To determine whether the given relationship forms a direct variation, we need to verify if the ratio [tex]\( \frac{y}{x} \)[/tex] remains constant for all pairs [tex]\((x, y)\)[/tex]. A direct variation exists if and only if [tex]\( \frac{y}{x} = k \)[/tex] for some constant [tex]\( k \)[/tex].
Given the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 0.50 \\
\hline
2 & 1.00 \\
\hline
3 & 1.50 \\
\hline
5 & 2.50 \\
\hline
\end{array}
\][/tex]
Let’s compute the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
1. For [tex]\( x = 1 \)[/tex] and [tex]\( y = 0.50 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{0.50}{1} = 0.50
\][/tex]
2. For [tex]\( x = 2 \)[/tex] and [tex]\( y = 1.00 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{1.00}{2} = 0.50
\][/tex]
3. For [tex]\( x = 3 \)[/tex] and [tex]\( y = 1.50 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{1.50}{3} = 0.50
\][/tex]
4. For [tex]\( x = 5 \)[/tex] and [tex]\( y = 2.50 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{2.50}{5} = 0.50
\][/tex]
We observe that the ratio [tex]\( \frac{y}{x} \)[/tex] is equal to [tex]\( 0.50 \)[/tex] in each case. Because the ratio remains the same for all pairs, this indicates a direct variation relationship.
Therefore, the given relationship forms a direct variation. The constant of variation [tex]\( k \)[/tex] is [tex]\( 0.50 \)[/tex].
