If the points with coordinates (a , b) and (c , d) lie on the the line with equation 2 y + 3 x = 4 and a - c = 3, then what is the value of d - b?

Use the coordinates of the two points on the line to find the slope m of the line. 

m = (d - b) / (c - a) 

The slope m of the line can also be calculated using the equation of the line rewritten in the form y = m x + k. 

2 y + 3 x = 4 can be written as 2y = - 3x + 4 

Divide all terms by 2 to obtain y = (-3/2) x + 2 

The slope m is equal to -3 / 2 

Equate the expression of the slope found above to -3 / 2 

-3 / 2 = (d - b) / (c - a) 

Use the above to find d - b 

d - b = -3 / 2 (c - a) = (-3 / 2)(-(a - c)) = 9 / 2 

Answer: 9 / 2