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Chapter 6: Problem 24

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

Short Answer

Expert verified
\( p = \frac{fq}{q - f} \)

Step by step solution

01

Start with the given equation

The given equation is \[ \frac{1}{p} + \frac{1}{q} = \frac{1}{f} \]. We need to solve for the variable \( p \).
02

Isolate the term involving \( p \)

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

Find a common denominator

Find a common denominator for the fractions on the right-hand side. The common denominator for \( \frac{1}{f} \) and \( \frac{1}{q} \) is \( fq \). So, rewrite the fractions: \[ \frac{1}{f} = \frac{q}{fq} \] and \[ \frac{1}{q} = \frac{f}{fq} \]. This gives us: \[ \frac{1}{p} = \frac{q - f}{fq} \]
04

Invert both sides

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

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Key Concepts

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

Isolating the Variable
When solving an equation for a specific variable, the first step is to isolate that variable on one side of the equation. In this case, we want to solve for the variable \( p \). The initial equation is: \[ \frac{1}{p} + \frac{1}{q} = \frac{1}{f} \].
We need to move all terms involving \( p \) to one side and everything else to the opposite side. To do this, subtract \( \frac{1}{q} \) from both sides: \[ \frac{1}{p} = \frac{1}{f} - \frac{1}{q} \]. This step ensures that the term with \( p \) is isolated, making it easier to solve for \( p \) in the upcoming steps.
Common Denominator
When dealing with fractions in an equation, finding a common denominator helps simplify the process. In our equation, we need to combine the fractions \( \frac{1}{f} \) and \( \frac{1}{q} \): \[ \frac{1}{p} = \frac{1}{f} - \frac{1}{q} \].
Identify the common denominator of \( f \) and \( q \), which is \( fq \). Rewrite each fraction with this common denominator: \[ \frac{1}{f} = \frac{q}{fq} \] and \[ \frac{1}{q} = \frac{f}{fq} \].
Now that they share a common denominator, we can rewrite the equation as: \[ \frac{1}{p} = \frac{q - f}{fq} \]. Simplifying the fractions in this way makes the next steps more straightforward.
Inverting Fractions
To finish solving for the variable \( p \), we need to remove it from the denominator. This is done by inverting both sides of the equation. From the previous step, we have: \[ \frac{1}{p} = \frac{q - f}{fq} \].
Invert both sides to get \( p \) on its own: \[ p = \frac{fq}{q - f} \].
Inverting both sides of a fraction means swapping the numerator and denominator, effectively 'flipping' the fraction. This step simplifies our equation and provides the solution for \( p \) in a clear, manageable form.

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