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Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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66 00:00

Difficulty:   65% (hard)

Question Stats: 54% (01:31) correct 46% (01:31) wrong based on 511 sessions

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Does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0

Originally posted by surupab on 10 Apr 2016, 08:38.
Last edited by Bunuel on 10 Apr 2016, 09:51, edited 1 time in total.
Renamed the topic and edited the question.
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Posts: 8290
Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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39
26
surupab wrote:
does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0

Hi,
ONLY line that will not cross x axis is the line parallel to x axis...
THe equation of line parallel to x axis is y=b..
so if ax+by = c has to be parallel to x- axis, a should be 0..

lets see the statements..

(1) b not equal to 0
This tells us that the line is not parallel to y-axis
Insuff

(2) ab>0
this tells us that neither of a or b is 0..
so the line crosses the x-axis
Suff

B
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Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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23
5
Another way:

If line ax+by=c does cross x axis then, there will be a point (z,0) on this line. Which means:

az=c. or z= (c/a). z should have real value and this will be possible when a is not equal to zero.

So our question now becomes: Is a = 0?

statement 1: b not equal to 0; This does not tell anything about a. NS

statement 2: ab>0. a is certainly not equal to 0. Sufficient.

(Note that if we put z = (c/a) in equation; and we know that a is not equal to zero; we get: (b)(y)=0; Now because statement (2) also says b is not equal to zero, so y will have to be zero. This is what we want and this proves the point that this line cuts x axis. )
##### General Discussion
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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i chose E which, i now realize is wrong.
Great solution chetan2u sir. Concept got clear.
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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2
Hi chetan2u,

I have one doubt. If a line passes through origin, then can we say that the line passes through x and y axis both...?
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Thanks and Regards,
Prakhar
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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PrakharGMAT wrote:
Hi chetan2u,

I have one doubt. If a line passes through origin, then can we say that the line passes through x and y axis both...?

Yes, Prakhar the line passing through ORIGIN crosses x-axis at x=0 and y-axis at y=0...
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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chetan2u wrote:
surupab wrote:
does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0

Hi,
ONLY line that will not cross x axis is the line parallel to x axis...
THe equation of line parallel to x axis is y=b..
so if ax+by = c has to be parallel to x- axis, a should be 0..

lets see the statements..

(1) b not equal to 0
This tells us that the line is not parallel to y-axis
Insuff

(2) ab>0
this tells us that neither of a or b is 0..
so the line crosses the x-axis
Suff

B

chetan2u My question is wrt Statement A

If it is clear that the line is not || to y axis that means the line will be || to x axis? Or can we assume that there is a possibility that the line cuts the x axis and still is not || to y axis?
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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Keats wrote:
chetan2u wrote:
surupab wrote:
does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0

Hi,
ONLY line that will not cross x axis is the line parallel to x axis...
THe equation of line parallel to x axis is y=b..
so if ax+by = c has to be parallel to x- axis, a should be 0..

lets see the statements..

(1) b not equal to 0
This tells us that the line is not parallel to y-axis
Insuff

(2) ab>0
this tells us that neither of a or b is 0..
so the line crosses the x-axis
Suff

B

chetan2u My question is wrt Statement A

If it is clear that the line is not || to y axis that means the line will be || to x axis? Or can we assume that there is a possibility that the line cuts the x axis and still is not || to y axis?

Only a line that is not parallel to both the axes, crosses both the axes at different points or at a single point i.e. origin.

Coming to this particular question, as mentioned above, we need to make sure both a and b are not equal to '0'

At a point (c/a,0) line becomes by=0. This requires a statement that b is also not equal to '0'. Statement II (ab>0) hints that both a and b are not 0.

Hence B.
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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PrakharGMAT wrote:
Hi chetan2u,

I have one doubt. If a line passes through origin, then can we say that the line passes through x and y axis both...?

This is a really clever thought!! I would say that, in the spirit of the question, to 'cross though an axis' means to have at least 1 value ABOVE the axis and at least 1 value BELOW the axis. Since every value of y=0 exists AT the axis, the line does not pass through it but merely exists AT it infinitely. That is my take anyways.

Here is a question: Does y = x^2 pass through the x-axis? I think that it does not, because 100% of it's values are at or above the X-axis. It has no values below the X-axis, and therefor it TOUCHES but does not cross? I really don't know, merely speculating.
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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17
12
surupab wrote:
Does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0

We are given a line with equation ax+by = c, where a,b and c are real constants, and we need to determine whether it crosses the x-axis. Let’s start by isolating y in our given equation:

ax+by = c

by = -ax + c

y = (-a/b)x + (c/b)

We now have the given equation in slope-intercept form; the slope is -a/b and the y-intercept is c/b.

