Problem 34 - College Algebra

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(f : \frac{1}{p}+\frac{1}{q}=\frac{1}{f}\)

Step-by-Step Solution

Answer

\( f = \frac{pq}{q + p} \).

Step 1: Review and Arrange the Equation

The equation given is \( \frac{1}{p} + \frac{1}{q} = \frac{1}{f} \). Our task is to solve for \( f \). To do this, we need to combine the fractions on the left side to create a single fraction.

Step 2: Find a Common Denominator

The fractions \( \frac{1}{p} \) and \( \frac{1}{q} \) have denominators \( p \) and \( q \), respectively. The common denominator for these fractions is \( pq \). We'll rewrite each fraction with this common denominator:\[ \frac{q}{pq} + \frac{p}{pq} = \frac{1}{f} \]

Step 3: Combine the Fractions

Now, add the two fractions on the left side, since they have a common denominator:\[ \frac{q + p}{pq} = \frac{1}{f} \]

Step 4: Solve for f

We can solve for \( f \) by taking the reciprocal of both sides of the equation:\[ f = \frac{pq}{q + p} \].This gives us the expression for \( f \) in terms of \( p \) and \( q \).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (like addition and multiplication). Understanding algebraic expressions is fundamental to solving equations, as they form the basis of more complex algebraic manipulations. In our exercise, the expression involves fractions with variables: \( \frac{1}{p} + \frac{1}{q} = \frac{1}{f} \). Here, \( p \), \( q \), and \( f \) are the variables, and our goal is to isolate \( f \), the variable of interest. By rearranging and simplifying the algebraic expression using skills like finding a common denominator, we can solve for \( f \) effectively. Becoming comfortable with identifying and manipulating algebraic terms is essential for successful algebraic problem-solving.

Fraction Manipulation

Fraction manipulation is a crucial skill in algebra that involves operations like addition, subtraction, multiplication, and division of fractions. In the given exercise, we need to add two fractional expressions: \( \frac{1}{p} \) and \( \frac{1}{q} \). To achieve this effectively, recognizing that we need a common denominator is key. This initial step leads us to rewrite each fraction, enabling proper simplification and eventually, solving for the variable. Mastery of fraction manipulation allows for fluid transitions between different forms, which is vital when working with complex expressions.

Common Denominator

Finding a common denominator is a necessary step when adding or subtracting fractions with different denominators. In our problem, the fractions \( \frac{1}{p} \) and \( \frac{1}{q} \) have distinct denominators, \( p \) and \( q \). To combine these fractions into a single expression, we determine their common denominator to be \( pq \). This allows us to rewrite the problem as \( \frac{q}{pq} + \frac{p}{pq} = \frac{1}{f} \). With a shared denominator, you can then easily add the numerators to streamline the equation. Understanding how to find and use a common denominatorhelps in efficiently combining and simplifying algebraic fractions.

Reciprocal of a Fraction

The reciprocal of a fraction is simply another fraction flipped upside down. If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This concept plays an essential role in finding solutions to equations, particularly when solving for a variable that is found in a denominator, as in our exercise. After adding and simplifying fractions, we arrive at: \( \frac{q + p}{pq} = \frac{1}{f} \). To solve for \( f \), we take the reciprocal of both sides, giving us \( f = \frac{pq}{q + p} \). Recognizing and understanding the reciprocal aids in straightforward manipulation and rearrangement of algebraic expressions.