Solving a system of equations using Euclid's Algorithm

Question

Find the general solution $(x, y)$ ∈ $\mathbb Z × \mathbb Z$ for the system of equations

$355x − 113y = 1$ ... eq$1$

$355x + 113y = 1$ ... eq$2$

You need one parameter $t ∈ \mathbb Z$ to describe the general solution.

My attempt:

Euclid's Algorithm: gcd($355,113$) $= 1$

$1 = 355(-7)+113(22)$

So for eq$2$ $(x,y)=(-7,22)$

And for eq$1$ $(x,y)= (-7,-22)$

So general solution is $(-7,22t)$, where $t$={$-1,1$}

Is my answer correct? If not, please how do we solve this?

Answer 1

The system has no solution. If there were integers $x$ and $y$ such that $355x-113y=355x+113y=1$, then$$2=1+1=355x-113y+355x+113y=710x.$$But there is no integer $x$ such that $710x=2$.