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In the xy plane, which of the statements below individually

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In the xy plane, which of the statements below individually [#permalink] New post 16 Dec 2019, 10:07
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In the xy plane, which of the statements below individually provide enough information to determine whether line z passes through the origin?

Indicate all such statements.

[A] The equation of line z is y = mx + b and b = 0.
[B] The sum of the slope and the y-intercept of line z is 0.
[C] For any point (a, b) on line z, |a| = |b|.
[Reveal] Spoiler: OA

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Re: In the xy plane, which of the statements below individually [#permalink] New post 16 Dec 2019, 10:10
OFFICIAL EXPLANATION:

Statement I tells you directly that b, the y-intercept, is equal to 0. Thus, the line passes through the origin.

For statement II, both the slope and the y-intercept could be 0, in which case line z is a horizontal line lying on the x-axis and therefore passes through the origin. Or, the slope and y-intercept could simply be opposites, such as 2 and -2. A line with a y-intercept of -2 and a slope of 2 would not pass through the origin. Therefore, this statement is not sufficient to determine whether line z passes through the origin.

As for statement III, since |a| = |b| must hold for every point on the line, then (0, 0) is a point on the line, since |0| = |0|.
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Re: In the xy plane, which of the statements below individually [#permalink] New post 22 Jun 2020, 17:52
huda wrote:
OFFICIAL EXPLANATION:

Statement I tells you directly that b, the y-intercept, is equal to 0. Thus, the line passes through the origin.

For statement II, both the slope and the y-intercept could be 0, in which case line z is a horizontal line lying on the x-axis and therefore passes through the origin. Or, the slope and y-intercept could simply be opposites, such as 2 and -2. A line with a y-intercept of -2 and a slope of 2 would not pass through the origin. Therefore, this statement is not sufficient to determine whether line z passes through the origin.

As for statement III, since |a| = |b| must hold for every point on the line, then (0, 0) is a point on the line, since |0| = |0|.


For statement 3, why did you consider |a| and |b| = 0?
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Re: In the xy plane, which of the statements below individually [#permalink] New post 23 Jun 2020, 06:00
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Farina wrote:
huda wrote:
OFFICIAL EXPLANATION:

Statement I tells you directly that b, the y-intercept, is equal to 0. Thus, the line passes through the origin.

For statement II, both the slope and the y-intercept could be 0, in which case line z is a horizontal line lying on the x-axis and therefore passes through the origin. Or, the slope and y-intercept could simply be opposites, such as 2 and -2. A line with a y-intercept of -2 and a slope of 2 would not pass through the origin. Therefore, this statement is not sufficient to determine whether line z passes through the origin.

As for statement III, since |a| = |b| must hold for every point on the line, then (0, 0) is a point on the line, since |0| = |0|.


For statement 3, why did you consider |a| and |b| = 0?


Key concept: If a point lies ON a line, then the coordinates of that point must SATISFY the equation of that line.

Statement 3: For any point (a, b) on line z, |a| = |b|
This tells us that all points on the line are such that the absolute value of the x-coordinate = the absolute value of the y-coordinate.
In other words, |x| = |y|
So, for example, since |-3| = 3, we know that the point (-3, 3) lies on line z
Likewise, since |0| = 0, we know that the point (0, 0) lies on line z
In other words, line z passes through the origin

Cheers,
Brent
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Re: In the xy plane, which of the statements below individually [#permalink] New post 23 Jun 2020, 16:41
Thank you for your reply. Just want to add that 0 is one possibility, the value could be any number right? in that case statement 3 shouldnt be the confirmed answer?

GreenlightTestPrep wrote:
Farina wrote:
huda wrote:
OFFICIAL EXPLANATION:

Statement I tells you directly that b, the y-intercept, is equal to 0. Thus, the line passes through the origin.

For statement II, both the slope and the y-intercept could be 0, in which case line z is a horizontal line lying on the x-axis and therefore passes through the origin. Or, the slope and y-intercept could simply be opposites, such as 2 and -2. A line with a y-intercept of -2 and a slope of 2 would not pass through the origin. Therefore, this statement is not sufficient to determine whether line z passes through the origin.

As for statement III, since |a| = |b| must hold for every point on the line, then (0, 0) is a point on the line, since |0| = |0|.


For statement 3, why did you consider |a| and |b| = 0?


Key concept: If a point lies ON a line, then the coordinates of that point must SATISFY the equation of that line.

Statement 3: For any point (a, b) on line z, |a| = |b|
This tells us that all points on the line are such that the absolute value of the x-coordinate = the absolute value of the y-coordinate.
In other words, |x| = |y|
So, for example, since |-3| = 3, we know that the point (-3, 3) lies on line z
Likewise, since |0| = 0, we know that the point (0, 0) lies on line z
In other words, line z passes through the origin

Cheers,
Brent

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Re: In the xy plane, which of the statements below individually [#permalink] New post 23 Jun 2020, 17:09
Expert's post
Farina wrote:
Thank you for your reply. Just want to add that 0 is one possibility, the value could be any number right? in that case statement 3 shouldnt be the confirmed answer?


A line is just a graphical representation of all possible solutions to an equation. That is, the x- and y-coordinates of every point on a line satisfy the equation of that line.
So, for example, the equation y = x + 1 has infinitely many solutions, including (0,1), (1,2), (2,3), (3.97, 4.97), etc.
For more on this concept watch: https://www.greenlighttestprep.com/modu ... /video/996

Likewise, the equation |x| = |y| also has infinitely many solutions. One of those solutions is (0, 0) since x = 0 and y = 0 satisfies the equation |x| = |y|
In fact any pair of values that satisfy the equation will be on the line.
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Re: In the xy plane, which of the statements below individually   [#permalink] 23 Jun 2020, 17:09
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