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Help simplifying 2y - x = 2xy into x in terms of y

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Help simplifying 2y - x = 2xy into x in terms of y  [#permalink]

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New post 26 Jan 2019, 11:33
2y - x = 2xy and x ≠ 0. If x and y are integers, which of the following could equal y?

(A) 2
(B) 1
(C) 0
(D) -1
(E) - 2

I understand how to get to the answer (D) but I can't figure out how to simplify the equation and feel like I am missing something. anyone know how to simplify 2y - x = 2xy into x in terms of y and can explain the steps? Thanks!
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Re: Help simplifying 2y - x = 2xy into x in terms of y  [#permalink]

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New post 26 Jan 2019, 11:53
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Hi HanktheTank,

We're given three pieces of information to work with:
1) 2Y - X = 2(X)(Y)
2) X ≠ 0
3) X and Y are both INTEGERS

We're asked which of the following could equal Y. You can approach this question in a number of different ways. Here's how you can put X in terms of Y:

2Y - X = 2(X)(Y)

To start, let's put all the Xs on the 'right side':

2Y = 2(X)(Y) + X

Next, factor out the X:

2Y = (X)(2Y + 1)

Now, isolate the X:

(2Y)/(2Y+1) = X

At this point, we can compare the 5 answer choices to this equation (try plugging each value in for Y and solve). Remember that X CANNOT be 0 and BOTH X and Y must be INTEGERS. Only one of the 5 answer choices will fit all of those restrictions.

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Re: Help simplifying 2y - x = 2xy into x in terms of y  [#permalink]

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New post 26 Jan 2019, 12:02
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It can be written as
x = 2y /(2y+1)

Then I think its fairly easy to sub y and see what gives an integer for x.
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Help simplifying 2y - x = 2xy into x in terms of y  [#permalink]

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New post 26 Jan 2019, 12:09
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For such algebra questions, please substitute answer options. In this question, please notice that all the answer options are possible values of integer y.
Rearrange the equation, 2y - x = 2xy
x = 2y/(2y + 1)

We know x and y are both Integers and x ≠ 0.

Now substitute values of y from the answer options, to check which of them provides integer value of x.

A. y= 2, then x = 4/5 Not Integer
(B) y = 1, then x = 2/3 Not Integer
(C) y = 0, then x = 0 Not Possible since x ≠ 0
(D) y = -1, then x = 2 Integer
(E) y = - 2, then x = 4/3 Not Integer

Hence, the correct answer is Option D. -1

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Re: Help simplifying 2y - x = 2xy into x in terms of y  [#permalink]

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New post 26 Jan 2019, 12:56
HanktheTank wrote:
2y - x = 2xy and x ≠ 0. If x and y are integers, which of the following could equal y?

(A) 2
(B) 1
(C) 0
(D) -1
(E) - 2

I understand how to get to the answer (D) but I can't figure out how to simplify the equation and feel like I am missing something. anyone know how to simplify 2y - x = 2xy into x in terms of y and can explain the steps? Thanks!


The key point here is to get every term that has a x in it on the same side. (If you want x in terms of y, put all of the xs on the same side. If you want y in terms of x, put all of the ys on the same side.)

It looks like this:

2y - x = 2xy
2y = 2xy + x
2y = x(2y + 1) (this is the most important step - since we have all of the xs together on one side, we can 'pull out' an x...)
x = 2y / (2y + 1)

And there's your simplified equation!
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Re: Help simplifying 2y - x = 2xy into x in terms of y   [#permalink] 26 Jan 2019, 12:56
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