Question: Does xy = Square of x?

Comment on Does xy = Square of x?

Hi i have a doubt on this solution. If given y^2 = xy in statement 1, that means xy is positive. So either both x and y are positive or both negative. If thats the case, i replace y with x which is our question, and i will still be able to conclude that x^2=xy. your inputs ?
gmat-admin's picture

It COULD mean that xy is positive, or it could mean that y = 0. If y = 0, then it could be the case that x^2=xy or it could be the case that x^2 does not equal xy. Thus, statement 1 is not sufficient.

Hi Brent,

Thank you for your thorough explanation in this video! However, one thing I'm not sure of is why it is not appropriate to divide both sides of the target question by x to get a rephrased target question of y=x. Could you possibly elaborate on why this approach is not correct if you get a moment?

Thanks,
bhoglund
gmat-admin's picture

Hi bhoglund,
We need to be super careful whenever we consider dividing both sides of an equation by a variable, since we might inadvertently divide by zero, which has unintended consequences.

For example, if x = 0 and y = 1, then it's safe to say that the equation xy = x² is true, since (0)(1) = 0²
However, if we take xy = x² and divide both sides by x, we get y = x, which is false (since 1 ≠ 0)

The only way we could safely take xy = x² and divide both sides by x is if we are certain that x does not equal 0. For example, if we were given information that said x > 0, then we'd know that x ≠ 0, in which case we could simplify the equation to be y = x.

Does that help?

Hi Brent,

That helps a lot. Thank you!

Hi Brent, great videos,thanks!
I have a question: if I leave the statement unchanged we get x^2 =? xy.
Stat1 says y^2 = xy. No info about x^2 so obviously not sufficient.
BUT stat2 can be translated as follows :
xy = (x^2 + y^2)/2
So this stat is sufficient only if x^2=y^2.
However we don't have this info in the statement BUT we can get it from stat1, and in such case, my reasoning is that the answer is C.
Where am I wrong ?
gmat-admin's picture

Hi Mike,

I agree that statement 2 can be rewritten as xy = (x^2 + y^2)/2

However, I'm not sure what you mean when you say "this stat is sufficient only if x^2=y^2"

Can you elaborate on this?

Stat2 is not sufficient by itself to say whether or not xy = x^2.
However, if x^2 = y^2, then xy = 2*x^2/2, then xy = x^2. But we had to use stat1 to complement stat2 to provide a deterministic answer so my answer was C. Am I wrong?
gmat-admin's picture

Ahh. Sorry for taking so long to get that.

You're right in that, if we rewrite statement 2 as xy = (x² + y²)/2, then for the answer to our target question to be YES, it must be the case that x² = y²

However, this is exactly what statement 2 is telling us. The hard part is seeing it.

Statement 2:x² - 2xy + y² = 0
Factor: (x - y)(x - y) = 0
Solve: x - y = 0
So, it must be the case that x = y
This ALSO means that x² = y²

I've missed that, thanks a lot!

Hi Brent,
For Statement 2, your rephrased x-y =0 as x=y. can we do the same for y-x=0 in statement 1? i.e. y=x? Thanks
gmat-admin's picture

Sure, for statement 1, we can take y-x = 0 and rewrite it as y=x.

Hi Brent, stmt 1 tells us y-x=0 and this should be sufficient to ans question,am I missing anything?
gmat-admin's picture

Be careful; statement 1 (y² - xy = 0) tells us that EITHER y-x = 0 OR y = 0

Here are two sets of values that satisfy the equation y² - xy = 0:
case a) x = 1 and y = 1. In this case xy = x²
case b) x = 1 and y = 0. In this case xy ≠ x²

Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Hi Brent

If I make the assumption that for xy to = x^2, x must = y, then statement 1 is sufficient since I can show that x=y if I solve it. i gather from your approach this is a false assumption, but I am struggling to see how this can not be the case since x^2 can only be (x)(x).

Am I making some common error here. The rest of your explanation makes sense, but I looked past it since I made the first assumption.

Kind regards
gmat-admin's picture

Many students will (incorrectly) take the equation xy = x² and conclude that y = x.

The rationale is that it's perfectly valid to divide both sides of an equation by the same value (in the above case, we are dividing both sides by x).

HOWEVER, this step is only valid as long as x DOES NOT equal zero. If x = 0, then we are essentially dividing both sides by 0, which inevitably causes problems.

Since we are not given any information telling us that x does not equal zero, we can't divide both sides of the equation by x.

Consider the case where x = 0 and y = 3
Notice that these values satisfy the given equation: xy = x²
Plug in the values to get: (0)(3) = (0)² PERFECT!!

HOWEVER, the same values (x = 0 and y = 3) do NOT satisfy the equation y = x
So, we can't say that the equation xy = x² is equivalent to the equation y = x

Does that help?

Cheers,
Brent

I have a problem with the evaluation of the first statement.
My approach was the following:
"Does xy = x^2?" is the same as the question "Does y = x?", because just in this case is x*y=x^2 => Therefore my conclusion for the first statement was, that its sufficient, because y^2=x*y (with the rephrased target question "is x=y?") => therefore x must be y, because otherwise the solution wouldn't be y^2

Can you tell me, why this approach is wrong?

