The number of pairs of integers $$(x,y)$$ satisfying $$x\geq y\geq-20$$ and $$2x+5y=99$$

We have 2x + 5y = 99 or $$x=\frac{\left(99-5y\right)}{2}$$

Now $$x\ge\ y\ \ge\ -20$$ ; So $$\frac{\left(99-5y\right)}{2}\ge\ y\ ;\ 99\ge7y\ or\ y\le\ \approx\ 14$$

So $$-20\le y\le14$$. Now for this range of "y", we have to find all the integral values of "x". As the coefficient of "x" is 2,

then (99 - 5y) must be even, which will happen when "y" is odd. However, there are only 17 odd values of "y" be -20 and 14.

Hence the number of possible values is 17.