For integers a, b, x, and y, ab + yb = xy - yb. If a - b = 0 and x + y

Question

For integers a, b, x, and y, ab + yb = xy - yb. If a - b = 0 and x + y = 0, what is the value of x?

(1) a = 3

(2) y = -3

Kudos for a correct  solution .

kdatt1991 · Accepted Answer

This was quite a tricky question for me, but I hope that I have been able to crack it! Here's my solution: From the information given in the question stem, I decided to let a = b (from a - b = 0) and y = -x (from x + y = 0) so that we can have fewer variables to solve for x. I also re-arranged/simplifed the original equation from ab + yb = xy - yb to ab + yb +yb - xy = 0....... which equals; ab + 2yb - xy = 0. This way we can substitute a = b and y = -x within our re-arranged equation. This gives us a*(a) + 2*(-x)*(a) - x*(-x) = 0............ this simplifies to a^2 - 2ax + x^2 = 0. Factorising a^2 - 2ax + x^2 = 0....... gives us (x-a)^2 = 0. Therefore x = a. (1) a = 3 SUFFICIENT - now that we know that x = a; means that x = 3. (2) y = -3 SUFFICIENT - we know that y = -x (from x + y = 0), therefore x = 3. I think it is D! Please consider giving me KUDOS if you felt this post was helpful and correct! or please enlighten me (in case my answer's incorrect) so that I can learn and improve from my mistakes! Thanks.

devilbart · Answer

a=b and y=-x

So, substituting in equation

a^2 - ax = -x^2 + ax
Or, x^2 + a^2 - 2ax = 0

(x-a)^2=0
x-a=0,
x=a

(1) Sufficient as x=a
(2) sufficient as x=-y

Answer D

Posted from my mobile device

EMPOWERgmatRichC · Answer

Hi All,

This DS question has an interesting pattern built into it. If you understand the concept behind "systems" and recognize the system of equations in the prompt, then you can take advantage of that pattern and avoid most of the "work."

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

Bunuel · Answer

VERITAS PREP OFFICIAL SOLUTION:

D. Statement (2) should be clearly sufficient, as if y = -3 and, as you know from the stimulus, x + y = 0, then x must be 3. Statement (1), however, may not look immediately sufficient - but the ease with which Statement (2) can be proven sufficient without using 2/3 of the equations in the stimulus should tip you off - Statement (1) deserves your attention!

If you rephrase the most complex of the stimulus equations, ab + yb = xy - yb, substituting a for b and -x for y, you have:

a^2−ax=−x^2+ax
This should start to look like a quadratic, and if you move everything to the left hand side of the equation, you now have:

a^2−2ax+x^2=0
A classic common algebraic equation, and given that you know that a = 3, you can make that:

9−6x+x^2=0
Rephrase that to:

x^2−6x+9=0
And you should see that this is the same thing as (x−3)^2=0, meaning that x = 3. Statement (1) is thus sufficient.

minwoswoh · Answer

Just by looking at the question stem, you realize something interesting. There are:
- 4 variables
- 3 equations

If any statement were to give you a new equation, then the information would be sufficient. And so do statements 1 and 2, hence answer choice D is the correct answer.
If you can tackle this and rephrase the question in your head, you can solve this problem within 30 seconds.

Is this the pattern you refer to, Rich?

PS: btw, I dislike the official solution by Verita s.

EMPOWERgmatRichC · Answer

Hi minwoswoh,

Yes, that IS the pattern. Very nicely done.

Unfortunately, not all 'system' questions on the GMAT will be this easy to solve. Some will be wordy PS questions that will actually involve several math 'steps' to solve, while certain DS questions that appear to be systems will actually NOT be systems. Keep a critical eye, and do the necessary work on the pad, and you'll be able to pick up all of those points.

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

chetan2u · Answer

Hi,
substiute a=b and x=-y in main equation..
 ab + yb = xy - yb.....
b^2+yb=-y^2-yb...(b+y)b=-y(y+b)... ..
 b(b+y)+y(y+b)=0...
or (b+y)^2=0.. 
so b=-y..
now we have all four values interlinked..a=b=x=-y
any one value will give us the answer..

lets see the statements..
(1) a = 3.. one value given ..suff

(2) y = -3.. again one value given..suff

D

ENGRTOMBA2018 · Answer

Given, ab + yb = xy - yb, a=b and x=-y, x=?

Simplifying the given equation you get, ab + yb = xy - yb ---> b^2+by=-y^2-by  -->  (b+y)^2=0  --->  b =-y

Thus, a=b=-y=x

Per statement 1, a=3. As shown above, x= a and as such this statement is sufficient.

Per statement 2, y=-3. Clearly sufficient as x=-y.

D is the correct answer.

