Final answer:
The value of b in the equation y = mx + b for the line perpendicular to y = 9 - \(\frac{1}{3}\)x that intersects at a point on the x-axis is 0.
Step-by-step explanation:
The lines in question, y = 9 - \(\frac{1}{3}\)x and y = mx + b, are said to be perpendicular. For two lines to be perpendicular in a two-dimensional plane, the product of their slopes must equal -1. The slope of the first line, given by the coefficient of x, is -\(\frac{1}{3}\). Therefore, the slope of the second line (m) must be the negative reciprocal of -\(\frac{1}{3}\), which is 3. Since the lines intersect on the x-axis, their point of intersection has a y-coordinate of 0. To find the value of b, we substitute m with 3 and y with 0 into the equation of the second line, yielding 0 = 3x + b.
As they intersect on the x-axis, we also know that x must equal 0 at the point of intersection. Plugging x = 0 into the equation, we get 0 = 3(0) + b, which simplifies to 0 = b. Therefore, the value of b is 0.