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Simplifying fractions is an essential skill in mathematics. It involves reducing the fraction to its simplest form. The fraction is simplified by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD. This process ensures the fraction is in its lowest terms, making it easier to understand and work with. In our example, after performing arithmetic operations on the numerator, we ended up with \(\frac{-18}{-9}\). Since both the numerator and denominator share a common factor, which is \(-9\), dividing them results in \(2\). Always remember, simplifying fractions helps in making calculations manageable and results straightforward.

Understanding the roles of the numerator and the denominator is crucial in fractions. The numerator is the top number in a fraction and represents how many parts we have. The denominator is the bottom number and shows the total number of equal parts the whole is divided into. For example, in the fraction \(\frac{a}{b}\), \(a\) is the numerator and \(b\) is the denominator. In our problem, the numerator started as \-15 - 6 + 3\, and the denominator was \-9\. After simplifying the numerator to \-18\, we proceeded to place it over the denominator, forming the fraction \(\frac{-18}{-9}\). The numerator and denominator's proper understanding helps break down and solve fractions smoothly.

Basic arithmetic operations are fundamental in simplifying expressions. These operations include addition, subtraction, multiplication, and division. In our given problem, the focus was on addition and subtraction within the numerator. Here’s a brief step-by-step on how to handle such expressions:

• Identify and perform operations inside the numerator: \-15 - 6 + 3\. This first simplifies to \-21\ after performing \-15 - 6\, then to \-18\ by performing \-21 + 3\.
• After simplifying the numerator, divide by the denominator, i.e., \(\frac{-18}{-9}\).
• Division of two negative numbers results in a positive outcome, thus \(\frac{-18}{-9} = 2\).

The careful step-by-step performance of these arithmetic operations is key to arriving at the correct solution. Practice them regularly to grasp the concepts fully.