Problem 76 - Introductory Algebra for College Students

Question

$$\text { Solve for } f: \frac{1}{p}+\frac{1}{q}=\frac{1}{f}$$

Step-by-Step Solution

Answer

After all the steps, \( f \), in terms of \( p \) and \( q \), is represented by the formula \( f = \frac{pq}{p + q} \). This is the simplified form of the initial formula.

Step 1: Rewrite the equation

The first step is to rewrite the equation in terms of \( f \). Currently, \( f \) is on the right side of the equation with other variables. The goal is to isolate \( f \). To do this, start by rewriting the formula \(\frac{1}{p} + \frac{1}{q} = \frac{1}{f}\) which can be written in the form of \( fb = a \), where \( a = \frac{1}{p} + \frac{1}{q} \) and \( b = 1 \).

Step 2: Solve for f

To obtain \( f \) alone on one side, divide both sides of the equation by \( b \) (which is 1 in this case). Thus, \( f = \frac{a}{b} \) or, when substituted back for the original variables, \( f = \frac{1}{\frac{1}{p} + \frac{1}{q}} \).

Step 3: Simplify the equation

The obtained result for \( f \) is correct but can be further simplified. Notice the form of the result: a fraction where the numerator is 1 and the denominator is a sum of fractions. When simplified, this can be rewritten as \( f = \frac{pq}{p+q} \).

Key Concepts

Understanding Rational Expressions

Rational expressions are fractions that have polynomials in both the numerator and the denominator. In this exercise, the equation \( \frac{1}{p} + \frac{1}{q} = \frac{1}{f} \) is a type of rational expression because it involves fractions where each term is a single rational number. These sorts of expressions are crucial in algebra because they embody relationships between quantities that can be manipulated and solved for unknown variables.

Dealing with equations involving rational expressions often requires finding a common denominator or manipulating the fractions to get to a form that is easy to solve. They can sometimes be confusing because they add layers of complexity compared to simpler linear equations. Still, understanding how they work helps in deciphering more complex mathematical problems.

In this specific problem, the rational expression is manipulated to get \( f \) by performing operations common to fractions and algebraic techniques.

Mastering Fraction Manipulation

Fraction manipulation is a key skill when working with rational expressions. It involves the application of basic arithmetic operations like addition, subtraction, multiplication, and division to fractions and is fundamental to isolating variables in algebraic equations.

For the given exercise, we encounter fractions in the form \( \frac{1}{p} \) and \( \frac{1}{q} \). To solve for \( f \), you need to add these fractions. Usually, this requires a common denominator but here a different approach is used by employing reciprocal rules directly, yielding \( f = \frac{1}{\frac{1}{p} + \frac{1}{q}} \).

Further simplification is achieved by dealing with the complex fraction. Simplifying complex fractions involves rewriting them to have a simple fraction. In this case, the expression \( \frac{pq}{p+q} \) is derived, simplifying our expression significantly by multiplying the numerator and the denominator by \( pq \) to help eliminate the inner fractions.

Key points for fraction manipulation include:

• Identifying common denominators when needed.
• Transforming fractions into a format that makes calculations more straightforward.
• Simplifying complex fractions to make equations easier to manage.
• Using reciprocal identities efficiently.

Solving Algebraic Equations

Algebraic equations like the one in this problem often require methodical steps for solving. These steps usually involve isolating the variable of interest—in this case, \( f \). To isolate \( f \), the entire equation is manipulated through algebraic operations.

Here, the original equation is rearranged as \( f = \frac{1}{\frac{1}{p} + \frac{1}{q}} \). Then further simplifying things involves transforming this expression into \( f = \frac{pq}{p+q} \). This final concise form is achieved through understanding and manipulating the expressions correctly.

The process generally includes:

• Rearranging terms to select the unknown variable.
• Applying operations such as multiplication or division across the entire equation to solve for the target variable.
• Simplifying the expression as much as possible to ensure it is in the simplest form.

Understanding and applying these strategies helps bring clarity to solving equations across various topics in algebra.