The discussion revolves around solving a system of equations involving square roots and logarithms in real numbers. The equations are symmetric in nature, and participants explore potential solutions and the implications of symmetry.
Participants generally agree that (3, 3, 3) is a solution, but there is disagreement regarding whether it is the only solution, with some suggesting that other solutions may exist due to the nature of the equations.
The discussion does not resolve the existence of other potential solutions, and the implications of symmetry are not fully explored in terms of their mathematical consequences.
Due to the symmetry, I assume that x = y = z.\text{Solve the following system of equations in real numbers:}
. . \begin{array}{ccc}\sqrt{x^2-2x+6}\cdot \log_{3}(6-y) &=&x \\<br /> \sqrt{y^2-2y+6}\cdot \log_{3}(6-z) &=& y \\<br /> \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)&=&z\end{array}
Symmetry only guarantees that any permutation of the values of x, y, z for a solution is also a solution.soroban said:Hello, jacks!
I agree with pickslides . . .
Due to the symmetry, I assume that x = y = z.
Then we have: .\sqrt{x^2-2x+6}\cdot \log_3(6-x) \:=\:x
By inspection, we see that: .x\,=\,3.