We aim to solve the system of equations:

1. $x^x + y^y = 31$
2. $x^y + y^x = 17$

This system involves variables raised to the power of themselves or each other. While it may not yield straightforward solutions, let's attempt to find possible values through reasoning and substitution.

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Step 1: Check for integer solutions.

1. Start with small integers for $x$ and $y$ to see if they satisfy both equations.

Case 1: $x = 2, y = 3$

• For $x^x + y^y$: $2^2 + 3^3 = 4 + 27 = 31$ ✅ Satisfies the first equation.

• For $x^y + y^x$: $2^3 + 3^2 = 8 + 9 = 17$ ✅ Satisfies the second equation.

Thus, $x = 2$ and $y = 3$ is a solution.

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Step 2: Verify symmetry.

If $x = 3$ and $y = 2$, due to symmetry, the same results hold:

• $x^x + y^y = 31$
• $x^y + y^x = 17$

Thus, $x = 3, y = 2$ is also a solution.

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Final Solutions

The solutions are: $(x, y) = (2, 3) \quad \text{or} \quad (x, y) = (3, 2)$

Would you like a detailed explanation of how to explore non-integer solutions or verify these results? Here are some related questions for exploration:

1. What happens if $x$ and $y$ are non-integer values?
2. How can we systematically find solutions for similar exponential equations?
3. Are there any conditions where no solutions exist for similar problems?
4. How does symmetry help in reducing the effort to solve such equations?
5. What tools or software can be used to solve non-linear equations like these efficiently?

Tip: For equations with exponents involving the variables themselves, symmetry and integer trials often simplify the process.