For \(x\) and \(y\) to be integers, \(a + b\) and \(b - a\) must both be even, so \(a\) and \(b\) must have the same parity.

["Title: Understanding When (a) and (b) Are Both Integers: The Key Role of Even Sums and Differences", "When working with integers and expressions involving sums and differences, a fundamental rule determines whether two numbers (a) and (b) can both be integers: (a + b) and (b - a) must both be even numbers. This condition ensures that (a) and (b) share the same parity—either both are even or both are odd.", "### Why Parity Matters in Integer Solutions", "Parity refers to whether a number is even (divisible by 2) or odd (leaving a remainder of 1 when divided by 2). If (a + b) is even, it means (a) and (b) are both even or both odd. Similarly, for (b - a) to be even, the subtraction must also yield an even result, reinforcing the same parity requirement.", "Mathematically, an even number can be written as (2k) for some integer (k), while an odd number is (2k + 1). Let’s explore:", "- If (a = 2m) (even), (b = 2n) (even), then:\n - (a + b = 2(m + n)) → even\n - (b - a = 2(n - m)) → even", "- If (a = 2m + 1) (odd), (b = 2n + 1) (odd), then:\n - (a + b = 2(m + n + 1)) → even\n - (b - a = 2(n - m)) → even", "In both cases, (a) and (b) are both even or both odd (same parity), satisfying the condition for integrality of expressions built from their sum and difference.", "### Consequences and Applications", "This parity rule simplifies checking feasibility in equations, Diophantine problems, and systems involving integer variables. For example:", "- If you solve (a + b = 10) and (b - a = 2), solving for (a = 4), (b = 6) confirms both numbers are even integers.", "Checking parity first avoids unnecessary attempts with incompatible parity that would violate the equations.", "### Summary", "For (a) and (b) to be integers when built from (a + b) and (b - a), both expressions must be even. This guarantees (a) and (b) share parity—both even or both odd. Understanding this rule helps streamline problem-solving in number theory and algebra involving integer solutions.", "---", "Key Takeaway:\n(a + b) and (b - a) are both even ⇔ (a) and (b) have the same parity ⇔ both are even or both are odd ⇔ integrality conditions are satisfied.", "This insight is crucial when working with integer solutions, Diophantine equations, and parity-based reasoning."]