To determine whether the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table is proportional, we need to check if the ratio [tex]\( \frac{y}{x} \)[/tex] is constant for all data points.
Let's calculate the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values provided in the table:
1. For the first pair [tex]\((x = 1, y = 2)\)[/tex]:
[tex]\[
\frac{y}{x} = \frac{2}{1} = 2.0
\][/tex]
2. For the second pair [tex]\((x = 3, y = 6)\)[/tex]:
[tex]\[
\frac{y}{x} = \frac{6}{3} = 2.0
\][/tex]
3. For the third pair [tex]\((x = 7, y = 14)\)[/tex]:
[tex]\[
\frac{y}{x} = \frac{14}{7} = 2.0
\][/tex]
Since the ratio [tex]\( \frac{y}{x} \)[/tex] is the same (2.0) for all the data points [tex]\((1, 2)\)[/tex], [tex]\((3, 6)\)[/tex], and [tex]\((7, 14)\)[/tex], we can conclude that the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is proportional.
Thus, the given table shows a proportional relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
