To determine which table of ordered pairs represents a proportional relationship, we must analyze if there is a consistent ratio [tex]\( \frac{y}{x} \)[/tex] for each table.
A proportional relationship indicates that the ratio of [tex]\( y \)[/tex] to [tex]\( x \)[/tex] remains constant across all pairs in the table.
Let's analyze each table step-by-step:
First Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
4 & 8 \\
\hline
7 & 11 \\
\hline
10 & 14 \\
\hline
\end{array}
\][/tex]
Calculate the ratio [tex]\( \frac{y}{x} \)[/tex]:
[tex]\[
\frac{8}{4} = 2, \quad \frac{11}{7} \approx 1.57, \quad \frac{14}{10} = 1.4
\][/tex]
Since the ratios are not constant, this table does not represent a proportional relationship.
Second Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
5 & 25 \\
\hline
7 & 49 \\
\hline
9 & 81 \\
\hline
\end{array}
\][/tex]
Calculate the ratio [tex]\( \frac{y}{x} \)[/tex]:
[tex]\[
\frac{25}{5} = 5, \quad \frac{49}{7} = 7, \quad \frac{81}{9} = 9
\][/tex]
Since the ratios are not constant, this table does not represent a proportional relationship.
Third Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
6 & 3 \\
\hline
10 & 5 \\
\hline
14 & 7 \\
\hline
\end{array}
\][/tex]
Calculate the ratio [tex]\( \frac{y}{x} \)[/tex]:
[tex]\[
\frac{3}{6} = 0.5, \quad \frac{5}{10} = 0.5, \quad \frac{7}{14} = 0.5
\][/tex]
Since the ratios are consistent and equal to [tex]\( 0.5 \)[/tex], this table represents a proportional relationship.
Fourth Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & 6 \\
\hline
8 & 11 \\
\hline
13 & 18 \\
\hline
\end{array}
\][/tex]
Calculate the ratio [tex]\( \frac{y}{x} \)[/tex]:
[tex]\[
\frac{6}{3} = 2, \quad \frac{11}{8} \approx 1.375, \quad \frac{18}{13} \approx 1.385
\][/tex]
Since the ratios are not constant, this table does not represent a proportional relationship.
Conclusion:
The third table represents a proportional relationship. Thus, the table with ordered pairs [tex]\((6, 3)\)[/tex], [tex]\((10, 5)\)[/tex], and [tex]\((14, 7)\)[/tex] is the one that shows a consistent ratio and hence demonstrates a proportional relationship.
