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

[Rathna]

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

1. x=0 and z=0
2. x=1 and y=1
3. y=1 and z=0
4. x=1 or y=0
5. x=1 or z=0

I was able to solve for x as
xy+z = x(y+z)
xy+z = xy +xz
xz = z
x =1

I randomly marked option 2 and moved on as i dint know how to evaluate z and y. Please Help.

OA is 5.

[Rathna]

Oops ....realized the mistake
xz = z

xz-z = 0
z(x-1)=0
x =1, z=0

[RonPurewal]

yes.

DO NOT DIVIDE BY VARIABLES.
if you have a variable that appears in all terms of some equation, then move the terms over to one side of the equation, and FACTOR OUT the common variable in question. if you divide by the variable, then you will be missing out on the possibility that the variable in question is zero.

in fact, even if you know that the variable is nonzero - in which case it's technically safe to divide by it on both sides - such division is still generally not a good idea, because it tends to build bad habits. you should only do so if you consciously acknowledge, upon each such division, "it's ok to do this because the variable is guaranteed to be nonzero, and i know that because of _______".

[dhushan]

Hi, I was just wondering why the answer couldn't be (A), as if you plug the variables into the equation you find that the left side equals the right side.

i.e.
xy + z = x(y+z)
(0)y + 0 = 0(y+0)
0 = 0

what's wrong with my logic here.

[anoo.anand]

what is wrong with option A ?

0 = 0 ?

[sushanta_debnath]

what is wrong with option A ?

0 = 0 ?

the question asks, what must be true, not what can be true.

1. X= 0 and Z=0: if X=0, then Z may be 0 or may not be 0; hence it can not satisfy the "must be" part of the question.

[Ben Ku]

Sushantha's reply is correct. When the question asks which MUST be true, then there are no other options that counter that condition.

The first option states: x=0 and z=0
Because of the conjunction AND, it means there are no solutions where x and z are not both equal to 0. This option is false because we can create counterexamples where this first option is NOT true; for example, x = 1 and z = 5 is also a solution that counters the first option.

Hope that helps.