Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Passing through $(4,-2)$ and perpendicular to the line whose equation is $y=\frac{1}{3}x+4$
Write an equation for the line in point-slope form.
(Simplify your answer. Use integers or fractions for any numbers in the equation.) Write an equation for the line in slope-intercept form.
(Simplify your answer. Use integers or fractions for any numbers in the equation.)

Step 1 ：The slope of the given line is $\frac{1}{3}$. The slope of a line perpendicular to this line would be the negative reciprocal of $\frac{1}{3}$, which is $-3$.

Step 2 ：The point-slope form of a line is $y-y_1=m(x-x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope of the line.

Step 3 ：The slope-intercept form of a line is $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

Step 4 ：We can use these formulas to find the equations of the line in point-slope form and slope-intercept form.

Step 5 ：Substitute $m=-3$, $x_1=4$, and $y_1=-2$ into the point-slope form, we get $y+2=-3(x-4)$.

Step 6 ：Rearrange the equation to slope-intercept form, we get $y=-3x+10$.

Step 7 ：$\boxed{{\displaystyle \text{The equation of the line in point-slope form is }y+2=-3(x-4)\text{ and in slope-intercept form is }y=-3x+10}}$