To identify the rule of the given table, we need to determine the relationship between \( x \) and \( y \).
Given the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & 7 \\
\hline
2 & 22 \\
\hline
3 & 37 \\
\hline
4 & 52 \\
\hline
5 & 67 \\
\hline
\end{tabular}
\][/tex]
We start by calculating the differences between consecutive \( y \)-values:
[tex]\[
\begin{aligned}
y_2 - y_1 &= 22 - 7 = 15, \\
y_3 - y_2 &= 37 - 22 = 15, \\
y_4 - y_3 &= 52 - 37 = 15, \\
y_5 - y_4 &= 67 - 52 = 15.
\end{aligned}
\][/tex]
The differences are all constant and equal to 15, indicating a linear relationship between \( x \) and \( y \).
Next, we can express this relationship as a linear equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
The constant difference (slope \( m \)) is 15. To find the y-intercept (\( b \)), we use one of the points from the table. Let's use the point \( (1, 7) \):
[tex]\[
y = 15x + b.
\][/tex]
Substituting \( x = 1 \) and \( y = 7 \) into the equation, we get:
[tex]\[
7 = 15(1) + b \implies 7 = 15 + b \implies b = 7 - 15 \implies b = -8.
\][/tex]
Therefore, the equation that describes the relationship between \( x \) and \( y \) is:
[tex]\[
y = 15x - 8.
\][/tex]
So, the rule for the table can be written as:
[tex]\[
15x - 8 = y.
\][/tex]
In the equation:
[tex]\[
15x - [\text{?}]= y,
\][/tex]
the number that goes in the green box is:
[tex]\[
\boxed{8}.
\][/tex]
