Problem 15 - Algebra for College Students

Question

$$\text { Solve each formula for the indicated variable.}$$ $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \text { for } R_{1}$$

Step-by-Step Solution

Answer

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

Step 1 - Combine the fractions on the right side

The given equation is \(\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} \). To combine the fractions on the right side, find a common denominator, which is \( R_{1} R_{2} \). The equation becomes: \[ \frac{1}{R} = \frac{R_{2} + R_{1}}{R_{1} R_{2}} \]

Step 2 - Take the reciprocal of both sides

To isolate \( R_{1} \), take the reciprocal of both sides to get: \[ \frac{R_{1} R_{2}}{R_{1} + R_{2}} = R \]

Step 3 - Solve for \( R_{1} \)

Multiply both sides of the equation by \( R_{1} + R_{2} \) to get: \[ R_{1} R_{2} = R (R_{1} + R_{2}) \] Next, distribute \( R \): \[ R_{1} R_{2} = R R_{1} + R R_{2} \] To solve for \( R_{1} \), get all \( R_{1} \) terms on one side: \[ R_{1} R_{2} - R R_{1} = R R_{2} \] Factor out \( R_{1} \) on the left side: \[ R_{1}(R_{2} - R) = R R_{2} \] Finally, divide both sides by \( R_{2} - R \): \[ R_{1} = \frac{R R_{2}}{R_{2} - R} \]

Key Concepts

Formula Manipulation

Formula manipulation is a fundamental skill in algebra that enables you to rearrange formulas to solve for a specific variable. In the given problem, you need to isolate the variable \( R_1 \) from the equation \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \). To do this, you combine fractions, take reciprocals, and apply algebraic operations such as multiplication and division. It's like solving a puzzle where you move pieces around to get the desired outcome.
Remember, always perform the same operation on both sides of the equation to keep it balanced.

Solving Equations

Solving equations involves finding the value of the unknown variables that make the equation true. For our example, after combining the fractions and taking the reciprocal of both sides, you're left with \( \frac{R_1 R_2}{R_1 + R_2} = R \).
To isolate \( R_1 \), multiply both sides by the denominator \( R_1 + R_2 \). This gives you \( R_1 R_2 = R (R_1 + R_2) \). From here, distribute \( R \) and then involve moving terms around and factoring to solve for the specific variable.

Reciprocals

Reciprocals are key in many algebra problems. A reciprocal of a number \( x \) is \( \frac{1}{x} \). In our problem, when you have \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \), taking the reciprocal on both sides helps to simplify and isolate one variable.
By taking the reciprocal of \( \frac{R_1 R_2}{R_1+R_2} = R \), you get \( R = \frac{R_1 R_2}{R_1 + R_2} \). The reciprocal operation often makes complex algebraic fractions easier to handle.

Factoring

Factoring is useful when you need to simplify expressions or solve equations. In the context of the given problem, after distributing \( R \) to get \( R_1 R_2 = R R_1 + R R_2 \), the next step is to get all the \( R_1 \) terms on one side so you can factor them out. This step involves writing it as \( R_1 R_2 - R R_1 = R R_2 \).
Factoring \( R_1 \) from the left-hand terms, you get \( R_1 (R_2 - R) = R R_2 \), which makes it easier to solve for \( R_1 \). Dividing both sides by \( R_2 - R \) finally gives you the solution \( R_1 = \frac{R R_2}{R_2 - R} \).
This technique of factoring ensures you systematically reduce and simplify equations.