can be thought of as the opposition to the flow of electrical current. The formula we are dealing with in this context is for the total resistance (\(R\)) in a parallel circuit. Parallel circuits have multiple paths for electricity to flow, and the total resistance is affected by each individual resistance in the circuit.
In a parallel circuit with just two resistors, \(R_1\) and \(R_2\), the total resistance can be calculated using the formula:

• \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \)

This equation represents how the inverse of the total resistance is the sum of the inverses of the individual resistances. Solving for one of the components, such as \(R_2\), requires some algebraic manipulation to isolate the variable.

Reciprocal in Equations

Reciprocals are an essential concept when dealing with fractions and ratios, especially in equations. The reciprocal of a number is simply 1 divided by that number. For instance, the reciprocal of \(R\) is \(\frac{1}{R}\). This idea is used in equations like the resistance formula to simplify and solve for variables.
While solving \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \) for \(R_2\), it's necessary to express \(R_2\) in terms of \(R\) and \(R_1\). By taking the reciprocal of both sides of the equation at the end, you can isolate \(R_2\). Here are the essential steps:

• Subtract \(\frac{1}{R_1}\) from both sides to get \(\frac{1}{R_2} = \frac{1}{R} - \frac{1}{R_1}\)
• Simplify and find a common denominator
• Take the reciprocal to solve for \(R_2\)

Common Denominator

Finding a common denominator is a fundamental step in algebra, especially when adding or subtracting fractions. A common denominator allows you to combine fractions by rewriting them with the same bottom part (denominator).
When you have different denominators, like \(R\) and \(R_1\), you must find a common multiple. In our resistance formula problem, the least common denominator of \(\frac{1}{R}\) and \(\frac{1}{R_1}\) is \(R \times R_1\). This helps combine the fractions:

• Convert \(\frac{1}{R}\) to \(\frac{R_1}{R \cdot R_1}\)
• Convert \(\frac{1}{R_1}\) to \(\frac{R}{R \cdot R_1}\)

Afterward, subtract the fractions, getting \(\frac{R_1 - R}{R \cdot R_1}\), which simplifies the right side of our equation successfully. This step is vital for calculating the reciprocal and isolating \(R_2\) in the last steps.