Write an equation for a line passing through the point \( (4,-2) \) that is parallel to the line \( 4 x+7 y=9 \). Then write a second equation for a line passing through the point \( (4,-2) \) that is perpendicular to the line \( 4 x+7 y=9 \). The equation of the parallel line is (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 equation of a line passing through the point \( (4,-2) \) that is parallel to the line \( 4x + 7y = 9 \), we first need to find the slope of the given line. Rearranging \( 4x + 7y = 9 \) into slope-intercept form (y = mx + b): \[ 7y = -4x + 9 \] \[ y = -\frac{4}{7}x + \frac{9}{7} \] The slope (m) is \( -\frac{4}{7} \). Since parallel lines have the same slope, our parallel line will also have a slope of \( -\frac{4}{7} \). Using the point-slope form \( y - y_1 = m(x - x_1) \): \[ y - (-2) = -\frac{4}{7}(x - 4) \] \[ y + 2 = -\frac{4}{7}x + \frac{16}{7} \] \[ y = -\frac{4}{7}x + \frac{16}{7} - 2 \] \[ y = -\frac{4}{7}x + \frac{16}{7} - \frac{14}{7} \] \[ y = -\frac{4}{7}x + \frac{2}{7} \] The equation of the parallel line is: \[ y = -\frac{4}{7}x + \frac{2}{7} \] For the line that is perpendicular to \( 4x + 7y = 9 \), we need the negative reciprocal of the slope \( -\frac{4}{7} \), which is \( \frac{7}{4} \). Using the point-slope form again: \[ y - (-2) = \frac{7}{4}(x - 4) \] \[ y + 2 = \frac{7}{4}x - 7 \] \[ y = \frac{7}{4}x - 7 - 2 \] \[ y = \frac{7}{4}x - 9 \] The equation of the perpendicular line is: \[ y = \frac{7}{4}x - 9 \]