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Sub 505 Level|   Algebra|   Coordinate Geometry|                     
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Bunuel
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Bunuel
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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Can y or x be integers?
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Bunuel
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yalembagrad
Can y or x be integers?

Line y = x + 1/2 contains points with integers as one of the coordinates but not both. For example, the line contains points (1, 1.5) and (2.5, 3) but it does not contain any point with (integer, integer).
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I first assumed that x or y either one has to be interger but thanks to your explanation.
Really helps
Bunuel
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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Hi can I ask for E how do we get (-10,5) ?
Abdul29
(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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BelisariusTirto
Hi can I ask for E how do we get (-10,5) ?
Abdul29
(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

That solution finds the x- and y-intercepts of the functions. For example, for E, y = x/2 + 5, the x-intercept is found by setting y = 0, which gives the point (-10, 0), and the y-intercept is found by setting x = 0, which gives the point (0, 5).

However, the solution is not the most precise, so I’d suggest reviewing the other ones.
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Appreciate it, thanks Bunuel. Will review the rest of the solutions.
Bunuel
BelisariusTirto
Hi can I ask for E how do we get (-10,5) ?
Abdul29
(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

That solution finds the x- and y-intercepts of the functions. For example, for E, y = x/2 + 5, the x-intercept is found by setting y = 0, which gives the point (-10, 0), and the y-intercept is found by setting x = 0, which gives the point (0, 5).

However, the solution is not the most precise, so I’d suggest reviewing the other ones.
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