Answer :
To determine which table of ordered pairs represents a proportional relationship, we need to check if the ratio [tex]\(\frac{y}{x}\)[/tex] is constant for all pairs in the table. Here's the detailed step-by-step solution for each table given:
1. First Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 3 \\ \hline -4 & 2 \\ \hline -5 & 1 \\ \hline \end{array} \][/tex]
Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[ \frac{3}{-3} = -1, \quad \frac{2}{-4} = -0.5, \quad \frac{1}{-5} = -0.2 \][/tex]
The ratios are not the same, so this table does not represent a proportional relationship.
2. Second Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -1 & 1 \\ \hline -3 & 3 \\ \hline -5 & 5 \\ \hline \end{array} \][/tex]
Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[ \frac{1}{-1} = -1, \quad \frac{3}{-3} = -1, \quad \frac{5}{-5} = -1 \][/tex]
The ratios are the same for all pairs, so this table represents a proportional relationship.
3. Third Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & -5 \\ \hline -4 & -7 \\ \hline -6 & -9 \\ \hline \end{array} \][/tex]
Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[ \frac{-5}{-2} = 2.5, \quad \frac{-7}{-4} = 1.75, \quad \frac{-9}{-6} = 1.5 \][/tex]
The ratios are not the same, so this table does not represent a proportional relationship.
4. Fourth Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & 0 \\ \hline -3 & -1 \\ \hline -4 & -2 \\ \hline \end{array} \][/tex]
Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[ \frac{0}{-2} = 0, \quad \frac{-1}{-3} = \frac{1}{3}, \quad \frac{-2}{-4} = \frac{1}{2} \][/tex]
The ratios are not the same, so this table does not represent a proportional relationship.
Therefore, only the second table of ordered pairs represents a proportional relationship.
1. First Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 3 \\ \hline -4 & 2 \\ \hline -5 & 1 \\ \hline \end{array} \][/tex]
Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[ \frac{3}{-3} = -1, \quad \frac{2}{-4} = -0.5, \quad \frac{1}{-5} = -0.2 \][/tex]
The ratios are not the same, so this table does not represent a proportional relationship.
2. Second Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -1 & 1 \\ \hline -3 & 3 \\ \hline -5 & 5 \\ \hline \end{array} \][/tex]
Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[ \frac{1}{-1} = -1, \quad \frac{3}{-3} = -1, \quad \frac{5}{-5} = -1 \][/tex]
The ratios are the same for all pairs, so this table represents a proportional relationship.
3. Third Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & -5 \\ \hline -4 & -7 \\ \hline -6 & -9 \\ \hline \end{array} \][/tex]
Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[ \frac{-5}{-2} = 2.5, \quad \frac{-7}{-4} = 1.75, \quad \frac{-9}{-6} = 1.5 \][/tex]
The ratios are not the same, so this table does not represent a proportional relationship.
4. Fourth Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & 0 \\ \hline -3 & -1 \\ \hline -4 & -2 \\ \hline \end{array} \][/tex]
Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each pair:
[tex]\[ \frac{0}{-2} = 0, \quad \frac{-1}{-3} = \frac{1}{3}, \quad \frac{-2}{-4} = \frac{1}{2} \][/tex]
The ratios are not the same, so this table does not represent a proportional relationship.
Therefore, only the second table of ordered pairs represents a proportional relationship.