To solve for the value of \( b \) (the vertical intercept) using the information provided, we need to recognize that it appears you are working with a linear equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
You've given two points:
- \( (1, 5) \)
- \( (3, 8) \)
From these points, we can calculate the slope \( m \) of the line:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 5}{3 - 1} = \frac{3}{2} = 1.5 \]
We can now use one of the points — let's use \( (1, 5) \), to find \( b \).
The linear equation can be set up as follows:
\[ y = mx + b \]
Substituting for \( m \) and the coordinates of point \( (1, 5) \):
\[ 5 = 1.5 \cdot 1 + b \]
Now, solve for \( b \):
\[ 5 = 1.5 + b \] \[ b = 5 - 1.5 \] \[ b = 3.5 \]
Thus, the value of \( b \) (the vertical intercept) is:
\[ \boxed{3.5} \]