Problem 58 - Intermediate Algebra

Question

Solve each formula for the specified variable. See Example 5. $$ S=\frac{n}{2}(f+l) \quad \text { for } n $$

Step-by-Step Solution

Answer

\( n = \frac{2S}{f+l} \)

Step 1: Understand the Formula

The formula given is for the sum of an arithmetic series: \( S = \frac{n}{2}(f + l) \). Here, \( S \) is the sum, \( n \) is the number of terms, \( f \) is the first term, and \( l \) is the last term.

Step 2: Multiply to Eliminate the Fraction

To clear the fraction, multiply both sides of the equation by 2: \[ 2S = n(f + l) \] This transformation eliminates the fraction by moving the denominator \( 2 \) to the left side.

Step 3: Isolate the Variable \( n \)

To solve for \( n \), divide both sides by \( (f + l) \): \[ n = \frac{2S}{f + l} \] This isolates \( n \) as the subject of the equation.

Key Concepts

Understanding Arithmetic Series

Arithmetic series consist of a sequence of numbers with a constant difference between consecutive terms. The formula for the sum of an arithmetic series is \( S = \frac{n}{2}(f+l) \), where:

• \(S\) is the total sum of the series.
• \(n\) represents the number of terms.
• \(f\) is the first term of the series.
• \(l\) is the last term of the series.

To further illustrate, consider a series like 2, 4, 6, 8. Here, the difference between each term is 2, making it arithmetic. Calculating how these terms add up relies on this sum formula. It's a handy way to find the total without having to add each term individually.

Mastering Algebraic Manipulation

Algebraic manipulation allows us to rearrange and modify equations to isolate a specific variable or to simplify expressions. In our example, the manipulation began by multiplying both sides of the equation by 2. This step eliminated the fraction, transforming the formula into:

• \(2S = n(f + l)\)

By multiplying, we move towards simplifying the equation. Different manipulations like using the distributive property, combining like terms, or expanding expressions are fundamental in algebraic operations.
These manipulations are crucial for performing operations needed to handle variables, rearranging equations to solve for different components, and making complex formulas more approachable.

The Art of Isolating Variables

Isolating variables is the process of rearranging an equation so that a particular variable stands alone on one side of the equation. This is essential for solving equations. In our exercise, the variable \(n\) was isolated by returning to our manipulated equation \(2S = n(f+l)\) and dividing both sides by \((f + l)\).This final maneuver perfectly isolates \(n\), resulting in:

• \(n = \frac{2S}{f + l}\)

By achieving this, we now have\(n\) as the subject of the formula, with its value clearly defined in terms of other known quantities. Similar methods are applied in countless mathematical problems to distill solutions and extract values efficiently. Understanding and practicing these techniques will enhance problem-solving skills in algebra.