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Simplifying algebraic fractions often starts with recognizing and factoring out common factors in the numerator or denominator. It's like noticing that two people in a room are both wearing blue jeans; that's an attribute they have in common. In algebra, we look for numerical or algebraic terms that appear in each term of a polynomial.

Consider the numerator of the given fraction, which is made up of the terms \(3 y^{2}\) and \-27 y\. The number \(3\) and the variable \(y\) are the 'blue jeans' here - they're what's shared between them. When we factor \(3y\) out, we're essentially gathering the common elements and putting them outside of the parentheses, leading to the expression \(3y(y - 9)\). This simplifies the process, setting us up to 'cancel out' these common factors against the denominator.

Once we've factored out the common factors, it's time to simplify or 'cancel out' these elements within the faction. It's akin to eliminating unnecessary tasks from your to-do list, streamlining what needs to be done. In the exercise, after factoring out the common factor \(3y\), we end up with \(3y(y - 9)/-3y\).

The \(3y\) factor is present in both the numerator and the denominator, much like having one umbrella for each person in a pair – one can be taken away and they'll both still be just as dry! When we cancel \(3y\) from both parts of the fraction, we're left with \-1(y - 9)\, a much simpler expression that's way easier to handle.

To finalize the simplification of algebraic fractions, we may need to distribute a negative sign, which we can think of as attaching a minus sign to each term that we're disbursing into. After cancelling the common factors, we're left with the expression \-1(y - 9)\.

Distributing the \-1\ means that we ‘give’ the negative sign to each inside term, like handing out minus-shaped chocolates to every term at a party – everybody gets one. As a result, \(y\) becomes \-1 \times y = -y\, and \(9\) becomes \-1 \times -9 = 9\. The final simplified form is the expression \-y + 9\, which is the clean, condensed essence of the original, more complex fraction.