Difference between revisions of "1986 AIME Problems/Problem 4"
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== See also == | == See also == | ||
{{AIME box|year=1986|num-b=3|num-a=5}} | {{AIME box|year=1986|num-b=3|num-a=5}} | ||
+ | * [[AIME Problems and Solutions]] | ||
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[Mathematics competition resources]] | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 14:38, 6 May 2007
Problem
Determine if
,
,
,
, and
satisfy the system of equations below.
![$\displaystyle 2x_1+x_2+x_3+x_4+x_5=6$](http://latex.artofproblemsolving.com/a/b/8/ab820c7ac6c3edef0bb989d92059fbc38d263ee1.png)
![$\displaystyle x_1+2x_2+x_3+x_4+x_5=12$](http://latex.artofproblemsolving.com/5/0/7/5078abc978de8c47927ba4e139a0a72ad5abea1a.png)
![$\displaystyle x_1+x_2+2x_3+x_4+x_5=24$](http://latex.artofproblemsolving.com/8/9/7/8976ad6980ddad409e011b98f18b3d9a0e16e431.png)
![$\displaystyle x_1+x_2+x_3+2x_4+x_5=48$](http://latex.artofproblemsolving.com/6/8/6/686a2b62d98be0c1221bf0ea3e75cc0375ac2cec.png)
![$\displaystyle x_1+x_2+x_3+x_4+2x_5=96$](http://latex.artofproblemsolving.com/b/a/e/baec7b4c76295d73b189a0940004cf6c3f1c3a06.png)
Solution
Adding all five equations gives us so
. Subtracting this from the fourth given equation gives
and subtracting it from the fifth given equation gives
, so our answer is
.
See also
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 | ||
All AIME Problems and Solutions |