2. Find the equation of the line which perpendicular to the line -2x + 3y = 32 and passes through the point

(-4,-8).

To find the equation of a line that is perpendicular to another line, we need to first determine the slope of the original line.

The given line is -2x + 3y = 32. To find its slope, we need to rearrange the equation into the form y = mx + b, where m represents the slope:

-2x + 3y = 32
3y = 2x + 32
y = (2/3)x + 32/3

From this equation, we can see that the slope of the given line is 2/3.

Since the line we want to find is perpendicular to the given line, the slope of the new line will be the negative reciprocal of 2/3. We can find this by flipping the fraction and changing the sign:

New slope = -1 / (2/3) = -3/2

Now we have the slope of the new line, but we still need to find the equation. We can use the point-slope form of a linear equation, which states that y - y1 = m(x - x1), where (x1, y1) is any point on the line, and m is the slope. We can plug in the values (-4, -8) for (x1, y1) and -3/2 for m:

y - (-8) = -3/2(x - (-4))

Simplifying:

y + 8 = -3/2(x + 4)
y + 8 = -3/2x - 6
y = -3/2x - 14

Therefore, the equation of the line that is perpendicular to -2x + 3y = 32 and passes through the point (-4, -8) is y = -3/2x - 14.