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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.

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 \(b>0\).

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.

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

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14 Aug 2013, 08:18

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....

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

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05 Sep 2014, 14:25

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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.

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

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10 May 2016, 23:48

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.

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: \(y - y_1 = m(x - x_1)\)

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

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12 May 2016, 21:48

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

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

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12 May 2016, 22:14

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

Please give kudos if you think above points help.
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