1. Problem: We need to find the condition for $x = y$ given that $x \leq y$ and $y \leq x$. 2. Formula and rules: The inequalities $x \leq y$ and $y \leq x$ imply that $x$ is less than or equal to $y$ and $y$ is less than or equal to $x$. 3. Intermediate work: If $x \leq y$ and $y \leq x$, then both must be equal because neither can be strictly less than the other. 4. Explanation: When two numbers satisfy $x \leq y$ and $y \leq x$ simultaneously, the only possibility is that $x = y$. This is a fundamental property of inequalities. 5. Final answer: The condition for $x = y$ given $x \leq y$ and $y \leq x$ is simply that $x$ equals $y$.