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.
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.
$\displaystyle x^2 + y^2 + z^2 = xy + xz + yz$
$\displaystyle \Rightarrow 2(x^2 + y^2 + z^2) = 2(xy + xz + yz)$
$\displaystyle \Rightarrow 2(x^2 + y^2 + z^2) - 2(xy + xz + yz) = 0$
$\displaystyle \Rightarrow (x^2 + y^2 - 2xy)+(x^2 + z^2 - 2xz) + (z^2 + y^2 - 2zy) = 0$
$\displaystyle \Rightarrow (x-y)^2+(x-z)^2 + (z-y)^2 = 0$
$\displaystyle \text{This forces }x=y=z.$
Mr F, can you please explain the validity of your method?Originally Posted by Mr.F