To simplify the expression [tex]\[\frac{1}{y+6} \div \frac{8-y}{3 y-24},\][/tex] we will follow a sequence of algebraic steps.
1. Rewrite the Division as Multiplication by the Reciprocal:
Division of a fraction is equivalent to multiplying by its reciprocal. Thus, we can rewrite the expression as:
[tex]\[
\frac{1}{y+6} \times \frac{3 y-24}{8-y}
\][/tex]
2. Simplify the Reciprocal:
We can simplify the expression [tex]\(\frac{3 y-24}{8-y}\)[/tex]. First, notice that we can factor out common terms in the numerator and attempt to simplify:
[tex]\[
3 y - 24 = 3(y - 8)
\][/tex]
Substituting this back in:
[tex]\[
\frac{3(y-8)}{8-y}
\][/tex]
Here, observe that [tex]\(8 - y\)[/tex] is the negation of [tex]\(y - 8\)[/tex]. Thus,
[tex]\[
\frac{3(y - 8)}{8 - y} = \frac{3(y - 8)}{-(y - 8)} = -3
\][/tex]
3. Substitute the Simplified Reciprocal Back into the Product:
Now substitute [tex]\(-3\)[/tex] into the product:
[tex]\[
\frac{1}{y+6} \times (-3)
\][/tex]
This multiplication becomes:
[tex]\[
-\frac{3}{y+6}
\][/tex]
Therefore, the simplest form of the expression representing the quotient is:
[tex]\[
-\frac{3}{y+6}
\][/tex]
