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Sagot :
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.
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.
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