To determine the relationship between the quantities in the given table, we can follow these steps:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
1 & 13 \\
\hline
2 & 26 \\
\hline
3 & 39 \\
\hline
4 & 52 \\
\hline
\end{tabular}
\][/tex]
First, let's examine the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] by calculating the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair of values:
1. For [tex]\( x = 1 \)[/tex], [tex]\( y = 13 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{13}{1} = 13
\][/tex]
2. For [tex]\( x = 2 \)[/tex], [tex]\( y = 26 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{26}{2} = 13
\][/tex]
3. For [tex]\( x = 3 \)[/tex], [tex]\( y = 39 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{39}{3} = 13
\][/tex]
4. For [tex]\( x = 4 \)[/tex], [tex]\( y = 52 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{52}{4} = 13
\][/tex]
In each case, the ratio [tex]\( \frac{y}{x} \)[/tex] is consistently 13. This indicates that for every value of [tex]\( x \)[/tex], [tex]\( y \)[/tex] is 13 times [tex]\( x \)[/tex].
Hence, the relationship between the quantities [tex]\( x \)[/tex] and [tex]\( y \)[/tex] shown in the table is:
[tex]\[
y = 13x
\][/tex]
So, the correct answer from the provided choices is:
The relationship between quantities is [tex]\( \times 13 \)[/tex].
