smc

The graph $(x+y-5)(2x-3y+5)=0$ is the combined graphs of $x+y-5=0$ and $2x-3y+5=0$. Likewise, the graph $(x-y+1)(3x+2y-12)=0$ is the combined graphs of $x-y+1=0$ and $3x+2y-12=0$. All these lines intersect at one point, $(2,3)$. Therefore, the answer is $\boxed{1}$.