Solve algebraically
2x -7y = 25
Hello,
I suppose you have to solve it in $\displaystyle \mathbb{Z}$ ...
Well, Bézout's identity let you write :
$\displaystyle \exists x',y' \in \mathbb{Z}$ such as $\displaystyle 2x'-7y'=1$ as 2 and 7 are relatively prime numbers.
So find a particular solution :
2*4-7*1=1
->2*100-7*25=25
x=100 and y=25 are particular solutions.
Then, write 2x-7y=2*100-7*25 (=25)
-> 2(x-100)=7(y-25)
As 2 and 7 are relatively prime numbers, 2 divides (y-25) and 7 divides (x-100)
We can say that y-25=2k --> 2(x-100)=14k --> x-100=7k
$\displaystyle S=\{\underbrace{100+7k}_{x} \ ; \ \underbrace{25+2k}_{y} \} \ , \ k \in \mathbb{Z}$