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Is the slope of the line that passes through the point (x, y) negative 01 Nov 2022, 01:30
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C
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35% (medium)

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67% (01:34) correct 33% (01:28) wrong based on 129 sessions

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Is the slope of the line that passes through the point (x, y) negative?

(1) The x-intercept of the line is a such that a < x
(2) Both x and y are less than 0

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C
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Is the slope of the line that passes through the point (x, y) negative 12 Dec 2022, 05:55
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ANS- C

to find the nature of the slope we have to determine whether the line goes right to left (slope +ve) or left to right (slope -ve)

stat1- no info about the nature of x and a
insuff

stat2- the line could be drawn from either side
insuff

together- line passes through (x,y) where x and y are less than 0 that is negative. and the x intercept a is less than x (which means it is also negative)
hence the line goes left to right, slope is negative.
suff
ps: draw a graph for better understanding!
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Is the slope of the line that passes through the point (x, y) negative 05 Jan 2023, 07:32
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egmat

please provide a detailed explanation
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Is the slope of the line that passes through the point (x, y) negative 10 Jan 2023, 05:27
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Quote:
Is the slope of the line that passes through the point (x, y) negative?

(1) The x-intercept of the line is a such that a < x
(2) Both x and y are less than 0

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AnujL
egmat

please provide a detailed explanation
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Hey AnujL,

Thanks for your query.

CONCEPT RECAP:
  1. Slope of a line (used in question stem): To find the slope of any line, we need the coordinates of two points on that line.
    1. If (x, y) and (a, b) are the two points on a line, then slope of that line = \(\frac{(y−b)}{(x−a)}\)
  2. x-intercept of a line (used in statement 1): The x-intercept of a line is the point where the line cuts the x-axis. At this point, the y-coordinate is 0.


SOLUTION:
What should be the first thing that we do once we see a DS question? - Analyse the question stem completely!

Question stem analysis:
All we are given in the question stem is a line that passes through point (x, y). We need to find whether the slope of this line is negative.

From Concept Recap - Point 1, we know that to find the slope of any line, we need two points on that line. And here we only have one point, (x, y). Since we can’t go further with this, let’s analyze the statements one by one.

Statement 1: “The x-intercept of the line is ‘a’ such that a < x”.
  • Using Concept Recap – Point 2 along with statement 1, we can infer that point (a, 0) lies on the line. (See how we just got another point on the line!)
  • Okay! So, now we have two points on the line - everything we need to calculate its slope.
    • Slope =\(\frac{(y−0)}{(x−a)}\) = \(\frac{(y)}{(x−a)}\)
  • Now, from statement 1, we also know that a < x.
    • Thus, (x – a) > 0 (the denominator of the slope is positive).
    • So, whether the slope is negative will depend on the value of ‘y’.
      • If y > 0, the slope will be positive,
      • and if y < 0, the slope will be negative.
So, even after knowing the two points, we still cannot tell whether the slope is negative. Thus, statement 1 alone is INSUFFICIENT and options A and D are rejected!

Statement 2: “Both x and y are less than 0”.
This information just tells us more about the same point, (x, y) that we have from the stem. But we cannot find the slope without one more point, right?
And therefore, statement 2 alone is INSUFFICIENT, and thus, option B is rejected, too!

IMPORTANT: Make sure you do not drag anything from statement 1 while analyzing statement 2. It is a common error and will surely take you to the wrong choice!

Statements 1 and 2 together:
Finally, since each statement alone is insufficient, let’s combine our conclusions from both the statements.
  • From statement 1, we inferred that the slope = \(\frac{(y)}{(x−a)}\), and (x – a) > 0.
  • From statement 2, we got that y < 0.
Recall that we only needed one more thing after statement 1 (the sign of y) to determine the sign of the slope. And then statement 2 gave us just that. We can conclude here itself, therefore, that the statements together are sufficient and thus, the answer is choice C.

But, just to take the Math through, let’s actually combine and see what we get for the slope:
  • Slope = \(\frac{(y)}{(x−a)}\)=\(\frac{(−)}{(+)}\) = (-)ve
We, thus, get a SURE YES to the question asked using both statements together.

Correct Answer: Option C


Hope this helps!


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Is the slope of the line that passes through the point (x, y) negative 12 Jan 2023, 08:41
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Bunuel
Is the slope of the line that passes through the point (x, y) negative?

(1) The x-intercept of the line is a such that a < x
(2) Both x and y are less than 0
Show more

Solution:
Pre Analysis:
  • We are asked if the slope of a line passing through a point \((x, y)\) is negative or not
  • This is a YES-NO question

Statement 1: The x-intercept of the line is a such that a < x
  • Accoridng to this statement, the line passes through another point that is \((a,0)\) which is on the x-axis
  • Now we have two points that the line passes through. \((x, y)\) and \((a,0)\)
  • So, the slope of the line \(=\frac{y-0}{x-a}=\frac{y}{x-a}\)
  • We know \(x>a\) which makes denominator \(x-a\) positive but we don't know the positive-negative nature of the numerator i.e., \(y\)
  • Thus, statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2: Both x and y are less than 0
  • Just with this information, we cannot get the slope of the line and neither can we know if it is positive or negative
  • Thus, statement 2 alone is also sufficient

Combining:
  • From statement 1, we get that slope \(=\frac{y}{x-a}\) and that \(x-a\) is positive
  • From statement 2, we get that x and y are both negative
  • Thus, after combining we can infer that slope \(=\frac{y}{x-a}=\frac{negative}{positive}=negative\)

Hence the right answer is Option C
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