Line l lies in the xy-plane and does not pass through the origin. What

Target question: What is the slope of line l?

Statement 1: The x-intercept of line \(\ell\) is twice the y-intercept of line l
Let k = the y-intercept of line l
This means 2k = the x-intercept of line l
If the y-intercept is k, then line l passes through the y-axis at the point (0, k)
If the x-intercept is 2k, then line l passes through the x-axis at the point (2k, 0)
Since (0, k) and (2k, 0) are both points on line l, we can apply the slope formula to these points to find the slope of line l.
We get: slope = (k - 0)/(0 - 2k) = k/(-2k) = -1/2
So, the slope of line l = -1/2
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The x-and y-intercepts of line l are both positive
If we're able to imagine different lines (with DIFFERENT SLOPES) that satisfy this condition, we'll quickly see that statement 2 is not sufficient. However, if we don't automatically see this, we can take the following approach...
There are many different cases that satisfy statement 2 yet yield different answers to the target question. Here are two:
Case a: the x-intercept is 1 and the y-intercept is 1, which means line l passes through (1, 0) and (0, 1). Applying the slope formula, we get: slope = (0 - 1)/(1 - 0) = -1
Case b: the x-intercept is 2 and the y-intercept is 1, which means line l passes through (2, 0) and (0, 1). Applying the slope formula, we get: slope = (0 - 1)/(2 - 0) = -1/2
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A
_________________

When I see something like this, I just start drawing. For me, it's easier to look at the slopes and compare them, versus trying to understand the slopes based on numbers and equations.

For statement 1, draw a couple of lines that have an x-intercept twice the y-intercept. Don't forget negatives (for instance, x-intercept of -2 and y-intercept of -1). You should notice that all of the slopes of these lines are equal.



Note that this is an example of a DS problem with a 'nice but not necessary' statement. Be very careful to analyze the statements each on their own before putting them together. It's nice to know that the slopes are both positive (statement 2), because it gives you a clearer picture of what's going on. But critically, it's not necessary to know that. You can answer the question even without it.
_________________

We need to determine the slope of line ℓ, given that it doesn’t pass through the origin.

Statement One Alone:

The x-intercept of line ℓ is twice the y-intercept of line ℓ.

We can let b = the y-intercept of line ℓ; thus, 2b = the x-intercept of line ℓ. Thus, the two points through which line ℓ passes are (2b, 0) and (0, b). With two points known, we can calculate the slope of line ℓ:

(b - 0)/(0 - 2b) = b/(-2b) = -½

Statement one alone is sufficient to answer the question.

Statement Two Alone:

The x- and y-intercepts of line ℓ are both positive.

Knowing that both the x- and y-intercepts of a line are positive does not allow us to determine the slope of the line. For example, the slope of the line with x-intercept = 1 and y-intercept = 2 will be different from the slope of the line with x-intercept = 1 and y-intercept = 3. Statement two alone is not sufficient to answer the question.

Answer: A
_________________

Answer: Option A

Video solution by GMATinsight



Get TOPICWISE: Concept Videos | Practice Qns 100+ | Official Qns 50+ | 100% Video solution CLICK.
Two MUST join YouTube channels : GMATinsight (1000+ FREE Videos) and GMATclub
_________________

Dear ccooley and Brent,

I have a question. You both consider the slop is negative., while it could be positive too. For example, the line could intersect the 'y' in point (0,1) and 'x' in point (-2,0). This line satisfies the condition too. What did not you take it into consideration?

Thanks

In your example, the x-intercept is -2 and the y-intercept is 1

However, statement 1 says that the x-intercept twice the y-intercept.
-2 is not twice 1

Cheers,
Brent
_________________

Thanks Brent. What I understand from Fact 1 is the that 'twice' means x-intercept 'double' the y-intercept regardless of any sign. It treated the intercept as distance from zero to the intercept regardless the sign. Where is the problem in my understanding?

Thanks in advance

This question requires no pen to paper. From 1 we know that slope is .5 regardless of the signs of the x and y-intercepts (2). 2 is basically irrelevant and insufficient without knowing the values. Hence A, 1 alone is sufficient.

I think you might be confusing the x- and y-intercepts with the DISTANCE from the origin.
An x-intercept of -2 is 2 units away from the origin (0,0) and a y-intercept of 1 is 1 units away from the origin.


Cheers,
Brent
_________________

y=ax+ b
x intercept mean y=0
x= -b/a
y intercept mean x=0
y =b
I have no
-b/a=2b

we can infer a, which is slope

dont draw anything.

Line ℓ lies in the xy-plane and does not pass through the origin. What is the slope of line ℓ ?

(1) The x-intercept of line ℓ is twice the y-intercept of line ℓ

(2) The x-and y-intercepts of line ℓ are both positive.

Guys - Are we not talking about absolute values of the x & y intercept? How can we infer that even the signs have to be same for the intercepts.

X intercept = -4 & y intercept = 2 - this will make the statement 1 insufficient.

the statement as such does refer to the magnitude only and not the signs .

@experts - please help.

hi...
when we talk of intercept and say y-intercept is 2, it means 2 and not -2..
the intercepts are never the absolute values but exact value..

we always say y- intercept or x- intercept is -2 and so on
_________________

Line L is in the form of Y= mx + b (Here, m is the slope)

St 1: Y-intercept = b so, x-intercept = 2b

In general, We know from the equation of any line
Y-intercept = b (when x=0)
x-intercept = -b/m (when y=0)

Now replace the value x=2b, we get 2b = -b/m or m = - 1/2 Sufficient

St 2: No specific value or no relation is given. Insufficient.

Ans A

First, try to break down the question stem as much as possible.
Assume: y=kx+b, we need to find out the value of k or k= (y-b)/x

Statement 1:
To find out the value of x (X-intercept), let y=0, x= -b/k;
To find out the value of y (Y-intercept), let x=0, y=b;
Given that x=2y, -b/k=2b, divide b on both sides of the equation, we get -1/k=2, and we find out the value of k. We don't even need to calculate and find out the exact value of k, we just know that we get a confirmed value of K, hence Statement 1 is SUFFICIENT.

(CANCEL OUT ANSWER CHOICES BCE, WE KEEP ANSWER CHOICES A&D)

Statement 2:
Given that x and y are both positive, we need to find out k= (y-b)/x;
We are unable to find a definite value of k, hence Statement 2 is NOT SUFFICIENT.

(ANSWER CHOICE IS A)

I don't think statement 1 can stand on its own.

Case 1: Slope of line AB where A(0,2) and B(4.0)= \(-\frac{1}{2}\)
Case 1: Slope of line AB where A(0,2) and B(-4.0)= \(\frac{1}{2}\)

But this is an official question and official answer says A is sufficient.

chetan2u VeritasKarishma Bunuel Can you please help me understand where am I wrong ?
_________________

Case in red: the x-intercept = -4, while the y-intercept = 2. -4 is not twice 2. BTW, this same doubt is addressed above.
_________________

I don't think that it's necessary to specify that the slope of both lines should be positive (or negative), as we have been told that the Y-intercept is 2 times the X-intercept. Thus, both intercept values have the same sign.

Hello sir

But we don't know where does the polarity of the x and y intercept lie, how can we be certain that they lie in the first quad. Slope can be 1/2 or -1/2.

Gag

I'm not 100% sure what you're asking.
That said, I believe you're inquiring about the x and y-intercepts, and whether they're positive or negative.

In my solution, I let k = the y-intercept, and I let 2k = the x intercept.
As you can see, k can be either positive or negative, which means I've accounted for both cases in my solution.

Does that answer your question?
_________________