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16 Sep 2014, 01:49
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
65% (02:13) correct
35%
(01:56)
wrong
based on 49
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For any numbers \(x\) and \(y\), \(x \diamond y = 2x - y - xy\). If \(x \diamond y = 0\), then which of the following CANNOT be \(y\)?
A. 3
B. 2
C. 0
D. \(-\frac{4}{3}\)
E. -2
A. 3
B. 2
C. 0
D. \(-\frac{4}{3}\)
E. -2
16 Sep 2014, 01:49
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Bookmarks
Official Solution:
For any numbers \(x\) and \(y\), \(x \diamond y = 2x - y - xy\). If \(x \diamond y = 0\), then which of the following CANNOT be \(y\)?
A. 3
B. 2
C. 0
D. \(-\frac{4}{3}\)
E. -2
We must determine which of the answer choices cannot be a value of \(y\).
If \(x \diamond y = 2x - y - xy\) and \(x \diamond y = 0\), we can combine equations to get: \(2x - y - xy = 0\). Because the problem tells us that one value for \(y\) is impossible, one answer choice will make the equation \(2x - y - xy = 0\) false.
Plug in each answer choice and see which makes the equation false.
A. If \(y = 3\), then \(2x - y - xy = 0\) becomes \(2x - 3 - x(3) = 0\). Combine like terms: \(-x - 3 = 0\), so \(x = -3\). Because \(x\) is a variable, it can be equal to any quantity. Therefore, this equation is not false.
B. If \(y = 2\), then \(2x - 2 - x(2) = 0\). The \(x\) terms cancel, leaving \(-2 = 0\). This is false, so B is correct. Double-check by making sure that no other answer yields a false equation.
C. If \(y = 0\), then \(2x - 0 - x(0) = 0\). This simplifies to \(2x = 0\), which is not false.
D. If \(y = -\frac{4}{3}\) then \(2x - (-\frac{4}{3}) - x(-\frac{4}{3}) = 0\). Combine like terms: \(\frac{10}{3}x + \frac{4}{3} = 0\). Isolate the \(x\) term: \(\frac{10}{3}x = -\frac{4}{3}\), so \(x = -\frac{2}{5}\). This is not false.
E. If \(y = -2\), then \(2x - (-2) - x(-2) = 0\). Combine like terms: \(4x +2 = 0\). Isolate the \(x\) term: \(4x = -2\), so \(x = -\frac{1}{2}\). This is not false, so only choice B yields a false equation.
Choice B is the correct answer.
Alternatively, we can solve this problem using algebra. Recall that a value can be impossible for a variable in two ways: if the value of the variable causes a division by 0 or if the value forces the equation to look for the square root of a negative number. Since there are no exponents or roots in this equation, the answer will likely involve dividing by zero. If plugging in a certain value for \(y\) causes a division by zero, we should look for \(y\) in the denominator of a fraction. Solving the equation for \(x\) may give us a fraction in terms of \(y\).
First factor out the \(x\) to get \(x(2 - y) - y = 0\).
Add \(y\) to both sides: \(x(2 - y) = y\).
Divide both sides by \(2 - y\) to get \(x = \frac{y}{2-y}\).
If \(y = 2\), the denominator becomes \(2 - 2 = 0\). Any value that makes a denominator equal to 0 cannot be a valid solution to an equation because dividing by 0 is undefined. Therefore, \(y\) cannot be equal to 2.
Answer: B
For any numbers \(x\) and \(y\), \(x \diamond y = 2x - y - xy\). If \(x \diamond y = 0\), then which of the following CANNOT be \(y\)?
A. 3
B. 2
C. 0
D. \(-\frac{4}{3}\)
E. -2
We must determine which of the answer choices cannot be a value of \(y\).
If \(x \diamond y = 2x - y - xy\) and \(x \diamond y = 0\), we can combine equations to get: \(2x - y - xy = 0\). Because the problem tells us that one value for \(y\) is impossible, one answer choice will make the equation \(2x - y - xy = 0\) false.
Plug in each answer choice and see which makes the equation false.
A. If \(y = 3\), then \(2x - y - xy = 0\) becomes \(2x - 3 - x(3) = 0\). Combine like terms: \(-x - 3 = 0\), so \(x = -3\). Because \(x\) is a variable, it can be equal to any quantity. Therefore, this equation is not false.
B. If \(y = 2\), then \(2x - 2 - x(2) = 0\). The \(x\) terms cancel, leaving \(-2 = 0\). This is false, so B is correct. Double-check by making sure that no other answer yields a false equation.
C. If \(y = 0\), then \(2x - 0 - x(0) = 0\). This simplifies to \(2x = 0\), which is not false.
D. If \(y = -\frac{4}{3}\) then \(2x - (-\frac{4}{3}) - x(-\frac{4}{3}) = 0\). Combine like terms: \(\frac{10}{3}x + \frac{4}{3} = 0\). Isolate the \(x\) term: \(\frac{10}{3}x = -\frac{4}{3}\), so \(x = -\frac{2}{5}\). This is not false.
E. If \(y = -2\), then \(2x - (-2) - x(-2) = 0\). Combine like terms: \(4x +2 = 0\). Isolate the \(x\) term: \(4x = -2\), so \(x = -\frac{1}{2}\). This is not false, so only choice B yields a false equation.
Choice B is the correct answer.
Alternatively, we can solve this problem using algebra. Recall that a value can be impossible for a variable in two ways: if the value of the variable causes a division by 0 or if the value forces the equation to look for the square root of a negative number. Since there are no exponents or roots in this equation, the answer will likely involve dividing by zero. If plugging in a certain value for \(y\) causes a division by zero, we should look for \(y\) in the denominator of a fraction. Solving the equation for \(x\) may give us a fraction in terms of \(y\).
First factor out the \(x\) to get \(x(2 - y) - y = 0\).
Add \(y\) to both sides: \(x(2 - y) = y\).
Divide both sides by \(2 - y\) to get \(x = \frac{y}{2-y}\).
If \(y = 2\), the denominator becomes \(2 - 2 = 0\). Any value that makes a denominator equal to 0 cannot be a valid solution to an equation because dividing by 0 is undefined. Therefore, \(y\) cannot be equal to 2.
Answer: B
bhatiavai
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GMAT Date: 12-03-2014
25 Jan 2015, 04:17
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Hi Bunuel,
I did the following and selected C as the answer.
2x-y-xy = 0
2x-y(1+x) = 0
2x = y(1+x)
2x/y = 1+x
Now since division by Zero is not defined y cannot be Zero.
I understand your solution,However I'm not able to figure out where I'm going wrong .
Many thanks
I did the following and selected C as the answer.
2x-y-xy = 0
2x-y(1+x) = 0
2x = y(1+x)
2x/y = 1+x
Now since division by Zero is not defined y cannot be Zero.
I understand your solution,However I'm not able to figure out where I'm going wrong .
Many thanks
