Writing equations of lines parallel and perpendicular to a given line... Consider the line \( -4 x-9 y=4 \). Find the equation of the line that is parallel to this line and passes through the point \( (-7,6) \). Find the equation of the line that is perpendicular to this line and passes through the point \( (-7,6) \). Note that the ALEKS graphing calculator may be helpful in checking your answer.

To find the equations of lines parallel and perpendicular to the given line \( -4x - 9y = 4 \), we first need to determine the slope of the given line. ### Step 1: Find the slope of the given line We can rewrite the equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope. Starting with the equation: \[ -4x - 9y = 4 \] We can isolate \( y \): \[ -9y = 4 + 4x \] \[ -9y = 4x + 4 \] \[ y = -\frac{4}{9}x - \frac{4}{9} \] From this, we see that the slope \( m \) of the given line is \( -\frac{4}{9} \). ### Step 2: Find the equation of the parallel line Lines that are parallel have the same slope. Therefore, the slope of the line we want to find is also \( -\frac{4}{9} \). We can use the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope. Using the point \( (-7, 6) \) and the slope \( -\frac{4}{9} \): \[ y - 6 = -\frac{4}{9}(x + 7) \] Now, we can simplify this equation: \[ y - 6 = -\frac{4}{9}x - \frac{28}{9} \] \[ y = -\frac{4}{9}x - \frac{28}{9} + 6 \] To combine the constants, convert 6 to a fraction with a denominator of 9: \[ 6 = \frac{54}{9} \] So, \[ y = -\frac{4}{9}x - \frac{28}{9} + \frac{54}{9} \] \[ y = -\frac{4}{9}x + \frac{26}{9} \] Thus, the equation of the line that is parallel to the given line and passes through the point \( (-7, 6) \) is: \[ y = -\frac{4}{9}x + \frac{26}{9} \] ### Step 3: Find the equation of the perpendicular line Lines that are perpendicular have slopes that are negative reciprocals of each other. The negative reciprocal of \( -\frac{4}{9} \) is \( \frac{9}{4} \). Using the point-slope form again with the point \( (-7, 6) \) and the slope \( \frac{9}{4} \): \[ y - 6 = \frac{9}{4}(x + 7) \] Now, simplify this equation: \[ y - 6 = \frac{9}{4}x + \frac{63}{4} \] \[ y = \frac{9}{4}x + \frac{63}{4} + 6 \] Convert 6 to a fraction with a denominator of 4: \[ 6 = \frac{24}{4} \] So, \[ y = \frac{9}{4}x + \frac{63}{4} + \frac{24}{4} \] \[ y = \frac{9}{4}x + \frac{87}{4} \] Thus, the equation of the line that is perpendicular to the given line and passes through the point \( (-7, 6) \) is: \[ y = \frac{9}{4}x + \frac{87}{4} \] ### Summary of Results - The equation of the parallel line is: \[ y = -\frac{4}{9}x + \frac{26}{9} \] - The equation of the perpendicular line is: \[ y = \frac{9}{4}x + \frac{87}{4} \]