To find the value of \( b \) (the vertical intercept) in the equation of the line that passes through the points \((1,5)\) and \((3,8)\), we first need to determine the slope (\( m \)) of the line using these two points.
Step 1: Calculate the slope (\( m \))
The slope formula is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((1,5)\) and \((3,8)\):
- \( x_1 = 1, y_1 = 5 \)
- \( x_2 = 3, y_2 = 8 \)
Now plug these values into the slope formula:
\[ m = \frac{8 - 5}{3 - 1} = \frac{3}{2} = 1.5 \]
Step 2: Use the point-slope form to find \( b \)
Now we have the slope (\( m = 1.5 \)). We can use the equation of a line in slope-intercept form \( y = mx + b \). We can use either of the points, but let's use \((1,5)\):
\[ 5 = 1.5 \cdot 1 + b \]
Step 3: Solve for \( b \)
Plugging in the values:
\[ 5 = 1.5 + b \]
To isolate \( b \), subtract \( 1.5 \) from both sides:
\[ b = 5 - 1.5 \] \[ b = 3.5 \]
Conclusion
Thus, the value of \( b \), or the vertical intercept, is
\[ \boxed{3.5} \]