Problem

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:

Solution

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 \(\boxed{y = -\frac{1}{9}x - \frac{13}{3}}\).

From Solvely APP
Source: https://solvelyapp.com/problems/46108/

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