Diophantine Equations (another)

Question

I have the following equation

$756x+630y = 2394$

and i am trying to find the general solution to it. I have done these type of questions numerous times and had no difficulty but I cant seem to do this.

What I have so far is

$756x+630y=2394$

$756-1(630)=126$

$630-5(126)=0$

I tried using the technique roll back but it dosent work.

Answer 1

What your calculation shows is that $126$ is the greatest common divisor of $756$ and $630$ and that $756-630=126$ and $6\times630-5\times756=0$. Since $2394/126=19$ is not fractional, we can solve the equation. $19(756-630)+k(6\times630-5\times756)=2394$.

Note that $6\times630-5\times756=0$ comes from substituting $756-630=126$ into $630-5\times126=0$.

Answer 2

You can divide all by $$126$$ and you have to solve $$6x+5y=19$$ and then you can write $$y=4-x-\frac{1+x}{5}$$ Substitute $$\frac{1+x}{5}=t$$ we get $$x=5t-1,y=5-6t$$

Answer 3

If x= 1 and y= 1 then 6x+ 5y= 1. So if x= 19 and y= -19, 6x+ 5y= 19. But x= 19- 5k, y= -19+ 6k is also a solution: 6(19- 5k)+ 5(-19+ 6k)= 6(19)- 6(5)k- 5(19)+ 5(6)k= 114-95= 19 for all k.