Date: 11/28/95 at 13:26:8 From: Anonymous Subject: Simultaneous equations Hi! x+y+z=1 x^2+y^2+z^2=2 x^3+y^3+z^3=3 The problem is to find the value of: x^4+y^4+z^4=? Thanks very much! Danny Bromberg email@example.com
Date: 11/28/95 at 16:6:57 From: Doctor Ken Subject: Re: Simultaneous equations Hello! I'll tell you right now, I think this is going to be a messy problem. What you DON'T want to do is try to solve for x, y, and z and then plug in. But nonetheless, I think it's kind of messy. Hm, how can I give you a hint ... Okay, try this: square the first equation. That gives us x^2 + y^2 + z^2 + 2(xy + xz + yz) = 1. Since the first three terms add to 2, we get 2 + 2(xy + xz + yz) = 1 xy + xz + yz = -1/2 Notice that we found the value of xy + xz + yz without ever finding the value of x, y, or z. That's pretty neat. See if you can do something similar to find out what x^4 + y^4 + z^4 is without finding x, y, and z. Good luck! -Doctor Ken, The Geometry Forum
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