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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
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woohoo921 wrote:
KarishmaB wrote:
nalinnair wrote:
In the xy-plane, lines k and l intersect at the point (1,1). Is the y-intercept of k greater than the y-intercept of l?

(1) The slope of k is less than the slope of l.
(2) The slope of l is positive.



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

https://www.gmatclub.com/forum/veritas- ... -vertices/
https://www.gmatclub.com/forum/veritas- ... line-gmat/


KarishmaB
If you have time, I would be so appreciative to understand how you may approach this algebraically. I think Brent's response with the graphs are helpful, but for some reason I am still struggling with wondering if there are more cases, so I am hoping to see if there is a way to prove it algebraically. Many thanks :)


The equation of a line is y = mx + c where m is the slope and c is the y intercept. Write the equations for the two lines passing through (1, 1) as:

\(1 = m_k + c_k\)
\(1 = m_l + c_l\)

So \(m_k + c_k = m_l + c_l\)

Stmnt 1:
Given: \(m_k < m_l\)
Then \(c_k > c_l\). Only then will the two sums be equal above.
Hence, sufficient alone.

Stmtn 2:
\(m_l > 0\)
Not sufficient to tell us the relation between the two y intercepts.

Answer (A)
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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
chetan2u wrote:
nalinnair wrote:
In the \(xy\)-plane, lines \(k\) and \(l\) intersect at the point \((1,1)\). Is the \(y\)-intercept of \(k\) greater than the \(y\)-intercept of \(l\)?

(1) The slope of \(k\) is less than the slope of \(l\).
(2) The slope of \(l\) is positive.



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


Hi,
I am unable to imagine one case for statement 1- What if line *l* has a positive slope and line k have a negative slope? Line L's y-intercept can be negative and line k's slope positive.
Can you please elaborate more on this?
Thank you
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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
Expert Reply
RenB wrote:
chetan2u wrote:
nalinnair wrote:
In the \(xy\)-plane, lines \(k\) and \(l\) intersect at the point \((1,1)\). Is the \(y\)-intercept of \(k\) greater than the \(y\)-intercept of \(l\)?

(1) The slope of \(k\) is less than the slope of \(l\).
(2) The slope of \(l\) is positive.



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


Hi,
I am unable to imagine one case for statement 1- What if line *l* has a positive slope and line k have a negative slope? Line L's y-intercept can be negative and line k's slope positive.
Can you please elaborate more on this?
Thank you



For a line passing through (1, 1), if the y intercept is 1, the slope of the line is 0. (try to draw this. It is parallel to x axis)
If y intercept is less than 1, slope of line is positive.
If y intercept is more than 1, slope of line is negative.

So if line l has positive slope and line k has negative slope, then (y intercept of line k) > (y intercept of line l).
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In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
KarishmaB")[quote="chetan2u wrote:
nalinnair wrote:
In the \(xy\)-plane, lines \(k\) and \(l\) intersect at the point \((1,1)\). Is the \(y\)-intercept of \(k\) greater than the \(y\)-intercept of \(l\)?

(1) The slope of \(k\) is less than the slope of \(l\).
(2) The slope of \(l\) is positive.



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


Hi,
I am unable to imagine one case for statement 1- What if line *l* has a positive slope and line k have a negative slope? Line L's y-intercept can be negative and line k's slope positive.
Can you please elaborate more on this?
Thank you



For a line passing through (1, 1), if the y intercept is 1, the slope of the line is 0. (try to draw this. It is parallel to x axis)
If y intercept is less than 1, slope of line is positive.
If y intercept is more than 1, slope of line is negative.

So if line l has positive slope and line k has negative slope, then (y intercept of line k) > (y intercept of line l).[/quote]
KarishmaB
Can you explain the above point with graph or with example facing difficulty in understanding
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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
Expert Reply
saby1410 wrote:
KarishmaB")[quote="chetan2u wrote:
nalinnair wrote:
In the \(xy\)-plane, lines \(k\) and \(l\) intersect at the point \((1,1)\). Is the \(y\)-intercept of \(k\) greater than the \(y\)-intercept of \(l\)?

