Write an equation of the line that is perpendicular to the given line and un point. through the given pol \( \begin{array}{ll}\text { 7. } y=-4 x+2 ;(0,2) & \text { 8. } y=\frac{2}{3} x+6 ;(-6,2)\end{array} \)

Sure! Let's find the equations of the lines that are **perpendicular** to the given lines and pass through the specified points. --- ### **Problem 7** **Given:** - **Original Line:** \( y = -4x + 2 \) - **Point:** \( (0, 2) \) **Steps to Find the Perpendicular Line:** 1. **Determine the slope of the original line.** - The given line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. - **Slope (\( m_1 \))** of the original line: \( -4 \). 2. **Find the slope of the perpendicular line.** - The slope of a line perpendicular to another is the **negative reciprocal** of the original slope. - **Slope (\( m_2 \))** of the perpendicular line: \( m_2 = -\frac{1}{m_1} = -\frac{1}{-4} = \frac{1}{4} \). 3. **Use the point-slope form to write the equation of the perpendicular line.** - **Point-Slope Form:** \( y - y_1 = m(x - x_1) \) - Plugging in \( m = \frac{1}{4} \) and the point \( (0, 2) \): \[ y - 2 = \frac{1}{4}(x - 0) \] - Simplifying: \[ y = \frac{1}{4}x + 2 \] **Answer:** \[ \boxed{y = \frac{1}{4}x + 2} \] --- ### **Problem 8** **Given:** - **Original Line:** \( y = \frac{2}{3}x + 6 \) - **Point:** \( (-6, 2) \) **Steps to Find the Perpendicular Line:** 1. **Determine the slope of the original line.** - The given line is in the slope-intercept form \( y = mx + b \). - **Slope (\( m_1 \))** of the original line: \( \frac{2}{3} \). 2. **Find the slope of the perpendicular line.** - The slope of a line perpendicular to another is the **negative reciprocal** of the original slope. - **Slope (\( m_2 \))** of the perpendicular line: \( m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2} \). 3. **Use the point-slope form to write the equation of the perpendicular line.** - **Point-Slope Form:** \( y - y_1 = m(x - x_1) \) - Plugging in \( m = -\frac{3}{2} \) and the point \( (-6, 2) \): \[ y - 2 = -\frac{3}{2}(x - (-6)) \\ y - 2 = -\frac{3}{2}(x + 6) \] - Simplifying: \[ y - 2 = -\frac{3}{2}x - 9 \\ y = -\frac{3}{2}x - 7 \] **Answer:** \[ \boxed{y = -\frac{3}{2}x - 7} \] --- These are the equations of the lines perpendicular to the given lines and passing through the specified points.