To determine which equation represents the line passing through the given points \((3, 1)\) and \((6, 6)\), we will follow these steps:
1. Calculate the slope of the line passing through the points \((3, 1)\) and \((6, 6)\):
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates of the points:
[tex]\[
m = \frac{6 - 1}{6 - 3} = \frac{5}{3}
\][/tex]
2. Determine the line equation using the slope \(m\) and one of the points.
A general form for the equation of a line with slope \(m\) passing through point \((x_1, y_1)\) is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substitute \(m = \frac{5}{3}\) and the point \((3, 1)\):
[tex]\[
y - 1 = \frac{5}{3}(x - 3)
\][/tex]
Simplifying this equation to check if it aligns with any of the options:
[tex]\[
y - 1 = \frac{5}{3}(x - 3)
\][/tex]
This matches exactly with option B:
[tex]\[
y - 1 = \frac{5}{3}(x - 3)
\][/tex]
Thus, the correct equation representing the line that passes through the points \((3, 1)\) and \((6, 6)\) is:
[tex]\[
\boxed{B. \ y - 1 = \frac{5}{3}(x - 3)}
\][/tex]
