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 ![\[(x+*)(y+*)(z+*)\]](http://latex.artofproblemsolving.com/a/6/4/a6457a8bfdcad15592339c36a4ade0435f73cfea.png)
where the blanks should be filled in with numbers.
Filling in as ![\[(x+4)(y+2)(z+1)\]](http://latex.artofproblemsolving.com/9/8/9/9899b92a51853d1719bfd7f46ad450b370bd6d39.png)
gives ![\[2x+4y+8z+xy+4yz+2xz+xyz+8\]](http://latex.artofproblemsolving.com/3/4/5/3457d3b6650194964ec0580904ee6d9dd8d8bf42.png)
, and all parts happen to be multiples of the given equations. After substitution, we get ![\[(x+4)(y+2)(z+1)=60\]](http://latex.artofproblemsolving.com/2/1/4/214126139aefd768498023f7e055d602a87076b5.png)
.
