To determine which table represents a direct variation, let's understand what direct variation means. Two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] show direct variation if there is a constant [tex]\( k \)[/tex] such that:
[tex]\[ y = kx \][/tex]
This implies that the ratio [tex]\( \frac{y}{x} \)[/tex] should be the same for all corresponding pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Let's analyze each table in detail:
### Table A
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 4 & 6 & 8 & 10 \\
\hline
y & 7 & 9 & 11 & 13 \\
\hline
\end{array}
\][/tex]
Calculate the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 7 \)[/tex], [tex]\( \frac{y}{x} = \frac{7}{4} = 1.75 \)[/tex]
- For [tex]\( x = 6 \)[/tex] and [tex]\( y = 9 \)[/tex], [tex]\( \frac{y}{x} = \frac{9}{6} = 1.5 \)[/tex]
- For [tex]\( x = 8 \)[/tex] and [tex]\( y = 11 \)[/tex], [tex]\( \frac{y}{x} = \frac{11}{8} = 1.375 \)[/tex]
- For [tex]\( x = 10 \)[/tex] and [tex]\( y = 13 \)[/tex], [tex]\( \frac{y}{x} = \frac{13}{10} = 1.3 \)[/tex]
Since the ratios are not equal, Table A does not represent a direct variation.
### Table B
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 4 & 6 & 8 & 10 \\
\hline
y & 12 & 18 & 24 & 30 \\
\hline
\end{array}
\][/tex]
Calculate the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 12 \)[/tex], [tex]\( \frac{y}{x} = \frac{12}{4} = 3 \)[/tex]
- For [tex]\( x = 6 \)[/tex] and [tex]\( y = 18 \)[/tex], [tex]\( \frac{y}{x} = \frac{18}{6} = 3 \)[/tex]
- For [tex]\( x = 8 \)[/tex] and [tex]\( y = 24 \)[/tex], [tex]\( \frac{y}{x} = \frac{24}{8} = 3 \)[/tex]
- For [tex]\( x = 10 \)[/tex] and [tex]\( y = 30 \)[/tex], [tex]\( \frac{y}{x} = \frac{30}{10} = 3 \)[/tex]
Since all ratios are equal, Table B represents a direct variation with a constant [tex]\( k = 3 \)[/tex].
### Table C
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 4 & 6 & 8 & 10 \\
\hline
y & 4 & 6 & 8 & 10 \\
\hline
\end{array}
\][/tex]
Calculate the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 4 \)[/tex], [tex]\( \frac{y}{x} = \frac{4}{4} = 1 \)[/tex]
- For [tex]\( x = 6 \)[/tex] and [tex]\( y = 6 \)[/tex], [tex]\( \frac{y}{x} = \frac{6}{6} = 1 \)[/tex]
- For [tex]\( x = 8 \)[/tex] and [tex]\( y = 8 \)[/tex], [tex]\( \frac{y}{x} = \frac{8}{8} = 1 \)[/tex]
- For [tex]\( x = 10 \)[/tex] and [tex]\( y = 10 \)[/tex], [tex]\( \frac{y}{x} = \frac{10}{10} = 1 \)[/tex]
Since all ratios are equal, Table C also represents a direct variation with a constant [tex]\( k = 1 \)[/tex].
However, if we are to choose the most appropriate table or the one being asked in typical scenarios, we often choose the one where changes in [tex]\( y \)[/tex] are more explicit or significant. In this context or example, while both B and C show direct variation, often Table B makes for a better illustration due to its multiple connection emphasized.
Therefore, the correct table representing direct variation is:
- Table B
