To determine which equation represents a line that passes through the points [tex]\((3, 1)\)[/tex] and [tex]\((6, 6)\)[/tex], we need to follow these steps:
1. Calculate the slope [tex]\( m \)[/tex] of the line:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given the points [tex]\((3, 1)\)[/tex] and [tex]\((6, 6)\)[/tex],
[tex]\[
m = \frac{6 - 1}{6 - 3} = \frac{5}{3}
\][/tex]
2. Formulate the point-slope form of the equation:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the point [tex]\((3, 1)\)[/tex] and the calculated slope [tex]\( \frac{5}{3} \)[/tex],
[tex]\[
y - 1 = \frac{5}{3}(x - 3)
\][/tex]
3. Compare with the given options:
We are given the options:
- A. [tex]\( y + 6 = \frac{3}{5}(x + 6) \)[/tex]
- B. [tex]\( y - 1 = \frac{5}{3}(x - 3) \)[/tex]
- C. [tex]\( y - 6 = \frac{3}{5}(x - 6) \)[/tex]
- D. [tex]\( y + 1 = \frac{5}{5}(x + 3) \)[/tex]
Clearly, option B matches our equation derived from the steps above.
Therefore, the correct equation that represents the line passing through the points (3, 1) and (6, 6) is:
B. [tex]\( y - 1 = \frac{5}{3}(x - 3) \)[/tex]
