If x and y are non-zero integers and |x| + |y| = 32, what is

[Pia]

If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0

(2) |x| - |y| = 16

if i solve it .. from statement 1, i get y =-16 and not y =8 as given in solution.. could you pls check:

x=-3y. this means x and y have opp signs
if x<0, y>0 : the equation becomes -3y+y=32 or -2y = 32 or y =-16

[JadranLee]

We can indeed conclude from statement (1) that

x=-3y, and that x and y have opp signs

However, when we substitute this into the equation |x| + |y| = 32, we get

|-3y| + |y| = 32
3|y| + |y| = 32
4|y| = 32
|y|=8

It looks like you forgot the absolute value signs in the equation |x| + |y| = 32.

-Jad

[Guest]

Jadran (or someone else) -

Do you mind posting the entire solution to the problem?

Thanks.

[StaceyKoprince]

I'm copying the official solution from our database b/c it sounded like the guest who asked for the solution hasn't seen our solution yet. If you actually have a question about our solution and need something to be explained differently, let me know.

* * *
Note that one need not determine the values of both x and y to solve this problem; the value of product xy will suffice.

(1) SUFFICIENT: Statement (1) can be rephrased as follows:

-4x - 12y = 0
-4x = 12y
x = -3y

If x and y are non-zero integers, we can deduce that they must have opposite signs: one positive, and the other negative. Therefore, this last equation could be rephrased as

|x| = 3|y|

We don’t know whether x or y is negative, but we do know that they have the opposite signs. Converting both variables to absolute value cancels the negative sign in the expression x = -3y.

We are left with two equations and two unknowns, where the unknowns are |x| and |y|:

|x| + |y| = 32
|x| - 3|y| = 0

Subtracting the second equation from the first yields

4|y| = 32
|y| = 8

Substituting 8 for |y| in the original equation, we can easily determine that |x| = 24. Because we know that one of either x or y is negative and the other positive, xy must be the negative product of |x| and |y|, or -8(24) = -192.

(2) INSUFFICIENT: Statement (2) also provides two equations with two unknowns:

|x| + |y| = 32
|x| - |y| = 16

Solving these equations allows us to determine the values of |x| and |y|: |x| = 24 and |y| = 8. However, this gives no information about the sign of x or y. The product xy could either be -192 or 192.

The correct answer is A.

[avishal]

Why statement 1 is sufficient:

statement 1: -4x - 12y = 0
=> x = -3y --- (a)

substituting the value in original expression: |-3y| + |y| = 32

so, either 3y + y = 32 (when y>0)
=> y = 8, x = -24 (using a)
=> xy = -192

or -3y - y = 32 (when y <0)
=> -4y = 32
=> y = -8, x = 24 (using a)
=> xy = -192

[brian]

Thanks avishal.

-Brian Lange
Instructor and Online Moderator

[shaan17]

In the explanations given in the MGMAT material it says one way of solving is "simply stripping the absolute-value bars off the expressions, and not changing anything else". In this case if we apply the above rule to the equation |-3y| + |y| = 32
we would get -3y + y =32 which would yield y= -16 and x= 48 which is incorrect when substituted in original equation. Again when we reverse the signs we get 3y - y =32 which again yields the wrong answer. In the explanation above this post for scenario one it is shown as 3y + y = 32 , if we are just removing the abs-value bars shouldn't it be -3y + y = 32. I was confused with this, please advise.

[StaceyKoprince]

This can be really confusing; part of the complication here is that you have a negative coefficient within one of the absolute values - that negative never changes, and it can also be dealt with before we start to figure out what the different scenarios are based on the variable (which could be either positive or negative). Another complication is that you have two absolute value signs in this equation, not just one.

So if I start with:
|-3y| + |y| = 32
I first need to manipulate (or pull out) that -3. Think of it this way:
|-3| * |y| + |y| = 32
That first item, -3 in an absolute value sign, is always going to be positive. So what I really have is:
3* |y| + |y| = 32
Now, do your pos/neg scenarios:
pos: 3y + y = 32
neg: -(3y + y) = 32
and so forth

Just remember, if you have any coefficients (numbers), you can (and should) pull them out of the absolute value sign first b/c those coefficients don't change. Only the variables could change.

[shaan17]

Awesome .... thanks so much Stacey ! :D

[StaceyKoprince]

you're welcome!

[abedinbhuiyan]

Dear Koprince,

We expect the ans like this way below:


BR

Farid

""
..................................
Solving these equations allows us to determine the values of |x| and |y|: |x| = 24 and |y| = 8. However, this gives no information about the sign of x or y. The product xy could either be -192 or 192.

The correct answer is A.""

[jnelson0612]

Dear Koprince,

We expect the ans like this way below:


BR

Farid

""
..................................
Solving these equations allows us to determine the values of |x| and |y|: |x| = 24 and |y| = 8. However, this gives no information about the sign of x or y. The product xy could either be -192 or 192.

The correct answer is A.""

Sorry, do you have a question?

[xanadian]

One can use graphical method to solve this.
Note that
XY = positive in 1st and 3rd quadrant
XY = negative in 2nd and 4th quadrant

|x| + |y| = 32 represents a square with vertices at ( -32, 0), (0,32), (-32,0) and (0,-32).

-4x - 12y = 0 represents a straight line which intersects the above mentioned square at two points. Since the slope of this line is negative and the Y intercept is zero, one can easily say that the points of intersection are in 2nd and 4th quadrant and will have the same value of XY. Sufficient.

|x| - |y| = 16 is a bit tricky to visualize. It's a shape which is symmetrical with respect to X and Y axis. It intersects the square at 4 points. Hence the product XY will have positive and negative values. Not sufficient.

PS: Does anyone know the name of the shape of the equation |x| - |y| = 16 ?

[tim]

it has no commonly accepted name, and i would hesitate to call it a "shape" at all. are you sure you have the right idea of what this graph looks like?

[CharlesP11]

Hello,

In the solution posted by Stacey I was confused by the following statement:

"Converting both variables to absolute value cancels the negative sign in the expression x = -3y."

I understand that x and y have opposite signs because x = -3y, but the principle of "converting" confused me. Is this a principle that is discussed in the Manhattan GMAT books or is relatively common in absolute value problems?

I really appreciate JadranLee's post, substitute x = -3y into the equation |x| + |y| = 32. I found this explanation is easier to understand.

Thanks JadranLee for the post and thank you MGMAT team for your assistance!

- Charles