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The distributive property is a core concept in algebra that helps to simplify expressions and equations, especially those involving parentheses. When you come across a term that multiplies directly into a parenthesis, the distributive property comes into play. In this exercise, we start with the equation \( G = 2b(r-1) \).

By applying the distributive property, we multiply \(2b\) by both terms inside the parentheses:

• First, multiply \(2b\) by \(r\), which gives \(2br\).
• Next, multiply \(2b\) by \(-1\), resulting in \(-2b\).

Putting these results together, the simplified form of the right side becomes \( G = 2br - 2b \).

This step is crucial as it helps to eliminate parentheses, making it easier to isolate and solve for the variable of interest. Understanding how to utilize the distributive property will aid in breaking down more complex algebraic problems.

Isolating a variable is all about getting the variable you are trying to solve for on one side of the equation, keeping it alone. In our problem, we need to isolate \(r\) from the equation \( G = 2br - 2b \).

The key is to perform operations that "undo" what has been done to the variable. This means eliminating any additional terms or coefficients on the same side as the variable.

• First, we see \(-2b\) on the right. To move it, add \(2b\) to both sides of the equation. This ensures the equation stays balanced.

The equation then changes to \( G + 2b = 2br \).

This step efficiently moves all terms involving \(r\) to one side, preparing for the final step, where we specifically isolate \(r\) to solve for it.

Algebraic manipulation refers to the techniques used to rearrange and simplify equations to solve for unknown variables. In the final step of our problem, we manipulate the equation \( G + 2b = 2br \) to isolate \(r\).

To do this, we need to get rid of the coefficient \(2b\) that is attached to \(r\). The simplest way to remove it is by dividing both sides of the equation by \(2b\):

• This gives \( r = \frac{G + 2b}{2b} \).

Through division, the equation is simplified to express \(r\) in terms of \(G\) and \(b\).

Mastering these algebraic techniques like addition, subtraction, multiplication, and division of terms, helps in efficiently solving algebraic equations. These skills are essential for both simple and complex mathematical problems across different fields of study.