Final answer:
The equation of the line that passes through the point (9, -2) and is perpendicular to the line y = 2/3x + 2 is y = -3/2x + 23/2.
Step-by-step explanation:
To write an equation of the line that is perpendicular to the line y = 2/3x + 2 and passes through the point (9, -2), we first need to identify the slope of the existing line. Here, the original line has a slope, or m, of 2/3. A line that is perpendicular to another will have a slope that is the negative reciprocal of the original slope. Therefore, the slope for our new line is -3/2, since the reciprocal of 2/3 is 3/2 and we change the sign to get the negative reciprocal.
Next, using the point-slope form of the equation of a line, which is y - y1 = m(x - x1) where (x1, y1) is a point on the line and m is the slope of the line, we plug in our point (9, -2) and our determined slope of -3/2. This gives us y - (-2) = -3/2(x - 9).
Simplifying this, we get:
y + 2 = -3/2x + 27/2
To put this in slope-intercept form, which is y = mx + b, we subtract 2 from both sides to isolate y, resulting in the equation:
y = -3/2x + 23/2
This is the equation of the line that is perpendicular to y = 2/3x + 2 and goes through the point (9, -2).