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Difference between revisions of "2013 AMC 10B Problems/Problem 11"

(Created page with "==Problem== Real numbers <math>x</math> and <math>y</math> satisfy the equation <math>x^2 + y^2 = 10x - 6y - 34</math>. What is <math>x+y</math>? <math> \textbf{(A)}\ 1 \qquad...")
 
m (moved 2013 AMC 10B Problems/Problem 10 to 2013 AMC 10B Problems/Problem 11: The content of this page is actually problem 11)
(No difference)

Revision as of 16:47, 21 February 2013

Problem

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

$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8$

Solution

If we complete the square after bringing the x and y terms to the other side, we get $(x-5)^2 + (y+3)^2 = 0$. Squares of real numbers are nonnegative, so we need both $(x-5)^2$ and $(y+3)^2$ to be $0$. This obviously only happens when $x = 5$ and $y = -3$. $x+y = 5 + (-3) = \boxed{\textbf{(B) }2}$

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