M08-15 : GMAT Club Tests

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askhere

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Re: M08-15 [#permalink]

01 Jun 2015, 22:36

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Mrtinhnv wrote:

Dear Bunuel!
I can find the slope of the first line is m and the slope of the second line is 1
But I don’t know How can we find value of a if we know the slope of two lines. You can more detail.
Thanks so much!

Slope of a line is a measure of its inclination with respect to the horizontal.
The more the inclination the more will be the slope, lesser the inclination less will be the slope.
In general, slope of a line is defined as m = tan θ , where θ is the angle between the line and the horizontal axis.
(Even though trigonometry is not involved in GMAT, if you understand the concept of slope it will help you in solving
problems such as these. In fact the formula for slope m = (y2-y1)/(x2-x1) is actually derived from m = tan θ)

So, if you know slope of a line, you know the angle with which its is inclined. [tan^-1 (m)]
Since the given problem is a data sufficiency problem you dont have to solve for anything. You just need to say whether the info given is sufficient to
answer the question.

See attached figure
From the question stem you know the slope of one line which means the angle at which this line is inclined (θ1)
From the statement 1 you know the slope of other line which means the angle at which this line is inclined (θ2)
The difference between these two angles will give you the angle at which they are intersected.

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slope.jpg [ 20.84 KiB | Viewed 38280 times ]

atl12688

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Re: M08-15 [#permalink]

25 Jun 2015, 17:29

I'm struggling to see how Slope of line two equals 1. Can anyone explain that part with some added detail, thank you!

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Bunuel

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Re: M08-15 [#permalink]

26 Jun 2015, 01:55

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atl12688 wrote:

I'm struggling to see how Slope of line two equals 1. Can anyone explain that part with some added detail, thank you!

If the equation of the line is given in the Point-intercept form: y=mx+b, then m is the slope.

Therefore, the slope of y=x−bm, is 1 (coefficient of x).

Check for more here: math-coordinate-geometry-87652.html

Hope it helps.

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Re: M08-15 [#permalink]

03 Jun 2019, 20:26

I think this is a high-quality question.

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Re: M08-15 [#permalink]

01 Jun 2020, 05:28

Bunuel wrote:

Official Solution:

The angle between the two lines depends on their slope (the same way as the angle between a line and x-axis depends on the slope of that line). We have equations of two lines $y=mx+b$ and $y=x-bm$, so the slope of the first line is $m$ and the slope of the second line is 1. Basically all we need to find is the value of $m$.

(1) $m=2$. Sufficient.

(2) $m=b$. Irrelevant information. Not sufficient.

Answer: A

VeritasKarishma, chetan2u

Can you pls explain how knowing that the slope is 1 & 2 here, helps us in finding a

Thanks in advance!

L

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Re: M08-15 [#permalink]

02 Jun 2020, 01:00

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GDT wrote:

Bunuel wrote:

Official Solution:

The angle between the two lines depends on their slope (the same way as the angle between a line and x-axis depends on the slope of that line). We have equations of two lines $y=mx+b$ and $y=x-bm$, so the slope of the first line is $m$ and the slope of the second line is 1. Basically all we need to find is the value of $m$.

(1) $m=2$. Sufficient.

(2) $m=b$. Irrelevant information. Not sufficient.

Answer: A

VeritasKarishma, chetan2u

Can you pls explain how knowing that the slope is 1 & 2 here, helps us in finding a

Thanks in advance!

Think about it - when you know the slope of a line, you basically know the angle it makes with the x axis.
So knowing m tells us the slope of both the lines i.e. the angle made by each with the x axis and that will stay the same.
It doesn't matter where they intersect. The angle made between them will just be the difference between the angles they make on the x axis.

Say line 1 makes an angle of 80 degrees with the x axis and line 2 makes an angle of 50 degrees with the x axis. The angle between them will be 30 degrees.

@ankhere has done a great job of showing this with a diagram here: https://gmatclub.com/forum/m08-183786.html#p1533190

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Re: M08-15 [#permalink]

02 Jun 2020, 02:42

VeritasKarishma wrote:

GDT wrote:

Bunuel wrote:

Official Solution:

The angle between the two lines depends on their slope (the same way as the angle between a line and x-axis depends on the slope of that line). We have equations of two lines $y=mx+b$ and $y=x-bm$, so the slope of the first line is $m$ and the slope of the second line is 1. Basically all we need to find is the value of $m$.

(1) $m=2$. Sufficient.

(2) $m=b$. Irrelevant information. Not sufficient.

Answer: A

VeritasKarishma, chetan2u

Can you pls explain how knowing that the slope is 1 & 2 here, helps us in finding a

Thanks in advance!

Think about it - when you know the slope of a line, you basically know the angle it makes with the x axis.
So knowing m tells us the slope of both the lines i.e. the angle made by each with the x axis and that will stay the same.
It doesn't matter where they intersect. The angle made between them will just be the difference between the angles they make on the x axis.

Say line 1 makes an angle of 80 degrees with the x axis and line 2 makes an angle of 50 degrees with the x axis. The angle between them will be 30 degrees.

