DS problem with equations as statements...Help!!!

[relentlesspursuito700plus]

I was trying to figure out the value of X^2 + y^2 using the following information:

x^2 + y^2 = 2xy+1.

That's obviously not enough.

x^2 + y^2 = 4-2xy.

Again, not enough.

So at this point, I've crossed off choices A, B, and D. Now I am left with C or E.

Since X^2 + Y^2 equals two different equations, I set them equal to each other.

So, 2xy + 1 = 4 - 2xy. This, when simplified, equals 4xy=3, which can be further simplified to xy = 3/4.

So then I took the 3/4 and plugged it into one of the equations. 2*(3/4) + 1 = x^2 + y^2. Just to be safe, I tested to see if 2*(3/4)+1 would equal 4-2*(3/4). And they do. So I picked C. But the answer given in this book was E.

The way it approach the problem was:

x^2 + Y^2 = 2xy + 1, therefore, (x^2) - 2xy + (y^2) = 1
Then it factored out (x-y)(x-y)=1 and concluded x-1 was either 1 or negative 1. Thus it concluded that because there are infinite possibilities for x and y, you can't determine x^2 + y^2 from this. I am not sure what this accomplished but that's what it did.

For statement 2, it basically did the same thing. It added 2xy to both sides and the factored out the left side of the quation to: (x+y)(x+y)=4 and said x +y could be either 2 or -2. Thus no answer can be derived.

Put the two together and still they are insufficient so the answer was E.

I don't get it. Did I miss something here? Isn't the actual answer 2.5? Why can't I set the
2xy+1 = 4-2xy? Did I miss something obvious?

[relentlesspursuito700plus]

Sorry, I didn't know I had to cite it.

alternative email account 00 at gmail

Thanks!!

[RonPurewal]

three pronged response:
1) you're right
2) source is wrong
3) you shouldn't trust this source anymore.

--

EVERYTHING you have done is absolutely correct, and, in fact, optimal! you have learned well.
you should NOT concern yourself with finding individual values of x and y if you can find the desired combinations immediately.

--

even if you do go through with the long-winded solution that's begun in the solution key, here's what happens:
if x + y is 2 and x - y is 1, then x is 1.5 and y is 0.5.
if x + y is -2 and x - y is 1, then x is -0.5 and y is -1.5.
if x + y is 2 and x - y is -1, then x is 0.5 and y is 1.5.
if x + y is -2 and x - y is -1, then x is -1.5 and y is -0.5.
in all four cases, x^2 + y^2 is 1.5^2 + 0.5^2 (note that there's no need to actually calculate the value of this expression), so it's still sufficient.

huzzah!