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Question 1159868: Line A is perpendicular to Line B. Determine
the equation of Line A if it passes through Line B
at (5, -1), where Line B is represented by the
equation x + 2y = 3.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
First, a solution using elementary methods; you should understand it and be able to solve similar problems by this method.
(1) Put the equation of B in slope-intercept form  to find the slope.
The slope is (-1/2).
(2) Find the slope of a line perpendicular to B -- it is the negative reciprocal of the slope of B: +(2/1) = 2
(3) Find the equation of line A knowing it has a slope of 2 and passes through (5,-1).
The equation is 
A graph of the two lines -- B red, A green....
Alternatively, a shortcut using ideas about vectors. This is a quick path to the answer you can use if a formal algebraic solution is not required.
Given line B with equation  , every line parallel to B will have an equation of the form  for some constant c; and every line perpendicular to B will have an equation of the form  for some constant c.
Without understanding vectors, what you need to see is that the coefficients of x and y have switched places, and one of them has changed signs. In this example, the coefficients 1 and 2 become 2 and -1.
So the equation of line A is ; and we can determine the constant c knowing that (x,y)=(5,-1) satisfies the equation.
The equation of A is 
That is a different form of the same equation found earlier, which was
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