@ankhere has done a great job of showing this with a diagram here: https://gmatclub.com/forum/m08-183786.html#p1533190

VeritasKarishma

Though this is a DS ques, but still if we were to find the value of angle a, will the following process be correct?

So, if m=1 then tan θ=1; θ=45 degree and for m=2 θ=90 degree, Is this correct? because tan 45 is 1 but tan 90 is undefined so how do we get angle for that

Thanks in advance!

L

KarishmaB

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Re: M08-15 [#permalink]

03 Jun 2020, 05:50

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GDT wrote:

VeritasKarishma wrote:

GDT wrote:

Bunuel wrote:

Official Solution:

The angle between the two lines depends on their slope (the same way as the angle between a line and x-axis depends on the slope of that line). We have equations of two lines $y=mx+b$ and $y=x-bm$, so the slope of the first line is $m$ and the slope of the second line is 1. Basically all we need to find is the value of $m$.

(1) $m=2$. Sufficient.

(2) $m=b$. Irrelevant information. Not sufficient.

Answer: A

VeritasKarishma, chetan2u

Can you pls explain how knowing that the slope is 1 & 2 here, helps us in finding a

Thanks in advance!

Think about it - when you know the slope of a line, you basically know the angle it makes with the x axis.
So knowing m tells us the slope of both the lines i.e. the angle made by each with the x axis and that will stay the same.
It doesn't matter where they intersect. The angle made between them will just be the difference between the angles they make on the x axis.

Say line 1 makes an angle of 80 degrees with the x axis and line 2 makes an angle of 50 degrees with the x axis. The angle between them will be 30 degrees.

@ankhere has done a great job of showing this with a diagram here: https://gmatclub.com/forum/m08-183786.html#p1533190

VeritasKarishma

Though this is a DS ques, but still if we were to find the value of angle a, will the following process be correct?

So, if m=1 then tan θ=1; θ=45 degree and for m=2 θ=90 degree, Is this correct? because tan 45 is 1 but tan 90 is undefined so how do we get angle for that

Thanks in advance!

When m = 1, tan θ = 1, θ = 45 degrees
When m = 2, tan θ = 2, θ = somewhere near 75-80 degrees (from the graph)

Attachment:

b0314c458ecaf2909d5e63fb0e619ed2.png [ 8.75 KiB | Viewed 3728 times ]

When θ = 90 degrees, tan θ is not defined.

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dave13

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Re: M08-15 [#permalink]

01 Mar 2022, 03:11

Bunuel wrote:

atl12688 wrote:

I'm struggling to see how Slope of line two equals 1. Can anyone explain that part with some added detail, thank you!

If the equation of the line is given in the Point-intercept form: y=mx+b, then m is the slope.

Therefore, the slope of y=x−bm, is 1 (coefficient of x).

Check for more here: https://gmatclub.com/forum/math-coordina ... 87652.html

Hope it helps.

KarishmaB checked other posts but still have questions...
how do we know that slope of y=x−bm is 1 ?

i equated to equations of two lines (since one of the points is the same for both lines)

so with statement one i got this
2x+b=-b2+x ----> so i get x=-b

no b is Y-intercept and x is x-intercept ... and now what

so how can it be sufficient ?

L

KarishmaB

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Re: M08-15 [#permalink]

01 Mar 2022, 04:20

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dave13 wrote:

Bunuel wrote:

atl12688 wrote:

I'm struggling to see how Slope of line two equals 1. Can anyone explain that part with some added detail, thank you!

If the equation of the line is given in the Point-intercept form: y=mx+b, then m is the slope.

Therefore, the slope of y=x−bm, is 1 (coefficient of x).

Check for more here: https://gmatclub.com/forum/math-coordina ... 87652.html

Hope it helps.

KarishmaB checked other posts but still have questions...
how do we know that slope of y=x−bm is 1 ?

i equated to equations of two lines (since one of the points is the same for both lines)

so with statement one i got this
2x+b=-b2+x ----> so i get x=-b

no b is Y-intercept and x is x-intercept ... and now what

so how can it be sufficient ?

The form of the equation of a line is
y = mx + c
Whatever is the co-efficient of x, it is the slope and c is the y intercept.

y = x - bm
is same as y = 1*x - bm
Here, co-effiicient of x is 1 so slope is 1 and -bm is the y intercept.

We solve the equations together when we want to find their point of intersection (the x and y co-ordinates of the point) but we don't care about that here. We just need the angle between the lines. The angle of a line with the X axis is completely defined by its slope. If we know the slope, we know the angle the line makes with the X axis.
Since statement 1 gives us the slope of both the lines, we know the angles made by each line with the X axis. Say if one line makes an angle of 75 degrees and the other makes an angle of 45 degrees, we know that the angle between them is 30 degrees.
So angle made by each line with the X axis is sufficient to give us the angle between the two lines.
That is why statement 1 is sufficient.

L

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Re: M08-15 [#permalink]

20 Aug 2023, 06:24

Expert Reply

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

gmatclubot

Re: M08-15 [#permalink]

20 Aug 2023, 06:24