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 each table.
Let's analyze each table:
Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline 0 & 10 \\
\hline 5 & 20 \\
\hline 10 & 30 \\
\hline
\end{array}
\][/tex]
In this table, the first pair is [tex]\((0, 10)\)[/tex]. Since the value of [tex]\(x\)[/tex] is zero, we cannot calculate [tex]\(\frac{y}{x}\)[/tex] for the first pair. Moving to the second pair:
[tex]\[
\frac{y}{x} = \frac{20}{5} = 4
\][/tex]
For the third pair:
[tex]\[
\frac{y}{x} = \frac{30}{10} = 3
\][/tex]
The ratio [tex]\(\frac{y}{x}\)[/tex] is not constant (it changes from 4 to 3), so this table 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]
For the first pair:
[tex]\[
\frac{y}{x} = \frac{10}{2} = 5
\][/tex]
For the second pair:
[tex]\[
\frac{y}{x} = \frac{20}{4} = 5
\][/tex]
For the third pair:
[tex]\[
\frac{y}{x} = \frac{30}{6} = 5
\][/tex]
The ratio [tex]\(\frac{y}{x}\)[/tex] is constant (5 for all pairs), so this table 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]
For the first pair:
[tex]\[
\frac{y}{x} = \frac{2}{1} = 2
\][/tex]
For the second pair:
[tex]\[
\frac{y}{x} = \frac{3}{2} = 1.5
\][/tex]
For the third pair:
[tex]\[
\frac{y}{x} = \frac{4}{3} \approx 1.33
\][/tex]
The ratio [tex]\(\frac{y}{x}\)[/tex] is not constant (it changes from 2 to 1.5 to 1.33), so this table 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]
For the first pair:
[tex]\[
\frac{y}{x} = \frac{4}{1} = 4
\][/tex]
For the second pair:
[tex]\[
\frac{y}{x} = \frac{10}{3} \approx 3.33
\][/tex]
For the third pair:
[tex]\[
\frac{y}{x} = \frac{13}{4} \approx 3.25
\][/tex]
The ratio [tex]\(\frac{y}{x}\)[/tex] is not constant (it changes from 4 to 3.33 to 3.25), so this table does not represent a proportional relationship.
Conclusion:
Among the given tables, only Table 2 represents a proportional relationship as the ratio [tex]\(\frac{y}{x}\)[/tex] is constant at 5.
