1986 AIME Problems/Problem 4

Problem

Determine $\displaystyle 3x_4+2x_5$ if $\displaystyle x_1$ , $\displaystyle x_2$ , $\displaystyle x_3$ , $\displaystyle x_4$ , and $\displaystyle x_5$ satisfy the system of equations below.

$\displaystyle 2x_1+x_2+x_3+x_4+x_5=6$ $\displaystyle x_1+2x_2+x_3+x_4+x_5=12$ $\displaystyle x_1+x_2+2x_3+x_4+x_5=24$ $\displaystyle x_1+x_2+x_3+2x_4+x_5=48$ $\displaystyle x_1+x_2+x_3+x_4+2x_5=96$

Solution

Adding all five equations gives us $6(x_1 + x_2 + x_3 + x_4 + x_5) = 6(1 + 2 + 4 + 8 + 16)$ so $x_1 + x_2 + x_3 + x_4 + x_5 = 31$ . Subtracting this from the fourth given equation gives $x_4 = 17$ and subtracting it from the fifth given equation gives $x_5 = 65$ , so our answer is $3\cdot17 + 2\cdot65 = 181$ .

1986 AIME (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1986 AIME Problems/Problem 4

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