Sash143 · Answer

Given: a-b=0 ..... i
 x+y=0 ...... ii
 ab+yb=xy-yb ..... iii

Stat (1) a=3 
 ==> b=3 (from i)
Substituting in iii we have 2V2E ( 2 Variables 2 Equations (iii &ii) ). Hence we can solve it. (1) Suff
 
Stat (2) y=-3
 ==> x=3 (from ii). (2) Suff

joshbeall · Answer

Hi,

After taking a few practice GMAT exams, it's been clear I'm not fast enough on the quant section. I am used to having more time to work through problems, and I'm struggling with my pacing and strategy for the quant section.

I was just working on the following practice problem, after doing some quick math and going with what seemed reasonable, I picked the wrong answer (C, both statements together).

I went back and tried going through the complete process of figuring out the answer, but it took me several minutes, which highlights my problem. To those of you that did well on the GMAT quant section, what's your strategy for tackling a question like this and getting it in time? Is it just about being able to grind out the math faster?

I started working my way through this problem and found that I found up with a squared value (in my case,  y^2 ), and so I immediately figured "insufficient, because we could have a positive or negative root."

Well, I was wrong, and when I went back and reworked the problem more thoroughly, I think I found my issue:

ab + yb = xy - yb
ab + 2yb = xy

a - b = 0
a = b

x - y = 0
x = -y

Now, we're letting a = 3 (according to statement (1)), and we know a = b and x = -y, so we'll make those substitutions on the first equation:

3 * 3 + 2 * y * 3 = (-y) * y
9 + 6y = (-1) y^2 
9 + 6y +  y^2  = 0

I looked at that for a 30 to 60 seconds and thought about how to factor it (maybe that's my problem?), and got:

(y + 3)(y + 3) = 0

And from that, I can see that y = -3, and with that you can solve for x. And a similar method could be used if you started with the value of y instead of a.

So, I'm left with two questions:

1) Did I solve this correctly? The Veritas prep material tells me that both statements are sufficient, but it doesn't provide the details, so while I reached a definite answer, I'm not positive I got that definite answer via the correct mathematical process.
2) Do I just need to drill these types of problems so I can get faster at them, or am I missing a faster way to tackle these problems?

-Josh

p.s. I'm not sure the difficulty level, the prep material I'm using doesn't indicate the difficulty level.

ENGRTOMBA2018 · Answer

DS is a very different and tricky beast to tame. You need to do the following things:

1. Understand the given statements, try to explain to yourself what is given
2. Then, explain in your own words what is asked Is it asking for a unique value or is it a yes/no type of DS question.
3. Analyse the statements given individually and to completion (this is where you made a mistake!)
4. Only when you are sure that statements are not sufficient on their own, go to combining the 2 statements.

Coming back to your question.

1. You are given that a,b,x,y are integers and ab+by=xy-by 
Always try to simplify any given alegbraic equations given to you to whatever extent you can.

For this particular one, you get, ab+2by=xy. Additionally, you are given x+y=0 ---> x=-y or y=-x (just because we need x). Also given to us is a-b=0 ---> a=b. Substitute these values in the given algebraic equation,  ab+2by=xy ---> a^2-2ax=-x^2 ---> a^2-2ax+x^2=0 ---> (a-x)^2=0 ---> a=x

Thus, after simplification you see that x=-y and x=a ---> any statement giving any of these 2 values will thus be sufficient.

You can not do anything more than what I have mentioned for this step.

2. The question is asking you about the value 'x'. Thus a statement giving you 1 and only 1 value will be sufficient. Otherwise, itll not be sufficient.

3. Per the analysis in step 1 above, a statement mentioning the value of a or y will be sufficient. Both statements individually provide both and hence both are sufficient ---> D is thus the correct answer.

4. No need for this step as both statements are sufficient on their own.

DS demands a lot of timed practice and making notes of typical traps in DS. Although this question did not require the use of the information about the 4 variables being integers, you should nonetheless always remember these small markers whenever you solve any DS question (or any GMAT question for that matter). Timing will be taken care of once you start recognising these things and applying the correct sequence of steps.

Hope this helps.

P.S.: Make sure to search for a question before you start a new thread. Posting rules are mentioned in the link in my signatures. Your question has already been discussed before. Topics merged.

whatsarc · Answer

Information from Question
a=b
x=-y 
b(a+y)=y(x-b)

Statement 1
y^2+6y+9=0 
Y=-3 , hence x = 3
Suff

Statement 2
y=-3, hence x=3
Suff

D

dhruvie · Answer

Answer: D - Each statement is alone sufficient.

Reason for Statement 1 : if we put b=a and y=(-x) in the given equation, then we get an equation independent of b and y. If we get the value of a then we can calculate the value of x.

Resulting equation:   a^2 + x^2 -2ax = 0   i.e.   (a-x)^2=0   hence  a=x=3

Reason for Statement 2 : Given that y=(-x), hence  x=3

For integers a, b, x, and y, ab + yb = xy - yb. If a - b = 0 and x + y

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GMAT Data Sufficiency (DS) Questions

For integers a, b, x, and y, ab + yb = xy - yb. If a - b = 0 and x + y

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards