#### \(x = \frac{11}{7}, y = \frac{9}{7}\)Question: Find the number of integer solutions to the equation $ x^2 + y^2 = 25 $, where $ x $ and $ y $ are integers.
["Finding Integer Solutions to ( x^2 + y^2 = 25 ): A Step-by-Step Breakdown", "When solving equations involving sums of squares, especially those constrained to integer values, it’s helpful to explore all possible integer pairs ( (x, y) ) that satisfy the equation. One classic example is ( x^2 + y^2 = 25 ), a circle centered at the origin with radius 5 in the coordinate plane. Here, we focus on uncovering how many integer solutions exist—points with whole number coordinates lying exactly on this circle.", "### Step 1: Understand the Range\nSince ( x^2 + y^2 = 25 ), both ( x^2 ) and ( y^2 ) must be non-negative and less than or equal to 25. This means the absolute value of any integer solution must satisfy:\n[\n|x| \leq 5 \quad \ ext{and} \quad |y| \leq 5\n]\nSo we only need to test integer values of ( x ) and ( y ) from (-5) to (5).", "### Step 2: Test Possible Values of ( x )\nWe evaluate ( x = -5, -4, \dots, 5 ) and check if ( 25 - x^2 ) is a perfect square:", "- If ( x = \pm 5 ): ( x^2 = 25 \Rightarrow y^2 = 0 \Rightarrow y = 0 ) → Solutions: ( (5, 0), (-5, 0) )\n- If ( x = \pm 4 ): ( x^2 = 16 \Rightarrow y^2 = 9 \Rightarrow y = \pm 3 ) → Solutions: ( (4, 3), (4, -3), (-4, 3), (-4, -3) )\n- If ( x = \pm 3 ): ( x^2 = 9 \Rightarrow y^2 = 16 \Rightarrow y = \pm 4 ) → Solutions: ( (3, 4), (3, -4), (-3, 4), (-3, -4) )\n- If ( x = \pm 2 ): ( x^2 = 4 \Rightarrow y^2 = 21 ), but 21 is not a perfect square\n- If ( x = \pm 1 ): ( x^2 = 1 \Rightarrow y^2 = 24 ), not a perfect square\n- If ( x = 0 ): ( y^2 = 25 \Rightarrow y = \pm 5 ) → Solutions: ( (0, 5), (0, -5) )", "### Step 3: List All Valid Integer Solutions\nFrom the above, the integer solutions are:\n[\n(5, 0), (-5, 0), (4, 3), (4, -3), (-4, 3), (-4, -3), (3, 4), (3, -4), (-3, 4), (-3, -4), (0, 5), (0, -5)\n]\nTotal count: 12 integer solutions", "### Step 4: Why This Matters – Integer Points on a Circle\nThis problem illustrates how Diophantine equations—polynomial equations seeking integer solutions—often reduce to finite, systematic checks over bounded domains. In geometry, such equations define circle lattice points; here, we found exactly 12 distinct pairs of integers whose squared distances from the origin equal 25.", "### Conclusion\nThe equation ( x^2 + y^2 = 25 ) has 12 integer solutions, consistently arising from all integer-coordinate points lying on a circle of radius 5. Whether studying number theory, geometry, or cryptography, understanding such solutions underpins deeper mathematical insights.", "---", "Keywords: ( x^2 + y^2 = 25 ), integer solutions, lattice points, Diophantine equations, number theory, coordinate geometry, 12 solutions", "Optimize this article for search engines with relevant questions, definitions, and structured sections that match common user intent: “find integer solutions to ( x^2 + y^2 = 25 )”, “number of integer pairs satisfying ( x^2 + y^2 = 25 )”, and related terms."]
