Problem 78 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the specified variable. \(S=\frac{n}{2}\left[2 a_{1}+(n-1) d\right]\) for \(a_{1} \quad\) (Mathematics)

Step-by-Step Solution

Answer

The solution for \(a_1\) is \(a_1 = \frac{S - \frac{n(n-1)}{2}d}{n}\).

Step 1: Distribute the fraction

Start by distributing the fraction \( \frac{n}{2} \) into the brackets: \[ S = \frac{n}{2} \cdot 2a_1 + \frac{n}{2} \cdot (n-1)d \] This simplifies to \[ S = na_1 + \frac{n(n-1)}{2}d \].

Step 2: Isolate the term with \(a_{1}\)

To solve for \(a_1\), first, we need to isolate the term that includes \(a_1\). Subtract \(\frac{n(n-1)}{2}d\) from both sides: \[ S - \frac{n(n-1)}{2}d = na_1 \].

Step 3: Solve for \(a_{1}\)

Divide every term by \(n\) to solve for \(a_1\): \[ a_1 = \frac{S - \frac{n(n-1)}{2}d}{n} \].

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics, often used to describe relationships through equations. It involves both the use of symbols and numbers to represent mathematical situations.
At its core, algebra focuses on operations and the rules for manipulating those operations to find unknowns or solve problems.

• Algebra uses variables, which serve as placeholders for unknown values.
• Equations in algebra often describe the equivalence between two mathematical expressions.
• Understanding algebra requires knowing how to work with both sides of an equation to maintain balance.

In the given exercise, algebra provides the tools to manipulate the equation in order to isolate the variable of interest, which in this case is \(a_1\). With algebraic operations such as simplifying and rearranging terms, you can solve for unknowns efficiently.

Formula Manipulation

Formula manipulation is essential when dealing with algebraic expressions and equations to find specific unknowns. It involves altering the formula's structure without changing its original meaning. This skill is crucial for adapting equations to solve for one or more variables.
When performing formula manipulation, keep in mind:

• Each manipulation must adhere to the rules of algebra to ensure the integrity of the equation.
• Common operations include distributing factors, combining like terms, and factoring.
• One goal is often to make complex expressions simpler or more manageable.

In the step-by-step solution, formula manipulation is evident when distributing the fraction \( \frac{n}{2} \) across terms in the brackets. This step is key to expressing the equation in a form where \(a_1\) can be isolated and solved.

Variable Isolation

Variable isolation is an important technique used to solve equations. The aim is to rearrange the equation so that the variable you are solving for is on one side of the equation by itself. This often involves several algebraic operations.
To achieve good variable isolation:

• Identify the variable you need to isolate.
• Use inverse operations to move other terms to the opposite side of the equation.
• Consistently apply operations to both sides of the equation to maintain equality.

In the provided exercise, isolating \(a_1\) is completed in steps. First, subtract \(\frac{n(n-1)}{2}d\) from both sides of the equation to remove those terms from the side with \(a_1\).
Then, divide every remaining term by \(n\) to leave \(a_1\) standing alone. This technique is useful across many domains, making complex problems much simpler by focusing on one part at a time.