Answer :
To determine which table of ordered pairs represents a proportional relationship, we must verify whether the ratios [tex]\(\frac{y}{x}\)[/tex] for each pair [tex]\( (x, y) \)[/tex] are consistent across all pairs within the same table.
Let’s analyze each table to check for a proportional relationship:
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 10 \\ \hline 5 & 20 \\ \hline 10 & 30 \\ \hline \end{array} \][/tex]
We need to check the ratios [tex]\(\frac{y}{x}\)[/tex]:
1. [tex]\(\frac{20}{5} = 4\)[/tex]
2. [tex]\(\frac{30}{10} = 3\)[/tex]
The ratios [tex]\( 4 \)[/tex] and [tex]\( 3 \)[/tex] are not equal. Therefore, Table 1 does not represent a proportional relationship.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 2 & 10 \\ \hline 4 & 20 \\ \hline 6 & 30 \\ \hline \end{array} \][/tex]
We need to check the ratios [tex]\(\frac{y}{x}\)[/tex]:
1. [tex]\(\frac{10}{2} = 5\)[/tex]
2. [tex]\(\frac{20}{4} = 5\)[/tex]
3. [tex]\(\frac{30}{6} = 5\)[/tex]
The ratios are consistent and all equal to [tex]\( 5 \)[/tex]. Therefore, Table 2 represents a proportional relationship.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 2 \\ \hline 2 & 3 \\ \hline 3 & 4 \\ \hline \end{array} \][/tex]
We need to check the ratios [tex]\(\frac{y}{x}\)[/tex]:
1. [tex]\(\frac{2}{1} = 2\)[/tex]
2. [tex]\(\frac{3}{2} = 1.5\)[/tex]
3. [tex]\(\frac{4}{3} = \frac{4}{3}\)[/tex] (approximately 1.33)
The ratios [tex]\( 2 \)[/tex], [tex]\( 1.5 \)[/tex], and [tex]\( \frac{4}{3} \)[/tex] are different. Therefore, Table 3 does not represent a proportional relationship.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 4 \\ \hline 3 & 10 \\ \hline 4 & 13 \\ \hline \end{array} \][/tex]
We need to check the ratios [tex]\(\frac{y}{x}\)[/tex]:
1. [tex]\(\frac{4}{1} = 4\)[/tex]
2. [tex]\(\frac{10}{3} \approx 3.33\)[/tex]
3. [tex]\(\frac{13}{4} \approx 3.25\)[/tex]
The ratios [tex]\( 4 \)[/tex], [tex]\( 3.33 \)[/tex], and [tex]\( 3.25 \)[/tex] are different. Therefore, Table 4 does not represent a proportional relationship.
By our analysis, the table of ordered pairs that represents a proportional relationship is:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 2 & 10 \\ \hline 4 & 20 \\ \hline 6 & 30 \\ \hline \end{array} \][/tex]
which is Table 2.
Let’s analyze each table to check for a proportional relationship:
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 10 \\ \hline 5 & 20 \\ \hline 10 & 30 \\ \hline \end{array} \][/tex]
We need to check the ratios [tex]\(\frac{y}{x}\)[/tex]:
1. [tex]\(\frac{20}{5} = 4\)[/tex]
2. [tex]\(\frac{30}{10} = 3\)[/tex]
The ratios [tex]\( 4 \)[/tex] and [tex]\( 3 \)[/tex] are not equal. Therefore, Table 1 does not represent a proportional relationship.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 2 & 10 \\ \hline 4 & 20 \\ \hline 6 & 30 \\ \hline \end{array} \][/tex]
We need to check the ratios [tex]\(\frac{y}{x}\)[/tex]:
1. [tex]\(\frac{10}{2} = 5\)[/tex]
2. [tex]\(\frac{20}{4} = 5\)[/tex]
3. [tex]\(\frac{30}{6} = 5\)[/tex]
The ratios are consistent and all equal to [tex]\( 5 \)[/tex]. Therefore, Table 2 represents a proportional relationship.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 2 \\ \hline 2 & 3 \\ \hline 3 & 4 \\ \hline \end{array} \][/tex]
We need to check the ratios [tex]\(\frac{y}{x}\)[/tex]:
1. [tex]\(\frac{2}{1} = 2\)[/tex]
2. [tex]\(\frac{3}{2} = 1.5\)[/tex]
3. [tex]\(\frac{4}{3} = \frac{4}{3}\)[/tex] (approximately 1.33)
The ratios [tex]\( 2 \)[/tex], [tex]\( 1.5 \)[/tex], and [tex]\( \frac{4}{3} \)[/tex] are different. Therefore, Table 3 does not represent a proportional relationship.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 4 \\ \hline 3 & 10 \\ \hline 4 & 13 \\ \hline \end{array} \][/tex]
We need to check the ratios [tex]\(\frac{y}{x}\)[/tex]:
1. [tex]\(\frac{4}{1} = 4\)[/tex]
2. [tex]\(\frac{10}{3} \approx 3.33\)[/tex]
3. [tex]\(\frac{13}{4} \approx 3.25\)[/tex]
The ratios [tex]\( 4 \)[/tex], [tex]\( 3.33 \)[/tex], and [tex]\( 3.25 \)[/tex] are different. Therefore, Table 4 does not represent a proportional relationship.
By our analysis, the table of ordered pairs that represents a proportional relationship is:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 2 & 10 \\ \hline 4 & 20 \\ \hline 6 & 30 \\ \hline \end{array} \][/tex]
which is Table 2.
