In the xy-plane, lines k and l intersect at the point (1,1). Is the y-

We can write the equation of the line in slope-intercept form as follow: [Where M is the slope and C is the y-intercept]

For Line K : Yk = Mk.Xk +Ck
For Line L : Yl = Ml.Xl + Cl

Now, since these lines intercept at (1,1), it must satisfy the equation

For Line K : 1 = Mk.1 +Ck
For Line L : 1 = Ml.1 + Cl

=> Mk + Ck = Ml + Cl ------ (equation I)

Now, From Statement 1 - Mk > Ml. Thus, for equation I to hold true. Y intercept of line k must be less than Y intercept of line l (Ck < Cl). Therefore Statement 1 is Sufficient.

Statement 2: Is not sufficient as it doesn't provide information regarding the slope of line L.

first line equation 1= m+c , second line equation 1 = n+b , question asks whether c>b

from equations

c = 1-m , b = 1-n , therefore we need relation between the 2 slopes to solve

from 1 m>n this (1-m) < ( 1-n) and therefore b>c... suff

from 2

we have only one slope and no relation between 2 slopes ( insuff)

A

I had a really hard time following and understanding the above discussion... Therefore I tried to use a more graphical approach and see how S(1) is sufficient!

We should consider 3 cases for S(1): Booth slopes are negative, booth slopes are positive, and slope \(K\) is negative and slope \(L\) is positive.

Case 1. Negative
\(Line K:\)\(y=-2x+3\)
\(Line L:\)\(y=-1x+2\)



Case 2. Positive
\(Line K:\)\(y=2x-1\)
\(Line L:\)\(y=3x-2\)



Case 3. \(K\) negative and \(L\) positive
\(Line K:\)\(y=-2x+3\)
\(Line L:\)\(y=3x-2\)


We can see in every case \(K\) has a higher y intercept than \(L\).

Therefore S(1) is sufficient and A is the answer.

Hi,

An easy question if you are aware of the slopes....

slope is the Increase in Y/ Increase in X.......... OR Vertical increase / Horizontal increase..

If BOTH increase with increase in each other, the SLOPE is Positive... But if Y decreases with increase in X, slope is -ive....
so for a point (1,1), a line intercepting above Y as 1, it is -ive and with increase in Y, the slope will have even lesser value...
At y=1, slope is 0 and at y<1, the slopw will keep increasing as the value of Y increases...

So the relation of y-intercept is dependent on slope.. lesser the slope higher is the intercept..

(1) The slope of \(k\) is less than the slope of \(l\).
As seen above y intercept of line l will be more than line k...
Suff

(2) The slope of \(l\) is positive.
we require to know slope of line k also
Insuff

A
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Thanks, chetan2u great response. I am trying to understand the relationship of the slope and y intercept. I think irrespective of the line is increasing or decreasing the slope defines lines steepness w.r.t the y -axis. Thus, as we increase the slope our y -intercept should increase. [considering both the line don't have a common intersecting point on the y-axis]. Please let me know if my understanding is correct?

Hi...

If you just take the absolute value, yes with change in x as constant, change in y will be as per your understanding..

The slope depends on change in value of both x and y.
Numerator has change in y, so more the change more the slope
Denominator has change in x, so less the change more the slope

But the slope can be of two types..
From left bottom to right top...... Both change in x and y will be POSITIVE, so slope is POSITIVE
From left top to bottom right......Change in y is NEGATIVE and change in x will be POSITIVE, so slope is -/+= NEGATIVE even if the absolute value or steepness is MORE
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please see the attachment,
let say Ml= -4
Mk= -6
then the slope of k is less than slope of L
y intercept K is greater than L

so I think we need both statement to confirm.
isn't it?

Bunuel or anyone, please correct me

If the slope of line L is -4 and the slope of line K is -6, the lines won't look like the way you've drawn. They will look like as below:




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Oh yea, I mean the red line is K
and the blue line is L
so then, even the slope L is larger than slope K,
y intercept of K is still larger than L
am I correct?
that means, statement 1 is not suff.
we also need to know if it is positive slope or negative

(1) says that the slope of \(k\) is less than the slope of \(l\). So, in my image BLUE line is K (slope -6) and the RED line is L (slope -4).

As you can see the y-intercept of K is greater than the y-intercept of L. For (1) the y-intercept of K will always be greater than the y-intercept of L.
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I agree with msurls, if for line K we have equation X= costant, how do you define the slope and therefore the intercept with Y (which doesn't exist) ? Do you consider the slope of a line parallel to Y axis as +infinite or (-) infinite...? In this case how can you say that 1) is suff, since you don't have any value of y-intercept for line k ?

A vertical line has no slope. Or put another way, for a vertical line the slope is undefined. This, by the way, does NOT mean that its slope is 0, horizontal lines has slope equal to 0.

Statement (1) says that "The slope of k is less than the slope of l". If any of the lines were vertical, their slopes would be undefined and it would not make sense to compare an undefined slope to anything. Thus, (1) implies that neither of the lines is vertical.
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Hope it helps.
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Given: In the xy-plane, lines k and l intersect at the point (1,1).

Target question: Is the y-intercept of k greater than the y-intercept of l?

Statement 1: The slope of k is less than the slope of l.
Let's examine 2 cases, and then make a generalization.

CASE A) the slope of line k is POSITIVE
For example, let's say line k has slope 1.

If line l has a greater slope (like a slope of 2), we can see that the y-intercept of k IS greater than the y-intercept of l
We can further see that if line l has a slope of 3, 4, 5, 6 etc , it will always be the case that the y-intercept of k IS greater than the y-intercept of l

CASE B) the slope of line k is NEGATIVE
For example, let's say line k has slope -1.5

If line l has a greater slope (like a slope of -0.6), we can see that the y-intercept of k IS greater than the y-intercept of l
We can further see that if line l has a slope that's greater than line k, it will always be the case that the y-intercept of k IS greater than the y-intercept of l

In both of the above cases, the answer to the target question will be the y-intercept of k IS greater than the y-intercept of l 00
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The slope of l is positive.
No information about line k
Statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

GMATPrepNow

I'm having difficulty in understanding the working of this question. By saying the slope is less, does that mean a slope of -1 is less than 1? I understand that one is positive and the other negative but the magnitude of the slope is equivalent and thus I would conclude that the slope is the same.

Lines with slopes 1 and -1 have the same STEEPNESS, but the slopes are different.
So, the SLOPE of -1 is less than the SLOPE of 1

Cheers,
Brent

Let line k be \(y-1 = m_1(x-1) => y=m_1x+(1-m_1)\)
Let line l be \(y-1 = m_2(x-1) => y =m_2x+(1-m_2)\)

Statement 1
\(m_1<m_2\)
\(=> 1-m_1>1-m_2\)
y intercept of k > y intercept of l
SUFFICIENT

Statement 2
m_2>0
NOT SUFFICIENT

IMO A

The equation of a line can be written as
\(y=mx+c\)
Since the 2 lines pass through (1,1), we can write this as
\(1=m+c\)
c is the y intercept as we get \(y=c\) when \(x=0\)
Hence
\(c=1-m\)

(1) The slope of k is less than the slope of l.
Hence c will always be greater for k -Sufficient

2) The slope of l is positive.
No info about l -Insufficient

A

Here are two posts that will help you understand the slope-intercept concept:

https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... -vertices/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2016/0 ... line-gmat/
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