To determine which of the given equations correctly represents the line passing through the points (3, 3) and (6, 5), we need to follow these steps:
1. Calculate the slope (m) of the line:
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
With the points [tex]\((3, 3)\)[/tex] and [tex]\((6, 5)\)[/tex]:
[tex]\[
m = \frac{5 - 3}{6 - 3} = \frac{2}{3}
\][/tex]
2. Use the point-slope form to determine the equation:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
Substituting one of the points [tex]\((3, 3)\)[/tex] into the equation with the calculated slope [tex]\(m = \frac{2}{3}\)[/tex]:
[tex]\[
y - 3 = \frac{2}{3}(x - 3)
\][/tex]
3. Compare with the given options:
Now, we'll check which of the given options matches the equation we derived:
[tex]\[
C. \quad y - 3 = \frac{3}{2}(x - 3) \quad \text{(Incorrect, wrong slope)}
\][/tex]
[tex]\[
D. \quad y - 3 = \frac{2}{3}(x - 3) \quad \text{(Correct, matches our equation)}
\][/tex]
[tex]\[
A. \quad y + 3 = \frac{3}{2}(x + 3) \quad \text{(Incorrect, wrong form and slope)}
\][/tex]
[tex]\[
B. \quad y + 3 = \frac{2}{3}(x + 3) \quad \text{(Incorrect, wrong form)}
\][/tex]
4. Conclusion:
The equation that correctly represents the line passing through the points (3, 3) and (6, 5) is:
[tex]\[
\boxed{y - 3 = \frac{2}{3}(x - 3)}
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{\text{D}}
\][/tex]
