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Given 3x + 2y = 6 and the point (6, -3):

Transform the equation into slope-intercept form:

3x + 2y = 6

3x - 3x + 2y = - 3x + 6

2y = -3x + 6

Divide both sides by 2:

2y/2 = (-3x + 6)/2

y = -3/2x + 6 (slope-intercept form)

Next, to find the line perpendicular to the original equation:

By definition, perpendicular lines have negative reciprocals. This means that if you multiply the slope of the original equation with the slope of perpendicular line, it will have a product of -1.

The slope of the original equation is -3/2. This means that the negative reciprocal must be 2/3. Therefore, the slope of the perpendicular line is 2/3.

Next, we’ll use the given point (6,-3) and slope (2/3) to find out what the y-intercept is:

y = mx + b

-3 = 2/3(6) + b

-3 = 4 + b

-3 - 4 = 4 - 4 + b

-7 = b

Therefore, the y-intercept is -7.

We can establish the linear equation of the line perpendicular to 3x + 2y = 6:

y = 2/3x - 7