Writing equations of lines parallel and perpendicular to a given line... Consider the line \( y=-\frac{7}{5} x-7 \) Find the equation of the line that is perpendicular to this line and passes through the point \( (7,-5) \). Find the equation of the line that is parallel to this line and passes through the point \( (7,-5) \). Note that the ALEKS graphing calculator mady be helpful in checking your answer.

To find the equations of lines that are parallel and perpendicular to the given line \( y = -\frac{7}{5} x - 7 \), we first need to identify the slope of the given line. ### Step 1: Identify the slope of the given line The equation of the line is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. From the equation \( y = -\frac{7}{5} x - 7 \), we see that the slope \( m \) is \( -\frac{7}{5} \). ### Step 2: Find the slope of the perpendicular line The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope \( m_{\perp} \) of the perpendicular line is: \[ m_{\perp} = -\frac{1}{-\frac{7}{5}} = \frac{5}{7} \] ### Step 3: Write the equation of the perpendicular line We need to find the equation of the line with slope \( \frac{5}{7} \) that passes through the point \( (7, -5) \). We can use the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1) \] Substituting \( m = \frac{5}{7} \), \( x_1 = 7 \), and \( y_1 = -5 \): \[ y - (-5) = \frac{5}{7}(x - 7) \] This simplifies to: \[ y + 5 = \frac{5}{7}(x - 7) \] Now, distribute \( \frac{5}{7} \): \[ y + 5 = \frac{5}{7}x - 5 \] Subtracting 5 from both sides gives: \[ y = \frac{5}{7}x - 10 \] ### Step 4: Find the slope of the parallel line The slope of a line that is parallel to the given line is the same as the slope of the original line, which is \( -\frac{7}{5} \). ### Step 5: Write the equation of the parallel line We need to find the equation of the line with slope \( -\frac{7}{5} \) that passes through the point \( (7, -5) \). Again, we use the point-slope form: \[ y - (-5) = -\frac{7}{5}(x - 7) \] This simplifies to: \[ y + 5 = -\frac{7}{5}(x - 7) \] Distributing \( -\frac{7}{5} \): \[ y + 5 = -\frac{7}{5}x + \frac{49}{5} \] Subtracting 5 (which is \( \frac{25}{5} \)) from both sides gives: \[ y = -\frac{7}{5}x + \frac{49}{5} - \frac{25}{5} \] This simplifies to: \[ y = -\frac{7}{5}x + \frac{24}{5} \] ### Final Answers - The equation of the line that is perpendicular to the given line and passes through the point \( (7, -5) \) is: \[ y = \frac{5}{7}x - 10 \] - The equation of the line that is parallel to the given line and passes through the point \( (7, -5) \) is: \[ y = -\frac{7}{5}x + \frac{24}{5} \]