Thinking about $ax + by = ab$ This is embarrassing because I know this is fairly simple, but I'm hitting a mental block and not having much luck with any references I'm aware of.
What properties does $ax + by = ab$ have? (a and b are integers)
I guess I'm sort of thinking...


*

*Is there always a solution? (I think so)

*How many solutions are there?

*Is there always a solution where neither x or y is zero? (not sure)


Does this sort of equation look familiar to anyone else? Does it remind you of anything? I feel like it must have some obvious properties.

Assuming $b\neq 0$, this is the straight line $y=-\frac{a}{b}x+a$. There are infinitely many $(x,y)$ pairs. If $b=0, a\neq 0$ then $x=0$, so $(0,y)$ where $y$ being any number is a solution. So again infinitely many solutions. So your 3rd statement is wrong.
