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 $y+2=-1/3x+4/3$, which simplifies to $y=-1/3x+4/3-2$, or $y=-1/3x+2/3$.