Need help with this math problem

Hey there I need some help with this problem

Find x^2 + y^2 if x,y http://data.artofproblemsolving.com/...67cdd0d075.gif N and xy + x + y = 71 and (x^2)y + x(y^2) = 880

I've been working on this for hours still no luck, hoping someone can show me how to solve this problem thanks.

Re: Need help with this math problem

Quote:

Originally Posted by

**gfbrd**

$\displaystyle (xy+x+y)^2=x^2y^2+x^2+y^2+2x^2y+2xy^2+2xy=(71)^2$

Re: Need help with this math problem

I did that already and got stuck a bit farther after that

what I did was substituted those x^2y and xy^2 with 880 to get

x^2(y^2) + x^2 + y^2 + 880 + 880 + 2xy = 71^2

and I noticed

x^2(y^2)((x+y)^2) = 880^2

I had a bit of trouble after that

Re: Need help with this math problem

What I did was to substitute u = xy and w = x + y

Then we have

u = 71 - x - y = 71 - w

and also

u = (880)/(x + y) = (880)/w

Now you can eliminate u and you'll have a quadratic you can solve for w.

Also, you may have noticed that x and y are symmetric. They appear in the same way in both equations, so if (a , b) is a solution, (b , a) should be also.

I assume that's enough for now? Write back please if you still have any trouble.

Re: Need help with this math problem

hey zhandele thanks for the hint but after doing what you told me I came up with 2 answers:

xy = 16 = 55 is this supposed to be like this or is there 1 solution? im not sure which one to pick but after substituting in both of them I get

x^2 + y^2 = 3493 = 646

yea I get 2 solutions here, let me know if this is correct or not thanks

Re: Need help with this math problem

Here's my understanding.

If x + y = 55 then xy = 16. This leads to two answers, but they're irrational. They involve the square root of 329, as I recall. I believe you're supposed to look for integers, so these answers you don't want.

If x + y = 16 then xy = 55. You can then solve for x = 5 or x = 11 and for y = 5 and x = 11. If you substitute back, I think you'll find either pair will work, so I suppose you can you have two answers.

The way I see it, these two answers aren't really different. That's what I meant when I said that x and y are symmetric. Maybe that wasn't the best way to put it. If you substitute x for y and y for x, nothing about the problem changes, it looks just the same. Did you try graphing the two lines and two hyperbolas? They're all symmetric about the line y = x.

If you were doing a story problem, where x and y stood for something in the real world, so somebody cared whether you used five or eleven of x, then I'd say they're different answers.

Perhaps that's why you're asked for x^2 + y^2. The sum is the same no matter which is the 5. BTW, I get

x^2 + y^2 = 146

How did you get that 646?

I'd say 146 is the answer you want.

Kindly, R

Re: Need help with this math problem

I got it now thanks for your help