1976 AHSME Problems/Problem 30
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. Filling in as 
gives 
, and all parts happen to be multiples of the given equations.
