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

Step 1 ：Given the line equation \(y = 9x - 8\).

Step 2 ：The slope of the given line is 9.

Step 3 ：A line parallel to this line will also have a slope of 9.

Step 4 ：We are asked to find the equation of the line that is parallel to the given line and passes through the point (6,-5).

Step 5 ：The equation of a line in slope-intercept form is \(y = mx + b\), where m is the slope and b is the y-intercept.

Step 6 ：We can find the y-intercept of the parallel line by substituting the point (6,-5) into this equation and solving for b, which gives us b = -59.

Step 7 ：So, the equation of the line that is parallel to the line \(y = 9x - 8\) and passes through the point (6,-5) is \(y = 9x - 59\).

Step 8 ：A line perpendicular to the given line will have a slope that is the negative reciprocal of the slope of the given line. So the slope of the perpendicular line is -1/9.

Step 9 ：We can find the y-intercept of the perpendicular line in the same way as we did for the parallel line, which gives us b = -13/3.

Step 10 ：So, the equation of the line that is perpendicular to the line \(y = 9x - 8\) and passes through the point (6,-5) is \(y = -\frac{1}{9}x - \frac{13}{3}\).

Step 11 ：Final Answer: The equation of the line that is parallel to the line \(y = 9x - 8\) and passes through the point (6,-5) is \(\boxed{y = 9x - 59}\). The equation of the line that is perpendicular to the line \(y = 9x - 8\) and passes through the point (6,-5) is \(\boxed{y = -\frac{1}{9}x - \frac{13}{3}}\).