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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 each table. This means that if you divide [tex]\( y \)[/tex] by [tex]\( x \)[/tex] for each pair in the table, the result should be the same for every pair (excluding pairs where [tex]\( x \)[/tex] is zero because division by zero is undefined).
Let's check each table one by one:
1. Table 1:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 0 & 10 \\ \hline 5 & 20 \\ \hline 10 & 30 \\ \hline \end{tabular} \][/tex]
- Pair (5, 20): [tex]\( \frac{20}{5} = 4 \)[/tex]
- Pair (10, 30): [tex]\( \frac{30}{10} = 3 \)[/tex]
The ratios 4 and 3 are not the same, so this table does not represent a proportional relationship.
2. Table 2:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 2 & 10 \\ \hline 4 & 20 \\ \hline 6 & 30 \\ \hline \end{tabular} \][/tex]
- Pair (2, 10): [tex]\( \frac{10}{2} = 5 \)[/tex]
- Pair (4, 20): [tex]\( \frac{20}{4} = 5 \)[/tex]
- Pair (6, 30): [tex]\( \frac{30}{6} = 5 \)[/tex]
All ratios are 5, so this table does represent a proportional relationship.
3. Table 3:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 1 & 2 \\ \hline 2 & 3 \\ \hline 3 & 4 \\ \hline \end{tabular} \][/tex]
- Pair (1, 2): [tex]\( \frac{2}{1} = 2 \)[/tex]
- Pair (2, 3): [tex]\( \frac{3}{2} = 1.5 \)[/tex]
- Pair (3, 4): [tex]\( \frac{4}{3} \approx 1.33 \)[/tex]
The ratios 2, 1.5, and approximately 1.33 are not the same, so this table does not represent a proportional relationship.
4. Table 4:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 1 & 4 \\ \hline 3 & 10 \\ \hline 4 & 13 \\ \hline \end{tabular} \][/tex]
- Pair (1, 4): [tex]\( \frac{4}{1} = 4 \)[/tex]
- Pair (3, 10): [tex]\( \frac{10}{3} \approx 3.33 \)[/tex]
- Pair (4, 13): [tex]\( \frac{13}{4} = 3.25 \)[/tex]
The ratios 4, approximately 3.33, and 3.25 are not the same, so this table does not represent a proportional relationship.
Thus, the table that represents a proportional relationship is Table 2.
Let's check each table one by one:
1. Table 1:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 0 & 10 \\ \hline 5 & 20 \\ \hline 10 & 30 \\ \hline \end{tabular} \][/tex]
- Pair (5, 20): [tex]\( \frac{20}{5} = 4 \)[/tex]
- Pair (10, 30): [tex]\( \frac{30}{10} = 3 \)[/tex]
The ratios 4 and 3 are not the same, so this table does not represent a proportional relationship.
2. Table 2:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 2 & 10 \\ \hline 4 & 20 \\ \hline 6 & 30 \\ \hline \end{tabular} \][/tex]
- Pair (2, 10): [tex]\( \frac{10}{2} = 5 \)[/tex]
- Pair (4, 20): [tex]\( \frac{20}{4} = 5 \)[/tex]
- Pair (6, 30): [tex]\( \frac{30}{6} = 5 \)[/tex]
All ratios are 5, so this table does represent a proportional relationship.
3. Table 3:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 1 & 2 \\ \hline 2 & 3 \\ \hline 3 & 4 \\ \hline \end{tabular} \][/tex]
- Pair (1, 2): [tex]\( \frac{2}{1} = 2 \)[/tex]
- Pair (2, 3): [tex]\( \frac{3}{2} = 1.5 \)[/tex]
- Pair (3, 4): [tex]\( \frac{4}{3} \approx 1.33 \)[/tex]
The ratios 2, 1.5, and approximately 1.33 are not the same, so this table does not represent a proportional relationship.
4. Table 4:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 1 & 4 \\ \hline 3 & 10 \\ \hline 4 & 13 \\ \hline \end{tabular} \][/tex]
- Pair (1, 4): [tex]\( \frac{4}{1} = 4 \)[/tex]
- Pair (3, 10): [tex]\( \frac{10}{3} \approx 3.33 \)[/tex]
- Pair (4, 13): [tex]\( \frac{13}{4} = 3.25 \)[/tex]
The ratios 4, approximately 3.33, and 3.25 are not the same, so this table does not represent a proportional relationship.
Thus, the table that represents a proportional relationship is Table 2.
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