To find the equation of the line that passes through the points \((0, 5)\) and \((-3, 17)\), we first need to find the slope \(m\) of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we can assign \((x_1, y_1) = (0, 5)\) and \((x_2, y_2) = (-3, 17)\). Plugging in these values:
\[ m = \frac{17 - 5}{-3 - 0} = \frac{12}{-3} = -4 \]
Now that we have the slope \(m = -4\), we can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Using the point \((0, 5)\) and the slope \(m = -4\):
\[ y - 5 = -4(x - 0) \]
This simplifies to:
\[ y - 5 = -4x \]
Now, solving for \(y\):
\[ y = -4x + 5 \]
Thus, the equation of the line in slope-intercept form is:
\[ \boxed{y = -4x + 5} \]