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 that passes through the point \( (8, -7) \) and is perpendicular to the line \( y = \frac{1}{5}x + 4 \), follow these steps: ### 1. Determine the Slope of the Given Line The equation of the given line is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. \[ y = \frac{1}{5}x + 4 \quad \Rightarrow \quad \text{Slope } (m_1) = \frac{1}{5} \] ### 2. Find the Slope of the Perpendicular Line Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope \( m_2 \) of the line perpendicular to the given line is: \[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{1}{5}} = -5 \] ### 3. Write the Equation in Point-Slope Form The **point-slope form** of a line is given by: \[ y - y_1 = m(x - x_1) \] Using the point \( (8, -7) \) and the slope \( m = -5 \): \[ y - (-7) = -5(x - 8) \\ y + 7 = -5(x - 8) \] **Point-Slope Form:** \[ y + 7 = -5(x - 8) \] ### 4. Convert to Slope-Intercept Form To convert the equation to **slope-intercept form** \( y = mx + b \), simplify the point-slope equation: \[ y + 7 = -5(x - 8) \\ y + 7 = -5x + 40 \\ y = -5x + 40 - 7 \\ y = -5x + 33 \] **Slope-Intercept Form:** \[ y = -5x + 33 \] **Final Answers:** - **Point-Slope Form:** \( y + 7 = -5(x - 8) \) - **Slope-Intercept Form:** \( y = -5x + 33 \)