Let $x,y,z$ is positive. Solve:
$\begin{cases}{x\left( {6 - y} \right) = 9}\\
{y\left( {6 - z} \right) = 9}\\
{z\left( {6 - x} \right) = 9}\end{cases}$
Given the system of equations:
\[
\begin{cases}
x(6 - y) = 9 \\
y(6 - z) = 9 \\
z(6 - x) = 9
\end{cases}
\]

1. **Express each variable in terms of the others:**
   From the first equation, we can solve for \( y \):
   \[
   x(6 - y) = 9 \implies 6x - xy = 9 \implies xy = 6x - 9 \implies y = \frac{6x - 9}{x}
   \]

   From the second equation, we can solve for \( z \):
   \[
   y(6 - z) = 9 \implies 6y - yz = 9 \implies yz = 6y - 9 \implies z = \frac{6y - 9}{y}
   \]

   From the third equation, we can solve for \( x \):
   \[
   z(6 - x) = 9 \implies 6z - zx = 9 \implies zx = 6z - 9 \implies x = \frac{6z - 9}{z}
   \]

2. **Substitute and simplify:**
   Substitute \( y = \frac{6x - 9}{x} \) into the second equation:
   \[
   \left(\frac{6x - 9}{x}\right)(6 - z) = 9
   \]
   Simplify:
   \[
   \frac{(6x - 9)(6 - z)}{x} = 9
   \]
   Multiply both sides by \( x \):
   \[
   (6x - 9)(6 - z) = 9x
   \]
   Expand and simplify:
   \[
   36x - 6xz - 54 + 9z = 9x
   \]
   Rearrange terms:
   \[
   36x - 6xz + 9z - 54 = 9x
   \]
   Combine like terms:
   \[
   27x - 6xz + 9z = 54
   \]

3. **Use symmetry and AM-GM inequality:**
   Notice that the equations are symmetric. We can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality to find a solution. The AM-GM inequality states that for non-negative real numbers \( a_1, a_2, \ldots, a_n \):
   \[
   \frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}
   \]
   Equality holds if and only if \( a_1 = a_2 = \cdots = a_n \).

   Apply AM-GM to the product \( x(6 - x)y(6 - y)z(6 - z) \):
   \[
   \left(\frac{x + (6 - x) + y + (6 - y) + z + (6 - z)}{6}\right)^6 \geq x(6 - x)y(6 - y)z(6 - z)
   \]
   Simplify the left-hand side:
   \[
   \left(\frac{6 + 6 + 6}{6}\right)^6 = 6^3 = 216
   \]
   Given \( x(6 - x)y(6 - y)z(6 - z) = 9^3 = 729 \), we have:
   \[
   216 \geq 729
   \]
   This is a contradiction unless \( x = y = z = 3 \).

4. **Verify the solution:**
   Substitute \( x = 3 \), \( y = 3 \), and \( z = 3 \) back into the original equations:
   \[
   3(6 - 3) = 9 \\
   3(6 - 3) = 9 \\
   3(6 - 3) = 9
   \]
   All equations are satisfied.

Thus, the solution is \( x = 3 \), \( y = 3 \), and \( z = 3 \).

The final answer is \( \boxed{ x = 3 } \), \( y = 3 \), and \( z = 3 \).