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If xy + z = x(y + z), which of the following must be true? A. x = 0 a

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If xy + z = x(y + z), which of the following must be true? A. x = 0 a [#permalink] New post   26 Jan 2022, 07:35
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If xy + z = x(y + z), which of the following must be true?

A. x = 0 and y = 0
B. x = 1 and y = 1
C. y = 1 and z = 0
D. x = 1 or y = 0
E. x = 1 or z = 0
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E
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Re: If xy + z = x(y + z), which of the following must be true? A. x = 0 a [#permalink] New post   26 Jan 2022, 08:06
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Carcass wrote:
If xy + z = x(y + z), which of the following must be true?

A. x = 0 and y = 0
B. x = 1 and y = 1
C. y = 1 and z = 0
D. x = 1 or y = 0
E. x = 1 or z = 0


Given: xy + z = x(y + z)
Expand the right side: xy + z = xy + xz
Subtract xy from both sides: z = xz
Subtract xz from both sides: z - xz = 0
Factor: z(1 - x) = 0

So, either z = 0 or x = 1
Answer: E
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Re: If xy + z = x(y + z), which of the following must be true? A. x = 0 a [#permalink] New post   03 May 2022, 23:59
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GreenlightTestPrep wrote:
Carcass wrote:
If xy + z = x(y + z), which of the following must be true?

A. x = 0 and y = 0
B. x = 1 and y = 1
C. y = 1 and z = 0
D. x = 1 or y = 0
E. x = 1 or z = 0


Given: xy + z = x(y + z)
Expand the right side: xy + z = xy + xz
Subtract xy from both sides: z = xz
Subtract xz from both sides: z - xz = 0
Factor: z(1 - x) = 0

So, either z = 0 or x = 1


Answer: E


Hi Brent,

if XZ=Z. Cant we cancel Z with us leaving with x=1.and from the answer choices. Since Z = XZ. Y needs to be one. As zx = z.

Please let me know if i am missing a point, With regards to cancellation of Z.

Thanks.
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Re: If xy + z = x(y + z), which of the following must be true? A. x = 0 a [#permalink] New post   04 May 2022, 05:45
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NickOP wrote:
Hi Brent,

if XZ=Z. Cant we cancel Z with us leaving with x=1.and from the answer choices. Since Z = XZ. Y needs to be one. As zx = z.

Please let me know if i am missing a point, With regards to cancellation of Z.

Thanks.


The key idea here is that dividing both sides of an equation by zero yields all kinds of problems.
You're suggesting that if we take xz = z, and divide both sides by z, we get x = 1, but that's only true if z ≠ 0.
If z = 0, then your conclusion is not necessarily true.
If z = 0, the equation xz = z become (x)(0) = 0. Notice that if we divide both sides by 0, we don't get x = 1.
In fact, if z = 0, then the resulting equation (x)(0) = 0 can have infinitely many solutions (e.g., x = 1, x = 2, x = 3, x = 4 . . . . )

Does that help?

Key takeaway: Before dividing both sides of an equation by a variable, we must make certain that we're not dividing by zero.
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Re: If xy + z = x(y + z), which of the following must be true? A. x = 0 a [#permalink] New post   04 May 2022, 14:54
GreenlightTestPrep wrote:
NickOP wrote:
Hi Brent,

if XZ=Z. Cant we cancel Z with us leaving with x=1.and from the answer choices. Since Z = XZ. Y needs to be one. As zx = z.

Please let me know if i am missing a point, With regards to cancellation of Z.

Thanks.


The key idea here is that dividing both sides of an equation by zero yields all kinds of problems.
You're suggesting that if we take xz = z, and divide both sides by z, we get x = 1, but that's only true if z ≠ 0.
If z = 0, then your conclusion is not necessarily true.
If z = 0, the equation xz = z become (x)(0) = 0. Notice that if we divide both sides by 0, we don't get x = 1.
In fact, if z = 0, then the resulting equation (x)(0) = 0 can have infinitely many solutions (e.g., x = 1, x = 2, x = 3, x = 4 . . . . )

Got it Brent. Thanks!

Does that help?

Key takeaway: Before dividing both sides of an equation by a variable, we must make certain that we're not dividing by zero.
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Re: If xy + z = x(y + z), which of the following must be true? A. x = 0 a [#permalink] New post   21 Mar 2025, 14:54
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Re: If xy + z = x(y + z), which of the following must be true? A. x = 0 a [#permalink]
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