To write the linear equation given the slope \( m = \frac{5}{6} \) and the point \( P(-6, -2) \), we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point and \( m \) is the slope.
In this case, \( (x_1, y_1) = (-6, -2) \) and \( m = \frac{5}{6} \).
Plugging in these values, we have:
\[ y - (-2) = \frac{5}{6}(x - (-6)) \]
This simplifies to:
\[ y + 2 = \frac{5}{6}(x + 6) \]
Now, to rearrange into slope-intercept form \( y = mx + b \), we distribute \( \frac{5}{6} \):
\[ y + 2 = \frac{5}{6}x + \frac{5}{6} \cdot 6 \]
Calculating \( \frac{5}{6} \cdot 6 = 5 \):
\[ y + 2 = \frac{5}{6}x + 5 \]
Next, subtract 2 from both sides:
\[ y = \frac{5}{6}x + 5 - 2 \]
Which simplifies to:
\[ y = \frac{5}{6}x + 3 \]
So the equation of the line in slope-intercept form is:
\[ \boxed{y = \frac{5}{6}x + 3} \]