SOLUTION: What is the equations of the line passing through (8,−2) and perpendicular to the line whose equation is x−7y−9=0

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Question 1092180: What is the equations of the line passing through (8,−2) and perpendicular to the line whose equation is x−7y−9=0
Found 2 solutions by josmiceli, greenestamps:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Find the slope of the given line
+x+-+7y+-+9+=+0+
+7y+=+x+-+9+
+y+=+%281%2F7%29%2Ax+-+9%2F7+
----------------------
Any line perpendicular to this line has
slope +m%5B2%5D+=+-1%2Fm%5B1%5D+ where
+m%5B1%5D+=+1%2F7+
+m%5B2%5D+=+-1%2F%281%2F7%29+
+m%5B2%5D+=+-7+
---------------------
(8,-2)
Use the general point-slope formula
+%28+y+-%28-2%29+%29+%2F+%28+x+-+8+%29+=+-7+
+%28+y+%2B+2+%29+%2F+%28+x+-+8+%29+=+-7+
+y+%2B+2+=+-7%2A%28+x+-+8+%29+
+y+%2B+2+=+-7x+%2B+56+
+y+=+-7x+%2B+54+
----------------------
check:
Does the line go through ( 8, -2 ) ?
+-2+=+-7%2A8+%2B+54+
+-2+=+-56+%2B+54+
+-2+=+-2+
OK

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

Here is a much faster way to find the answer to a problem like this. It uses something called a dot product that you will learn about when you study vectors. The key is this:

Given a line with equation

ax%2Bby+=+c (where c can be any constant)

the equation of any line perpendicular to the given line will have an equation of the form

bx-ay+=+d (where d can be any constant)

Notice that the coefficients on x and y have changed places, and one of them has changed sign.

So in your problem, given the equation

x-7y=9

the equation of any line perpendicular to the given line is of the form

7x%2By+=+c

Then, since you know the coordinates of one point on the line, (8,-2), you can determine the value of c in the equation.

7%288%29%2B%28-2%29+=+56-2+=+54

So the equation of the line perpendicular to the given line and passing through the given point is

7x%2By+=+54