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HMMT 二月 2002 · 代数 · 第 7 题

HMMT February 2002 — Algebra — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. The real numbers x, y, z, w satisfy 2 x + y + z + w = 1 x + 3 y + z + w = 2 x + y + 4 z + w = 3 x + y + z + 5 w = 25 . Find the value of w .
解析
  1. The real numbers x, y, z, w satisfy 2 x + y + z + w = 1 x + 3 y + z + w = 2 x + y + 4 z + w = 3 x + y + z + 5 w = 25 . Find the value of w . Solution: 11 / 2 . Multiplying the four equations by 12 , 6 , 4 , 3 respectively, we get 24 x + 12 y + 12 z + 12 w = 12 6 x + 18 y + 6 z + 6 w = 12 4 x + 4 y + 16 z + 4 w = 12 3 x + 3 y + 3 z + 15 w = 75 . Adding these yields 37 x + 37 y + 37 z + 37 w = 111, or x + y + z + w = 3. Subtract this from the fourth given equation to obtain 4 w = 22, or w = 11 / 2.