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In the xy-plane, if line k has negative slope and passes

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In the xy-plane, if line k has negative slope and passes  [#permalink]

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In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 02 Jul 2012, 02:47
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SOLUTION

In the xy-plane, if line k has negative slope and passes through the point (-5,r), is the x-intercept of line k positive?

This question can be done with graphic approach (just by drawing the lines) or with algebraic approach.

Algebraic approach:

Equation of a line in point intercept form is \(y=mx+b\), where: \(m\) is the slope of the line, \(b\) is the y-intercept of the line (the value of \(y\) for \(x=0\)), and \(x\) is the independent variable of the function \(y\).

We are told that slope of line \(k\) is negative (\(m<0\)) and it passes through the point (-5,r): \(y=mx+b\) --> \(r=-5m+b\).

Question: is x-intercept of line \(k\) positive? x-intercep is the value of \(x\) for \(y=0\) --> \(0=mx+b\) --> is \(x=-\frac{b}{m}>0\)? As we know that \(m<0\), then the question basically becomes: is \(b>0\)?.

(1) The slope of line \(k\) is -5 --> \(m=-5<0\). We've already known that slope was negative and there is no info about \(b\), hence this statement is insufficient.

(2) \(r>0\) --> \(r=-5m+b>0\) --> \(b>5m=some \ negative \ number\), as \(m<0\) we have that \(b\) is more than some negative number (\(5m\)), hence insufficient, to say whether \(b>0\).

(1)+(2) From (1) \(m=-5\) and from (2) \(r=-5m+b>0\) --> \(r=-5m+b=25+b>0\) --> \(b>-25\). Not sufficient to say whether \(b>0\).

Answer: E.

Graphic approach:

If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

When we take both statement together all we know is that slope is negative and that it crosses some point in II quadrant (-5, r>0) (this info is redundant as we know that if the slope of the line is negative, the line WILL intersect quadrants II). Basically we just know that the slope is negative - that's all. We can not say whether x-intercept is positive or negative from this info.

Below are two graphs with positive and negative x-intercepts. Statements that the slope=-5 and that the line crosses (-5, r>0) are satisfied.

\(y=-5x+5\):
Image

\(y=-5x-20\):
Image

For more on Coordinate Geometry check: http://gmatclub.com/forum/math-coordina ... 87652.html


Answer: E.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 11 May 2016, 00:48
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Bunuel wrote:
In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0

Diagnostic Test
Question: 39
Page: 26
Difficulty: 650

I will not repeat the solutions above because I did the same way.

I will just repeat the most important takeaway: When slope is negative then both intercepts will have equal sign but when the slope is positive the intercepts will have opposite signs. Remember this and bully your way through these type of questions.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 02 Jul 2012, 03:53
1
Hi,

Difficulty level: 650

With reference to below diagram:
Attachment:
r.jpg
r.jpg [ 5.36 KiB | Viewed 26375 times ]

Even after using (1) & (2), we can't find the x-intercept.

Thus, Answer (E).

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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 06 Jul 2012, 03:30
SOLUTION

In the xy-plane, if line k has negative slope and passes through the point (-5,r), is the x-intercept of line k positive?

This question can be done with graphic approach (just by drawing the lines) or with algebraic approach.

Algebraic approach:

Equation of a line in point intercept form is \(y=mx+b\), where: \(m\) is the slope of the line, \(b\) is the y-intercept of the line (the value of \(y\) for \(x=0\)), and \(x\) is the independent variable of the function \(y\).

We are told that slope of line \(k\) is negative (\(m<0\)) and it passes through the point (-5,r): \(y=mx+b\) --> \(r=-5m+b\).

Question: is x-intercept of line \(k\) positive? x-intercep is the value of \(x\) for \(y=0\) --> \(0=mx+b\) --> is \(x=-\frac{b}{m}>0\)? As we know that \(m<0\), then the question basically becomes: is \(b>0\)?.

(1) The slope of line \(k\) is -5 --> \(m=-5<0\). We've already known that slope was negative and there is no info about \(b\), hence this statement is insufficient.

(2) \(r>0\) --> \(r=-5m+b>0\) --> \(b>5m=some \ negative \ number\), as \(m<0\) we have that \(b\) is more than some negative number (\(5m\)), hence insufficient, to say whether \(b>0\).

(1)+(2) From (1) \(m=-5\) and from (2) \(r=-5m+b>0\) --> \(r=-5m+b=25+b>0\) --> \(b>-25\). Not sufficient to say whether \(b>0\).

Answer: E.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 06 Jul 2012, 03:32
SOLUTION

In the xy-plane, if line k has negative slope and passes through the point (-5,r), is the x-intercept of line k positive?

(1) The slope of line k is -5
(2) r>0

Graphic approach:

If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

When we take both statement together all we know is that slope is negative and that it crosses some point in II quadrant (-5, r>0) (this info is redundant as we know that if the slope of the line is negative, the line WILL intersect quadrants II). Basically we just know that the slope is negative - that's all. We can not say whether x-intercept is positive or negative from this info.

