In the rectangular coordinate system, line k passes through the point

Question

In the rectangular coordinate system, line k passes through the point (n, −1). Is the slope of line k greater than zero?

(1) Line k passes through the origin.
(2) Line k passes through the point (1, n + 2).

DS35210.01

shridhar786 · Accepted Answer

In the rectangular coordinate system, line k passes through the point (n, −1). Is the slope of line k greater than zero?

STATEMENT (1) Line k passes through the origin.
line k passes through (0,0) and (n,-1)
-- the slope of line k =  \frac{-1-0}{n-0}  =  \frac{-1}{n}  (slope of line passing through (x1,y1) and (x2,y2) =  \frac{y2-y1}{x2-x1} )

we don't know the value of n so we cant find the slope
INSUFFICIENT

STATEMENT (2) Line k passes through the point (1, n + 2).
line k passes through (1,n+2) and (n,-1)
slope =  \frac{-1-n-2}{n-1} 
we dont know value of n so we cant find slope
INSUFFICIENT

combining both statements together
--we know line passes through (0,0) and (1,n+2)
the slope of the line k = n+2
equation of line k -- (y-0) = (n+2)(x-0) -- y =(n+2)x

line k passes through (n,-1)
-1 = (n+2)n 
 n^2 +2n+1 = 0
 (n+1)^2  = 0
n = -1
slope of line = n+2 = -1+2 = 1
Is the slope of line k greater than zero?---YES

SUFFICIENT

C is the answer

ScottTargetTestPrep · Answer

Solution:

We need to determine whether the slope of line k is positive.

Statement One Only:

Line k passes through the origin. 
 
Since line k passes through the origin i.e., (0, 0), we can calculate the slope of line k as:
 
(-1 - 0)/(n - 0) = -1/n
 
However, since we don’t know the value of n, we can’t determine whether the slope of line k is positive. For example, if n is negative, then -1/n is positive. However, if n is positive, then -1/n is negative.
 
 Statement Two Only:

Line k passes through the point (1, n + 2).
 
Since line k passes through the point (1, n + 2), we can calculate the slope of line k as:
 
(-1 - (n + 2))/(n - 1) = (-n - 3)/(n - 1) = (n + 3)/(1 - n)
 
However, since we don’t know the value of n, we can’t determine whether the slope of line k is positive. For example, if n is 0, then (n + 3)/(1 - n) = 3 is positive. However, if n is 2, then (n + 3)/(1 - n) = -5 is negative.
 
 Statements One and Two Together:
  
From statement one, we see that the slope of line k is -1/n and from statement two, we see that the slope is (n + 3)/(1 - n). Since the slope of any line must be unique, we have:
 
-1/n = (n + 3)/(1 - n)
 
-1(1 - n) = n(n + 3)
 
n - 1 = n^2 + 3n
 
n^2 + 2n + 1 = 0
 
(n + 1)^2 = 0
 
n + 1 = 0
 
n = -1
 
Since n = -1 and the slope of line k is -1/n, the slope is -1/(-1) = 1. We see that the slope of line k is positive. Statements one and two together are sufficient to answer the question.
 
 Answer: C

Jsound996 · Answer

We know that line k passes through (n, -1) so if we graph that, we should get this: (figure 1)
We are trying to figure out if the Slope is positive.

1.) This means line K goes through (0,0).
We have 2 scenarios (Figure 2)
 If K>0 
 If K<0 
Insufficient

2.) Refer to Figure 3:
This means we could have a point anywhere on line X = 1 (Pink Line)
Once again, we have 2 scenarios that could happen, where K can be both negative and positive.
Insufficient

1+2 Refer to Figure 4.
K has to be greater than 0 in order to intersect the origin and cross both y = -1 and x= 1.
From statement 2, because (1, n+2), you can't have a negative slope that also intersects the origin.
That means both 1 and 2 are enough.

C is the Answer!

Akp880 · Answer

In the rectangular coordinate system, line k passes through the point (n, −1). Is the slope of line k greater than zero?

(1) Line k passes through the origin.
(2) Line k passes through the point (1, n + 2).

Easiest Explanation:

As per S1-

line k passes through (0,0) and (n,-1)
Hence equation of line will be 
y=-nx (with slope=-n)

We can't comment whether slope is positive or negative as value of 'n'is still unknown.Hence insufficient.

As per S2-

line k passes through (1,n+2) and (n,-1)

Similar to S1 ,here again We can't comment whether slope is positive or negative as value of 'n'is still unknown.Hence insufficient.

Now combining both S1& S2

You have points (0,0) (n,-1) from which line k passes

y=-nx is the equation
(slope=-n)

In S2 already given that line passes through (1,n+2)

Put this in above derived line equation y= -nx,you will get-

n+2=-n*1
n=-1

Hence slope(-n) of line k= 1 (slope>0)

C is the correct answer.

Award kudos if you like the explanation.?

Regards,
Atul Pandey

Posted from my mobile device

AK1307 · Answer

Bunuel  can you please help with a solution? I can't understand how to combine statement 1 and 2. Thanks.

