Problem Solving

if XY + Z = X (Y+Z), which of the following must be true?

x= 0 and z = 0

x=1 and y = 1

y=1 and z = 0

x=1 or y = 0

x1 or z = 0

OA: E

My thoughts:

[spoiler]Turn this into the equation XY + Z = XY + XZ

Subtrack XY from both sides

Z = XZ

Divide Z by both sides

X = 1

This is where I get confused.

At this point, I replugged X into XY to get 1Y + Z = 1Y + 1Z.

everything cancels out. How does possibly give me Z=1? [/spoiler]

You did it right up to a point. You got that:

XZ = Z - but you're not supposed to immediately divide by Z on both sides. What if Z = 0? Then the division by Z would not be possible, since division by 0 is undefined.

My approach would be to subtract Z from both sides to get:

XZ - Z = 0

Z(X - 1) = 0

You can easily tell that there are two options for this one. In order for the product of two numbers (in this case, it's Z and X - 1) to be zero, at least one of those two numbers has to be zero. "At least" basically means "OR": you can have either X - 1 = 0 OR Z = 0. There's your answer.

XY + Z = X (Y+Z)
xy+ z = xy + xz
z = xz

or

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

hence z = 0 , x = 1.
IMO E.

I have a problem with this task... It seems like any answer here would be correct

Lattefah84 wrote:I have a problem with this task... It seems like any answer here would be correct

i was under the same impression, but a closer look at the question reveals it all. The question says which of the following must be true. All the choices can be true.

Dana has done a great job of explaining the question and the OA

DanaJ wrote:XZ = Z - but you're not supposed to immediately divide by Z on both sides. What if Z = 0? Then the division by Z would not be possible, since division by 0 is undefined.

My approach would be to subtract Z from both sides to get:

XZ - Z = 0

Z(X - 1) = 0

You can easily tell that there are two options for this one. In order for the product of two numbers (in this case, it's Z and X - 1) to be zero, at least one of those two numbers has to be zero. "At least" basically means "OR": you can have either X - 1 = 0 OR Z = 0. There's your answer.

So, to help you out, i'll explain why the other answer choices are incorrect.

Keep in mind that the question asks "which MUST be true and NOT which CAN be true"
We'll use the factored expression that Dana has derived -> Z(X-1) =0

Choice (A): x= 0 and z = 0
z =0 makes the whole expression Z(X-1) equal to 0 irrespective of the value of X
Therefore, if z =0, x can be any value, it can be 1,10,-1,5000 etc. While it can also be 0, it is not necessarily and therefore this answer choice is eliminated.

Choice (B): x=1 and y = 1
-> x = 1 makes the expression Z(1-1) which will equal 0 irrespective of the value of y, which is not feature in our expression. Therefore this answer choice is also eliminated

Choice (C): y=1 and z = 0
again, y does not feature so we cannot say that this is true. Eliminated

Choice (D): x=1 or y = 0
OR means that we have to consider the two independently
x=1 -> z(1-1) which equals 0, so this is fine
y=0 -> who knows? this doesn't figure in our expression. Not fine! Eliminated!

Choice (E): x=1 or z = 0
Only choice that has not been eliminated, so we should select this and proceed.
x=1 -> z(1-1) which equals 0, so this is fine
z = 0 -> 0(x-1) which equals 0, so this is also fine. BINGO!

Hope this helps mate.

if XY + Z = XY +XZ

Z= XZ

Z(1-x) =0

which means either z =0 or x =1

Hence E

sars72 wrote:

Lattefah84 wrote:I have a problem with this task... It seems like any answer here would be correct

i was under the same impression, but a closer look at the question reveals it all. The question says which of the following must be true. All the choices can be true.

Dana has done a great job of explaining the question and the OA

DanaJ wrote:XZ = Z - but you're not supposed to immediately divide by Z on both sides. What if Z = 0? Then the division by Z would not be possible, since division by 0 is undefined.

My approach would be to subtract Z from both sides to get:

XZ - Z = 0

Z(X - 1) = 0

You can easily tell that there are two options for this one. In order for the product of two numbers (in this case, it's Z and X - 1) to be zero, at least one of those two numbers has to be zero. "At least" basically means "OR": you can have either X - 1 = 0 OR Z = 0. There's your answer.

So, to help you out, i'll explain why the other answer choices are incorrect.

Keep in mind that the question asks "which MUST be true and NOT which CAN be true"
We'll use the factored expression that Dana has derived -> Z(X-1) =0

Choice (A): x= 0 and z = 0
z =0 makes the whole expression Z(X-1) equal to 0 irrespective of the value of X
Therefore, if z =0, x can be any value, it can be 1,10,-1,5000 etc. While it can also be 0, it is not necessarily and therefore this answer choice is eliminated.

Choice (B): x=1 and y = 1
-> x = 1 makes the expression Z(1-1) which will equal 0 irrespective of the value of y, which is not feature in our expression. Therefore this answer choice is also eliminated

Choice (C): y=1 and z = 0
again, y does not feature so we cannot say that this is true. Eliminated

Choice (D): x=1 or y = 0
OR means that we have to consider the two independently
x=1 -> z(1-1) which equals 0, so this is fine
y=0 -> who knows? this doesn't figure in our expression. Not fine! Eliminated!

Choice (E): x=1 or z = 0
Only choice that has not been eliminated, so we should select this and proceed.
x=1 -> z(1-1) which equals 0, so this is fine
z = 0 -> 0(x-1) which equals 0, so this is also fine. BINGO!

Hope this helps mate.

So, we should find the choice that makes the same conclusion? Hm, I didn't think in that way. Thank you very much!