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.