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Chapter 14: Problem 89

Solve. \(\frac{1}{F}=\frac{1}{m}+\frac{1}{p},\) for \(p\) (A formula from optics)

Short Answer

Expert verified
The short answer is: To solve the formula \(\frac{1}{F} = \frac{1}{m} + \frac{1}{p}\) for \(p\), you can perform the following steps: 1. Combine the fractions on the right-hand side with a common denominator of \(mp\). 2. Simplify the equation. 3. Cross-multiply. 4. Simplify the equation further and solve for \(p\). The solution is given by: \[p = \frac{m}{F - 1}\]

Step by step solution

01

Make a common denominator

To solve this equation for \(p\), we first need to combine the fractions on the right-hand side. To do this, we need to find a common denominator, which is \(mp\), since \(m\) and \(p\) are both in the denominators. So, we can rewrite the right-hand side as follows: \[\frac{1}{m} + \frac{1}{p} = \frac{p}{mp} + \frac{m}{mp}\]
02

Simplify the right-hand side

Now that the fractions have a common denominator, we can add them together by simply adding the numerators: \[\frac{1}{F} = \frac{p+m}{mp}\]
03

Swap the left-hand and right-hand sides

We will swap the sides of the equation so that the variable of interest, \(p\), is on the left side. This makes the equation easier to read and work with: \[\frac{p+m}{mp} = \frac{1}{F}\]
04

Cross-multiply

To get \(p\) out of the denominator, we need to cross-multiply. This means we will multiply both sides by the product of their denominators, \(mpF\): \[mpF\left(\frac{p+m}{mp}\right)= mpF\left(\frac{1}{F}\right)\]
05

Simplify the equation and solve for p

Now we can simplify the equation by canceling out the denominators: \[(p+m)=pF\] To isolate \(p\), we can move the term containing \(p\) to one side of the equation: \[m = pF - p\] Now, we can factor out \(p\) from the terms on the right-hand side: \[m = p(F - 1)\] Finally, to solve for \(p\), we just need to divide both sides of the equation by \((F - 1)\): \[p = \frac{m}{F - 1}\] So, the object distance \(p\) can be expressed in terms of the focal length \(F\) and the image distance \(m\) as: \[p = \frac{m}{F - 1}\]

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Key Concepts

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

Solving Equations
Understanding how to solve equations is vital in algebra since it involves finding the value of an unknown variable that makes the equation true. In the given exercise, the main goal is to solve for the variable \( p \) in the equation \( \frac{1}{F} = \frac{1}{m} + \frac{1}{p} \). The process of solving equations usually involves rearranging terms and performing operations, such as addition, subtraction, multiplication, and division, to isolate the variable of interest.
  • In the initial steps, simplifying the equation by finding a common denominator for the fractions makes it easier to deal with them as a single entity.
  • Following that, operations like cross-multiplying are utilized to eliminate fractions by transforming the equation into a simpler equivalent form.
  • The final move involves isolating the variable \( p \), which might include several algebraic manipulations like factoring and simplifying expressions.
Remember, maintaining each operation's balance (by performing the same operation on both sides of the equation) is key to arriving at an accurate solution.
Fraction Operations
Working with fractions is a core aspect of algebra, and understanding fraction operations is necessary to solve equations, especially when they involve variables. In the original exercise, the fractions \( \frac{1}{m} \) and \( \frac{1}{p} \) needed to be combined before proceeding with the solution. Here, the operation utilized is finding a common denominator.
  • The least common denominator (LCD) for the fractions \( \frac{1}{m} \) and \( \frac{1}{p} \) is \( mp \) because it is a product that both denominators divide into without remainder.
  • Once the common denominator is defined, each fraction is rewritten to have this common denominator and then combined. This results in a single fraction, simplifying further algebraic operations.
Simplifying fractions by combining them into a single term allows for easier manipulation and simplifies further steps in solving equations.
Cross Multiplication
Cross multiplication is a technique often used to rid equations of fractions, making them easier to work with. It involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. In the exercise, after combining the fractions and simplifying, the equation was set as \( \frac{p+m}{mp} = \frac{1}{F} \).
  • To eliminate the fractions, we perform cross multiplication, which involves multiplying both sides by the complete cross product of denominators, essentially multiplying each side by \( mpF \).
  • This operation allows the cancelation of the denominators, leaving a simple equation where algebraic manipulation to isolate \( p \) is straightforward.
  • This step transforms the problem into one involving basic algebraic operations, making the process of finding the solution more manageable.
Cross multiplication is a powerful tool in solving equations involving fractions, simplifying complex fractional expressions into standard algebraic forms.

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