# Math Help - Number of Interger value of x and y.

1. ## Number of Interger value of x and y.

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,

2. All you need to do is find one solution, and you have all of them for this. Let $(x_0,y_0)$ be a solution to $ax+by=c$.

Then all other solutions will be of the form:

$x=x_0 + b\left(\frac{k}{gcd(a,b)}\right)$
$y=y_0 - a\left(\frac{k}{gcd(a,b)}\right)$

Where k runs through the integers positive and negative.

So in this case, inspections shows (-1,1) is a solution, so the others are :

$x=-1 + 7(\frac{k}{1})=7k-1$
$y=1 - 4(\frac{k}{1})=1-4k$

I trust that you can take it from here