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Math Expert V
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Which of the following lines in the xy-plane does not contai  [#permalink]

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Question Stats: 76% (01:00) correct 24% (01:08) wrong based on 505 sessions

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

Which of the following lines in the xy-plane does not contain any point with integers as both coordinates?

(A) y = x
(B) y = x + 1/2
(C) y = x + 5
(D) y = x*1/2
(E) y = x/2 + 5

Problem Solving
Question: 83
Category: Algebra; Arithmetic Substitution; Operations with rationa l numbers
Page: 72
Difficulty: 600

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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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(A) y = x: Solving for x & y intercepts, we get (0, 0) Integer
(B) y = x + 1/2: Solving for x & y intercepts, we get (-1/2, 1/2) Non-Integer (Correct)
(C) y = x + 5: Solving for x & y intercepts, we get (-5, 5) Integer
(D) y = x*1/2: Solving for x & y intercepts, we get (0, 0) Integer
(E) y = x/2 + 5: Solving for x & y intercepts, we get (-10, 5) Integer
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Math Expert V
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Posts: 55271
Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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SOLUTION

Which of the following lines in the xy-plane does not contain any point with integers as both coordinates?

(A) y = x
(B) y = x + 1/2
(C) y = x + 5
(D) y = x*1/2
(E) y = x/2 + 5

The answer is y=x+1/2, since for any integer value of x, y becomes integer and a half: .

Answer: B.
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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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(B) y = x + 1/2 -> for integer values of x, y is not an integer. If x = 1/2, 3/2 etc (non-integers), y is an integer. x and y cannot both be integers.
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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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Step 1- Keep X =0 and solve for Y

Step 2- Keep Y=0 and Solve for X..

B--- give non integer i.e x= -1/2 and Y= 1/2
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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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SOLUTION

Which of the following lines in the xy-plane does not contain any point with integers as both coordinates?

(A) y = x
(B) y = x + 1/2
(C) y = x + 5
(D) y = x*1/2
(E) y = x/2 + 5

The answer is y=x+1/2, since for any integer value of x, y becomes integer and a half: .

Answer: B.
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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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Just Plot the line for each option . We will get a clear ans as B .
How to plot ?
to get y , put x=0
and to get x , put y=o and then join the points to get the line .
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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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For option D & E; What values to consider for x, y. Is it ok to consider values as 0??
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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

Which of the following lines in the xy-plane does not contain any point with integers as both coordinates?

(A) y = x
(B) y = x + 1/2
(C) y = x + 5
(D) y = x*1/2
(E) y = x/2 + 5

We see that the point (1,1) is on the line y = x (answer choice A); (1, 6) on y = x + 5 (C); (2, 1) on y = x*1/2 (D); and (2, 6) on y = x/2 + 5 (E). Thus, the only line that won’t have any point with integers as both coordinates is y = x + 1/2 (answer choice B). The reason is simple: if x is an integer, then y can’t be an integer, since the sum of an integer and 1/2 will never be an integer. Similarly, if y is an integer, then x can’t be an integer, since the difference of an integer and 1/2 (notice that x = y - 1/2) can never be an integer.

Answer: B
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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

Which of the following lines in the xy-plane does not contain any point with integers as both coordinates?

(A) y = x
(B) y = x + 1/2
(C) y = x + 5
(D) y = x*1/2
(E) y = x/2 + 5

Problem Solving
Question: 83
Category: Algebra; Arithmetic Substitution; Operations with rationa l numbers
Page: 72
Difficulty: 600

GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
1. Please provide your solutions to the questions;
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!

can someone explain what the question is asking about? and what is the logic behind solution ? i dont get.

i know this formula y = mx + b where

m is slope

b is y-intercept
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buuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuump  VP  D
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Re: Which of the following lines in the xy-plane does not contai  [#permalink]

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Bunuel wrote:
SOLUTION

Which of the following lines in the xy-plane does not contain any point with integers as both coordinates?

(A) y = x
(B) y = x + 1/2
(C) y = x + 5
(D) y = x*1/2
(E) y = x/2 + 5

The answer is y=x+1/2, since for any integer value of x, y becomes integer and a half: .

Answer: B.

Bunuel, bro how are you would be great if you could visualize solution have a great weekend Senior SC Moderator V
Joined: 22 May 2016
Posts: 2760
Which of the following lines in the xy-plane does not contai  [#permalink]

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dave13 wrote:
Bunuel wrote:
SOLUTION

Which of the following lines in the xy-plane does not contain any point with integers as both coordinates?

