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

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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.

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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]
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]
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 [ 59.21 KiB | Viewed 831 times ]

Now see if this makes sense:
Attachment:

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

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]
1
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saby1410 wrote:
KarishmaB

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

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?
Re: In the xy-plane, lines k and l intersect at the point (1,1). Is the y- [#permalink]
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