To write the equation of a line that passes through the points \((3, -2)\) and \((6, 2)\), we can start by calculating the slope (m) of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of the points \((x_1, y_1) = (3, -2)\) and \((x_2, y_2) = (6, 2)\):
\[ m = \frac{2 - (-2)}{6 - 3} = \frac{2 + 2}{6 - 3} = \frac{4}{3} \]
Now that we have the slope \(m = \frac{4}{3}\), we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Using the point \((3, -2)\) as \((x_1, y_1)\):
\[ y - (-2) = \frac{4}{3}(x - 3) \]
Simplifying this gives:
\[ y + 2 = \frac{4}{3}(x - 3) \]
Now we can rewrite it in standard form. First, we can eliminate the fraction by multiplying both sides by 3:
\[ 3(y + 2) = 4(x - 3) \]
Expanding both sides:
\[ 3y + 6 = 4x - 12 \]
Now, we can rearrange it into standard form \(Ax + By = C\):
\[ 4x - 3y = 18 \]
Thus, the equation of the line in standard form is:
\[ 4x - 3y = 18 \]