(A) y = x
(B) y = x + 1/2
(C) y = x + 5
(D) y = x*1/2
(E) y = x/2 + 5

The answer is y=x+1/2, since for any integer value of x, y becomes integer and a half: .

Answer: B.

Bunuel, bro how are you would be great if you could visualize solution have a great weekend dave13 , I am no Bunuel. Maybe I can help anyway.

The equations are in the form that you note above, y = mx + b
If there is no +b, it just means that b = 0
(we could still write option A, e.g., as y = (1)x + 0, in which m= 1 and b= 0)
If the equation of a line does not have a (+ b), then it passes through the origin
because both the x- and y- intercepts are 0.

We could manipulate that equation: set x = 0, then y=0, find the two intercepts: are the coordinates integers?
It's faster just to plug values in for x and find y. Bunuel noticed that in B,
every time we plug an integer in for x, we cannot get an integer for y.
That fact plays out below.

• Approach #1: Find exceptions to NEVER to eliminate answers

We essentially have a never question:
which of these lines represented by these equations
will never give (integer, integer) for any point (x, y) on the line ?

With "never," if we can find one instance in which (x,y) is (integer, integer)
from the equation in an answer option, then we can eliminate that answer.

To get rid of an answer, "input" an integer for x into the equation.
Pick strategically for x because we are also trying to make y an integer.

A) y = x
Let x = 0. Then y = 0
Let x = 1. Then y = 1.
We now have two points (0,0) and (1,1) in which x and y are both integers. Eliminate A.

B) y = (x + $$\frac{1}{2}$$) = (x + 0.5)
Let x = 0. Then y = 0.5
Let x = 1. Then y = 1.5
Let x = 2. Then y = 2.5
Let x = -1. Then y = (-1 + $$\frac{1}{2}$$) = - $$\frac{1}{2}$$ = -0.5
When I plug in an integer for x, I cannot seem to get an integer for y.
I have (0, 0.5), (1, 1.5), (2, 2.5) and (-1, -0.5). Hold this answer.

C) y = x + 5
Let x = 0. Then y = 5
Let x = 1. Then y = 6
We have two points on that line: (0, 5) and (1, 6).
Both (x, y) coordinates are integers. We need them not to be integers. Eliminate C.

D) y = x * $$\frac{1}{2}$$ => pick values for x that are integers divisible by 2
(I have to use an integer for x, and I am trying to make y an integer, too, so that I can eliminate this answer).

Let x = 0. Then y = 0
Let x = 2. Then y = (2 * $$\frac{1}{2}$$) = 1
Let x = 4. The y = (4 * $$\frac{1}{2}$$) = 2
We have three points on the line: (0,0) and (2,1) and (4,2)
All of those (x,y) coordinates are integers. Eliminate D.

(E) y = $$\frac{x}{2}$$ + 5. (Integer + integer) = integer,
so to get the first value, x, an integer, I need to make x divisible by 2.
Let x = 0. Then y = ($$\frac{x}{2}$$ + 5) = (0 + 5) = 5
Let x = 2. Then y = ($$\frac{2}{2}$$ + 5) = (1 + 5) = 6
Let x = 4. Then y = ($$\frac{4}{2}$$ + 5) = (2 + 5) = 7
Points on this line: (0,5) and (2, 6) and (4, 7). All integers. Eliminate E.

The answer must be B.
I couldn't think of any integral (integer) value of x
to which I could add $$\frac{1}{2}$$ and get an integer for y.

It makes sense; if x must be an integer, adding $$\frac{1}{2}$$ to an integer
is never going to get me another integer
for the value of y. Instead I will get 1.5 or 2.5 or 3.5 and so on.

• Approach #2: T-charts (and graphing)
You wanted a visual. Okay.

Same as above, only I drew little T-charts.
I pick an integer for x or a value that will give me an integer for x,
I run it through the equation, and I get a y-value.

LHS of T-chart: choose and input integers for the variable x, then "run" the number through the equation
RHS of T-chart: the output is the value of y

In the diagram below, the graphs of the lines give a visual representation
of the equation of the line. Make a T-chart. Plot two or three coordinate pairs. Connect the coordinates.

(We could, though, look only at the T-charts. We can see in those charts
whether x and y are both integers.)

Answer B yields y-values that are not integers.
If x is an integer, y cannot be.
The points on the line indicated by the equation in B
will never both be integers.

Hope that helps. Attachments graphed lines 3.28.2019.jpg [ 147.37 KiB | Viewed 375 times ]

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