HMMT 二月 2009 · 冲刺赛 · 第 10 题

HMMT February 2009 — Guts Round — Problem 10

[ 7 ] Let a , b , and c be real numbers. Consider the system of simultaneous equations in x and y : ax + by = c − 1 ( a + 5) x + ( b + 3) y = c + 1 Determine the value(s) of c , in terms of a , such that the system has a solution for any a and b .

解析

[ 7 ] Let a , b , and c be real numbers. Consider the system of simultaneous equations in variables x and y : ax + by = c − 1 ( a + 5) x + ( b + 3) y = c + 1 Determine the value(s) of c in terms of a such that the system always has a solution for any a and b . 2 a +5 Answer: 2 a/ 5 + 1. (or ) 5 ( ) a b Solution: We have to only consider when the determinant of is zero. That is, when a +5 b +3 b = 3 a/ 5. Plugging in b = 3 a/ 5, we find that ( a + 5)( c − 1) = a ( c + 1) or that c = 2 a/ 5 + 1.