To determine the relationship between \( y \) and \( x \) in the given table, we can observe the pattern in the values. It appears that \( y \) is a multiple of \( x \). Specifically, we can see that \( y \) is 6 times \( x \). Therefore, the equation relating \( y \) to \( x \) is \( y = 6x \).
We are given a table of values for \( x \) and \( y \):
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 6 \\
\hline
2 & 12 \\
\hline
3 & 18 \\
\hline
4 & 24 \\
\hline
\end{array}
\]
By examining the table, we notice that as \( x \) increases, \( y \) increases proportionally. This suggests a linear relationship between \( x \) and \( y \).
To find the exact relationship, we can observe the ratio of \( y \) to \( x \):
\[
\frac{y}{x} = \frac{6}{1} = 6, \quad \frac{12}{2} = 6, \quad \frac{18}{3} = 6, \quad \frac{24}{4} = 6
\]
This consistent ratio indicates that \( y \) is 6 times \( x \). Therefore, the relationship can be expressed as:
\[
y = 6x
\]
We can verify this relationship by substituting the values of \( x \) from the table into the equation \( y = 6x \):
\[
\begin{align_}
\text{For } x = 1, & \quad y = 6 \cdot 1 = 6 \\
\text{For } x = 2, & \quad y = 6 \cdot 2 = 12 \\
\text{For } x = 3, & \quad y = 6 \cdot 3 = 18 \\
\text{For } x = 4, & \quad y = 6 \cdot 4 = 24 \\
\end{align_}
\]
The calculated values of \( y \) match the values given in the table, confirming that the relationship \( y = 6x \) is correct.
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