# System of equations

#### jacks

##### Well-known member
Solve the following system of equations in real numbers:

$\sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x$

$\sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y$

$\sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z .$

#### pickslides

##### Member
Have you looked at (3,3,3) being a solution, not sure myself but it might work given the base of those logs.

#### soroban

##### Well-known member
Hello, jacks!

I agree with pickslides . . .

$$\text{Solve the following system of equations in real numbers:}$$

. . $$\begin{array}{ccc}\sqrt{x^2-2x+6}\cdot \log_{3}(6-y) &=&x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z) &=& y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)&=&z\end{array}$$

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: .$$x\,=\,3.$$

#### CaptainBlack

##### Well-known member
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: .$$x\,=\,3.$$

Symmetry only guarantees that any permutation of the values of x, y, z for a solution is also a solution.

Obviously x=y=z=3 is a solution, but symmetry alone does not force us to conclude that it is the only solution.

CB

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