2013 AMC 12B Problem 6

Below is the professionally curated solution for Problem 6 of the 2013 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 12B solutions, or check the answer key.

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Concepts:completing the square

Difficulty rating: 1370

Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34.$ What is $x+y?$

$1$

$2$

$3$

$6$

$8$

Solution:

Rearranging gives $x^2 - 10x + 25 + y^2 + 6y + 9 = 0,$ that is $(x-5)^2 + (y+3)^2 = 0.$ Hence $x = 5$ and $y = -3,$ so $x + y = 2.$ Thus, the correct answer is B.

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