Lines n and p lie in the xy-plane. Is the slope of line n less than th

Algebraic approach:

Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p?

We have two lines: \(y_n=m_1x+b_1\) and \(y_p=m_2x+b_2\). Q: \(m_1<m_2\) true?

(1) Lines n and p intersect at the point (5,1) --> \(1=5m_1+b_1=5m_2+b_2\) --> \(5(m_1-m_2)=b_2-b_1\). Not sufficient.

(2) The y-intercept of line \(n\) is greater than the y-intercept of line \(p\) --> y-intercept is value of \(y\) for \(x=0\), so it's the value of \(b\) --> \(b_1>b_2\) or \(b_2-b_1<0\). Not sufficient.

(1)+(2) \(5(m_1-m_2)=b_2-b_1\), as from (2) \(b_2-b_1<0\) (RHS), then LHS (left hand side) also is less than zero \(5(m_1-m_2)<0\) --> \(m_1-m_2<0\) --> \(m_1<m_2\). Sufficient.

Answer: C.

For more on this topic check Coordinate Geometry Chapter of Math Book: math-coordinate-geometry-87652.html

Hope it helps.
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Graphic approach:

Lines n and p lie in the xy plane. Is the slope of line n less than the slope of line p?

(1) Lines n and p intersect at (5,1)
(2) The y-intercept of line n is greater than y-intercept of line p

The two statements individually are not sufficient.

(1)+(2) Note that a higher absolute value of a slope indicates a steeper incline.

Now, if both lines have positive slopes then as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line p is steeper hence its slope is greater than the slope of the line n:



If both lines have negative slopes then again as the y-intercept of line n (blue) is greater than y-intercept of line p (red) then the line n is steeper hence the absolute value of its slope is greater than the absolute value of the slope of the line p, so the slope of n is more negative than the slope of p, which means that the slope of p is greater than the slope of n:



So in both cases the slope of p is greater than the slope of n. Sufficient.

Answer: C.


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Bunuel,

What if line p has a negative y intercept but line n has a positive intercept? Wouldn't that give the oposite answer?

If line p has a negative y-intercept then its slope is positive and it will still be more than the slope of n, with positive y-intercept (if the slope of n will be positive than p will still be steeper than n, and if the slope of n is negative it obviously will be less than positive slope of p). Consider first image and rotate line n (blue) so that it to have positive y-intercept and you'll easily see the answer.

Check the following links for similar questions, where different scenarios are considered:
in-the-xy-plane-is-the-slope-of-line-l-greater-than-the-126941.html
if-the-slopes-of-the-line-l1-and-l2-are-of-the-same-sign-is-126759.html
slopes-of-m-and-n-124025.html

Also check Coordinate Geometry chapter of Math Book for theory on this subject: math-coordinate-geometry-87652.html

Hope it helps.
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Bunuel,

In here - \(y_n=m_1x+b_1\) and \(y_p=m_2x+b_2\). Q: \(m_1<m_2\) true?

Why have you chosen different variables for the y?

Shouldnt the two equations be y=m1x+b1 and y=m2x+b2? We always form the equation from the basic form of y=mx+c wherein we substitute the values of m and c. And if that is the case, we can get the answer from statement II only.

I know I am missing something but I am not clear as to why you have picked different variables for y but not for x.

n and p are subscripts of y's, not variables.

\(y=m_1x+b_1\) is equation of line n.
\(y=m_2x+b_2\) is equation of line p.

I used subscripts simply to distinguish one equation from another.
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If that is the case then, from the above equations

we get b1=y-m1x and b2=y-m2x

Now from statement 2 we know that b1>b2...

therefore, y-m1x >y-m2x

which gives (m1-m2)x>0

So it can be proved from statement 2 only that m1>m2

Where am I going wrong?

The y-intercept is the value of \(y\) for \(x=0\). You should substitute x=0 into both equations.

So, the y-intercept of line n is b1 and the y-intercept of line p is b2, from (2) we only have that b1>b2.
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Experts

Please reply for my doubt.....
for below attached fig ...getting 2 diff.. answers.....

Responding to a pm:

I am not sure I understand why you say you are getting two different answers. In both diagrams, the slope of n is less than the slope of p. We are comparing actual values of the slopes, not just the absolute values.
So say slope of n is -2 in both cases. Slope of p in the first diagram will be -1/2 and slope of p in the second diagram would be about 1.
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Answer: Option C

Check the cases as per color coding

Here is a great resource to understand slope. You can plug in values and see how things change:
https://www.desmos.com/calculator/nuokqfhfxi

Hope this helps

We need to determine whether the slope of line n is less than the slope of line p.

Statement One Alone:

Lines n and p intersect at the point (5,1).

If the two lines intersect at a point, they are not parallel and hence their slopes are not equal (unless they are identical lines). So the slope of one line must be greater than the slope of the other line. However, we can’t determine which line has the greater slope. Statement one alone is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

The y-intercept of line n is greater than the y-intercept of line p.

Knowing that the y-intercept of one line is greater than the y-intercept of the other does not allow us to determine which line has the greater slope.

For example, line n could have a y-intercept 2 and slope 3, and line p could have a y-intercept 1 and slope 2. In this case, line p has the lesser slope. However, it’s also possible that line n could have a y-intercept 2 and slope 2, and line p could have a y-intercept 1 and slope 3. In which case, line n has the lesser slope. Statement two alone is not sufficient. Eliminate answer choice B.

Statements One and Two Together:

Knowing the point where the two lines intersect and the relationship of the y-intercept of each line allows us to determine which line has the lesser slope.

Even though we don’t know the actual y-intercept of each line, we know that the y-intercept of line n is greater than that of line p. So we can let the y-intercept of line n be b, and that of line p be c where b > c.

Thus, line n passes through (0, b), and line p passes through (0, c). Both lines also pass through (5, 1). Let’s calculate their slopes:

Slope of line n = (1 – b)/(5 – 0) = (1 – b)/5

Slope of line p = (1 – c)/(5 – 0) = (1 – c)/5

Now let’s determine whether (1 – b)/5 < (1 – c)/5.

Is (1 – b)/5 < (1 – c)/5 ?

Is 1 – b < 1 – c ?

Is –b < –c ?

Is b > c ?

Since, from the information in statement two, we know that b is greater than c, we have answered the question: the slope of line n is indeed less than that of line p.

Answer: C
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Am I right in my understanding that since the question asks for whether the slope of line n is greater than the slope of p, we need to consider only actual values and not absolute values.

If the question had asked for whether line n is steeper than line p, then we would have had to consider absolute values and in this second case, the answer will actually have been E?

Bunuel or Karishma or other experts please clarify.

Yes, the question asks whether \(m_1<m_2\) is true (not whether \(|m_1|<|m_2|\) is true). Every solution on the previous two pages answers exactly this question.
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What if the slope of n is negative and the slope of p is positive?

This would contradict the second statement: "The y-intercept of line n is greater than the y-intercept of line p".
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If you try to gauge the difficulty of this question, would it be a 400-level question, 500-level, 600-level, etc?

Coordinate plane questions tend to be most difficult for me right now. I'm working on my ability to solve these types. So, it would be helpful to be able to gauge the degree of difficulty for problems that trouble me.

Hello

The questions here on GmatClub are tagged as 'Sub 600' or '600-700' or '700' levels.

If you look at this question, this is tagged as '600-700' level question. (Question tags are visible on the question post only).

You can also see there that this is a 45% difficulty level (medium) based on 1029 sessions, out of which 62% got it correct and 38% wrong (These stats are till now, as of this comment of mine, but will change as more and more people solve this question using timer)