Difference between revisions of "1976 AHSME Problems/Problem 30"
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| − | Noticing the variables and them being multiplied together, we try to find a good factorization. After trying a few, we stumble upon something in the form of <math>(x+ | + | Noticing the variables and them being multiplied together, we try to find a good factorization. After trying a few, we stumble upon something in the form of <math>(x+*)(y+*)(z+*(</math> where the blanks should be filled in with numbers. Filling in as <math>(x+4)(y+2)(z+1)</math> gives <math>2x+4y+8z+xy+4yz+2xz+xyz</math>, and all parts happen to be multiples of the given equations. |
Revision as of 14:17, 7 May 2020
Problem 30
How many distinct ordered triples 
satisfy the equations
![\[x+2y+4z=12\]](http://latex.artofproblemsolving.com/e/5/3/e53f6d4f56a9fd1f01ffe12c1a29b46a0f06c1dd.png)
![\[xy+4yz+2xz=22\]](http://latex.artofproblemsolving.com/6/a/3/6a38bcbbdb99991b176c33335bd1a64cf6e13970.png)
![\[xyz=6\]](http://latex.artofproblemsolving.com/b/6/2/b62793631f6681941ffdba570d7401ab0f39f1e9.png)
Solution
Noticing the variables and them being multiplied together, we try to find a good factorization. After trying a few, we stumble upon something in the form of 
where the blanks should be filled in with numbers. Filling in as 
gives 
, and all parts happen to be multiples of the given equations.