(1) The slope of \(k\) is less than the slope of \(l\).
(2) The slope of \(l\) is positive.



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


Hi,
I am unable to imagine one case for statement 1- What if line *l* has a positive slope and line k have a negative slope? Line L's y-intercept can be negative and line k's slope positive.
Can you please elaborate more on this?
Thank you



For a line passing through (1, 1), if the y intercept is 1, the slope of the line is 0. (try to draw this. It is parallel to x axis)
If y intercept is less than 1, slope of line is positive.
If y intercept is more than 1, slope of line is negative.

So if line l has positive slope and line k has negative slope, then (y intercept of line k) > (y intercept of line l).

KarishmaB
Can you explain the above point with graph or with example facing difficulty in understanding[/quote]


First check this:
Attachment:
Screenshot 2023-04-26 at 9.39.02 PM.png
Screenshot 2023-04-26 at 9.39.02 PM.png [ 59.21 KiB | Viewed 831 times ]


Now see if this makes sense:
Attachment:
Screenshot 2023-04-26 at 9.48.12 PM.png
Screenshot 2023-04-26 at 9.48.12 PM.png [ 31.83 KiB | Viewed 849 times ]
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In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
KarishmaB

Thanks for your explaination


can you explain one more thing in a solution given by BrentGMATPrepNow
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

i do have doubt when slope is positive(0 to 90 degrees) increases in that case y intercept also increases but solution by brentgmat prepnow I'm unable to get why k will have higher y intercept
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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
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saby1410 wrote:
KarishmaB

Thanks for your explaination


can you explain one more thing in a solution given by BrentGMATPrepNow
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

i do have doubt when slope is positive(0 to 90 degrees) increases in that case y intercept also increases but solution by brentgmat prepnow I'm unable to get why k will have higher y intercept


Your one point is fixed (1, 1). The line must through that point. So think of a line rooted at point (1, 1) but free to rotate.
Say it is the green line in the second diagram above. It passes through (0, -0.5) too right now. What happens if you turn it up so that it passes through (0, 0)? The y intercept has increased but the slope has reduced. Keep increasing the y intercept so that it overlaps with the black line. Its y intercept has increased a whole lot and its slope has reduced and become negative. So lower the slope, higher the y intercept.
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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
KarishmaB wrote:
saby1410 wrote:
KarishmaB

Thanks for your explaination


can you explain one more thing in a solution given by BrentGMATPrepNow
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

i do have doubt when slope is positive(0 to 90 degrees) increases in that case y intercept also increases but solution by brentgmat prepnow I'm unable to get why k will have higher y intercept


Your one point is fixed (1, 1). The line must through that point. So think of a line rooted at point (1, 1) but free to rotate.
Say it is the green line in the second diagram above. It passes through (0, -0.5) too right now. What happens if you turn it up so that it passes through (0, 0)? The y intercept has increased but the slope has reduced. Keep increasing the y intercept so that it overlaps with the black line. Its y intercept has increased a whole lot and its slope has reduced and become negative. So lower the slope, higher the y intercept.



THANKS KarishmaB . your way of explaining things is very easy. thanks alot
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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
nalinnair wrote:
In the xy-plane, lines k and l intersect at the point (1,1). Is the y-intercept of k greater than the y-intercept of l?

(1) The slope of k is less than the slope of l.
(2) The slope of l is positive.


What if one of the lines is parallel to Y axis. In such a case there would not be a y-intercept for the given line. What would be the solution in this circumstance?
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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
 
chetan2u wrote:
nalinnair wrote:
In the \(xy\)-plane, lines \(k\) and \(l\) intersect at the point \((1,1)\). Is the \(y\)-intercept of \(k\) greater than the \(y\)-intercept of \(l\)?

(1) The slope of \(k\) is less than the slope of \(l\).
(2) The slope of \(l\) is positive.


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

Hi,

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

Can I consider this as a principle valid for all cases or only applicable to this question?
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Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
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