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.
