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Given (x^2 + y^2 = 25) and (x - y = 3), we can solve for (x + y) by adding (x + y) to both sides of the second equation and then squaring both sides of the resulting equation. This gives us ((x + y)^2 = (x - y)^2 + 4xy). Substituting (x^2 + y^2 = 25) and (x - y = 3) into this equation, we get ((x + y)^2 = 9 + 4(25) = 109). Taking the square root of both sides, we get (x + y = \pm\sqrt{109}). Since (x + y) must be positive, we have (x + y = \sqrt{109}). Therefore, the answer is (\sqrt{109}), which is approximately equal to 10.3.