Find all integer solutions to $70x$ + $28y$ = $518$
Attempt:
By reducing the equation we get, $5x$ + $2y$ = $37$
Since, $5(9) + 2(-4) = 37$
So, solutions are $(x,y) = (9,-4)$
And gcd(2,5) is 1
The full set of integer solutions is:
$(x,y):x = 9-4k , y = 2 - 5 k, k$$\in\Bbb Z$
If $(x_0,y_0)$ is a particular integer solution to $5x+2y=37$, all integer solutions are given by $x=x_0-2t$, $y=y_0+5t$, where $t$ ranges over the integers.
You had found the particular solution $(x_0,y_0)=(9,-4)$. Thus all solutions are given by $x=9-2t$, $y=-4+5t$.
