91Ó°ÊÓ

Algebraic manipulation involves the process of rearranging equations and expressions to isolate a desired variable. Here, we use algebraic techniques to solve for a specific variable in a formula. It typically takes several steps:

1. Identifying and understanding the original equation.
2. Using operations to simplify the equation, such as combining like terms or distributing constants across terms.
3. Rearranging terms to isolate the variable of interest.

In the given exercise, we start with the formula \( S = \frac{a - a r^{n}}{1 - r} \) and aim to solve for the variable \(a\). This involves manipulating the equation through multiplication, distribution, and factoring.

Isolating variables means getting a variable alone on one side of the equation. This is crucial for solving equations because it allows us to find the value of the unknown.

In the exercise:
1. We begin with \( S = \frac{a - a r^{n}}{1 - r} \) and aim to solve for \(a\).
2. To eliminate the fraction, multiply both sides by the denominator: \[ S(1 - r) = a - a r^{n} \].
3. By distributing \(S\) on the left side, we have: \[ S - Sr = a - a r^{n} \].
4. Next, factor out \(a\) on the right-hand side: \[ S - Sr = a(1 - r^{n}) \].
5. Finally, isolate \(a\) by dividing both sides by \(1 - r^n\): \[ a = \frac{S - Sr}{1 - r^{n}} \].

By isolating the variable, we make it easier to solve for its value.

Factoring is the process of breaking down an expression into a product of simpler expressions. This technique is often used to simplify equations or solve for variables.

In the current problem, we look to factor out the variable \(a\) from the expression \(a - a r^n\).
1. We rewrite the right-hand side as \(a(1 - r^n)\).
2. This helps us combine terms and ultimately isolate \(a\).

Factoring is a powerful tool in algebra that can transform complex expressions into more manageable forms. For example, going from \(a - a r^n\) to \(a(1 - r^n)\) simplifies the process of solving for \(a\).

Using these techniques effectively facilitates the process of solving equations.