Determine 3x_4+2x_5 if x_1, x_2, x_3, x_4, and x_5 satisfy the system 2x_1+x_2+x_3+x_4+x_5&=6, x_1+2x_2+x_3+x_4+x_5&=12, x_1+x_2+2x_3+x_4+x_5&=24, x_1+x_2+x_3+2x_4+x_5&=48, x_1+x_2+x_3+x_4+2x_5&=96.

1986 AIME Problem 4

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Determine

3x_4+2x_5

x_1,

x_2,

x_3,

x_4,

and

x_5

satisfy the system

\begin{aligned} 2x_1+x_2+x_3+x_4+x_5&=6,\\ x_1+2x_2+x_3+x_4+x_5&=12,\\ x_1+x_2+2x_3+x_4+x_5&=24,\\ x_1+x_2+x_3+2x_4+x_5&=48,\\ x_1+x_2+x_3+x_4+2x_5&=96. \end{aligned}

Answer

Let

S=x_1+x_2+x_3+x_4+x_5

Each equation has the form

S+x_i=

a constant

Put

S=x_1+x_2+x_3+x_4+x_5.

The five equations say that

S+x_i

equals

6,

12,

24,

48,

96,

respectively. Adding them gives

\begin{aligned} 6S&=6+12+24+48+96\\ &=186, \end{aligned}

S=31.

Hence

x_4=48-31=17

and

x_5=96-31=65.

The requested value is

3(17)+2(65)=181.