Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Passing through \( (9,-7) \) and perpendicular to the line whose equation is \( y=\frac{1}{2} x+5 \)

To find the equation of the line that **passes through the point \( (9, -7) \)** and is **perpendicular** to the line with equation \( y = \frac{1}{2}x + 5 \), follow these steps: ### 1. Determine the Slope of the Given Line The given line is in **slope-intercept form**: \[ y = \frac{1}{2}x + 5 \] Here, the **slope (\(m_1\))** is: \[ m_1 = \frac{1}{2} \] ### 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: \[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{1}{2}} = -2 \] ### 3. Use the Point-Slope Form 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 - \( m \) is the slope Plugging in the point \( (9, -7) \) and the slope \( m_2 = -2 \): \[ y - (-7) = -2(x - 9) \] Simplify: \[ y + 7 = -2(x - 9) \] This is the **point-slope form** of the desired line. ### 4. Convert to Slope-Intercept Form To express the equation in **slope-intercept form** (\( y = mx + b \)), solve for \( y \): \[ y + 7 = -2(x - 9) \] Distribute the slope on the right side: \[ y + 7 = -2x + 18 \] Subtract 7 from both sides to solve for \( y \): \[ y = -2x + 18 - 7 \] \[ y = -2x + 11 \] This is the **slope-intercept form** of the desired line. ### **Final Answers** - **Point-Slope Form:** \[ y + 7 = -2(x - 9) \] - **Slope-Intercept Form:** \[ y = -2x + 11 \]