Does line Ax + By + C = 0 (A is not 0) intersect the x-axis : GMAT Data Sufficiency (DS)
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# Does line Ax + By + C = 0 (A is not 0) intersect the x-axis

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Does line Ax + By + C = 0 (A is not 0) intersect the x-axis [#permalink]

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02 Dec 2007, 02:59
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Does line Ax + By + C = 0 (A is not 0) intersect the x-axis on the negative side?

(1) BA < 0.
(2) AC > 0.

M18-13
[Reveal] Spoiler: OA

Last edited by Bunuel on 13 Nov 2013, 01:03, edited 2 times in total.
Renamed the topic, edited the question, added the OA and moved to DS forum.
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Re: 18.13 X axis [#permalink]

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02 Dec 2007, 06:06
bmwhype2 wrote:
Does line Ax + By + C = 0 (A is not 0) intersect the x-axis on the negative side?

1. BA <0> 0

Getting E.

Ax + By + C = 0
By = -Ax - C (cannot divide by B just yet since B could be 0)

Stat 1:
Tells us that B is not 0 and that A and B have the same sign.
y = - (A/B)x - C
To find x, y = 0:
-(A/B)x = C
x = -(B/A) * C
B/A will have the same sign therefore -(B/A) will be negative which makes me think that the answer to the stem is yes. However, what if C = 0? The answer to the stem is no. Insuff.

Stat 2:
Tells us that A & C have opposite signs. I don't think that this alone helps us in determine the answer. Insuff.

Together:
If A is +ve and C is -ve then x intercept is +ve
If A is -ve and C is +ve then x intercept is -ve
Insuff.
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Re: 18.13 X axis [#permalink]

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02 Dec 2007, 07:40
AX + BY + C = 0

Y = -AX/B – C/B

Assume that this line intersects x axis at (-n, 0), where n is a +ve integer. As it is given that the given equation interest the x-axis on –ve side.

Then
0 = AN/B – C/B

AN = C
A/C = N here n is +ve integer
So either A> 0 & C > 0 or C< 0 & A<0> 0

So 2 alone is sufficient. Ans is B
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02 Dec 2007, 08:29
im also getting B.

Equation ends up y = (-Ax-C)/B

We are interested in the x intercept, so set the equation above to 0, and you end up with:

C=-Ax

Statement 1: tells us either B<0 or A<0. If A<0>0, so either A and C are both positive, or A and C are both negative

If you consider either case, and plug into x = -(C/A), you always end up with a negative x.

Sufficient.
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Re: Does line Ax + By + C = 0 (A is not 0) intersect the x-axis [#permalink]

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12 Nov 2013, 14:38
i was doing this on the gmatclub tests and i cannot figure out why all we need is x = -c/a

"So, the x-intercept of line ax+by+c=0 is x=−c/a."
I plugged in 0 so ax+ by+ c = 0
then y = ( -ax - c ) / b

was i supposed to think of this question like this
ax + b (0) + c = 0
ax + c = 0
x = -c/a

then use that equation to figure out what x is???

(when i was doing this before viewing the solution, i assumed that we would need a/b to solve because -a/b * x.)
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Re: Does line Ax + By + C = 0 (A is not 0) intersect the x-axis [#permalink]

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12 Nov 2013, 22:23
@laserglare

Yes, to find the x-intercept of a line, the point where it intersects the x-axis, we set the y-coordinate to 0. You correctly replaced y as 0 in the equation Ax+By+C=0, which gave you the x-intercept of -C/A.

Here 2 alone is sufficient because if AC>0, then we have either both A and C are positive or both A and C are negative, in both scenarios -C/A is negative, meaning the x-intercept is negative or intersects the x-axis to the left of the origin.