Below are two graphs with positive and negative x-intercepts. Statements that the slope=-5 and that the line crosses (-5, r>0) are satisfied.

\(y=-5x+5\):
Attachment:
1.png
1.png [ 9.73 KiB | Viewed 29084 times ]

\(y=-5x-20\):
Attachment:
2.png
2.png [ 10.17 KiB | Viewed 29073 times ]


For more on Coordinate Geometry check: math-coordinate-geometry-87652.html

Answer: E.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 12 Jul 2012, 23:33
what addition information should we have to make it sufficient.. ???
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 14 Aug 2013, 09:18
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sanjoo wrote:
what addition information should we have to make it sufficient.. ???


By regarding both statements:
For any r greater than 25 the x intercept of k is positive
For any r smaller than 25 it is negative.

This is the information which is missing for answer C to be correct.

Why 25?
--> Because for a slope of -5, you have to go down 25 steps if you want to go 5 steps to the right.
So if r was 25, the x intercept would be 0....
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 14 Aug 2013, 11:01
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i solved like this-

line passes thru (-5,r). Eqn r= -5m + c. since slope is -ve, if c is +ve , x intercept = +ve. otherwise -ve.

statement 1. m=-5. r= 25 +c . c= r-25. c could be +ve or -ve depending on r.

statement 2. r> 0. c = r+5m. c could be +ve or -ve depending on value of m.

together- c= r-25; r>0. c could still be +ve or - ve . Hence x intercept could be +ve or -ve.

Hence ans = E

Is it ok?
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 28 Apr 2015, 06:59
1
Line (d): y = kx + b
Passed (-5;r) <=> r = -5k + b
Question when y=0, x>0? <=> x= (r-b)/k >0?
(1) k= -5
x= (r-b)/-5: insufficiant.
(2) r>0: insufficient.
(1)+(2): r=25+b>0 <=>b>-25 <=>b= -25+|m|
y=0; x= (r+|m|-25)/-5?: insufficiant.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 16 Mar 2016, 05:44
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Bunuel wrote:
In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0

Diagnostic Test
Question: 39
Page: 26
Difficulty: 650


Responding to a pm:

Check out this post first: http://www.veritasprep.com/blog/2011/09 ... -vertices/

Slope of a line = - y intercept/x intercept

If slope is negative, x intercept will be positive when y intercept is positive too. So we need to know the sign of the y intercept.

(1) The slope of line k is –5.
Does not matter what the actual slope is. We already know it is negative. We need the sign of y intercept.

(2) r > 0
The line passes through (-5, r) where r is positive. This just gives one point through which the line passes. The y intercept could still be positive or negative as shown by the two diagrams in Bunuel's first post above.

Using both also we don't know the sign of the y intercept.

Answer (E)
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 11 May 2016, 03:09
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wings.ap wrote:
Bunuel wrote:
In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0

Diagnostic Test
Question: 39
Page: 26
Difficulty: 650

I will not repeat the solutions above because I did the same way.

I will just repeat the most important takeaway: When slope is negative then both intercepts will have equal sign but when the slope is positive the intercepts will have opposite signs. Remember this and bully your way through these type of questions.


TIPS ON SLOPE AND QUADRANTS:

1. If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

2. If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefore if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.

3. Every line (but the one crosses origin OR parallel to X or Y axis OR X and Y axis themselves) crosses three quadrants. Only the line which crosses origin \((0,0)\) OR is parallel to either of axis crosses only two quadrants.

4. If a line is horizontal it has a slope of \(0\), is parallel to X-axis and crosses quadrant I and II if the Y intersect is positive OR quadrants III and IV, if the Y intersect is negative. Equation of such line is y=b, where b is y intersect.

5. If a line is vertical, the slope is not defined, line is parallel to Y-axis and crosses quadrant I and IV, if the X intersect is positive and quadrant II and III, if the X intersect is negative. Equation of such line is \(x=a\), where a is x-intercept.

6. For a line that crosses two points \((x_1,y_1)\) and \((x_2,y_2)\), slope \(m=\frac{y_2-y_1}{x_2-x_1}\)

7. If the slope is 1 the angle formed by the line is \(45\) degrees.

8. Given a point and slope, equation of a line can be found. The equation of a straight line that passes through a point \((x_1, y_1)\) with a slope \(m\) is: \(y - y_1 = m(x - x_1)\)

Check for more here: math-coordinate-geometry-87652.html


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Re: In the xy plane, if line k has negative slope and passes through the  [#permalink]

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New post 12 May 2016, 22:48
Given: (x, y) = (-5, r)

St1: Slope of line k = -5
To find the x intercept let the other co-ordinate be (x, 0)
(r - 0)/(-5 - x) = -5
r = 25 + 5x
x = (r - 25)/5
x intercept is positive if r > 25 and negative if r < 25
Not Sufficient

St2: r > 0 --> Clearly insufficient

Combining St1 and St2: r may still be < 25 or > 25
Not Sufficient

Answer: E
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In the xy plane, if line k has negative slope and passes through the  [#permalink]

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New post 12 May 2016, 23:14
Using statement 2 only definitely there can not be any solution as negtive slope can be of high degree or low degree so if we imagine a line passing from a point in 2nd quardent, we can conclude that statement 2 is not sufficient.

