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

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

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.

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.)

@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.

Dabral

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.

Answer (B)

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.

Answer is B

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.

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.