Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Passing through \( (8,-7) \) and perpendicular to the line whose equation is \( y=\frac{1}{5} x+4 \) 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. \( \square \) (Simplify your answer. Use integers or fractions for any numbers in the equation.)

To find the equation of the line passing through the point \( (8, -7) \) and perpendicular to the line given by \( y = \frac{1}{5} x + 4 \), we first determine the slope of the new line. Since the given line has a slope of \( \frac{1}{5} \), the slope of the line that is perpendicular to it will be the negative reciprocal. Therefore, the slope of the new line is \( -5 \). **Point-slope form**: The point-slope form of a line is given by the equation: \[ y - y_1 = m(x - x_1) \] Plugging in the point \( (8, -7) \) and the slope \( -5 \), we have: \[ y - (-7) = -5(x - 8) \] This simplifies to: \[ y + 7 = -5(x - 8) \] **Slope-intercept form**: To convert this to slope-intercept form \( y = mx + b \), we distribute the \( -5 \): \[ y + 7 = -5x + 40 \] Now, bring \( 7 \) to the other side: \[ y = -5x + 40 - 7 \] Simplifying further gives us: \[ y = -5x + 33 \] So, the equations are: Point-slope form: \( y + 7 = -5(x - 8) \) Slope-intercept form: \( y = -5x + 33 \)