Cheers,
Fabi
gmat-admin's picture

Hi Fabi234,

The problem with your solution is that the equation xy = x² is NOT the same as the equation y = x

Please see my post above your post (in response to willem.wap@gmail.com)

If you have any follow-up questions, please don't hesitate to ask.

Cheers,
Brent

Hi Brent,

I'm having a difficult time understanding the so called "laws/rules" behind quadratic formulas and whether an answer is deemed sufficient or not. Before I go on, I do hope my question(s) make sense, and if you have trouble understanding, kindly let me know and I will further clarify.

To begin with, in this particular example. You simplify the initial formula: xy=x^2 into xy-x^2=0 then into x(y-x)=0

Whereas, I initially simplified the formula as:
1) xy=x^2
2) 0=-x^2-xy
3) 0=x(x-y)
Right off the bat, I get confused in terms of not knowing whether or not our approach is equivalent because x(y-x)=0 seems very different from x(x-y)=0. However they both stem from the same initial equation (That being xy=x^2).

Furthermore, lets use your simplified formula of x(y-x)=0.
This tells us that either x=0 or y=x or x=-y

Given statement 1) y^2-xy=0, which becomes y(y-x)=0.
We get a similar equation to our initial equation of x(y-x)=0. HOWEVER, where I get confused is when you state (*in this and other quadratic videos) that we cannot be certain that x equals the given value. For instance, in this case both our initial equation and statement one give us (y-x)

Question stem tells us: x(y-x)=0
Statement 1 tells us: y(y-x)=0

Does this not lead to the same solution of y=x or x=-y for both instances? I am aware that in the QS formuala, x can also equal 0 and in Statement 1, y can also equal 0. However, given the fact these are Quadratics, is it not known that we are expected to have TWO SEPARATE formulas that both apply to the equation? As a result, could you please provide more detail as to why this is not sufficient?

Apologies for the lengthy post. You do an excellent job in explaining these concepts that leave me overthinking.

Also, for a completely off-topic question, I would like to know whether your video modules are ordered in the sense that we should go through the modules from top to bottom. Or does it not matter?

Thanks a million.
gmat-admin's picture

It might help to recognize that, when we have an equation with 2 variables, the possible solutions must be in the form: x = something AND y = something.

So, for example, the equation x(y-x) = 0 can have several solutions including:
x = 0 and y = 1
x = 0 and y = 8
x = 0 and y = -6
x = 0 and y = 0
x = 4 and y = 4
x = 13 and y = 13
x = -7 and y = -7
etc
As we can see, x(y-x) = 0 when x = 0 OR when x = y

STATEMENT 1: y(y-x) = 0
What are some possible solutions here? Here are a few:
x = 3 and y = 0
x = 0 and y = 0
x = -6 and y = 0
x = 11 and y = 0
x = 4 and y = 4
x = 12 and y = 12
x = -7 and y = -7
etc
As we can see, y(y-x) = 0 when y = 0 OR when x = y

TARGET QUESTION: Does x(y-x) = 0?
For the solution x = 3 and y = 0, the answer to the target question is NO
For the solution x = 12 and y = 12, the answer to the target question is YES

So, statement 1 is not sufficient.

ASIDE: When you simplified the target question, you got: Does 0 = x(x-y)?
This has the exact same meaning as my rephrasing (Does x(y-x) = 0?)
Notice that, in both of our equations, the answer to the target question will be YES, when x = 0 OR when x = y

Does that help?

Cheers,
Brent

Thank you for your response Brent. I am still a little unclear on this subject, but I think more practice questions will help me apply/solidify this concept.

I failed to spot the pattern you used for this. So I tried to rationalise my way through testing numbers - 0 and 1

(1) x = 0 y = 1 equals to x^2
x = 1 y = 0 does not equal x^2 so insufficient

(2) I got the pattern of (x-y) = 0 in which case x must = y This was sufficient to answer the prompt

Is my approach valid? It seems a lot tedious compared to yours and I did make a mistake along the way and had to recheck my work
gmat-admin's picture

Testing values is a good approach for statement 1.
However, you made a small mistake when you chose the values x = 0 and y = 1
These values do not satisfy statement 1 (y² - xy = 0), so you'll have to find a different pair of values to test.

Cheers,
Brent

Wouldn't it be easier to rephrase the target question to "does y=x?"
gmat-admin's picture

Great question!

The problem with that approach is that xy = x² and x = y are not equivalent equations.

By dividing both sides of the equation by x (to get y = x), you were inadvertently (potentially) dividing by zero.

Notice that the equation xy = x² is satisfied if y = x OR if x = 0

For example, x = 0 and y = 1, is a solution to the equation xy = x²
So, the equation can be satisfied even if x and y are not equal

Likewise, x = 3 and y = 3, is also a solution to the equation xy = x²
In this case, it is the case that x and y are equal.

Does that help?

Cheers,
Brent

Well, I was actually thinking that in order for xy = x², then y would need to equal x, since x² = x². I wasn't intending to do any division of x (and I saw the previous comments of you addressing that ;-) ). I then proceeded to answer the question correctly. However, when I saw your explanation in the video afterwards, it seemed longer than for what it took me. I was left wondering if there was some sort of flaw in my approach.

Statement 1 said y=0 or x=y, which didn't give us a definitive answer. Statement 2 said x=y in both expressions (when factored), which gave a definitive answer.

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