Lets take k=-5
y-r=-5(x+5)
y-r=-5x-25
y/(r-25)+x((r-25)/5)=1

x intercept is (r-25)/5 so now it depends on the value of r whether x intercept is positive or negative. Statement 2 says r>0 which is not sufficient as if r<25 then intercept will be negative if r>25 intercept will be positive. So no comments even we use both statement.

Hence answer is E

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In the xy-plane, if line k has negative slope and passes through  [#permalink]

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New post 07 Dec 2017, 10:52
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GMATMadeeasy wrote:
In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0


We need to determine whether the x-intercept of line k is positive, given that line k has a negative slope and passes through the point (-5, r).

We can let the slope of line k be m and the x-intercept of line k be a; that is, line k passes through the point (a, 0). Using the slope formula m = (y_2 - y_1)/(x_2 - x_1), we have:

m = (0 - r)/(a -(-5))= -r/(a + 5)

Statement One Alone:

The slope of line k is -5.

Using the information in statement one, we can say the following:

-r/(a + 5) = -5

a + 5 = r/5

a = r/5 - 5

Since we don't know the value of r, we cannot determine the value of a.

For example, if r = 5, a = -4, which is negative. However, if r = 50, a = 5, which is positive. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

r > 0

Knowing r is positive does not give us enough information to determine whether a, the x-intercept of line k, is positive. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Looking at our work from statement one, and keeping in mind that r > 0 from statement two, we see that a can be positive or negative.

For example, if r = 5 then a = -4, which is negative. However, if r = 50 then a = 5, which is positive. The two statements together are still not sufficient to answer the question.

Answer: E
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In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 13 Mar 2018, 03:27
Bunuel chetan2u


Quote:
TIPS ON SLOPE AND QUADRANTS:

1. If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

2. If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefore if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.


The highlighted text refers to line passing through origin. Is my understanding correct?

For the text in red, I drew few lines considering x-intercept as (x,0) and y-intercept as (0,y). Is this optimum way to visualize?
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 13 Mar 2018, 03:45
adkikani wrote:
Bunuel chetan2u


Quote:
TIPS ON SLOPE AND QUADRANTS:

1. If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

2. If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefore if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.


The highlighted text refers to line passing through origin. Is my understanding correct?

For the text in red, I drew few lines considering x-intercept as (x,0) and y-intercept as (0,y). Is this optimum way to visualize?


No. The above properties are true for all lines, not only for those which pass through the origin.
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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 15 Nov 2018, 10:43
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Bunuel wrote:
In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0


Target question: Is the x-intercept of line k positive?

When I scan the two statements, I can see that, if r is POSITIVE, then the point (-5, r) can be very close to the x-axis or the point (-5, r) can be very far from the x-axis.
Given this, let's go straight to the statements COMBINED, and see if we can find two cases that yield DIFFERENT answers to the target question.

CASE A: If r =1, then the point (-5, r) is very close to the x-axis, and we get something like this:
Image
In this case, the answer to the target question is NO, the x-intercept of line k is NOT positive


CASE B: If r =50, then the point (-5, r) is far from the x-axis, and we get something like this:
Image
In this case, the answer to the target question is YES, the x-intercept of line k IS positive

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

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Re: In the xy-plane, if line k has negative slope and passes  [#permalink]

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New post 10 Jun 2019, 09:41
VeritasKarishma wrote:
Bunuel wrote:
In the xy-plane, if line k has negative slope and passes through the point (-5,r) , is the x-intercept of line k positive?

(1) The slope of line k is -5.
(2) r > 0

Diagnostic Test
Question: 39
Page: 26
Difficulty: 650


Responding to a pm:

Check out this post first: http://www.veritasprep.com/blog/2011/09 ... -vertices/

Slope of a line = - y intercept/x intercept

If slope is negative, x intercept will be positive when y intercept is positive too. So we need to know the sign of the y intercept.

(1) The slope of line k is –5.
Does not matter what the actual slope is. We already know it is negative. We need the sign of y intercept.

(2) r > 0
The line passes through (-5, r) where r is positive. This just gives one point through which the line passes. The y intercept could still be positive or negative as shown by the two diagrams in Bunuel's first post above.

Using both also we don't know the sign of the y intercept.

Answer (E)



Hi,

Just to add here we are told that, slope is negative and slope is (-5) This means as that x increases by 1 y decreases by 5. So Points on this line would be (-4, r-5), (-3, r-10), (-2, r-15),(-1, r-20),(0, r-25)

So we see that at x=0 y is r-25. Now since we know that if slope is negative both the x intercept and y intercept have same sign. So if r-25<0 then x intercept is negative or if r-25>0 then x intercept is positive

So what we rally have to know is that if r> 25 or r<25.

So we see that Both the statements are insufficient to conclude if x intercept is positive/ negative
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Re: In the xy-plane, if line k has negative slope and passes   [#permalink] 10 Jun 2019, 09:41
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