2013 AMC 12B Problem 6

Problem 6 of 25EasierAlgebra

Real numbers

x

and

y

satisfy the equation

x^2 + y^2 = 10x - 6y - 34.

What is

x+y?

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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,

x + y = 2.

Thus, the correct answer is B.

Concepts: completing the square

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.