To determine whether [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we need to see if there is a constant [tex]\( k \)[/tex] such that [tex]\( y = kx \)[/tex] for all given pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Given the pairs:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -2 \\
\hline
3 & -6 \\
\hline
5 & -10 \\
\hline
\end{array}
\][/tex]
1. First, we check the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] by calculating [tex]\( k = \frac{y}{x} \)[/tex] for the first pair:
[tex]\[
k = \frac{-2}{1} = -2
\][/tex]
2. Next, we verify if the same [tex]\( k \)[/tex] holds for the second pair:
[tex]\[
k = \frac{-6}{3} = -2
\][/tex]
3. We again check [tex]\( k \)[/tex] for the third pair:
[tex]\[
k = \frac{-10}{5} = -2
\][/tex]
Since [tex]\( k = -2 \)[/tex] is consistent across all the given pairs, [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], and the constant of variation is [tex]\( -2 \)[/tex].
The equation of direct variation is:
[tex]\[
y = -2x
\][/tex]
Thus, the correct answer is:
[tex]\[
\text{yes; } k = -2 \text{ and } y = -2x
\][/tex]
