To find the equation of the line passing through the points M(4, 3) and N(7, 12), we first need to calculate the slope \(m\) of the line using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the coordinates of points M and N, where \(M(4, 3)\) corresponds to \((x_1, y_1)\) and \(N(7, 12)\) corresponds to \((x_2, y_2)\):
\[ m = \frac{12 - 3}{7 - 4} = \frac{9}{3} = 3 \]
Now that we have the slope \(m = 3\), we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
We can use point M(4, 3) for \((x_1, y_1)\):
\[ y - 3 = 3(x - 4) \]
Distributing the slope on the right side:
\[ y - 3 = 3x - 12 \]
Next, we add 3 to both sides to put the equation in slope-intercept form \(y = mx + b\):
\[ y = 3x - 12 + 3 \] \[ y = 3x - 9 \]
In the slope-intercept form \(y = mx + b\), we can see that \(b = -9\).
Thus, the value of \(b\) is
\[ \boxed{-9} \]