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Question 463036: Which choice is the equation of a line that passes through the point (–2, 8) and is perpendicular to the line represented by this equation?
5x − 2y = −10
A. 2x− 5y = 26
B. 5x + 2y = 10
C. 2x + 5y = 36
D. −2x + 5y = −13
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you start with the equation of 5x - 2y = -10.
if you subtract 5x from both sides of this equation you get -2y = -5x - 10.
if you divide both sides of this equation by -2, you get y = (5/2)x + 5.
your slope is (5/2)
The slope of a perpendicular line would be equal to the negative reciprocal of (5/2) which would be equal to -(2/5).
the line would be passing through the point (x,y) = (-2,8).
you can use the point slope form of the equation to derive the equation as follows:
the point slope form is y-y1 = m*(x-x1)
(x1,y1) is the point that the line is passing through.
you get y - 8 = -(2/5) * (x + 2)
this becomes y - 8 = -(2/5)*x - (4/5)
add 8 to both sides of this equation and you get y = -(2/5)x + 8 - (4/5) which becomes y = (-2/5)x + (36/5)
multiply both sides of this equation by 5 and you get 5y = -2x + 36.
add 2x to both sides of this equation and you get 2x + 5y = 36.
that matches selection c.
there might be a simpler way of looking at this.
your equation is 5x - 2y = -10
the general form of this equation in standard form is ax + by = c
this means that a is the coefficient of x and b is the coefficient of y.
the slope is given by the formula -a/b
in your equation, a is 5 and b is -2.
the slope is therefore -5/-2 which is equal to 5/2.
the line perpendicular to this would have a slope that is the negative reciprocal of that, so the slope of the perpendicular line has to be -2/5.
plug this slope back into the standard form of the equation of ax + by = c and you get 2x + 5y = c because -a is equal to -2 which makes a = 2, and b = 5
because the equation passed through the point (-2,8), you then simply replace x with -2 and y with 8 to get 2(-2) + 5*8 = -4 + 40 = 36.
this makes the value of c = 36 and your equation becomes 2x + 5y = 36.
this also leads to selection c.
you can do it either way.
converting to slope intercept form (y = mx + b) where m is the slope and b is the y intercept is how its normally done, but if you are working from the standard form of the equation and remember that slope is equal to -b/a, then you can derive it the alternate way as well.
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