To find the equation of the line that passes through the point \((-3, -1)\) and is perpendicular to the line given by the equation \(y = -\frac{1}{9}x - 6\), we first need to determine the slope of the given line.
The slope of the line \(y = -\frac{1}{9}x - 6\) is \(-\frac{1}{9}\).
Since the line we want to find is perpendicular to this line, its slope will be the negative reciprocal of \(-\frac{1}{9}\). The negative reciprocal is calculated as follows:
\[ \text{slope} = -\frac{1}{(-\frac{1}{9})} = 9 \]
Now, we have the slope of the line we want to find, which is \(9\), and we also have the point \((-3, -1)\) that the line passes through.
We can use the point-slope form of the line equation, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \(m\) is the slope and \((x_1, y_1)\) is the point through which the line passes. Plugging in the values, we have:
- \(m = 9\)
- \(x_1 = -3\)
- \(y_1 = -1\)
Substituting these values into the point-slope form:
\[ y - (-1) = 9(x - (-3)) \]
This simplifies to:
\[ y + 1 = 9(x + 3) \]
Now, distributing the \(9\):
\[ y + 1 = 9x + 27 \]
Next, we isolate \(y\) by subtracting \(1\) from both sides:
\[ y = 9x + 27 - 1 \] \[ y = 9x + 26 \]
Thus, the equation of the line that passes through \((-3, -1)\) and is perpendicular to the line \(y = -\frac{1}{9}x - 6\) is:
\[ \boxed{y = 9x + 26} \]