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# The slope of line n in the xy-plane is not 0 and the y-inter

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Joined: 21 Oct 2013
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The slope of line n in the xy-plane is not 0 and the y-inter  [#permalink]

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Updated on: 12 Jan 2014, 05:14
2
00:00

Difficulty:

65% (hard)

Question Stats:

62% (02:19) correct 38% (02:48) wrong based on 62 sessions

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The slope of line n in the xy-plane is not 0 and the y-intercept of line n in the xy-plane is not 0. The equation of line n is y = mx + b, where m and b are constants. What is the value of m + b?

(1) The y-intercept is equal to the square of the x-intercept.
(2) The line goes through the point (−8, −16).

OE
(1): By definition, y-intercept of a line in form y = mx + b is equal to b. x-intercept is value of x when y = 0.
y = mx + b → 0 = mx + b → mx = −b → x = -(b/m)
Thus, y-intercept is b & x-intercept is –(b/m).
Since y-intercept is square of x-intercept, we can create equation: b = (-b/m)^2 → m^2*b= b^2
Since we are told in question stem that b does not equal 0, we can divide both sides by b. m^2= b
Since we know that b is equal to m^2, we can substitute that value into equation of line n.
y = mx + b → y = mx + m^2
"what is value of m + b?" Different values of m lead to different values for b.
So different values for m lead to different values for m + b.
For example,
if m = 1, then b = 1^2 = 1 → m + b = 1 + 1 = 2
If m = 2, then b = m^2 = 4 → m + b = 2 + 4 = 6
Insufficient.

(2): In order to determine equation of a line, it's necessary to have either two points, or one point and slope. Therefore, it is impossible to derive an equation of a line from simply one point on line. There are infinitely many different lines that go through point (−8, −16). So there are infinitely many different possible values of m + b.
Insufficient.

Combined: From (1), equation of line n is of form y = mx + m^2, where m is slope and m≠ 0.
(2) says that line goes through point (−8, −16). Remember that any paired point in form (x, y) represents a pair of values that could satisfy equation. In other words, first term in ordered pair can substitute in for x, and second term can be substituted in for y. Substituting −8 for x and −16 for y, we have:
−16 = m(−8) + m^2 → −16 = −8m + m^2 → 16 − 8m + m^2 = 0
With quadratics, it's difficult to tell sufficiency without solving.
This is because sometimes quadratics yield a single repeated root, and sometimes they yield two different values.
m^2 − 8m + 16 = (m − 4)(m − 4) = (m − 4)^2
If (m − 4)^2 = 0, then m = 4
As b = m^2 → b = 4^2 → b = 16
Able to determine m + b = 4 + 16 = 20
Sufficient.

Hi, I want to know if we have simpler solution for this question, please.

Originally posted by goodyear2013 on 12 Jan 2014, 03:56.
Last edited by Bunuel on 12 Jan 2014, 05:14, edited 1 time in total.
Renamed the topic and edited the question.
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Joined: 02 Sep 2009
Posts: 50000
Re: The slope of line n in the xy-plane is not 0 and the y-inter  [#permalink]

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12 Jan 2014, 06:49
2
1
goodyear2013 wrote:
The slope of line n in the xy-plane is not 0 and the y-intercept of line n in the xy-plane is not 0. The equation of line n is y = mx + b, where m and b are constants. What is the value of m + b?

(1) The y-intercept is equal to the square of the x-intercept.
(2) The line goes through the point (−8, −16).

OE
(1): By definition, y-intercept of a line in form y = mx + b is equal to b. x-intercept is value of x when y = 0.
y = mx + b → 0 = mx + b → mx = −b → x = -(b/m)
Thus, y-intercept is b & x-intercept is –(b/m).
Since y-intercept is square of x-intercept, we can create equation: b = (-b/m)^2 → m^2*b= b^2
Since we are told in question stem that b does not equal 0, we can divide both sides by b. m^2= b
Since we know that b is equal to m^2, we can substitute that value into equation of line n.
y = mx + b → y = mx + m^2
"what is value of m + b?" Different values of m lead to different values for b.
So different values for m lead to different values for m + b.
For example,
if m = 1, then b = 1^2 = 1 → m + b = 1 + 1 = 2
If m = 2, then b = m^2 = 4 → m + b = 2 + 4 = 6
Insufficient.

(2): In order to determine equation of a line, it's necessary to have either two points, or one point and slope. Therefore, it is impossible to derive an equation of a line from simply one point on line. There are infinitely many different lines that go through point (−8, −16). So there are infinitely many different possible values of m + b.
Insufficient.

Combined: From (1), equation of line n is of form y = mx + m^2, where m is slope and m≠ 0.
(2) says that line goes through point (−8, −16). Remember that any paired point in form (x, y) represents a pair of values that could satisfy equation. In other words, first term in ordered pair can substitute in for x, and second term can be substituted in for y. Substituting −8 for x and −16 for y, we have:
−16 = m(−8) + m^2 → −16 = −8m + m^2 → 16 − 8m + m^2 = 0
With quadratics, it's difficult to tell sufficiency without solving.
This is because sometimes quadratics yield a single repeated root, and sometimes they yield two different values.
m^2 − 8m + 16 = (m − 4)(m − 4) = (m − 4)^2
If (m − 4)^2 = 0, then m = 4
As b = m^2 → b = 4^2 → b = 16
Able to determine m + b = 4 + 16 = 20
Sufficient.

Hi, I want to know if we have simpler solution for this question, please.

The slope of line n in the xy-plane is not 0 and the y-intercept of line n in the xy-plane is not 0. The equation of line n is y = mx + b, where m and b are constants. What is the value of m + b?

(1) The y-intercept is equal to the square of the x-intercept. The y-intercept is the value of y for x=0, so b, and the x-intercept is the value of x for y=0, so -b/m. Thus we are given that $$b=\frac{b^2}{m^2}$$ --> reduce by b, which we know is not 0 to get: $$m^2=b$$. Not sufficient to get the value of m+b.

(2) The line goes through the point (−8, −16). Clearly insufficient: one point of a line cannot give neither the slope nor the y-intercept of it.

(1)+(2) From (2) we have that $$-16=-8m+b$$ and from (1) we have that $$m^2=b$$ --> substitute: $$-16=-8m+m^2$$ --> $$(m-4)^2=0$$ --> $$m=4$$ --> $$b=m^2=16$$ --> $$m+b=20$$. Sufficient.

Hope it helps.
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Re: The slope of line n in the xy-plane is not 0 and the y-inter  [#permalink]

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22 Oct 2017, 02:44
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Re: The slope of line n in the xy-plane is not 0 and the y-inter &nbs [#permalink] 22 Oct 2017, 02:44
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