Problem 81 - Algebra and Trigonometry

Question

\(79-92\) 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

\( R_1 = \frac{R \times R_2}{R_2 - R} \)

Step 1: Isolate the Fraction with \( R_1 \)

Start by isolating the term involving \( \frac{1}{R_1} \). To do that, subtract the term \( \frac{1}{R_2} \) from both sides of the equation. This gives us: \[ \frac{1}{R} - \frac{1}{R_2} = \frac{1}{R_1} \]

Step 2: Find a Common Denominator

In order to combine the terms on the left-hand side, find a common denominator for \( \frac{1}{R} \) and \( \frac{1}{R_2} \). The common denominator for \( R \) and \( R_2 \) is \( R \times R_2 \). Rewrite the fractions: \[ \frac{R_2}{R \times R_2} - \frac{R}{R \times R_2} = \frac{1}{R_1} \]

Step 3: Combine the Fractions

Now subtract the numerators over the common denominator: \[ \frac{R_2 - R}{R \times R_2} = \frac{1}{R_1} \]

Step 4: Reciprocate Both Sides

Reciprocate both sides to solve for \( R_1 \): \[ R_1 = \frac{R \times R_2}{R_2 - R} \]

Step 5: Verify the Solution

Verify the solution by checking that plugging \( \frac{R \times R_2}{R_2 - R} \) back into the original equation simplifies correctly, confirming the solution is accurate.

Key Concepts

Equation Manipulation

In algebra, equations are like puzzles that can be rearranged to make them easier to solve. Manipulating equations involves playing with the structure without changing their meaning. It's about moving terms around, combining or breaking down parts; essentially, doing whatever it takes to isolate the piece you're interested in.

When you have an equation like \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \), the goal is to make it look the way you want, focusing on isolating one variable. Equation manipulation mainly involves basic operations like addition, subtraction, multiplication, and division across both sides. These operations are used to isolate the term that contains the variable you're solving for. As long as what you do to one side, you also do to the other, the equality is maintained.

Mastering equation manipulation simplifies complex problems into manageable ones.

Isolating Variables

Isolating a variable means getting it alone on one side of the equation. It's like peeling away layers of an onion until you uncover the center. In our case, the target is \( R_1 \).
To isolate \( \frac{1}{R_1} \), we first need to get rid of \( \frac{1}{R_2} \) from the equation. So, we subtract \( \frac{1}{R_2} \) from both sides. This operation leaves us with \( \frac{1}{R} - \frac{1}{R_2} = \frac{1}{R_1} \).

• Think of \( \frac{1}{R_1} \) as a treasure you need to unveil by clearing away everything else.
• Each step must logically follow the previous until \( R_1 \) is alone.

Keep rearranging until the variable of interest stands independently.

Common Denominator

Fractions with different denominators can be tricky. It's like trying to compare apples to oranges. To make them compatible, we find a common denominator.For the fractions \( \frac{1}{R} - \frac{1}{R_2} \), we need them to speak the same 'language' of a common base.

• Multiply the denominator of each fraction so they become the same. In our example, the common denominator of \( R \) and \( R_2 \) is \( R \times R_2 \).
• Rewrite: \( \frac{R_2}{R \times R_2} - \frac{R}{R \times R_2} \)

This step prepares us to easily combine the fractions by ensuring both parts fit nicely together under one roof, the same denominator.

Fraction Operations

Once we have a common denominator, handling fractions becomes straightforward. We're now ready to combine them when their bases align.For \( \frac{R_2}{R \times R_2} - \frac{R}{R \times R_2} \), you just subtract the numerators, keeping the common denominator unchanged.

• Numerator: subtract \( R \) from \( R_2 \), resulting in \( R_2 - R \).
• Denominator remains \( R \times R_2 \).

Thus, everything contracts to \( \frac{R_2 - R}{R \times R_2} = \frac{1}{R_1} \).Reciprocating both sides then gives us the solution for \( R_1 \). Fraction operations allow for elegant solutions to what first seems convoluted, by breaking it down into simple numeric interactions.