1. The problem is to understand and analyze the equation $xux = yuy$. 2. This equation involves variables $x$, $y$, $u$ and possibly represents a relation or equality between two expressions. 3. To solve or simplify, we need to understand the operations involved. Assuming $x$, $y$, and $u$ are variables and the operation is multiplication, the equation is $x \times u \times x = y \times u \times y$. 4. We can rewrite the equation as $xux = yuy$ or $x u x = y u y$. 5. If $u$ is invertible (non-zero), multiply both sides on the left by $u^{-1}$ and on the right by $u^{-1}$ to isolate $x$ and $y$: $$u^{-1} (x u x) u^{-1} = u^{-1} (y u y) u^{-1}$$ 6. Simplifying, we get: $$ (u^{-1} x) (u x u^{-1}) = (u^{-1} y) (u y u^{-1})$$ 7. Without additional context or constraints, this is the simplified form. 8. If $u$ commutes with $x$ and $y$, then $xux = yuy$ implies $x^2 u = y^2 u$ and if $u \neq 0$, then $x^2 = y^2$. 9. Therefore, $x = \pm y$. Final answer: Under the assumption that $u$ commutes with $x$ and $y$ and $u \neq 0$, the solutions satisfy $x = \pm y$.