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For all numbers a and b, the operation ⊕ is defined by a ⊕ b
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14 Mar 2019, 10:43
14 Mar 2019, 10:43
Expert Reply
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Question Stats:
59% (01:41) correct
40% (01:57) wrong based on 44 sessions
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For all numbers \(a\) and \(b\), the operation \(⊕\) is defined by \(a ⊕ b = a^2 – ab\). If \(xy ≠ 0\), then which of the following can be equal to zero?
I. \(x ⊕ y\)
II. \(xy ⊕ y\)
III. \(x ⊕ (x + y)\)
A. II only
B. I and II only
C. I and III only
D. II and III only
E. All of the above
I. \(x ⊕ y\)
II. \(xy ⊕ y\)
III. \(x ⊕ (x + y)\)
A. II only
B. I and II only
C. I and III only
D. II and III only
E. All of the above
Re: For all numbers a and b, the operation ⊕ is defined by a ⊕ b
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20 Mar 2019, 15:21
20 Mar 2019, 15:21
Expert Reply
First, let's convert each statement into normal math using the template a \(a⊕b = a^{2} - ab\):
\(x⊕y = x^{2} - xy\)
\(xy⊕y = (xy)^{2} - (xy)y\)
\(x⊕(x + y) = x^{2} - x(x+y)\)
Which of these could equal 0? Well, if x = 1 and y = 1, then the first two expressions will equal 0.
Now, let me slightly rewrite the last expression: \(x(x) - x(x+y)\). Since both terms will have to be identical for the difference to be 0, it must be the case that x = x + y. However, this can only be true if y = 0, and we're told that neither variable is 0. So only expressions I and II work.
\(x⊕y = x^{2} - xy\)
\(xy⊕y = (xy)^{2} - (xy)y\)
\(x⊕(x + y) = x^{2} - x(x+y)\)
Which of these could equal 0? Well, if x = 1 and y = 1, then the first two expressions will equal 0.
Now, let me slightly rewrite the last expression: \(x(x) - x(x+y)\). Since both terms will have to be identical for the difference to be 0, it must be the case that x = x + y. However, this can only be true if y = 0, and we're told that neither variable is 0. So only expressions I and II work.
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Re: For all numbers a and b, the operation ⊕ is defined by a ⊕ b
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14 Jun 2019, 10:27
14 Jun 2019, 10:27
Carcass wrote:
For all numbers \(a\) and \(b\), the operation \(⊕\) is defined by \(a ⊕ b = a^2 – ab\). If \(xy ≠ 0\), then which of the following can be equal to zero?
I. \(x ⊕ y\)
II. \(xy ⊕ y\)
III. \(x ⊕ (x + y)\)
A. II only
B. I and II only
C. I and III only
D. II and III only
E. All of the above
I. \(x ⊕ y\)
II. \(xy ⊕ y\)
III. \(x ⊕ (x + y)\)
A. II only
B. I and II only
C. I and III only
D. II and III only
E. All of the above
We're asked to determine which of the expressions COULD equal zero.
So, let's set each expression equal to zero and see if we can solve it.
I. \(x ⊕ y\)
We get: \(x ⊕ y = 0\)
Apply formula to get: \(x^2 - xy = 0\)
Factor: \(x(x - y) = 0\)
If x = 1 and y = 1, then we get: \(1(1 - 1) = 0\)
Works!!
Check the answer choices....eliminate A and D (since they say that \(x ⊕ y\) CANNOT equal 0)
II. \(xy ⊕ y\)
We get: \(xy ⊕ y = 0\)
Apply formula to get: \((xy)^2 - (xy)(y) = 0\)
Simplify to get: \(x^2y^2 - xy^2 = 0\)
Factor: \(xy^2(x-1)= 0\)
If x = 1 and y = 1, then we get: \((1)(1^2)(1-1)= 0\)
Works!!
Check the answer choices....eliminate C (since is say that \(xy ⊕ y\) CANNOT equal 0)
III. \(x ⊕ (x + y)\)
We get: \(x ⊕ (x + y)=0\)
Apply formula to get: \(x^2 - x(x+y) = 0\)
Expand to get: \(x^2 - x^2 - xy = 0\)
Simplify to get: \(-xy = 0\)
Since we're told that \(xy ≠ 0\), it is IMPOSSIBLE for \(x ⊕ (x + y)\) to equal 0
Answer: B
Cheers,
Brent
