Final answer:
The equation of the line that passes through (-9, 12) and is perpendicular to the line y = 3x + 6 is y = -\frac{1}{3}x + 9.
Step-by-step explanation:
The question asks to find the equation of a line that is perpendicular to the line given by y = \frac{3}{1}x + 6 and passes through the point (-9, 12). The slope of the given line is 3, which means that the slope of a line perpendicular to it will be the negative reciprocal, which is -\frac{1}{3}. Now, using the point-slope form y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point through which the line passes, we can substitute m = -\frac{1}{3}, x1 = -9, and y1 = 12 to get the equation of the perpendicular line.
Therefore, the equation of the perpendicular line is:
y - 12 = -\frac{1}{3}(x + 9).
To find the y-intercept, simplify the equation:
y - 12 = -\frac{1}{3}x - 3 -> y = -\frac{1}{3}x + 9.
The final equation of the line passing through (-9, 12) and perpendicular to y = 3x + 6 is y = -\frac{1}{3}x + 9.