Write an equation for a line passing through the point \( (3,-2) \) that is parallel to the line \( 3 x+4 y=5 \). Then write a second equation for a line passing through the point \( (3,-2) \) that is perpendicular to the line \( 3 x+4 y=5 \). The equation of the parallel line is \( y=-\frac{3}{4} x+\frac{1}{4} \). (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.) The equation of the perpendicular line is (Simplify your answer. Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.)

To find the equations of the lines as specified, let's follow the steps below: ### 1. Parallel Line You've already correctly found the equation of the parallel line passing through the point \( (3, -2) \): \[ y = -\frac{3}{4}x + \frac{1}{4} \] ### 2. Perpendicular Line To find the equation of the line perpendicular to \( 3x + 4y = 5 \) and passing through \( (3, -2) \), follow these steps: 1. **Find the slope of the original line:** Rewrite \( 3x + 4y = 5 \) in slope-intercept form (\( y = mx + b \)): \[ 4y = -3x + 5 \implies y = -\frac{3}{4}x + \frac{5}{4} \] So, the slope (\( m_1 \)) of the original line is \( -\frac{3}{4} \). 2. **Determine the slope of the perpendicular line:** The slope of a line perpendicular to the original line is the negative reciprocal of \( m_1 \): \[ m_2 = \frac{4}{3} \] 3. **Use the point-slope form to find the equation:** Using the point \( (3, -2) \) and the slope \( m_2 = \frac{4}{3} \): \[ y - (-2) = \frac{4}{3}(x - 3) \] Simplify: \[ y + 2 = \frac{4}{3}x - 4 \implies y = \frac{4}{3}x - 6 \] ### Final Answer - **Equation of the parallel line:** \[ y = -\frac{3}{4}x + \frac{1}{4} \] - **Equation of the perpendicular line:** \[ y = \frac{4}{3}x - 6 \]