Consider the line $-2 x-9 y=5$
Find the equation of the line that is parallel to this line and passes through the point $(-8,6)$.
Find the equation of the line that is perpendicular to this line and passes through the point $(-8,6)$.
Equation of parallel line:
Equation of perpendicular line:

Step 1 ：The given line equation is \(-2x - 9y = 5\).

Step 2 ：The slope of a line given by the equation \(ax + by = c\) is \(-\frac{a}{b}\). So, the slope of the given line is \(-\frac{-2}{-9} = \frac{2}{9}\).

Step 3 ：A line parallel to another line will have the same slope. So, the slope of the line parallel to the given line is also \(\frac{2}{9}\).

Step 4 ：The equation of a line in slope-intercept form is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. We can find the y-intercept by substituting the coordinates of the given point and the slope into this equation.

Step 5 ：The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, the slope of the line perpendicular to the given line is \(-\frac{9}{2}\).

Step 6 ：We can find the equation of the line perpendicular to the given line in the same way as we did for the parallel line.

Step 7 ：The equation of the line that is parallel to the line \(-2 x-9 y=5\) and passes through the point \((-8,6)\) is \(y = \frac{2}{9}x + \boxed{7.78}\) (rounded to two decimal places).

Step 8 ：The equation of the line that is perpendicular to the line \(-2 x-9 y=5\) and passes through the point \((-8,6)\) is \(y = -\frac{9}{2}x - \boxed{30}\).