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