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Chapter 7: Problem 22

Solve each formula for the specified variable. $$ \frac{1}{p}+\frac{1}{q}=\frac{1}{f} ; q $$

Short Answer

Expert verified
q = \frac{fp}{p - f}

Step by step solution

01

- Rearrange the equation

Start by rearranging the given equation to isolate the terms involving the variable you need to solve for. The initial equation is: \[ \frac{1}{p} + \frac{1}{q} = \frac{1}{f} \] To isolate \( \frac{1}{q} \), subtract \( \frac{1}{p} \) from both sides: \[ \frac{1}{q} = \frac{1}{f} - \frac{1}{p} \]
02

- Simplify the expression

Find a common denominator to combine the fractions on the right-hand side: The common denominator for \(f\) and \(p\) is \(fp\): \[ \frac{1}{q} = \frac{p - f}{fp} \]
03

- Take the reciprocal

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

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Key Concepts

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

Solving for a Variable
When solving for a variable, we focus on isolating the unknown variable on one side of the equation. The goal is to make the variable the subject of the formula. Let's understand this through the given exercise: We start with the equation: \[ \frac{1}{p} + \frac{1}{q} = \frac{1}{f} \] We need to solve for 'q'. To begin, subtract \( \frac{1}{p} \) from both sides to isolate \( \frac{1}{q} \): \[ \frac{1}{q} = \frac{1}{f} - \frac{1}{p} \] By rearranging terms and focusing on isolating the variable, we get closer to expressing 'q' in terms of 'p' and 'f'. This technique鈥攔earranging and isolating鈥攊s foundational for solving equations in algebra.
Common Denominator
Combining fractions often requires finding a common denominator. A common denominator is a shared multiple of the denominators of the fractions you are working with. This technique simplifies the process of adding or subtracting fractions. In our exercise, we need a common denominator to combine \( \frac{1}{f} \) and \( \frac{1}{p} \). The common denominator for 'f' and 'p' is 'fp'. The right side of the equation becomes: \[ \frac{1}{q} = \frac{p - f}{fp} \] By rewriting the fractions with a common denominator, we can easily perform arithmetic operations on the fractions. Always seek the simplest common multiple, as it makes calculations more straightforward.
Reciprocal
A reciprocal of a number is simply one divided by that number. Understanding reciprocals is crucial in solving equations that involve fractions. In the exercise, once we combined the fractions, we needed to solve for 'q'. We had: \[ \frac{1}{q} = \frac{p - f}{fp} \] To isolate 'q', we take the reciprocal of both sides: \[ q = \frac{fp}{p - f} \] Taking the reciprocal effectively 'flips' the fraction, which moves the variable in the denominator to the numerator. This step is critical when dealing with fractions in algebraic equations. Remember, reciprocals help us transition between different forms of expressions, making complex equations easier to handle.

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Most popular questions from this chapter

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