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Re: Does line Ax + By + C = 0 (A is not 0) intersect the x-axis [#permalink]

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12 Nov 2013, 22:29
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laserglare wrote:
i was doing this on the gmatclub tests and i cannot figure out why all we need is x = -c/a

"So, the x-intercept of line ax+by+c=0 is x=−c/a."
I plugged in 0 so ax+ by+ c = 0
then y = ( -ax - c ) / b

was i supposed to think of this question like this
ax + b (0) + c = 0
ax + c = 0
x = -c/a

then use that equation to figure out what x is???

(when i was doing this before viewing the solution, i assumed that we would need a/b to solve because -a/b * x.)

Given Ax + By + C = 0 is the equation of a line. You need to figure out whether it intersects x axis on the negative side i.e. in the second quadrant. You want to know that when the line crosses the x axis (if it does), is x co-ordinate negative there? When does a line cross the x axis? When its y co-ordinate is 0. So how will you know the point where the line crosses the x axis?
You put y = 0.
Ax + B*0 + C = 0
x = -C/A
So when y = 0, x = -C/A

We want to know whether this x cor-ordinate (-C/A) is negative. It will be negative when C/A is positive i.e. both C and A will have the same sign (either both positive or both negative)
Statement 2 tells you that C and A have the same sign (since their product is positive). Hence it is enough alone.

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Re: Does line Ax + By + C = 0 (A is not 0) intersect the x-axis [#permalink]

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13 Nov 2013, 01:04
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bmwhype2 wrote:
Does line Ax + By + C = 0 (A is not 0) intersect the x-axis on the negative side?

(1) BA < 0.
(2) AC > 0.

M18-13

Does line Ax + By + C = 0 (A is not 0) intersect the x-axis on the negative side?

$$ax+by+c=0$$ is equation of a line. Note that the line won't have interception with x-axis when $$a=0$$ (and $$c\neq{0}$$): in this case the line will be $$y=-\frac{c}{b}$$ and will be parallel to x -axis.

Now, in other cases (when $$a\neq{0}$$) x-intercept of a line will be the value of $$x$$ when $$y=0$$, so the value of $$x=-\frac{c}{a}$$. Question basically asks whether this value is negative, so question asks is $$-\frac{c}{a}<0$$? --> is $$\frac{c}{a}>0$$? --> do $$c$$ and $$a$$ have the same sign?

(1) BA < 0. Not sufficient as we can not answer whether $$c$$ and $$a$$ have the same sign.
(2) AC > 0 --> $$c$$ and $$a$$ have the same sign. Sufficient.

Check more on this topic here: math-coordinate-geometry-87652.html

Hope it helps.
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Re: Does line Ax + By + C = 0 (A is not 0) intersect the x-axis [#permalink]

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25 Nov 2015, 08:14
Is the x intercept of the line negative? From the given equation: x = -by/a - c/a. At x intercept of this line: y = 0 and x = - c/a.

Question reformulated: Is - c/a a negative value?

Statement 1: gives no information about c, therefore the sign of - c/a cannot be determined.

Statement 2: ac > 0. Therefore a and c have the same sign, and either both are negative or both are positive. In either case c/a becomes a positive value and - c/a is becomes a negative value, therefore the x intercept of the line is a negative value.

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Re: Does line Ax + By + C = 0 (A is not 0) intersect the x-axis [#permalink]

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26 Nov 2015, 07:11
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Does line Ax + By + C = 0 (A is not 0) intersect the x-axis on the negative side?

(1) BA < 0.
(2) AC > 0.

We want to know whether in Ax+C=0, Ax=-C, x=-C/A, -C/A<0. If we multiply -A^2 on both sides, we are multiplying negative number, so the inequality sign flips.
So -C/A<0? --> CA>0?
Condition 2 answers this 'yes' and the answer becomes (B).

Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.
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Re: Does line Ax + By + C = 0 (A is not 0) intersect the x-axis   [#permalink] 26 Nov 2015, 07:11
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# Does line Ax + By + C = 0 (A is not 0) intersect the x-axis

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