Question

A line passes through the points in this table.
\begin{tabular}{|c|c|}
\hline\( x \) & \( y \) \\
\hline 7 & 31 \\
\hline 8 & 23 \\
\hline 9 & 15 \\
\hline 10 & 7 \\
\hline
\end{tabular}
What is the slope of the line?
Write your answer as an integer or simplified fraction.

Answer 1

To find the slope of the line that passes through the given points, we can use the formula for the slope \( m \):

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

We will use two points from the table. Let’s take the points \( (7, 31) \) and \( (8, 23) \).

Here, \( (x_1, y_1) = (7, 31) \) and \( (x_2, y_2) = (8, 23) \). Now we can substitute these values into the slope formula:

\[
m = \frac{23 - 31}{8 - 7} = \frac{-8}{1} = -8
\]

Thus, the slope of the line is

\[
\boxed{-8}
\]

We can verify this by calculating the slopes using another pair of points, such as \( (8, 23) \) and \( (9, 15) \):

Take \( (x_1, y_1) = (8, 23) \) and \( (x_2, y_2) = (9, 15) \):

\[
m = \frac{15 - 23}{9 - 8} = \frac{-8}{1} = -8
\]

Lastly, we'll check \( (9, 15) \) and \( (10, 7) \):

Take \( (x_1, y_1) = (9, 15) \) and \( (x_2, y_2) = (10, 7) \):

\[
m = \frac{7 - 15}{10 - 9} = \frac{-8}{1} = -8
\]

Thus, in all pairs of points, we find that the slope remains consistent. Therefore, the final answer for the slope of the line is

\[
\boxed{-8}
\]