Thread: Need help with solving algebraically

1. Need help with solving algebraically

Solve algebraically

2x -7y = 25

2. Hello,

I suppose you have to solve it in $\mathbb{Z}$ ...

Well, Bézout's identity let you write :

$\exists x',y' \in \mathbb{Z}$ such as $2x'-7y'=1$ 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

$S=\{\underbrace{100+7k}_{x} \ ; \ \underbrace{25+2k}_{y} \} \ , \ k \in \mathbb{Z}$