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.
