SOLUTION: As hard as I try, I have been unable to understand this lesson, much less answer anything correctly. This is an example homework question: Write an equation of the line containi

Question 534370: As hard as I try, I have been unable to understand this lesson, much less answer anything correctly. This is an example homework question:
Write an equation of the line containing the given point and perpendicular to the given line. Express your answer in the form of y=mx+b
(2, -6); 8x + 5y = 6
I know that I'm supposed to subtract 8x from both sides (to solve for y), and I am able to complete the first step correctly:
5y = -8x + 6
It then tells me to divide each side by 5 (this is where my answer does not match up with the correct homework answer) and I get:
y = 8 6
- _ x + _
5 5
The correct answer is:
y = 8 29
- __ x - __
5 4

What am I doing wrong???
Found 2 solutions by Alan3354, solver91311:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Write an equation of the line containing the given point and perpendicular to the given line. Express your answer in the form of y=mx+b
(2,-6); 8x + 5y = 6
---------------
Find the slope of the line.
To do that, put it in slope-intercept form. That means solve for y.
8x + 5y = 6
5y = -8x + 6
y = (-8/5)x + 6/5
The slope, m, is the coefficient of x in this form.
m = -8/5
----------
The slope of lines perpendicular is the negative inverse.
m = 5/8
------------
Use y = mx + b and the point (2,-6) to find b
-6 = (5/8)*2 + b
b = -29/4
-----------
For the line perpendicular, m = 5/8 and b = -29/4
--> y = (5/8)x - 29/4

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

There are only two ways to derive an equation of a line. You either need two points that are on the line or one point that is on the line and the slope of the line.

So let's look at this problem starting at the end and then work backwards.

Your goal is to write an equation of a line and you are given two pieces of information. One is a point on the desired line and the other is the equation of a line that is perpendicular to the one you want. Finding a second point to define the desired line is not possible with the information given, but the equation of the given line will tell us the slope of THAT line. The fortunate circumstance is that the slopes of perpendicular lines are related in a very special way. Perpendicular lines have slopes that are negative reciprocals of one another. That is to say if is the slope of a line, then is the slope of any line perpendicular to it.

The process you started, namely solving the given equation for is the first step because once a two variable linear equation is solved for the vertical axis variable () in terms of the other variable (), the coefficient on is the slope of the line. Then to get the slope of the desired line, just flip the fraction and change the sign.

Given the point and the line , write an equation of a line through the given point perpendicular to the given line.

Step 1: Solve the given equation for :

Which is what you had. But you weren't done.

Step 2: Determine the slope of the given line, which is simply the coefficient on , namely , and then compute the slope for the desired line. Flip the fraction and change the sign. .

Step 3: Use the point-slope form of an equation of a line to write the equation of the desired line:

where are the coordinates of the given point and is the given/calculated slope. So:

Doing the arithmetic:

John

My calculator said it, I believe it, that settles it

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