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

(1) The x-intercept of line \(\ell\) is twice the y-intercept of line \(\ell\) (2) The x-and y-intercepts of line \(\ell\) are both positive

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

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

(1) The x-intercept of line \(\ell\) is twice the y-intercept of line \(\ell\) (2) The x-and y-intercepts of line \(\ell\) are both positive

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.
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Chelsey Cooley | Manhattan Prep Instructor | Seattle and Online

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?

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

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?

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

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21 Jun 2017, 03:35

AbdurRakib wrote:

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

(1) The x-intercept of line \(\ell\) is twice the y-intercept of line \(\ell\) (2) The x-and y-intercepts of line \(\ell\) are both positive

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.

Last edited by rulingbear on 22 Jun 2017, 18:34, edited 1 time in total.

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

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.

Re: Line l lies in the xy-plane and does not pass through the origin. What [#permalink]

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22 Jun 2017, 14:39

GMATPrepNow wrote:

Mo2men wrote:

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

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.

Re: Line l lies in the xy-plane and does not pass through the origin. What [#permalink]

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28 Oct 2017, 06:55

victory47 wrote:

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.

Hi Victory,

This is a clear explanation, however for some reason I have troubles getting to the point where -b/a = 2b. Can you explain how you got there step by step?

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

(1) The x-intercept of line \(\ell\) is twice the y-intercept of line \(\ell\) (2) The x-and y-intercepts of line \(\ell\) are both positive

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
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