## To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.

Given that the equation of the given line is 3x + 6y = 9, we can rewrite it in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

To convert the given equation to slope-intercept form, let's solve it for y:

6y = -3x + 9

y = (-3/6)x + 9/6

Simplifying, we get:

y = (-1/2)x + 3/2

So, the slope of the given line is -1/2.

To find the slope of the line perpendicular to this, we take the negative reciprocal of the slope:

Perpendicular slope = -1 / (-1/2)

Perpendicular slope = 2

Now that we have the slope of the line perpendicular to the given line, we can use the point-slope form of a linear equation to find its equation. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Given that the line passes through (-4, 2) and has a slope of 2, we can substitute these values into the point-slope form:

y - 2 = 2(x - (-4))

Simplifying, we get:

y - 2 = 2(x + 4)

y - 2 = 2x + 8

y = 2x + 10

Therefore, the equation of the line that passes through (-4,2) and is perpendicular to 3x + 6y = 9 is y = 2x + 10.