# proof for an equation

• Apr 28th 2008, 03:46 AM
gfreeman75
proof 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, 04:25 AM
mr fantastic
Quote:

Originally Posted by gfreeman75
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.

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, 06:32 AM
Isomorphism
Quote:

Originally Posted by gfreeman75
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.

$\displaystyle x^2 + y^2 + z^2 = xy + xz + yz$

$\displaystyle \Rightarrow 2(x^2 + y^2 + z^2) = 2(xy + xz + yz)$

$\displaystyle \Rightarrow 2(x^2 + y^2 + z^2) - 2(xy + xz + yz) = 0$

$\displaystyle \Rightarrow (x^2 + y^2 - 2xy)+(x^2 + z^2 - 2xz) + (z^2 + y^2 - 2zy) = 0$

$\displaystyle \Rightarrow (x-y)^2+(x-z)^2 + (z-y)^2 = 0$

$\displaystyle \text{This forces }x=y=z.$

Quote:

Originally Posted by Mr.F
Well, I'll probably get jumped on here but here goes .......

• Apr 28th 2008, 06:36 AM
Jameson
Quote:

Originally Posted by Isomorphism
$\displaystyle x^2 + y^2 + z^2 = xy + xz + yz$

$\displaystyle \Rightarrow 2(x^2 + y^2 + z^2) = 2(xy + xz + yz)$

$\displaystyle \Rightarrow 2(x^2 + y^2 + z^2) - 2(xy + xz + yz) = 0$

$\displaystyle \Rightarrow (x^2 + y^2 - 2xy)+(x^2 + z^2 - 2xz) + (z^2 + y^2 - 2zy) = 0$

$\displaystyle \Rightarrow (x-y)^2+(x-z)^2 + (z-y)^2 = 0$

$\displaystyle \text{This forces }x=y=z.$

Very succinct. It took me at least twice as many lines when I tried working it out.
• Apr 28th 2008, 09:38 AM
gfreeman75
That's a pretty elegant solution to my problem, Isomorphism. Thank you.
• Apr 28th 2008, 01:08 PM
mr fantastic
Quote:

Originally Posted by Isomorphism
$\displaystyle x^2 + y^2 + z^2 = xy + xz + yz$

$\displaystyle \Rightarrow 2(x^2 + y^2 + z^2) = 2(xy + xz + yz)$

$\displaystyle \Rightarrow 2(x^2 + y^2 + z^2) - 2(xy + xz + yz) = 0$

$\displaystyle \Rightarrow (x^2 + y^2 - 2xy)+(x^2 + z^2 - 2xz) + (z^2 + y^2 - 2zy) = 0$

$\displaystyle \Rightarrow (x-y)^2+(x-z)^2 + (z-y)^2 = 0$

$\displaystyle \text{This forces }x=y=z.$