To find the slope of the line that goes through the points [tex]\((-1, 4)\)[/tex] and [tex]\( (14, -2) \)[/tex], we use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, [tex]\((x_1, y_1) = (-1, 4)\)[/tex] and [tex]\((x_2, y_2) = (14, -2)\)[/tex].
First, we calculate the change in [tex]\(y\)[/tex] (denoted as [tex]\(\Delta y\)[/tex]):
[tex]\[ \Delta y = y_2 - y_1 = -2 - 4 = -6 \][/tex]
Next, we calculate the change in [tex]\(x\)[/tex] (denoted as [tex]\(\Delta x\)[/tex]):
[tex]\[ \Delta x = x_2 - x_1 = 14 - (-1) = 14 + 1 = 15 \][/tex]
Now, we substitute [tex]\(\Delta y\)[/tex] and [tex]\(\Delta x\)[/tex] into the slope formula:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{-6}{15} \][/tex]
We then simplify the fraction to its lowest terms:
[tex]\[ m = -\frac{6}{15} = -\frac{2}{5} \][/tex]
Therefore, the slope of the line that goes through the points [tex]\((-1, 4)\)[/tex] and [tex]\( (14, -2) \)[/tex] is:
[tex]\[ \boxed{-\frac{2}{5}} \][/tex]
However, based on the given options, the closest and most simplified form corresponds to:
D. [tex]\(-\frac{6}{15}\)[/tex]
