jmc

Though we can solve this by substitution, an easier way to solve the problem is to simply add the two equations. Then, $$ab+5a+2b=-10⟹ab+5a+2b+10=0.$$ This factors! Thus, $(a+2)(b+5)=0$, from which it follows that at least one of the two statements $a=-2$, $b=-5$ is true. In the latter case, we obtain $a=\boxed{2}$, which yields the largest possible value of $a$.