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HMMT 二月 2006 · 代数 · 第 2 题

HMMT February 2006 — Algebra — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Find all real solutions ( x, y ) of the system x + y = 12 = y + x.
解析
  1. Find all real solutions ( x, y ) of the system x + y = 12 = y + x. ( ) ( ) √ √ √ √ 1 + 3 5 1 − 3 5 1 − 3 5 1 + 3 5 Answer: (3 , 3), ( − 4 , − 4), , , , 2 2 2 2 2 2 Solution: We have x + y = y + x which can be written as ( x − y )( x + y − 1) = 0. The 2 case x = y yields x + x − 12 = 0, hence ( x, y ) = (3 , 3) or ( − 4 , − 4). The case y = 1 − x √ √ 1 ± 1+44 1 ± 3 5 2 2 yields x + 1 − x − 12 = x − x − 11 = 0 which has solutions x = = . 2 2 The other two solutions follow.