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Manager  Joined: 10 Feb 2011
Posts: 104
If line l passes through point (m, n), is the slope of the  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 54% (02:12) correct 46% (01:53) wrong based on 142 sessions

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If line l passes through point (m,– n), is the slope of the line negative?

(1) The line passes through point (–m, n).
(2) mn is negative.
Manager  Joined: 07 Jun 2010
Posts: 76
Re: 179. If line l passes through point (m,– n), is the slope of  [#permalink]

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E. There are an infinite number of lines of every possible slope that can pass through a point. You need a second point to determine the slope of a line and neither question helps.
Manager  Joined: 10 Feb 2011
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Re: 179. If line l passes through point (m,– n), is the slope of  [#permalink]

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but we got 2 points here: (m,-n) and (-m, n) throw which passes a line L, no?...
Retired Moderator Joined: 20 Dec 2010
Posts: 1572
Re: 179. If line l passes through point (m,– n), is the slope of  [#permalink]

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179. If line l passes through point (m,– n), is the slope of the line negative?
(1) The line passes through point (–m, n).
(2) mn is negative.

Use sample numbers and test:

1:
Case: I
Let's put some values for m and n
m=1
n=-1

so{m,-n} = {1,1}

statement tells us: line also passes through; {-m,n} = {-1,1}

Slope of a line passing through {1,1} and {1,-1} would be undefined.
(y2-y1)/(x2-x1) = -1-1/1-1 = -2/0 = undefined

Case II:
m=-1
n=1

so{m,-n} = {-1,-1}

statement tells us: line also passes through; {-m,n} = {1,-1}

Slope of a line passing through {-1,-1} and {1,-1} would be.
(y2-y1)/(x2-x1) = -1+1/1+1 = 0/2 = 0. Neither +ve nor -ve.

Case III:
m=1
n=1
{m,-n} = {1,-1}
{-m,n} = {-1,1}
Slope = 1+1/-1-1 = 2/-2 = -1 Negative.

case IV:
m=-1
n=-1
{m,-n} = {-1,1}
{-m,n} = {1,-1}
Slope = -1-1/1+1 = -2/2=-1 Negative

Not sufficient.

2. mn is -ve.
In the above sample set; for caseI and case2, mn is -ve and they both are yielding different signs for slopes.
Not sufficient.

Using both; caseI and caseII from statement1 will result in different types of slopes. Not Sufficient.

Ans: "E"
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Re: 179. If line l passes through point (m,– n), is the slope of  [#permalink]

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Hi

For Case I

statement tells us: line also passes through; {-m,n} = {-1,1}

Slope of a line passing through {1,1} and {1,-1} would be undefined.
(y2-y1)/(x2-x1) = -1-1/1-1 = -2/0 = undefined

n = -1, right ?

And the denominator is not equal to 0, not sure if I'm making a mistake in reading this ?
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Retired Moderator Joined: 20 Dec 2010
Posts: 1572
Re: 179. If line l passes through point (m,– n), is the slope of  [#permalink]

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subhashghosh wrote:
Hi

For Case I

statement tells us: line also passes through; {-m,n} = {-1,1}

Slope of a line passing through {1,1} and {1,-1} would be undefined.
(y2-y1)/(x2-x1) = -1-1/1-1 = -2/0 = undefined

n = -1, right ?

And the denominator is not equal to 0, not sure if I'm making a mistake in reading this ?

I am not sure what are you trying to ask!!!

Let me rephrase few things in caseI:

1:
Case: I

m and n can literally have any value;
Let's use the following values for m and n

m=1
n=-1

so what is {m,-n}

m=1
-n = -(-1) = +1

So; the line passes through (1,1), say point P

statement tells us: line also passes through;
{-m,n}
m=1; -m = -1
n=-1

Line also passes through (1,-1), say point Q

Slope of a line passing through two points P(1,1)=(x1,y1) and Q(1,-1)=(x2,y2) can be defined as;

m = (y2-y1)/(x2-x1)

For the above two points P and Q; what are our x1,y1,x2,y2

x1=1
y1=1
x2=1
y2=-1

(y2-y1)/(x2-x1) = -1-1/1-1 = -2/0 = if 0 is in denominator; the slope becomes undefined. Means; no slope.

You can see that this line that we are talking about passes through (1,1) and (1,-1). It is a line parallel to y axis. There is no slanting in the line and thus has no slope.
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Re: 179. If line l passes through point (m,– n), is the slope of  [#permalink]

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banksy wrote:
179. If line l passes through point (m,– n), is the slope of the line negative?
(1) The line passes through point (–m, n).
(2) mn is negative.

The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ on a line, the slope $$m$$ of the line is: $$m=\frac{y_2-y_1}{x_2-x_1}$$

If line l passes through point (m,– n), is the slope of the line negative?

