Is the line l given by ax+by=c perpendicular to the line 2x+7y=3?

Question

Is the line  l  given by  ax+by=c  perpendicular to the line  2x+7y=3 ?

(1)  2a+7b=0 
(2) Line  l  is parallel to the line  2y-7x=3

GMATinsight · Answer

ax+by=c
i.e. y = -(a/b)x+(c/b)
i.e. Slope,  m_1 = (-\frac{a}{b})

2x+7y=3 
i.e.  y = -(2/7)x+(3/7) 
i.e. Slope,  m_2 = (-\frac{2}{7})

for perpendicular lines,  m_1*m_2 = -1

i.e.  (-\frac{a}{b})*(-\frac{2}{7}) = -1 
i.e.  (\frac{a}{b}) = -\frac{7}{2}

Question: Is  (\frac{a}{b}) = -\frac{7}{2} ?

Statement 1 :  2a+7b=0

i.e.  (\frac{a}{b}) = -\frac{7}{2}

SUFFICIENT

Statement 2 : Slope of line l = (-a/b) = slope of 2y-7x=3

i.e.  (-a/b) = 7/2

i.e.  (\frac{a}{b}) = -\frac{7}{2}

SUFFICIENT

Answer: Option D

Nups1324 · Answer

Hi GMATinsight , I'm confused as to how statement 1 is sufficient! 2a+7b=0 (2 and 7 are the values of the variables aren't they? Since the original equation is in the form ax+by=c) So the y=mx+b form becomes 7=-2a/b + 0 doesn't it? Or is the equation 2a+7b=0 considered independent? Please help me understand if I'm missing something. Thank you

Posted from my mobile device

GMATinsight · Answer

Nups1324 the equation 2a+7b=0 considered independent.

2a+7b=0 i.e. 2a = -7b i.w. a/b = -7/2

fmcgee1 · Answer

Shouldn't the perpendicular be the negative reciprocal of 2x+7y=3? So it would be 7/2, not -7/2.

dave13 · Answer

yeah, i agree, cause if slope is -2/7 then perpendicular line is -1 divided by -2/7 which is 7/2 GMATinsight

GMATinsight · Answer

dave13 
 fmcgee1

Can you please highlight the step that you find faulty in my explanation???

I rechecked my solution and it seems perfect.

dave13 · Answer

GMATinsight i know if slope of one line is m, then slope of perpendicular line should be -1/m now if slope if m_= (-\frac{2}{7}) then perpendicular line should have slope -1 / -2/7 = \frac{7}{2} and you write Question: Is (\frac{a}{b}) = -\frac{7}{2} ? Why its negative ?

TheUltimateWinner · Answer

Hi  IanStewart ,
Can you share your thought, please? I need short cut way to save time!

IanStewart · Answer

If the slope of one line is m, the slope of a perpendicular line will be -1/m. We can rewrite the equation 2x + 7y = 3 in the normal form y = mx + b, and find its slope. We can do the same thing with the equation in Statement 2 to find the slope of line L. So we'll definitely be able to answer the question using Statement 2; if we know the slopes of both lines, we'll know if they are negative reciprocals of each other. We don't need to calculate both of them.

For Statement 1 though, you need to do some work. Rewriting ax + by = c in the standard form, we find by = -ax + c, and y = (-a/b)x + c/b (if you want to save time, you can ignore "c" term, because it has no effect on the slope, only on the y-intercept). So the slope of L is -a/b. From Statement 1, -2a = 7b, so -a/b = 7/2. So Statement 1 lets us find the slope of L, which is all we needed to do. So the answer is D.

Nups1324 · Answer

Hi fmcgee1 and dave13 , I think I got your doubt and here is a short explanation. The reason slope of a/b is negative is because we know a/b has a negative sign. Check the explanation by GMATinsight again and you'll come to know. If not then I'm attaching a working. Refer to it and you'll be clear. Thank you

Posted from my mobile device

IanStewart · Answer

I haven't read the post you're quoting, but if you rewrite the equation ax + by = c so it looks like "y = mx + p", where m is the slope, you'll get y = (-a/b)x + something else that doesn't matter. So the slope of the line is -a/b. It seems confusing to me to rewrite the question as "Is a/b = -7/2?", even though that is correct. It makes more sense to me to write the question as "Is the slope equal to 7/2?" which becomes "Is -a/b = 7/2?", since -a/b is the slope of the line.

Is the line l given by ax+by=c perpendicular to the line 2x+7y=3?

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Is the line l given by ax+by=c perpendicular to the line 2x+7y=3?

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