Sure, let's go through this step by step.
1. Using y = mx + b, find the slope of the provided line. y = 3/2x + 6, means m is 3/2 in y = mx+ b, (the coefficient of x).
2. The slope of any line perpendicular to this line should be the negative reciprocal of the original slope. The reciprocal of 3/2 is 2/3, and then we take the negative of this to give the perpendicular slope. So, the slope of the line perpendicular to y = 3/2x + 6 is -2/3.
3. Now, we also know that our new line has to pass through the point (-3,4). We can use the point-slope form of the equation of the line, which is y - y1 = m(x - x1), where m is the slope (in this case, -2/3), and (x1, y1) is the point through which the line passes (in this case, (-3,4)).
4. Now, let's find the y-intercept (b in y = mx + b). Using the formula y = mx + b, b = y - mx and substituting slopes and coordinates of point, we'll get b = 4 - ((-2/3) * -3) = 2.
5. Thus, the equation of the line we are looking for is y = -2/3x + 2.
And that's your final answer! So, the equation of a line perpendicular to y = 3/2x + 6 and passing through the point (-3,4) is y = -2/3x + 2.