To determine the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table, we need to analyze how [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] changes.
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 3 & 4 & 6 & 8 & 12 \\
\hline
y & 16 & 12 & 8 & 6 & 4 \\
\hline
\end{array}
\][/tex]
One way to understand this is by finding the product of each pair [tex]\( (x, y) \)[/tex]. If [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex], then the product [tex]\( x \cdot y \)[/tex] should be constant.
Let's calculate the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for each pair:
1. For [tex]\( x = 3 \)[/tex] and [tex]\( y = 16 \)[/tex]:
[tex]\[
3 \cdot 16 = 48
\][/tex]
2. For [tex]\( x = 4 \)[/tex] and [tex]\( y = 12 \)[/tex]:
[tex]\[
4 \cdot 12 = 48
\][/tex]
3. For [tex]\( x = 6 \)[/tex] and [tex]\( y = 8 \)[/tex]:
[tex]\[
6 \cdot 8 = 48
\][/tex]
4. For [tex]\( x = 8 \)[/tex] and [tex]\( y = 6 \)[/tex]:
[tex]\[
8 \cdot 6 = 48
\][/tex]
5. For [tex]\( x = 12 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[
12 \cdot 4 = 48
\][/tex]
Since the product [tex]\( x \cdot y \)[/tex] is constant (48) for all pairs, we can conclude that [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex].
Therefore, the correct statement of variation is:
C. [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex]