BrentGMATPrepNow · Answer

Given : Line k passes through the point (n, −1)

Target question :    Is the slope of line k greater than zero?

Key concept: If points  (a, b)  and  (c, d)  both lie on a line, then the slope of that line  = \frac{d-b}{c-a}

Statement 1 : Line k passes through the origin.  
In other words, line k passes through the point  (0,0) 
Since we also know line k passes through the point  (n, −1) , the slope of line k  = \frac{(-1)-0}{n-0}=\frac{-1}{n}  
So, if  n = 1 , the slope of line k  = \frac{-1}{1} = -1 , which means the answer to the target question is  NO. 
Conversely, if  n = -1 , the slope of line k  = \frac{-1}{-1} = 1 , which means the answer to the target question is  YES. 
Since we can’t answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : Line k passes through the point (1, n + 2). 
Since we also know line k passes through the point  (n, −1) , the slope of line k  = \frac{(n+2)-(-1)}{1-n}=\frac{n+3}{1-n}  
So, if  n = 2 , the slope of line k  = \frac{2+3}{1-2} = -5 , which means the answer to the target question is  NO. 
Conversely, if  n = 0 , the slope of line k  = \frac{0+3}{1-0} = 3 , which means the answer to the target question is  YES. 
Since we can’t answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
 Key concept: The slope between any two points on a line will always be the same.  
The following three points all lie on line k: (n, −1), (0, 0) and (1, n + 2)
This means that slope between (n, −1) and (0, 0) = the slope between (0, 0) and (1, n + 2)

In other words:  \frac{n - 0}{(-1) - 0} = \frac{(n+2)-0}{1-0}

Simplify:  \frac{n}{-1} = \frac{n+2}{1}

Cross multiply: (-1)(n+2) = (n)(1)
Expand: -n - 2 = n
Solve:  n = -1 
Now that we know the value of n, we COULD perform all of the calculations necessary to determine the slope of line k (and thus answer the target question).
Since we could answer the target question with certainty, the combined statements are sufficient.

Answer: C

CrackverbalGMAT · Answer

In the rectangular coordinate system, line k passes through the point (n, −1). Is the slope of line k greater than zero?

(1) Line k passes through the origin.
(2) Line k passes through the point (1, n + 2).

This question clearly tests the understanding of the slope of the line.

If a line passes through the points (x1,y1) and (x2,y2) , then the slope of the line is  \frac{(y2-y1)}{(x2-x1)} .

Also, a few traps are hidden in the question. Here it's given that line k passes through the point (n,-1). Remember,  'n' could be any real number  i.e positive, negative, or zero. We cannot assume that n is a positive value.

We are asked to find  if the slope of line K is positive or not .

(1) Line k passes through the origin.

It's already given in the question that the line passes through (n,-1). From St1, it is given the line passes through origin i.e (0,0).

Since the line passes through (0,0)and (n,-1), the slope of line k =  \frac{(y2-y1)}{(x2-x1)}  =  \frac{(-1-0)}{(n-0)}  =  \frac{-1}{n}

Is the slope of line k is positive?   It depends on the value of n .

If n is a negative value, the slope of the line is positive and if n is a positive value, the slope will be negative. Since we don't have a definite YES/ NO to the Question stem, Statement 1 alone is not sufficient.

(2) Line k passes through the point (1, n + 2).

We know about the 2 points the line k passes through i.e (n,-1) and (1, n + 2)

Slope of the line k =  \frac{(n+2 --1)}{(1-n)} =  \frac{(n+3)}{(1-n)} .
Here 'n' will determine the sign of the slope in this case.

For example: n= -1, Slope = -1+3/1--1 = 2/2 = 1 ==> Slope is positive
n=2, slope = 2+3/1-2 = 5/-1 = -5 ==> Slope is negative

Based on the value of n, the slope of the line k could be positive or negative . Hence Statement 2 alone is not sufficient.

The next step is to combine both statements.

We figured out that the slope of line k from St 1 is  \frac{ -1}{n}  . From statement 2, we found that the slope of the same line is   \frac{(n+3)}{(1-n)}

Since both the slopes represent the same line k, we can equate both.

\frac{ -1}{n}   =   \frac{(n+3)}{(1-n)}

n-1 =  n^2  + 3n

n^2  + 2n + 1 = 0

(n+1)^2  = 0

n+1 =0 ===> n =-1
On solving, we found the value of n as -1.

Therefore , the slope of line k =  \frac{ -1}{n}   = -1/-1 = 1 (positive)

Is the slope of line k greater than zero?   YES

We are getting a definite answer to the Question stem by combining both statements. Hence, Option C is the correct answer.

Thanks,
Clifin J Francis,
GMAT Mentor

Iamcomingarv · Answer

Hi How is (n+1)^2 = 0 , becoming N+1=0. Should we not take squareroot and make them absolute value?  CrackverbalGMAT

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In the rectangular coordinate system, line k passes through the point

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In the rectangular coordinate system, line k passes through the point

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