absolute value equations

[StellaL608]

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

(1) -4x – 12y = 0

(2) |x| – |y| = 16

Hi,

I ran across this problem in the CAT and am very confused by the explanation given below for why the first statement is sufficient.

1. The statements in bold seem to conflict with each other.

Can someone please reconcile those statements? If the negative expression cancels out, then how do we know the product of |x| and |y| is negative?

(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.

[Sage Pearce-Higgins]

Sorry for the delay in replying here. Thanks for writing your questions so clearly.

First of all, absolute value problems are really confusing. I often get mixed up when I'm doing algebra involving absolute value, but I find that it's often clearer if I pick a few numbers as examples. Take the first bold sentence:

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.

Let's pick some example: if y is 1, then x is -3; if y is 2, then x is -6. Alternatively, if y is -1, then x is 3. This might help you see that, according to statement 1, x and y have different signs.

The other important thing to remember about absolute values is that they are always positive or zero. So we don't have to worry about the negative sign in front of the 3y. Try running through those examples and you might see how this works.

I assume that the next part of the explanation is okay. Usually we would solve equations for variables like x and y, but here solve the equations for |x| and |y|. Sure, we don't know if x and y are positive or negative, the solutions tell us that |y| = 8 (so y is +8 or -8) and |x| = 24 (so x is +24 or -24). Now for your second bold sentence:

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.

Remember that absolute values are always positive or zero. This means that |x| multiplied by |y| is going to be 192. But since we know that one of x or y is negative and the other is positive, then x multiplied by y is going to be -192.

Whew! This is a good problem to tackle and get some insight into absolute value and understand the key points. However, remember to have a "Plan B" - if you saw a similar problem in the future, how would you approach it if you were short of time and needed to guess?