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Factorization is an algebraic technique used to simplify expressions and solve equations. It involves breaking down an expression into a product of its factors. In the context of the given exercise, we use factorization to simplify the equation.

Consider the original equation:

• \(C = A + Ar\)

Both terms on the right, \(A\) and \(Ar\), have a common factor, which is \(A\). This step requires us to rearrange the expression by pulling out this common factor. This simplifies our work and is a useful first step in solving many algebraic equations. By factoring \(A\), the expression becomes:

• \(A(1 + r) = C\)

Factoring makes it easier to manipulate the equation in subsequent steps, especially when you need to isolate a particular variable.

Isolation of variables is crucial in solving equations. It involves rearranging the equation to express one variable solely in terms of other variables or constants. Once a term is factored, like in our given equation, isolating the desired variable becomes straightforward.

After factorization, we have:

• \(A(1 + r) = C\)

To isolate \(A\), divide both sides by the term \(1 + r\). This process of isolating \(A\) involves using the inverse operation of multiplication, which is division.

By dividing both sides of the equation:

• \(A = \frac{C}{1 + r}\)

This step allows us to explicitly determine the value of \(A\) if \(C\) and \(r\) are known. This is an essential skill, notably for solving equations, and it strengthens your problem-solving abilities in algebra.

Solving equations involves finding the value of unknown variables that satisfy the equation. This requires a series of steps to manipulate the given equation into a simpler form where the solution can be easily identified.

In our example:

• The equation we start with is \(C = A + Ar\).
• By factoring and isolating the variable, we simplify it to \(A = \frac{C}{1 + r}\).

This process of solving begins with simplifying the equation and ends with isolating the variable, allowing for explicit solutions. Understanding how to solve equations is fundamental because it is a skill that applies in various mathematical fields.

When dealing with more complex equations, these concepts—factorization, isolation, and solving—become invaluable, enabling a systematic approach to finding solutions.