Statement One Alone:

b not equal to 0

Since we are not provided any information about the slope of the line, statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

ab>0

Since ab > 0, neither a nor b can be 0. This means that the slope of the line is not equal to zero. Recall that if the slope of a line is 0, the line is horizontal and will not cross the x-axis unless the line is the x-axis itself. Thus, regardless of the values of a or b, the line will, at some point, cross the x-axis.

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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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JeffTargetTestPrep wrote:
surupab wrote:
Does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0

We are given a line with equation ax+by = c, where a,b and c are real constants, and we need to determine whether it crosses the x-axis. Let’s start by isolating y in our given equation:

ax+by = c

by = -ax + c

y = (-a/b)x + (c/b)

We now have the given equation in slope-intercept form; the slope is -a/b and the y-intercept is c/b.

Statement One Alone:

b not equal to 0

Since we are not provided any information about the slope of the line, statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

ab>0

Since ab > 0, neither a nor b can be 0. This means that the slope of the line is not equal to zero. Recall that if the slope of a line is 0, the line is horizontal and will not cross the x-axis unless the line is the x-axis itself. Thus, regardless of the values of a or b, the line will, at some point, cross the x-axis.

Perfect. Thanks. I was thinking that if a line passes through Origin, it can't be passing X-Axis and accordingly thought I need a solution wherein I can have any non-zero value for Y-Intercept, but I guess thinking only along the SLOPE Concept helps. Thanks JeffTargetTestPrep
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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Another way:

If line ax+by=c does cross x axis then, there will be a point (z,0) on this line. Which means:

az=c. or z= (c/a). z should have real value and this will be possible when a is not equal to zero.

So our question now becomes: Is a = 0?

statement 1: b not equal to 0; This does not tell anything about a. NS

statement 2: ab>0. a is certainly not equal to 0. Sufficient.

(Note that if we put z = (c/a) in equation; and we know that a is not equal to zero; we get: (b)(y)=0; Now because statement (2) also says b is not equal to zero, so y will have to be zero. This is what we want and this proves the point that this line cuts x axis. )

You have done it elegantly and I can make quite some sense out of it. On the first go, I just could not draw conclusions out of these statements. But, your way of reducing the question to checking if a = 0 simplifies the case.Thank You adiagr
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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chetan2u wrote:
surupab wrote:
does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0

Hi,
ONLY line that will not cross x axis is the line parallel to x axis...
THe equation of line parallel to x axis is y=b..
so if ax+by = c has to be parallel to x- axis, a should be 0..

lets see the statements..

(1) b not equal to 0
This tells us that the line is not parallel to y-axis
Insuff

(2) ab>0
this tells us that neither of a or b is 0..
so the line crosses the x-axis
Suff

B

Nice solution! Do you have an idea about what one should know with regard to the general form equation for a straight line (ax+by+c=0)? Only y=mx+c is covered in the OG.
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Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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JeffTargetTestPrep wrote:
surupab wrote:
Does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0

We are given a line with equation ax+by = c, where a,b and c are real constants, and we need to determine whether it crosses the x-axis. Let’s start by isolating y in our given equation:

ax+by = c

by = -ax + c

y = (-a/b)x + (c/b)

We now have the given equation in slope-intercept form; the slope is -a/b and the y-intercept is c/b.

Statement One Alone:

b not equal to 0

Since we are not provided any information about the slope of the line, statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

ab>0

Since ab > 0, neither a nor b can be 0. This means that the slope of the line is not equal to zero. Recall that if the slope of a line is 0, the line is horizontal and will not cross the x-axis unless the line is the x-axis itself. Thus, regardless of the values of a or b, the line will, at some point, cross the x-axis.

Hi Jeff,

Thank you for explanation, could the formula be parabola and not cross the x-axis?
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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Is this the right approach, please comment:

y=-(a/b)x+(c/b)

For this line to cross x axis, put y=0 ---> equation becomes x=c/a and therefore, knowing about a is important, which is satisfied by B
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Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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Ohh it was simple, I chose E. I didn't understand in the first place what exactly "cross" mean. Never used "cross" for "intersect" .
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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[quote="surupab"]Does the line with equation ax+by = c, where a,b and c are real constants, cross the x-axis?

(1) b not equal to 0
(2) ab>0[/quote]

If line passes through x axis, the y cordinate at intersection of X axis should be zero.
Rewritten, Y = (c-aX)/b
If Y = 0
(c-aX)/b = 0
Hence,
1. b should not be equal to 0
And c-aX = 0
X= c/a
2. a should not be equal to 0

From Statement 2 we can see that the product of ab is nit equal to 0, hence we know 1 and 2 are true.

[size=80][b][i]Posted from my mobile device[/i][/b][/size]
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Re: Does the line with equation ax+by = c, where a,b and c are real consta  [#permalink]

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Great explanation JeffTargetTestPrep Re: Does the line with equation ax+by = c, where a,b and c are real consta   [#permalink] 15 Jul 2019, 01:59
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