First, we need to find the slope of the line given by the equation 3x + y = 6.
To do this, we put the equation in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. When 3x + y = 6 is transformed to slope-intercept form, it becomes y = -3x + 6. From this, we can see that the slope of the given line is -3.
Next, we remember an important principle: two lines are parallel if and only if their slopes are equal. Since we're looking for a line that's parallel to the given line, our line also has a slope of -3.
We're also told that our line passes through the point (-1, -6). With this information and the slope, we are able to use the point-slope formula to write the equation of the line. The point-slope formula is y - y1 = m(x - x1), where m is the slope, and (x1, y1) is the known point on the line.
Substituting our values of m = -3, x1 = -1, and y1 = -6 into the point-slope formula, we get y - (-6) = -3(x - (-1)).
Solving for y will give us the equation of the line in slope-intercept form:
y = -3x - 3 - 6 (multiplied -3 by x and -1, then subtracted 6 from both sides)
Our final equation is y = -3x - 9.
So, the equation of the line that passes through (-1, -6) and is parallel to the line whose equation is 3x + y = 6 is y = -3x - 9.