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Identify the rule for this table.

[tex]15x - [?] = y[/tex]

Enter the number that goes in the green box.
\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}

Sagot :

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]