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,

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- Aug 13th 2009, 10:12 PMa69356Number 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, - Aug 13th 2009, 10:28 PMGamma
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