I have this equation here:

x^2 + y^2 + z^2 = xy + xz + yz

I need to somehow prove that the equation is trueonlywhen all three variables are equal, i.e., x = y = z.

I would appreciate any advice or suggestions on how to accomplish this.

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- Apr 28th 2008, 04:46 AMgfreeman75proof for an equation
I have this equation here:

x^2 + y^2 + z^2 = xy + xz + yz

I need to somehow prove that the equation is true**only**when all three variables are equal, i.e., x = y = z.

I would appreciate any advice or suggestions on how to accomplish this. - Apr 28th 2008, 05:25 AMmr fantastic
Well, I'll probably get jumped on here but here goes .......

The equation can be re-arranged in two ways:

x(x - y) + y(y - z) + z(z - x) = 0 .... (1)

x(x - z) + y(y - x) + z(z - y) = 0 ....(2)

Compare and equate coefficients of x, y and z:

x - y = x - z => y = z.

y - z = y - x => x = z.

Therefore x = y = z is the only solution. - Apr 28th 2008, 07:32 AMIsomorphism
- Apr 28th 2008, 07:36 AMJameson
- Apr 28th 2008, 10:38 AMgfreeman75
That's a pretty elegant solution to my problem, Isomorphism. Thank you.

- Apr 28th 2008, 02:08 PMmr fantastic