Difference between revisions of "2003 AMC 10A Problems/Problem 13"
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Which means that x = 0.5, y = 3.5, and z = 16. Therefore, xyz = (0.5)(3.5)(16) = 28 | Which means that x = 0.5, y = 3.5, and z = 16. Therefore, xyz = (0.5)(3.5)(16) = 28 | ||
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== See Also == | == See Also == | ||
Revision as of 17:24, 21 November 2007
Problem
The sum of three numbers is 
. The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?
Solution
Let the numbers be 
, 
, and 
in that order.
Therefore, the product of all three numbers is 
Alternatively, we can set up the system in matrix form:
is equivalent to
Or, in matrix form
To solve this matrix equation, we can rearrange it thus:
Solving this matrix equation by using inverse matrices and matrix multiplication yields
Which means that x = 0.5, y = 3.5, and z = 16. Therefore, xyz = (0.5)(3.5)(16) = 28
