If xy+z = x(y+z) which of the following must be true?

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

The key word here is must.
So, for example, consider answer choice A. While it's possible that x = 0 and z = 0, it need not be the case.
For example, x=1, y=1 and z=1 is a solution to the equation. So, this eliminates A.

The solution . . .
Given: xy+z = x(y+z)
Expand: xy+z = xy + xz
Subtract xy from both sides: z = xz
Rearrange: xz - z = 0
Factor: z(x-1) = 0

This tells us that z = 0 or x = 1
Answer: E

Cheers,
Brent

Hi All,

Notice how the answer choices to this question each involve two values and either an "and" or "or" qualifier. We can use those values against the prompt to determine what MUST be true.

To start, many of the answer choices include X=1, so let's see what happens when that 'condition' is met...

IF... X=1

XY + Z = X(Y+Z)
1(Y) + Z = 1(Y+Z)
Y + Z = Y + Z

X=1 certainly appears to be enough, on its own, to make the equation 'balance out' - so we wouldn't necessarily need any other restrictions on the values of Y or Z. There might be some other condition though that would also make the equation balance out, so we should take a good look at Answers D and E.

IF... Y=0

XY + Z = X(Y+Z)
X(0) + Z = X(0+Z)
X + Z = XZ

That isn't necessarily true (in fact, it's almost never true), so we can eliminate Answer D

IF... Z=0

XY + Z = X(Y+Z)
X(Y) + 0 = X(Y+0)
XY = XY

This is also balanced out, so this is another option.

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

kulksnikhil wrote: ↑

Wed Nov 15, 2006 6:39 pm

Q:

if xy+z = x(y+z), which of the following must be true,

1. x=0 & z = 0
2. x=1 & y=1
3. y=1 & z=0
4. x=1 OR y=0
5. x=1 OR z=0

I find all of the above to be true. Unfortunately I did not note down the answer choices given.. can anybody help solving this problem?

thanks,
Nikhil

Simplifying, we have:

xy + z = xy + xz

z = xz

xz - z = 0

z(x - 1) = 0

z = 0 or x = 1

Answer: 5