How many integer value of x and y are there such that
4x + 7y = 3 where IxI < 500 and IyI < 500?
Any help would be appreciated.
Thanks,
All you need to do is find one solution, and you have all of them for this. Let $\displaystyle (x_0,y_0)$ be a solution to $\displaystyle ax+by=c$.
Then all other solutions will be of the form:
$\displaystyle x=x_0 + b\left(\frac{k}{gcd(a,b)}\right)$
$\displaystyle y=y_0 - a\left(\frac{k}{gcd(a,b)}\right)$
Where k runs through the integers positive and negative.
So in this case, inspections shows (-1,1) is a solution, so the others are :
$\displaystyle x=-1 + 7(\frac{k}{1})=7k-1$
$\displaystyle y=1 - 4(\frac{k}{1})=1-4k$
I trust that you can take it from here