To find the equation of the line that passes through the points (5, -6) and (15, 2), we first need to calculate the slope (m) of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1) = (5, -6)\) and \((x_2, y_2) = (15, 2)\). Plugging in the values, we have:
\[ m = \frac{2 - (-6)}{15 - 5} = \frac{2 + 6}{15 - 5} = \frac{8}{10} = \frac{4}{5} \]
Now that we have the slope, we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Using point \((5, -6)\):
\[ y - (-6) = \frac{4}{5}(x - 5) \] \[ y + 6 = \frac{4}{5}(x - 5) \] \[ y + 6 = \frac{4}{5}x - 4 \] \[ y = \frac{4}{5}x - 4 - 6 \] \[ y = \frac{4}{5}x - 10 \]
So the equation of the line is:
\[ y = \frac{4}{5}x - 10 \]
Hence, the correct choice from the options you provided is:
\[ \boxed{y = \frac{4}{5} x - 10} \]