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Which of the following lines is perpendicular to 4x+5y=9

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by BTGmoderatorLU » Wed Jan 16, 2019 2:37 pm

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Source: Manhattan Prep

Which of the following lines is perpendicular to 4x+5y=9 on the xy plane?
$$A.\ y=\frac{5}{4}x+2$$
$$B.\ y=\frac{-5}{4}x+9$$
$$C.\ y=-4x+\frac{9}{5}$$
$$D.\ y=\frac{4}{5}x+\frac{-4}{5}$$
$$E.\ y=\frac{-4}{5}x$$
The OA is A
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by Jay@ManhattanReview » Wed Jan 16, 2019 7:54 pm
BTGmoderatorLU wrote:Source: Manhattan Prep

Which of the following lines is perpendicular to 4x+5y=9 on the xy plane?
$$A.\ y=\frac{5}{4}x+2$$
$$B.\ y=\frac{-5}{4}x+9$$
$$C.\ y=-4x+\frac{9}{5}$$
$$D.\ y=\frac{4}{5}x+\frac{-4}{5}$$
$$E.\ y=\frac{-4}{5}x$$
The OA is A
Note that two lines are perpendicular to each other if the product of their slopes is 1.

Let's first find out the slope of the given line 4x + 5y = 9. Let's transform it into y = mx + c form, where m is the slope of the line.

Thus, 4x+5y=9 => y = (-4/5)x + 9. Thus, m = -4/5

Say the slope of the line which is perpendicular to 4x+5y=9 is p.

Thus, mp = - 1. Thus, p = -1/m = -1/(-4/5) = 5/4.

The equation of the required line would be y = px + d, where p = slope of the line and d = y-intercept (We are not bothered about it).

The line presented by option A has a slope of 5/4, the correct answer.

The correct answer: A

Hope this helps!

-Jay
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by fskilnik@GMATH » Thu Jan 17, 2019 3:33 am
BTGmoderatorLU wrote:Source: Manhattan Prep

Which of the following lines is perpendicular to 4x+5y=9 on the xy plane?
$$A.\ y=\frac{5}{4}x+2 \,\,\,\,\,B.\ y=\frac{-5}{4}x+9\,\,\,\,\,C.\ y=-4x+\frac{9}{5}\,\,\,\,\,D.\ y=\frac{4}{5}x+\frac{-4}{5}\,\,\,\,\,E.\ y=\frac{-4}{5}x$$
$$4x + 5y = 9\,\,\,\,\, \Leftrightarrow \,\,\,\,\,y = - {4 \over 5}x + {9 \over 5}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{slope}} = - {4 \over 5}\,\,\,\,\,\,\left[ {{\rm{oblique}}\left( * \right)} \right]$$
$$?\,\,\,:\,\,\,\left( * \right)\,\,\, \Rightarrow \,\,\,{\rm{slop}}{{\rm{e}}_{{\rm{altern}}}} = {5 \over 4}\,\,\,$$
$$\left( {\rm{A}} \right)\,\,\,{\rm{slop}}{{\rm{e}}_{\left( {\rm{A}} \right)}} = {5 \over 4}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{done}}!$$


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Scott@TargetTestPrep » Mon Jan 21, 2019 5:48 pm
BTGmoderatorLU wrote:Source: Manhattan Prep

Which of the following lines is perpendicular to 4x+5y=9 on the xy plane?
$$A.\ y=\frac{5}{4}x+2$$
$$B.\ y=\frac{-5}{4}x+9$$
$$C.\ y=-4x+\frac{9}{5}$$
$$D.\ y=\frac{4}{5}x+\frac{-4}{5}$$
$$E.\ y=\frac{-4}{5}x$$
The OA is A
We need to first express the given line in slope-intercept form, which is y = mx + b, where m is the slope of the line:

5y = -4x + 9

y = (-4/5)x + 9/5

We see that the slope of the given line is -4/5. Since the slopes of two perpendicular lines are negative reciprocals of each other, then a line perpendicular to the given line will have a slope of 5/4. The line whose equation is given in choice A has that slope.

Answer: A

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