If 2xy + z = 9, what is the value of the positive integer z?

[BTGmoderatorDC]

If 2xy + z = 9, what is the value of the positive integer z?

(1) xyz - z^2 = 0

(2) x + y - 3z = -5

OA A

Source: Manhattan Prep

[Brent@GMATPrepNow]

If 2xy + z = 9, what is the value of the positive integer z?

(1) xyz - z² = 0

(2) x + y - 3z = -5

Given: 2xy + z = 9

Target question: What is the value of the positive integer z?

Statement 1: xyz - z² = 0
Factor to get: z(xy - z) = 0
This means that EITHER z = 0 OR xy - z = 0
HOWEVER, we're told that z is a POSITIVE integer, which means z ≠0
So, it must be the case that xy - z = 0

We know have:
2xy + z = 9
xy - z = 0

Multiply both sides of the bottom equation to get:
2xy + z = 9
2xy - 2z = 0

Subtract the bottom equation from the top equation to get: 3z = 9, which means z = 3
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x + y - 3z = -5
We also know that 2xy + z = 9

ASIDE: for statement 1, I was able to eliminate the x and y variables, because they appeared together in the terms 2xy and xy
In statement 2, we aren't as lucky.
As such, this system of 2 equations and 3 variables in not enough information to solve for z.
Statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

[deloitte247]

Statement 1
$$xyz-z^2=0$$
$$z\left(xy-z\right)=0$$
$$z=0\ or\ z=xy$$
Since we are looking for positive integers
$$z=xy$$
$$2xy+z=9\ where\ xy=z$$
$$2z+z=9$$
$$3z=9$$
z = 3
statement 1 is SUFFICIENT.

Statement 2
$$x+y-3z=-5$$
The value of 2 cannot be obtained as we still have unknown variable x and y left in the equation.
Statement 2 is INSUFFICIENT.
Statement 1 alone is SUFFICIENT.

$$answer\ is\ Option\ A$$