(1) The line passes through point (–m, n) --> $$slope=\frac{n-(-n)}{-m-m}=-\frac{n}{m}$$, so the question becomes: is $$-\frac{n}{m}<0$$? or do $$m$$ and $$n$$ have the same sign, but we don't know that. Not sufficient.

(2) mn is negative --> $$m$$ and $$n$$ have the opposite signs --> point (m, -n) is either in I or in III quadrant, though as we have only one point lines passing through it can have negative as well as positive slope. Not sufficient.

(1)+(2) As from (2) $$m$$ and $$n$$ have the opposite signs the from (1) $$slope=-\frac{n}{m}>0$$ and the answer to the question is NO. Sufficient.

Without any algebra:

If line l passes through point (m,– n), is the slope of the line negative?

(1) The line passes through point (–m, n). Two cases:

A. If m and n are both positive then point (m, -n)=(positive, negative) is in IV quadrant and the second point (-m, n)=(negative, positive) is in II quadrant line passing these two points will have negative slope;

B. If m and n have the opposite signs, for example m positive and n negative, (m, -n)=(positive, positive) is in I quadrant and the second point (-m, n)=(negative, negative) is in III quadrant, line passing these two points will have positive slope (if it's vise-versa, meaning if m is negative and n positive, then we'll still have the same quadrants: (m, -n)=(negative, negative) is in III quadrant and the second point (-m, n)=(positive, positive) is in I quadrant, line passing these two points will have positive slope). Not sufficient.

(2) mn is negative --> m and n have the opposite signs --> point (m, -n) is either in I quadrant in case (m, -n)=(positive, positive) or in III quadrant in case (m, -n)=(negative, negative), though as we have only one point lines passing through it can have negative as well as positive slope. Not sufficient.

(1)+(2) As from (2) m and n have the opposite signs then we have the case B from (1), whihc means that the slope is positive. Sufficient.

Check Coordinate Geometry chapter of Math Book for more: math-coordinate-geometry-87652.html
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Re: 179. If line l passes through point (m,– n), is the slope of  [#permalink]

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fluke wrote:
subhashghosh wrote:
Hi

For Case I

statement tells us: line also passes through; {-m,n} = {-1,1}

Slope of a line passing through {1,1} and {1,-1} would be undefined.
(y2-y1)/(x2-x1) = -1-1/1-1 = -2/0 = undefined

n = -1, right ?

And the denominator is not equal to 0, not sure if I'm making a mistake in reading this ?

I am not sure what are you trying to ask!!!

Let me rephrase few things in caseI:

1:
Case: I

m and n can literally have any value;
Let's use the following values for m and n

m=1
n=-1

so what is {m,-n}

m=1
-n = -(-1) = +1

So; the line passes through (1,1), say point P

statement tells us: line also passes through;
{-m,n}
m=1; -m = -1
n=-1

Line also passes through (1,-1), say point Q

Slope of a line passing through two points P(1,1)=(x1,y1) and Q(1,-1)=(x2,y2) can be defined as;

m = (y2-y1)/(x2-x1)

For the above two points P and Q; what are our x1,y1,x2,y2

x1=1
y1=1
x2=1
y2=-1

(y2-y1)/(x2-x1) = -1-1/1-1 = -2/0 = if 0 is in denominator; the slope becomes undefined. Means; no slope.

You can see that this line that we are talking about passes through (1,1) and (1,-1). It is a line parallel to y axis. There is no slanting in the line and thus has no slope.

Please ignore both my comments above; they contain calculation errors.
case I:
P should be (1,1) and Q (-1,-1)

I realized my mistake after Bunuel's explanation.
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Re: If line l passes through point (m, n), is the slope of the  [#permalink]

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banksy wrote:
If line l passes through point (m,– n), is the slope of the line negative?

(1) The line passes through point (–m, n).
(2) mn is negative.

Let's see

Is slope negative?

Passes through (m,-n) so actually we don't know much. Note that since 'n' is a variable it could be <0 and (m,-(-n) could as well be in the I st quadrant. So don't fall for (m,-n) being in the IV quadrant necessarily

Back to the question

Statement 1

Now if it passes through both (m,-n) and (-m,n) then the line can be either a positive line that is going from quadrant I to III or a negative line going from quadrant II to IV

Insuff

Statement 2

mn<0, this tells us that (m,n) have opposite signs. Therefore, point (m,-n) is either on the II or IV quadrant. But we know nothing about the slope of the line

Together

Since (m,-n) is on the II or IV quadrant then we have the second case in which the line has a negative slope passing through both II and IV quadrant

Hence answer is C

Hope it helps
Cheers!
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Posts: 13321
Re: If line l passes through point (m, n), is the slope of the  [#permalink]

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_________________ Re: If line l passes through point (m, n), is the slope of the   [#permalink] 25 Aug 2018, 09:30
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