I see that the system of equations involves only two variables, so I can try to eliminate one of them and solve for the other. One way to do that is to use the second equation to express $b$ in terms of $a$, and then substitute it into the first equation. So, from the second equation, I get that $b = -\frac{10}{a}$. Substituting this into the first equation, I get $5a + 2\left(-\frac{10}{a}\right) = 0$. Multiplying everything by $a$, I get $5a^2 - 20 = 0$. Adding $20$ to both sides, I get $5a^2 = 20$. Dividing by $5$, I get $a^2 = 4$. Taking the square root of both sides, I get $a = \pm 2$. But I want the greatest possible value of $a$, so I choose $a = 2$. To check my answer, I can plug it back into the original equations and see if they are satisfied. If $a = 2$, then $b = -\frac{10}{a} = -5$. Then, $5a + 2b = 5(2) + 2(-5) = 0$, which is true. And, $ab = 2(-5) = -10$, which is also true. So, $a = 2$ and $b = -5$ are the solutions to the system of equations.


What is the greatest possible value of $a$ in the system of equations $5a + 2b = 0$ and $ab = -10$?