which points satisfies the problem below?
$\displaystyle XYZ+XZ+XY-YZ-11X-3Y-Z+9=0$
Ah, how the road to mathematical truth is paved in blood...
streethot:
There are more solutions than you should care to count. A simple calculator program can tell you that...
1,-1,-1
1,-1,0
1,-1,1
-1,2,2
0,2,1
1,-1,-2
1,-1,2
-1,3,1
0,1,3
0,3,0
etcetera ad nauseam
As you know, Diophantine equations are notoriously difficult to solve. There may be an infinite number of solutions readily found by number-crunching.
What exactly is it that you want to know about this equation? A proof of whether or not there is an infinitude of points? A formula generating a set of solutions?
Here's a small part:
If you rearrange the equation you'll get the following:
$\displaystyle Z=\frac{11X+3Y-XY-9}{(X-1)(Y+1)}$
When X=1, and Y=-1 we get $\displaystyle Z=\frac{0}{0}$, in other words $\displaystyle 0 \cdot Z = 0$
So, any Z will work for X=1 and Y=-1
Some other brute force stuff:
When you set one variable to equal zero, there's only a finite number of solutions that will work (tried for X = 0 and Z = 0, Y = 0 will probably behave the same way)
I don't think there's one general formula for the solution though, because of the (1,-1,Z) case ...