Question

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

Answer 1

$2x+5y=99$
when $y=-19,x=97$;
since $x≥y;$ the maximum value of y is $13$ and corresponding value of $x$ is $17$.
We know that the solutions of $y$ are in arithmetic progression with common difference $2$.
Here $a=-19$, $d=2,t_n=13$
$t_n=a+(n-1)d$
$-19+(n-1)(2)=13$
$(n-1)2=32 ⇒ n=17$

Hence number of pairs integers is $17$