Find the equation of the line perpendicular to the line \( y = 3x + 2 \) and passing through the point \( (4, -2) \).

Step 1 ：Step 1: Find the slope of the given line. The slope of the line \( y = 3x + 2 \) is 3.

Step 2 ：Step 2: Find the slope of the line perpendicular to the given line. The slope of a line perpendicular to a line with slope \( m \) is \( -1/m \). So, the slope of the line we are trying to find is \( -1/3 \).

Step 3 ：Step 3: Use the point-slope form of a line to find the equation of the line. The point-slope form of a line is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Substituting \( m = -1/3 \), \( x_1 = 4 \), and \( y_1 = -2 \) gives us \( y - (-2) = -1/3 (x - 4) \).

Step 4 ：Step 4: Simplify the equation obtained in step 3 to get the equation of the line in slope-intercept form. This gives us \( y + 2 = -1/3 x + 4/3 \), which simplifies to \( y = -1/3 x + 4/3 - 2 \), or \( y = -1/3 x + 2/3 \).