To find the slope of the line passing through a set of points, we can use the formula for the slope (m) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
In the given table:
[tex]\[
\begin{array}{rrrrr}
x & -7 & -6 & -5 & -4 \\
\hline
y & 21 & 17 & 13 & 9
\end{array}
\][/tex]
We can select any two points to calculate the slope. For simplicity, we will use the first point [tex]\((-7, 21)\)[/tex] and the last point [tex]\((-4, 9)\)[/tex].
First, we find the differences in the [tex]\(y\)[/tex]-values and [tex]\(x\)[/tex]-values between these two points:
[tex]\[ \Delta y = y_2 - y_1 = 9 - 21 = -12 \][/tex]
[tex]\[ \Delta x = x_2 - x_1 = -4 - (-7) = -4 + 7 = 3 \][/tex]
Now, using the slope formula:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{-12}{3} = -4 \][/tex]
Thus, the slope of the line that contains these points is:
[tex]\[ \boxed{-4} \][/tex]
