Answer :
To determine which table of ordered pairs represents a proportional relationship, we need to check if each pair of [tex]\((x, y)\)[/tex] values in each table has a consistent ratio [tex]\( \frac{y}{x} \)[/tex]. This ratio should be the same for all pairs in the table.
Let's investigate each table:
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 4 & 8 \\ \hline 7 & 11 \\ \hline 10 & 14 \\ \hline \end{array} \][/tex]
For the pairs [tex]\((4, 8), (7, 11),\)[/tex] and [tex]\((10, 14):\)[/tex]
- Ratio for [tex]\((4, 8)\)[/tex]: [tex]\( \frac{8}{4} = 2 \)[/tex]
- Ratio for [tex]\((7, 11)\)[/tex]: [tex]\( \frac{11}{7} \approx 1.571 \)[/tex]
- Ratio for [tex]\((10, 14)\)[/tex]: [tex]\( \frac{14}{10} = 1.4 \)[/tex]
The ratios are not consistent.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 5 & 25 \\ \hline 7 & 49 \\ \hline 9 & 81 \\ \hline \end{array} \][/tex]
For the pairs [tex]\((5, 25), (7, 49),\)[/tex] and [tex]\((9, 81):\)[/tex]
- Ratio for [tex]\((5, 25)\)[/tex]: [tex]\( \frac{25}{5} = 5 \)[/tex]
- Ratio for [tex]\((7, 49)\)[/tex]: [tex]\( \frac{49}{7} = 7 \)[/tex]
- Ratio for [tex]\((9, 81)\)[/tex]: [tex]\( \frac{81}{9} = 9 \)[/tex]
The ratios are not consistent.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 6 & 3 \\ \hline 10 & 5 \\ \hline 14 & 7 \\ \hline \end{array} \][/tex]
For the pairs [tex]\((6, 3), (10, 5),\)[/tex] and [tex]\((14, 7):\)[/tex]
- Ratio for [tex]\((6, 3)\)[/tex]: [tex]\( \frac{3}{6} = 0.5 \)[/tex]
- Ratio for [tex]\((10, 5)\)[/tex]: [tex]\( \frac{5}{10} = 0.5 \)[/tex]
- Ratio for [tex]\((14, 7)\)[/tex]: [tex]\( \frac{7}{14} = 0.5 \)[/tex]
The ratios are consistent.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 3 & 6 \\ \hline 8 & 11 \\ \hline 13 & 18 \\ \hline \end{array} \][/tex]
For the pairs [tex]\((3, 6), (8, 11),\)[/tex] and [tex]\((13, 18):\)[/tex]
- Ratio for [tex]\((3, 6)\)[/tex]: [tex]\( \frac{6}{3} = 2 \)[/tex]
- Ratio for [tex]\((8, 11)\)[/tex]: [tex]\( \frac{11}{8} = 1.375 \)[/tex]
- Ratio for [tex]\((13, 18)\)[/tex]: [tex]\( \frac{18}{13} \approx 1.385 \)[/tex]
The ratios are not consistent.
### Conclusion
The table that represents a proportional relationship is:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 6 & 3 \\ \hline 10 & 5 \\ \hline 14 & 7 \\ \hline \end{array} \][/tex]
Let's investigate each table:
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 4 & 8 \\ \hline 7 & 11 \\ \hline 10 & 14 \\ \hline \end{array} \][/tex]
For the pairs [tex]\((4, 8), (7, 11),\)[/tex] and [tex]\((10, 14):\)[/tex]
- Ratio for [tex]\((4, 8)\)[/tex]: [tex]\( \frac{8}{4} = 2 \)[/tex]
- Ratio for [tex]\((7, 11)\)[/tex]: [tex]\( \frac{11}{7} \approx 1.571 \)[/tex]
- Ratio for [tex]\((10, 14)\)[/tex]: [tex]\( \frac{14}{10} = 1.4 \)[/tex]
The ratios are not consistent.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 5 & 25 \\ \hline 7 & 49 \\ \hline 9 & 81 \\ \hline \end{array} \][/tex]
For the pairs [tex]\((5, 25), (7, 49),\)[/tex] and [tex]\((9, 81):\)[/tex]
- Ratio for [tex]\((5, 25)\)[/tex]: [tex]\( \frac{25}{5} = 5 \)[/tex]
- Ratio for [tex]\((7, 49)\)[/tex]: [tex]\( \frac{49}{7} = 7 \)[/tex]
- Ratio for [tex]\((9, 81)\)[/tex]: [tex]\( \frac{81}{9} = 9 \)[/tex]
The ratios are not consistent.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 6 & 3 \\ \hline 10 & 5 \\ \hline 14 & 7 \\ \hline \end{array} \][/tex]
For the pairs [tex]\((6, 3), (10, 5),\)[/tex] and [tex]\((14, 7):\)[/tex]
- Ratio for [tex]\((6, 3)\)[/tex]: [tex]\( \frac{3}{6} = 0.5 \)[/tex]
- Ratio for [tex]\((10, 5)\)[/tex]: [tex]\( \frac{5}{10} = 0.5 \)[/tex]
- Ratio for [tex]\((14, 7)\)[/tex]: [tex]\( \frac{7}{14} = 0.5 \)[/tex]
The ratios are consistent.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 3 & 6 \\ \hline 8 & 11 \\ \hline 13 & 18 \\ \hline \end{array} \][/tex]
For the pairs [tex]\((3, 6), (8, 11),\)[/tex] and [tex]\((13, 18):\)[/tex]
- Ratio for [tex]\((3, 6)\)[/tex]: [tex]\( \frac{6}{3} = 2 \)[/tex]
- Ratio for [tex]\((8, 11)\)[/tex]: [tex]\( \frac{11}{8} = 1.375 \)[/tex]
- Ratio for [tex]\((13, 18)\)[/tex]: [tex]\( \frac{18}{13} \approx 1.385 \)[/tex]
The ratios are not consistent.
### Conclusion
The table that represents a proportional relationship is:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 6 & 3 \\ \hline 10 & 5 \\ \hline 14 & 7 \\ \hline \end{array} \][/tex]
