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Chapter 1: Problem 67

Solve for the indicated variable. $$ \frac{1}{f}=\frac{1}{p}+\frac{1}{q} \text { for } p $$

Short Answer

Expert verified
The solution for \( p \) is \(p = \frac{fq}{q - f}\).

Step by step solution

01

Start with the given equation

The equation given is \[ \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \]
02

Isolate the term with the variable

Subtract \( \frac{1}{q} \) from both sides of the equation to isolate \( \frac{1}{p} \):\[ \frac{1}{f} - \frac{1}{q} = \frac{1}{p} \]
03

Combine the fractions on the left-hand side

Find a common denominator to combine the fractions on the left-hand side. The common denominator for \(\frac{1}{f}\) and \(\frac{1}{q}\) is \(fq\):\[\frac{q - f}{fq} = \frac{1}{p} \]
04

Take the reciprocal

To solve for \(p\), take the reciprocal of both sides:\[ p = \frac{fq}{q - f} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
Isolating variables means getting the variable you want to solve for all by itself on one side of the equation. This often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same amount. For example, in the exercise \[ \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \], we need to isolate \( \frac{1}{p} \). Subtracting \( \frac{1}{q} \) from both sides allows us to do that: \[ \frac{1}{f} - \frac{1}{q} = \frac{1}{p} \]. By manipulating the equation this way, we make it easier to focus on solving for the variable \( p \).
Using a Common Denominator
When combining fractions, a common denominator is crucial. This means finding a shared multiple between the denominators of the fractions you want to add or subtract. In the equation \[ \frac{1}{f} - \frac{1}{q} = \frac{1}{p} \], the common denominator for \( \frac{1}{f} \) and \( \frac{1}{q} \) is \( fq \). Using this, we can write: \[ \frac{q - f}{fq} = \frac{1}{p} \]. Now, both fractions on the left side are expressed with the same bottom number, making it easier to work with them. This step is important because it consolidates the fractions into a single term.
Understanding Reciprocals
Reciprocals are numbers that, when multiplied together, equal 1. They are essentially 'flipped' versions of fractions. The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). In our equation, once we have \[ \frac{q - f}{fq} = \frac{1}{p} \], taking the reciprocal of both sides gets us closer to solving for \( p \): \[ p = \frac{fq}{q - f} \]. By flipping both sides, we effectively bring \( p \) out of the denominator and into a more workable position in the numerator.
Manipulating Fractions
Manipulating fractions often involves finding common denominators, adding or subtracting, and simplifying. In the exercise, we manipulate fractions to isolate \( p \). Starting from \[ \frac{1}{f} - \frac{1}{q} = \frac{1}{p} \], we identify that the common denominator is \( fq \), which helps us combine the fractions: \[ \frac{q - f}{fq} = \frac{1}{p} \]. Finally, manipulating fractions allows us to take the reciprocal, resulting in \( p = \frac{fq}{q - f} \). Breaking down the task of manipulating fractions step by step makes it less daunting and easier to understand.

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Most popular questions from this chapter

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