To determine which table of ordered pairs represents a proportional relationship, let's review the concept of proportionality. A relationship between two quantities [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is proportional if there is a constant ratio (k) such that [tex]\( \frac{y}{x} = k \)[/tex] for all pairs [tex]\((x, y)\)[/tex]. In mathematical terms, we'll check if [tex]\( \frac{y}{x} \)[/tex] is the same for all values in each table.
Let's go through each table one by one:
### Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
4 & 8 \\
\hline
7 & 11 \\
\hline
10 & 14 \\
\hline
\end{array}
\][/tex]
Calculate the ratios:
[tex]\[
\frac{8}{4} = 2, \quad \frac{11}{7} \approx 1.57, \quad \frac{14}{10} = 1.4
\][/tex]
The ratios are not the same, so Table 1 does not represent a proportional relationship.
### Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
5 & 25 \\
\hline
7 & 49 \\
\hline
9 & 81 \\
\hline
\end{array}
\][/tex]
Calculate the ratios:
[tex]\[
\frac{25}{5} = 5, \quad \frac{49}{7} = 7, \quad \frac{81}{9} = 9
\][/tex]
The ratios are not the same, so Table 2 does not represent a proportional relationship either.
### Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
6 & 3 \\
\hline
10 & 5 \\
\hline
14 & 7 \\
\hline
\end{array}
\][/tex]
Calculate the ratios:
[tex]\[
\frac{3}{6} = 0.5, \quad \frac{5}{10} = 0.5, \quad \frac{7}{14} = 0.5
\][/tex]
The ratios are the same, so Table 3 does represent a proportional relationship.
### Table 4:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & 6 \\
\hline
8 & 11 \\
\hline
13 & 18 \\
\hline
\end{array}
\][/tex]
Calculate the ratios:
[tex]\[
\frac{6}{3} = 2, \quad \frac{11}{8} \approx 1.375, \quad \frac{18}{13} \approx 1.385
\][/tex]
The ratios are not the same, so Table 4 does not represent a proportional relationship either.
After checking all the tables, we find that Table 3 represents a proportional relationship.
